CleanEnergyWIKI - User contributions [en]

Color and Chromaticity

2022-03-01T20:16:56Z

Cmditradmin: /* RGB Color Mixing Schema */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Luminescence Phenomena|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Basics of Light|Return to Basics of Light Menu]]</td>

</tr>
</table>

There are many terms that describe color. In addition to descriptive terms there are corresponding quantitative terms.

== Light absorption ==
[[Image:Excitation.jpg|thumb|300px|Electron configuration after absorption]]
When a molecule interacts with light and energy is absorbed, the molecule is said be excited and a transition occurs which can take the molecule from an initial state to a higher energy state

Within the one-electron approximation, this is described by the promotion of an electron from a filled orbital to an unfilled orbital (in the case of diamagnetic materials).
The difference in energy between those levels, (the excited state and the ground state), gives the energy of the photons that can be absorbed.

Several parameters can be used to characterize this transition, including the energy of the incident radiation required for the efficient absorption of the light and the inherent ability of the molecules to absorb radiation of the appropriate energy
by the Planck relation:

:<math>\Delta E_{ge} = E_{excited state}-E_{ground state} = h\nu\,\!</math>

where hv is the energy of the photon corresponding to the energy gap between the states.
The energy is reported in several units; the following is helpful for translating between some common units one comes across in the literature:

'''1 eV = 23.06 kcal/mol = 8065 cm-1 = 1240 nm'''

== Common Color Description==

Our perception of color is determined by what wavelengths of radiation reach our eye and the sensitivity of the receptors in our eye to various colors
The eye has rods and cones containing chromophores which convert light into electrical impulse that the brain uses to perceive images. This the opposite of what you see in light emitting diodes in which electricity causes emission of light.

The rods function under low intensity conditions and provide images in shades of black, grey, and white
This is referred to as scotopic vision

The cones process images of high intensity in color which is referred to as photopic vision.
Cones come in three varieties which correspond roughly to blue, green, and red sensitivities; if all three cones are simultaneously excited, then the image will appear white.

See [[wikipedia:Color Color]]

See also [http://www.glenbrook.k12.il.us/gbssci/phys/Class/light/lighttoc.html Physics Classroom tutorial]
 

=== Complementary Colors ===

[[Image:Complementarycolors.png|thumb|400px|This graphic shows what color will be perceived when a material absorbs in certain regions of the visible spectrum.]]
If wavelengths of light from a certain region of the spectrum are absorbed by a material, then the material will appear to be the complementary color. Thus, for instance, if violet light with wavelength of 400nm is absorbed, the material will look yellow. If the material absorbs blue you will see the color orange.
{| class="wikitable"
|-
! Color absorbed
! Color seen

|-
| Violet
| Yellow

|-
| Blue
| Orange
|-
| Green
| Red
|-
| Yellow
| Violet
|-
| Orange
| Blue
|}

Note that green is not indicated in the figure; this is because materials that appear green actually absorb in the red and the blue (i.e., about 650 nm and 425 nm)
band shape and color

Our ability to perceive very small differences in color is rather extraordinary; for instance, two solutions which appear to have virtually identical absorption spectra, with minute differences in their tails, can be recognized as clearly different hues. Very small changes in the shape of an absorption band (not only the position) will cause materials to appear different shades

=== Bright and Dull ===

[[Image:Brightanddull.jpg|thumb|300px|A sharp absorption peak results in the perception of a saturated color.]]
In general, colors that we perceive as brilliant and bright have strong narrow absorption bands whereas dull colors tend to have weaker and broader absorption bands.

 

=== Hue and Saturation ===

The '''hue''' is that aspect of color usually associated with terms such as red, orange, yellow, and so forth.Hue distinguishes the color purity of the dominant color (i.e. red from yellow). The position of absorption maxima largely determines this property.

'''saturation''' (also known as chroma, or tone) refers to relative purity; when a pure, vivid, strong shade of red is mixed with a variable amount of white, weaker or paler reds are produced, each having the same hue but a different saturation; such paler colors are called unsaturated colors. You can define the amount of saturation of a given using a chromaticity diagram. For example, suppose you had a red color and you slowly increased the amount of blue and green light reaching the eye, then the mixture of the red, blue and green would contribute to the perception of white. White plus red would give pink. The hue would not have been altered, but the saturation would be lower

Light of any given combination of hue and saturation can have a variable '''brightness''' (also called intensity, lightness, or value), which depends on the total amount of light energy present. Lightness of a color is changed by varying the intensity of all three primary colors by the same amount. For example, if the intensity of a red were increased it would appear brown.

=== RGB Color Mixing Schema===
All colors can be create by the addition of the primary colors. Use this Flash application to explore color mixing.

<div id="Flash">RBB Color Mixing Simulation</div>
  <swf width="600" height="500">http://depts.washington.edu/cmditr/mediawiki/images/f/fc/Colorbox.swf</swf>

https://flash.puffin.com/depts.washington.edu/cmditr/mediawiki/images/f/fc/Colorbox.swf

Engineers and designers have very specific requirements for light emitting and light absorbing materials. They frequently use a color measurement system called tristimulus to precisely specify any possible color, even those that can not be described with a simple wavelength.

== Tristimulus measurement and chromaticity diagrams ==

The tristimulus color measurement system is based on visually matching a color under standardized conditions against the three primary colors, red, green, and blue; the three results are expressed as X, Y, and Z, respectively, and are called '''tristimulus''' values

These values specify not only color but also visually perceived reflectance, since they are calculated in such a way that the Y value equals a sample's reflectivity (39.1 percent in this example) when visually compared to a standard white surface by a standard (average) viewer under average daylight.

The tristimulus values of the emerald-green pigment of Figure 6 are X = 22.7, Y = 39.1, and Z = 31.0[[Image:Emeraldabsorb.jpg|thumb|200px|Reflectance of emerald green color]]

The tristimulus values can also be used to determine the visually perceived dominant spectral wavelength (which is related to the hue) of a given sample; the dominant wavelength of the emerald-green pigment is 511.9 nm:

it is based on the values x, y, and z,

Where

'''x = X/(X + Y + Z)'''

'''y = Y/(X + Y + Z)'''

'''z = Z/(X + Y + Z)'''

 
[[Image:Cie_chromaticity_diagram_wavelength.png|thumb|300px|]]
Note that x + y + z = 1; thus if two values are known, the third can always be calculated and the z value is usually omitted thus, the x and y values together constitute the chromaticity of a sample light and dark colors that have the same chromaticity (and are therefore plotted at the same point on the two-dimensional chromaticity diagram) can be distinguished in a third dimension (by their luminance or visually perceived brightness).

White light is x= 1/3, y = 1/3 and z= 1/3. This is achromatic point. Pure grays and black are the same hue as white light but vary only in the magnitude of their luminance. Occasionally colors will be also be described using luminance as well.

So for the goal of LED makers is to make a white light with x and y values close to 1/3.

Around the horseshoe shaped periphery are the pure saturated colors , beginning with 400nm (violet) and going around to 700 nm (red). Those are the colors of the visible spectrum. The straight line across the bottom are colors that come from the non-spectral mixing of violet and red, they do not correspond to a single wavelength.
 
[[Image:300px-CIExy1931_twocolors.png|thumb|300px|]]

=== Plotting CIE values ===

By plotting the calculated x = 0.245 and y = 0.421 of the emerald-green pigment at point E on the chromaticity diagram and extending a line through it from the achromatic point W to the saturated spectral boundary, it is possible to determine the dominant wavelength of the pigment color, 511.9 nm. Emerald green is not a pure color. But it can be made by mixing the pure color with wavelength 511.9nm with white light.

The color of the pigment is the visual equivalent of adding white light and light of 511.9 nm in amounts proportional to the lengths '''n''' (the distance between points '''E''' and '''W''') and '''m''' (the distance between '''E''' and the point of the dominant wavelength) in the figure. The saturation or purity equals 100n/(m + n) percent - in this case, 22.8 percent. A purity of 100 percent corresponds to a pure saturated spectral color and 0 percent to the achromatic colors (white, gray, and black)

Another example, a red apple marked '''R''' on the diagram. If you connect the line through '''w''' and '''R''' it intersects the bottom line which are not pure spectral colors. In this case this shade of red must be defined in terms of the complementary color on the opposite side of the achromatic point.

The dominant color designation is then obtained by extrapolating the line in the opposite direction to a saturated spectral color it is given as "complementary dominant wavelength 495 nm" or 495c. The color of this apple is therefore the visual equivalent of a mixture of white light and the 495c saturated purple-red in the intensity ratio of the distances p to q shown in the figure with a purity of 100p/(p + q) percent.

 
=== Incandescent light sources ===

[[Image:533px-PlanckianLocus.png|thumb|300px|]]
Light from incandescent sources falls on the solid curve marked with temperatures in this figure, following the sequence saturated red to saturated orange to unsaturated yellow to white to unsaturated bluish white for an infinite temperature.

The points A, B, and C on the curve are CIE standard illuminants that approximate, respectively, a 100-watt incandescent filament lamp at a color temperature of about 2,850 K, noon sunlight (about 4,800 K), and average daylight (about 6,500 K)

The color of daylight changes over the course of a day. LED designers could make the color of their devices change during the day to better match the daylight experience. Some white lights feel “warmer “ or “colder “depending on the color balance. LEDs will have the same descriptives.

== External Links ==

*[http://hyperphysics.phy-astr.gsu.edu/hbase/vision/colper.html#c2 Hyperphysics materials about chromaticity]
[[category:Light]]
{{author
|AuthorFullName= Bredas, Jean-Luc
|AuthorName=Bredas}}
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Luminescence Phenomena|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Basics of Light|Return to Basics of Light Menu]]</td>

</tr>
</table>

Color and Chromaticity

2022-03-01T20:06:25Z

Cmditradmin:

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Luminescence Phenomena|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Basics of Light|Return to Basics of Light Menu]]</td>

</tr>
</table>

There are many terms that describe color. In addition to descriptive terms there are corresponding quantitative terms.

== Light absorption ==
[[Image:Excitation.jpg|thumb|300px|Electron configuration after absorption]]
When a molecule interacts with light and energy is absorbed, the molecule is said be excited and a transition occurs which can take the molecule from an initial state to a higher energy state

Within the one-electron approximation, this is described by the promotion of an electron from a filled orbital to an unfilled orbital (in the case of diamagnetic materials).
The difference in energy between those levels, (the excited state and the ground state), gives the energy of the photons that can be absorbed.

Several parameters can be used to characterize this transition, including the energy of the incident radiation required for the efficient absorption of the light and the inherent ability of the molecules to absorb radiation of the appropriate energy
by the Planck relation:

:<math>\Delta E_{ge} = E_{excited state}-E_{ground state} = h\nu\,\!</math>

where hv is the energy of the photon corresponding to the energy gap between the states.
The energy is reported in several units; the following is helpful for translating between some common units one comes across in the literature:

'''1 eV = 23.06 kcal/mol = 8065 cm-1 = 1240 nm'''

== Common Color Description==

Our perception of color is determined by what wavelengths of radiation reach our eye and the sensitivity of the receptors in our eye to various colors
The eye has rods and cones containing chromophores which convert light into electrical impulse that the brain uses to perceive images. This the opposite of what you see in light emitting diodes in which electricity causes emission of light.

The rods function under low intensity conditions and provide images in shades of black, grey, and white
This is referred to as scotopic vision

The cones process images of high intensity in color which is referred to as photopic vision.
Cones come in three varieties which correspond roughly to blue, green, and red sensitivities; if all three cones are simultaneously excited, then the image will appear white.

See [[wikipedia:Color Color]]

See also [http://www.glenbrook.k12.il.us/gbssci/phys/Class/light/lighttoc.html Physics Classroom tutorial]
 

=== Complementary Colors ===

[[Image:Complementarycolors.png|thumb|400px|This graphic shows what color will be perceived when a material absorbs in certain regions of the visible spectrum.]]
If wavelengths of light from a certain region of the spectrum are absorbed by a material, then the material will appear to be the complementary color. Thus, for instance, if violet light with wavelength of 400nm is absorbed, the material will look yellow. If the material absorbs blue you will see the color orange.
{| class="wikitable"
|-
! Color absorbed
! Color seen

|-
| Violet
| Yellow

|-
| Blue
| Orange
|-
| Green
| Red
|-
| Yellow
| Violet
|-
| Orange
| Blue
|}

Note that green is not indicated in the figure; this is because materials that appear green actually absorb in the red and the blue (i.e., about 650 nm and 425 nm)
band shape and color

Our ability to perceive very small differences in color is rather extraordinary; for instance, two solutions which appear to have virtually identical absorption spectra, with minute differences in their tails, can be recognized as clearly different hues. Very small changes in the shape of an absorption band (not only the position) will cause materials to appear different shades

=== Bright and Dull ===

[[Image:Brightanddull.jpg|thumb|300px|A sharp absorption peak results in the perception of a saturated color.]]
In general, colors that we perceive as brilliant and bright have strong narrow absorption bands whereas dull colors tend to have weaker and broader absorption bands.

 

=== Hue and Saturation ===

The '''hue''' is that aspect of color usually associated with terms such as red, orange, yellow, and so forth.Hue distinguishes the color purity of the dominant color (i.e. red from yellow). The position of absorption maxima largely determines this property.

'''saturation''' (also known as chroma, or tone) refers to relative purity; when a pure, vivid, strong shade of red is mixed with a variable amount of white, weaker or paler reds are produced, each having the same hue but a different saturation; such paler colors are called unsaturated colors. You can define the amount of saturation of a given using a chromaticity diagram. For example, suppose you had a red color and you slowly increased the amount of blue and green light reaching the eye, then the mixture of the red, blue and green would contribute to the perception of white. White plus red would give pink. The hue would not have been altered, but the saturation would be lower

Light of any given combination of hue and saturation can have a variable '''brightness''' (also called intensity, lightness, or value), which depends on the total amount of light energy present. Lightness of a color is changed by varying the intensity of all three primary colors by the same amount. For example, if the intensity of a red were increased it would appear brown.

=== RGB Color Mixing Schema===
All colors can be create by the addition of the primary colors. Use this Flash application to explore color mixing.

<div id="Flash">RBB Color Mixing Simulation</div>
  <swf width="600" height="500">http://depts.washington.edu/cmditr/mediawiki/images/f/fc/Colorbox.swf</swf>

Https://flash.puffin.com/depts.washington.edu/cmditr/mediawiki/images/f/fc/Colorbox.swf

Engineers and designers have very specific requirements for light emitting and light absorbing materials. They frequently use a color measurement system called tristimulus to precisely specify any possible color, even those that can not be described with a simple wavelength.

== Tristimulus measurement and chromaticity diagrams ==

The tristimulus color measurement system is based on visually matching a color under standardized conditions against the three primary colors, red, green, and blue; the three results are expressed as X, Y, and Z, respectively, and are called '''tristimulus''' values

These values specify not only color but also visually perceived reflectance, since they are calculated in such a way that the Y value equals a sample's reflectivity (39.1 percent in this example) when visually compared to a standard white surface by a standard (average) viewer under average daylight.

The tristimulus values of the emerald-green pigment of Figure 6 are X = 22.7, Y = 39.1, and Z = 31.0[[Image:Emeraldabsorb.jpg|thumb|200px|Reflectance of emerald green color]]

The tristimulus values can also be used to determine the visually perceived dominant spectral wavelength (which is related to the hue) of a given sample; the dominant wavelength of the emerald-green pigment is 511.9 nm:

it is based on the values x, y, and z,

Where

'''x = X/(X + Y + Z)'''

'''y = Y/(X + Y + Z)'''

'''z = Z/(X + Y + Z)'''

 
[[Image:Cie_chromaticity_diagram_wavelength.png|thumb|300px|]]
Note that x + y + z = 1; thus if two values are known, the third can always be calculated and the z value is usually omitted thus, the x and y values together constitute the chromaticity of a sample light and dark colors that have the same chromaticity (and are therefore plotted at the same point on the two-dimensional chromaticity diagram) can be distinguished in a third dimension (by their luminance or visually perceived brightness).

White light is x= 1/3, y = 1/3 and z= 1/3. This is achromatic point. Pure grays and black are the same hue as white light but vary only in the magnitude of their luminance. Occasionally colors will be also be described using luminance as well.

So for the goal of LED makers is to make a white light with x and y values close to 1/3.

Around the horseshoe shaped periphery are the pure saturated colors , beginning with 400nm (violet) and going around to 700 nm (red). Those are the colors of the visible spectrum. The straight line across the bottom are colors that come from the non-spectral mixing of violet and red, they do not correspond to a single wavelength.
 
[[Image:300px-CIExy1931_twocolors.png|thumb|300px|]]

=== Plotting CIE values ===

By plotting the calculated x = 0.245 and y = 0.421 of the emerald-green pigment at point E on the chromaticity diagram and extending a line through it from the achromatic point W to the saturated spectral boundary, it is possible to determine the dominant wavelength of the pigment color, 511.9 nm. Emerald green is not a pure color. But it can be made by mixing the pure color with wavelength 511.9nm with white light.

The color of the pigment is the visual equivalent of adding white light and light of 511.9 nm in amounts proportional to the lengths '''n''' (the distance between points '''E''' and '''W''') and '''m''' (the distance between '''E''' and the point of the dominant wavelength) in the figure. The saturation or purity equals 100n/(m + n) percent - in this case, 22.8 percent. A purity of 100 percent corresponds to a pure saturated spectral color and 0 percent to the achromatic colors (white, gray, and black)

Another example, a red apple marked '''R''' on the diagram. If you connect the line through '''w''' and '''R''' it intersects the bottom line which are not pure spectral colors. In this case this shade of red must be defined in terms of the complementary color on the opposite side of the achromatic point.

The dominant color designation is then obtained by extrapolating the line in the opposite direction to a saturated spectral color it is given as "complementary dominant wavelength 495 nm" or 495c. The color of this apple is therefore the visual equivalent of a mixture of white light and the 495c saturated purple-red in the intensity ratio of the distances p to q shown in the figure with a purity of 100p/(p + q) percent.

 
=== Incandescent light sources ===

[[Image:533px-PlanckianLocus.png|thumb|300px|]]
Light from incandescent sources falls on the solid curve marked with temperatures in this figure, following the sequence saturated red to saturated orange to unsaturated yellow to white to unsaturated bluish white for an infinite temperature.

The points A, B, and C on the curve are CIE standard illuminants that approximate, respectively, a 100-watt incandescent filament lamp at a color temperature of about 2,850 K, noon sunlight (about 4,800 K), and average daylight (about 6,500 K)

The color of daylight changes over the course of a day. LED designers could make the color of their devices change during the day to better match the daylight experience. Some white lights feel “warmer “ or “colder “depending on the color balance. LEDs will have the same descriptives.

== External Links ==

*[http://hyperphysics.phy-astr.gsu.edu/hbase/vision/colper.html#c2 Hyperphysics materials about chromaticity]
[[category:Light]]
{{author
|AuthorFullName= Bredas, Jean-Luc
|AuthorName=Bredas}}
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Luminescence Phenomena|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Basics of Light|Return to Basics of Light Menu]]</td>

</tr>
</table>

MediaWiki:Sidebar

2021-12-07T21:02:48Z

Cmditradmin:

Editing Hints

2020-08-31T21:41:12Z

Cmditradmin: /* MATH */

=== MATH ===
<math>0</math>

<math>\phi_n(\kappa) =
\frac{1}{4\pi^2\kappa^2} \int_0^\infty
\frac{\sin(\kappa R)}{\kappa R}
\frac{\partial}{\partial R}
\left [ R^2\frac{\partial D_n(R)}{\partial R} \right ] \,dR</math>

https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Functions.2C_symbols.2C_special_characters

===Links===
*[http://matdl.org/repository/index.php name] creates a link to the words to the right of the space after the link

=== Youtube ===
<nowiki>{{#ev:youtube|pm1yzlwaWAU}}</nowiki>
{{#ev:youtube|pm1yzlwaWAU}}
<nowiki><embed_document width="100%" height="600">http://photonicswiki.org/images/8/88/Stein_qdot_CEI_product.pdf</embed_document></nowiki>

<embed_document width="100%" height="600">http://cleanenergywiki.org/images/8/88/Stein_qdot_CEI_product.pdf</embed_document>

Editing Hints

2020-08-31T21:39:14Z

Cmditradmin: /* MATH */

=== MATH ===
<math>0</math>

<math>\phi_n(\kappa) =
\frac{1}{4\pi^2\kappa^2} \int_0^\infty
\frac{\sin(\kappa R)}{\kappa R}
\frac{\partial}{\partial R}
\left [ R^2\frac{\partial D_n(R)}{\partial R} \right ] \,dR</math>

https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Functions.2C_symbols.2C_special_characters

*[http://matdl.org/repository/index.php name] creates a link to the words to the right of the space after the link

=== Youtube ===
<nowiki>{{#ev:youtube|pm1yzlwaWAU}}</nowiki>
{{#ev:youtube|pm1yzlwaWAU}}
<nowiki><embed_document width="100%" height="600">http://photonicswiki.org/images/8/88/Stein_qdot_CEI_product.pdf</embed_document></nowiki>

<embed_document width="100%" height="600">http://cleanenergywiki.org/images/8/88/Stein_qdot_CEI_product.pdf</embed_document>

Graetzel or Dye Sensitized Solar Cell

2020-06-17T15:56:19Z

Cmditradmin:

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Luminescent Solar Concentrator|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Solar Materials|Return to Solar Materials Menu]]</td>
<td style="text-align: right; width: 33%">[[Photonic Crystal|Next Topic]]</td>

</tr>
</table>
This dye sensitized solar cell, also known as a Graetzel cell uses a thin film of titanium dioxide which has been ground to a fine powder (nanocrystalline) to increase its reactive surface area. The TiO2 is sandwiched between two glass slides that are coated with conductive and transparent indium tin oxide (ITO). The TiO2 is impregnated with some kind of colored dye, in this case anthocyanin from raspberry juice, which is the chemical which first traps the solar energy and passes the charge to the TiO2. Finally the space between the slides is filled with an liquid electrolyte solution of potassium iodide which serves to transport charge (by way of a redox reaction) from the bottom electrode to the dye to complete the circuit.
[[File:500px-Dye_Sensitized_Solar_Cell_Scheme.png|thumb]]
The efficiency research cells is 11-15%, this berry demo cell is likely <1% efficient.

{{#ev:youtube|Qbsl1NP5uZI}}

Links

[https://en.wikipedia.org/wiki/Dye-sensitized_solar_cell wikipedia article]

[https://coatings.specialchem.com/selection-guide/complete-guide-on-titanium-dioxide More information about TiO2]

Graetzel or Dye Sensitized Solar Cell

2020-06-17T15:54:45Z

Cmditradmin:

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Luminescent Solar Concentrator|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Solar Materials|Return to Solar Materials Menu]]</td>
<td style="text-align: right; width: 33%">[[Photonic Crystal|Next Topic]]</td>

</tr>
</table>
This dye sensitized solar cell, also known as a Graetzel cell uses a thin film of titanium dioxide which has been ground to a fine powder (nanocrystalline) to increase its reactive surface area. The TiO2 is sandwiched between two glass slides that are coated with conductive and transparent indium tin oxide (ITO). The TiO2 is impregnated with some kind of colored dye, in this case anthocyanin from raspberry juice, which is the chemical which first traps the solar energy and passes the charge to the TiO2. Finally the space between the slides is filled with an liquid electrolyte solution of potassium iodide which serves to transport charge (by way of a redox reaction) from the bottom electrode to the dye to complete the circuit.
[[File:500px-Dye_Sensitized_Solar_Cell_Scheme.png|thumb]]
The efficiency research cells is 11-15%, this berry demo cell is likely <1% efficient.

{{#ev:youtube|Qbsl1NP5uZI}}

Links

[https://en.wikipedia.org/wiki/Dye-sensitized_solar_cell]

More information about TiO2
[https://coatings.specialchem.com/selection-guide/complete-guide-on-titanium-dioxide]

Main Page

2020-06-16T23:09:16Z

Cmditradmin: /* Organic Light Emitting Diodes */

[[File:CEI_logo_tag_color.360x105.jpg]] 
[http://cei.washington.edu Link to the Clean Energy Institute]

== Clean Energy ==
===Energy Basics===
*[[Energy Basics]]
*[[Survey of Renewables]]
*[[Clean Energy Policy]]

===Solar Basics===
[[Image:Solar_tracker_in_Lixouri.jpg|thumb|200px|Solar tracker]]
*[[Electromagnetic Radiation]]
*[[Solar Technologies]]
*[[Solar Cell Introduction|Solar Cell Introduction]]
*[[Physics of Solar Cells]]
*[[Band-gap]]
 

===Solar Materials===
[[Image:Perovskite_ABO3.jpg|thumb|200px|Perovskite]]
*[[Silicon Solar]]
*[[Perovskites]]
*[[CIGS, CIS, CdTe]]
*[[Quantum Dots]]
*[[Luminescent Solar Concentrator]]
*[[Graetzel or Dye Sensitized Solar Cell]]
*[[Photonic Crystal]]
*[[2D Materials]]
 

'''Organic Solar Cells'''
[[Image:Opvtestcells.png|thumb|200px|OPV Test Cells]]
*[[Major Processes in Organic Solar Cells]]
*[[Organic Heterojunctions in Solar Cells]]
*[[Energy vs Charge Transfer at Heterojunctions]]
*[[Current OPV Research Directions]]
 

== Energy Storage ==
=== Introduction ===
[[Image:UET.jpg|thumb|200px|Unienergy Flow Battery]]
*[[Storage Basics]]
*[[Battery Basics]]

*[[Hydrogen Policy]]
*[[Data Center Energy Management]]

=== Battery Types ===
*[[Coin Cell Battery]]
*[[Lithium Ion Battery]]
*[[Redox Flow Battery]]
*[[Supercapacitor]]
=== Battery Materials ===
*[[Solid State Electrolyte]]
 
===Battery Research Tools===
*[[MACCOR Battery Tester]]
*[[Current Voltage Instrument Automation]]

== Grid Integration ==
[[Image:EGRIDSubregionmap.jpg|thumb|200px|US Power Grids]]
*[[Smart Grid]]
*[[Grid Integration of Renewables]]
*[[MatPower Simulation of Submodular Optimization for Voltage Control in Power Systems]]
 

== Photonics Core Concepts and Applications ==
[http://depts.washington.edu/cmditr/media/Photonics.html Concept Map CMDITR]

[[Image:Wordle3.png|thumb|center|600px|This graphic was created by processing the CMDITR 2009 annual report in the Wordle program. The larger the word the more times it appeared in the text.]]
 

===Basics of Light ===
[[Image:Snells_law_wavefronts.gif|thumb|150px|]]
*[[Propagation, Reflection and Refraction]]
*[[Dispersion and Scattering of Light]]
*[[Diffraction of Light]]

*[[Luminescence Phenomena]]
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[[Image:Military_laser_experiment.jpg|thumb|200px|]]

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[[Image:Abs Emis stokes.png|thumb|200px|]]
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[[Image:rubrene.png|thumb|150px|]]
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[[Image:PNNL_Light_Lab_041.jpg|thumb|200px|Blue phosphorescent OLED developed by Pacific Northwest National Laboratory.]]
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===Quantum Mechanical and Perturbation Theory of Polarizability===
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[[Image:Tpa_concentrated.png|thumb|100px|]]
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[[Image:Si_waveguide_em.jpg‎|thumb|200px|]]
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== Research Equipment, Devices and Techniques ==
These wiki pages include training videos produced at University of Washington, University of Arizona, Georgia Tech, and Norfolk State University. These videos can also be accessed through the [http://www.youtube.com/cmditr | CMDITR YouTube channel]
 
[[Image:PES.jpg|right|200px|link=Photoelectron Spectrometer XPS and UPS]]

=== Characterization ===
'''Composition'''
*[[Photoelectron Spectrometer XPS and UPS]]
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[[Image:Sirion sem.png|thumb|200px| ]]
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'''Properties'''
[[Image:Zscan.png|thumb|200px| ]]
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[[Image:glovebox.png|thumb|200px| ]]
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'''In Development'''
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*SPM

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MatPower Simulation of Submodular Optimization for Voltage Control in Power Systems

2020-06-08T22:29:33Z

Cmditradmin:

This video introduces a voltage control framework for power systems.

The simulation codes and instructions are provided at https://github.com/cdlzp/voltage
{{#ev:youtube|3iue7NI--EQ}}

Download introductory slides

http://photonicswiki.org/images/7/7c/Submod-opt.pdf

MatPower Simulation of Submodular Optimization for Voltage Control in Power Systems

2020-06-08T22:26:51Z

Cmditradmin:

This video introduces a voltage control framework for power systems.

The simulation codes and instructions are provided at https://github.com/cdlzp/voltage
{{#ev:youtube|/3iue7NI--EQ}}

Download introductory slides

http://photonicswiki.org/images/7/7c/Submod-opt.pdf

MatPower Simulation of Submodular Optimization for Voltage Control in Power Systems

2020-06-08T22:25:28Z

Cmditradmin:

This video introduces a voltage control framework for power systems.

The simulation codes and instructions are provided at https://github.com/cdlzp/voltage

Download introductory slides

https://youtu.be/3iue7NI--EQ

http://photonicswiki.org/images/7/7c/Submod-opt.pdf

Current Voltage Instrument Automation

2020-06-04T22:57:12Z

Cmditradmin:

Current Voltage Instrument Automation

2020-06-04T22:56:25Z

Cmditradmin: Created page with "https://youtu.be/7yBcCm-d3M8"

https://youtu.be/7yBcCm-d3M8

Main Page

2020-06-04T22:55:04Z

Cmditradmin: /* Energy Storage */

Grazing-Incidence Small-Angle X-ray Scattering (GISAXS)

2020-06-04T18:17:15Z

Cmditradmin:

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: center; width: 33%">[[Main_Page#Research Equipment, Devices and Techniques|Return to Research Tool Menu]]</td>

</tr>
</table>

Grazing incident small angle x-ray scattering (GISAXS) is a technique that is based on scattering theory operated in reciprocal space. Unlike microscopy techniques, such as scanning electron microscopy (SEM), transmission electron microscopy (TEM) and atomic force microscopy (AFM), GISAXS could give results reflecting statistical averaged information of the samples. Thus, it is usually coupled with microscopy to provide information of the sample as a whole[1]. As a non-invasive technique, it has been widely used in the area of polymer science, mainly because the polymer chain orientation on the substrate could be interpreted from the scattering profile. This information is valuable for organic electronics applications as how the chains orientate would greatly affect the charge transport within the devices.

Nowadays, organic electronic devices have generated many attentions because conjugated polymers are light, flexible, economic and scalable. For devices like organic photovoltaics (OPVs) and organic field effect transistors (OFETs), the active layers are thin films made of conjugated polymers. The charge transport in those thin film layers are strongly affected by the morphology and chain orientations of the polymers. For example, due to the device geometry, the charge transport in the vertical direction is favorable for organic solar cell operations, thus, “face-on” orientation of the polymer chains is usually preferred. Whereas in organic field effect transistors, the charge transport is in the lateral direction, so the polymers are usually processed such that the chains are “edge-on” and the p-p stacking direction is pointing from one electrode to another. The orientation of the chains will be discussed later. The charge transport in a polymer chain is dependent on the delocalization of the charge carriers by alternating single and double bonds. Although the transport along the backbone of the chain is fast, they are greatly dependent on the length of the polymer chain, which is in the nanometer scale range. Beyond that range, the charge transport needs to rely on hopping or other mechanism to another chain that is much slower. Along p-p stacking direction, charges transport slower than that along the backbone. However, the polymer p-p stacking could persist much longer than the polymer chain length[2]. This is beneficial for organic electronics applications. Thus how the chains orientate are very important in thin films. As the aromatic rings (thiophene rings in the case of P3HT) contains the p-bond, they will form attractive, non-covalent bonds[3]. The p-p bonds are also favorable for charge transport and they usually have a characterization length in the real space. In the reciprocal space, which is the space GISAXS is probing, those p-p bonds length will show up in certain q positions if they are repeating periodically for a long distance with a large amount.

X-ray scattering is a technique that is well suited for the characterization of chain orientations. The origin of the scattering comes from the difference of the electronic cloud density of the samples. Thus large atomic number elements scatter x-rays better than smaller ones. The small q-range data collected by small angle x-ray scattering in reciprocal space corresponds to long distance in the real space. The reciprocal space is the Fourier transform of the real space and is very important in the diffraction theory to study the periodic structure[4]. GISXAS data could be interpreted the same way as small angle x-ray scattering (SAXS) experiment. Instead of having transmission geometry as SAXS, GISAXS relied on reflection of x-ray from the samples, which enables the thin film characterization, as more sample volume could be probed. The typical experiment setup is shown in figure 1[4]. The incident angle a is defined by the angle between the incident beam and the sample plane. The reflection of the beam results in a 2-D scattering profile with a direction perpendicular to the sample plane (qz), which is also called out-of-plane direction, and along the sample plane (q//), namely in-plane direction, in which different information could be interpreted.

[[File:GISAXS.png|thumb|Figure 1. GISAXS experiment geometry[4]]]

.

The analysis of the GISAXS scattering profile is within the framework of distorted-wave Born approximation (DWBA)[5]. This is used to account for the redistribution of the electric field intensity of the incident beam after hitting the sample surface at various heights and the re-scattering of the beam[6]. Take poly(3-hexylthiophene) (P3HT) as an example, which would crystalize into unit cells with different chain orientations like the ones shown in figure 2. Figure 2(a) is the schematic drawing of edge-on orientation, with p-p stacking direction parallel to the substrate, whereas figure 2(b) is the face-on orientation of the polymer chains, with p-p stacking distances perpendicular to the substrate. Those different orientations resulted in very different scattering profiles[7] like the ones shown in figure 3. For different polymers, the characterization scattering peak position is different. Here we use P3HT to get an understanding of how the chains orientation affect the scattering profile. For P3HT, there are two main characterization distances that will show up in the scattering profile: The lamellae peaks and p-p stacking peaks. Lamella is a plate-like structure with open space in between[8]. For P3HT, the lamellae structure are consist of alkyl side chains. The side chains of P3HT make them solution processable, but they are electrically insulating. In figure 3(a), the high intensity peak at qz=~0.4 Å-1 in the out-of-plane orientation corresponds to the parallel lamellae, which has a repeating distance of ~ 1.6nm, whereas the high intensity peak at qz=~1.5 Å-1 in the in-plane orientation indicates an edge-on p-p stacking orientation, which has a repeating distance of ~0.39 nm in real space[7]. In figure 3 (b), the lamellae peak appears in the in-plane direction and the p-p stacking peak in the out-of-plane orientation.

http://gisaxs.com/images/thumb/5/5e/Cell3-edge_on2.png/200px-Cell3-edge_on2.png
http://gisaxs.com/images/thumb/2/2d/Cell3-face_on2.png/200px-Cell3-face_on2.png

Figure 2. P3HT crystalized into (a) edge-on and (b) face-on orientations [7].

http://gisaxs.com/images/thumb/e/e4/P3ht-generic_giwaxs.png/300px-P3ht-generic_giwaxs.png
http://gisaxs.com/images/thumb/1/16/P3ht-face-on_giwaxs.png/300px-P3ht-face-on_giwaxs.png

Figure 3. GISAXS scattering profile of P3HT film with predominantly (a) edge-on orientation and (b) face-on orientation[7].

Reference:

* DM Smilgies http://staff.chess.cornell.edu/~smilgies/gisaxs/GISAXS.php (Links to an external site.)
* Veaceslav Coropceanu, et al., Chem. Rev. 2007, 107, 926.
* https://en.wikipedia.org/wiki/Stacking_(chemistry)
* https://en.wikipedia.org/wiki/Reciprocal_lattice
* Lecture notes in physics 776. “Applications of synchrotron light to scattering and diffraction in materials and life sciences”
* http://www.amercrystalassn.org/documents/2014%20Meeting/gisaxs_jiang_ACA.pdf (Links to an external site.)
* http://gisaxs.com/index.php/Example:P3HT_orientation_analysis (Links to an external site.)
* https://en.wikipedia.org/wiki/Lamella_(anatomy) (Links to an external site.)

Electrochemical Impedence Spectroscopy EIS

2020-06-04T18:07:37Z

Cmditradmin: /* Nyquist Plots */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

[[File:eis2.png|thumb|Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.]
] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b) [[File:eis3.png|thumb|Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.]]

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
[[File:eis3.png|thumb|Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.]]

This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb|Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

 
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T18:07:00Z

Cmditradmin: /* Equivalent Electrical Circuit Elements */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

[[File:eis2.png|thumb|Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.]
] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b) [[File:eis3.png|thumb|Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.]]

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
[[File:eis3.png|thumb|Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.]]

This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

 
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T18:06:16Z

Cmditradmin: /* EEC Model Example */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

[[File:eis2.png|thumb]Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.
] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b) [[File:eis3.png|thumb]]
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
[[File:eis3.png|thumb|Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.]]

This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

 
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T18:05:33Z

Cmditradmin: /* Bode Plots */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

[[File:eis2.png|thumb]Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.
] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b) [[File:eis3.png|thumb]]
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
[[File:eis3.png|thumb]]
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

 
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T18:03:19Z

Cmditradmin: /* Bode Plots */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

[[File:eis2.png|thumb]Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.
] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b) [[File:eis3.png|thumb]]
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
[[File:eis3.png|thumb]]
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb|left|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

 
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T18:02:41Z

Cmditradmin: /* Bode Plots */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

[[File:eis2.png|thumb]Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.
] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b) [[File:eis3.png|thumb]]
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
[[File:eis3.png|thumb]]
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb|left|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T18:01:03Z

Cmditradmin: /* Bode Plots */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

[[File:eis2.png|thumb]Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.
] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b) [[File:eis3.png|thumb]]
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
[[File:eis3.png|thumb]]
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T18:00:30Z

Cmditradmin: /* Bode Plots */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

[[File:eis2.png|thumb]Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.
] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b) [[File:eis3.png|thumb]]
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
[[File:eis3.png|thumb]]
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T17:58:51Z

Cmditradmin: /* Equivalent Electrical Circuit Elements */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

[[File:eis2.png|thumb]Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.
] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b) [[File:eis3.png|thumb]]
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
[[File:eis3.png|thumb]]
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb]]

Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T17:57:47Z

Cmditradmin: /* Equivalent Electrical Circuit Elements */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

[[File:eis2.png|thumb]] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b) [[File:eis3.png|thumb]]
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
[[File:eis3.png|thumb]]
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb]]

Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T17:56:27Z

Cmditradmin: /* EEC Model Example */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b)
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
[[File:eis3.png|thumb]]
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb]]

Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T17:55:38Z

Cmditradmin: /* Bode Plots */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b)
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
 

[[File:bode2.png|thumb]]

Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T17:55:10Z

Cmditradmin: /* Bode Plots */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b)
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.

[[File:bode2.png|thumb]]

Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T17:54:42Z

Cmditradmin: /* Bode Plots */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b)
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
[[File:bode2.png|thumb]]

Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 

==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

File:Bode2.png

2020-06-04T17:53:40Z

Cmditradmin:

File:Eis3.png

2020-06-04T17:52:17Z

Cmditradmin:

File:Eis2.png

2020-06-04T17:51:59Z

Cmditradmin:

File:Eis1.png

2020-06-04T17:51:42Z

Cmditradmin:

Electrochemical Impedence Spectroscopy EIS

2020-06-04T17:39:48Z

Cmditradmin: /* Impedance Basics */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) Z=(V(t))/(I(t))=(V_o cos⁡(ωt))/(I_o cos⁡(ωt+φ))=Z_o (cos⁡(ωt))/(cos⁡(ωt+φ))

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b)
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 
==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Electrochemical Impedence Spectroscopy EIS

2020-06-04T17:38:21Z

Cmditradmin: /* Nyquist Plots */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) <math>Z= \frac{V\left (t)}{I\left (t\right)}= \frac{V_{o} \cos⁡\left (\omega t\right)}{I_{o} \cos ⁡\left ( \omega t + \varphi\right)}=Z_{o} \frac{ \cos\left⁡(\omega t\right)}{\cos⁡\left(\omega t+\varphi\right)}</math> 

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b)
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:nyquist2.png|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 
==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

File:Nyquist2.png

2020-06-04T17:37:13Z

Cmditradmin:

Electrochemical Impedence Spectroscopy EIS

2020-06-04T17:35:45Z

Cmditradmin: /* Nyquist Plots */

==== Impedance Basics ====
Electrochemical Impedance Spectroscopy (EIS) is a frequency domain measurement made by applying a sinusoidal perturbation, often a voltage, to a system. The impedance at a given frequency is related to processes occurring at timescales of the inverse frequency (e.g. f=10 Hz, t=0.1 s). Although many other electrochemical measurements focus on driving a system far from equilibrium with potential sweeps or steps, such as cyclic voltammetry or chronoamperometry, EIS uses small perturbations. These small departures from equilibrium are assumed to have a linear response (Fig. 1), thus greatly simplifying the frequency analysis; however, linearization of physical models can lead to indistinguishable cases, as will be discussed in the Nonlinear EIS (NLEIS) section. Furthermore, operating conditions for relevant devices (e.g. batteries and fuel cells) are often far from equilibrium and exhibit nonlinear behavior. Practically speaking, EIS is performed by sweeping through a wide range of frequencies at a single perturbation amplitude. As instrumentation has improved over the last several decades, frequencies ranging from 10-4 to 108 Hz are attainable, thus allowing for the study of both fast kinetic and slow transport process1.

[[http://photonicswiki.org/images/c/ca/EIS_and_NLEIS_Wiki.pdf | Download PDF with all figures]]

[[File:Curvolt.png|thumb|Fig 1. Generalized current-voltage curve; inset shows the principle of linear approximation for small perturbations.]]

Broadly defined, impedance is the opposition of an electrical system to the flow of electric current and carries units of Ohms, Ω. It reduces to resistance under the following conditions: (1) there is no phase shift in current under an applied potential, and vice versa (2) all potentials and currents can be used (i.e. there is no saturation potential) (3) the impedance is not a function of frequency2. Under these conditions, the well-known Ohm’s Law applies: 

(1) V=IR or R=V/I 

Where V is the voltage in V, I is the current in A, and R is the resistance in Ω. However, in EIS both V and I are time dependent, sinusoidal functions. A single frequency potential input with amplitude, Vo, and radial frequency, ω, can be expressed as: 

(2) V(t)=V_o cos⁡(ωt) 

Note that EIS data is usually discussed in terms of linear frequency, f, with units of Hz. The conversion is ω=2πf. In a real system, the current output will have some phase shift, φ, and an amplitude, Io, expressed as: 

(3) <math>V_{\left (t \right )}=V_{o}\cos \left ( \omega T+\varphi \right )</math> 

For a generalized expression of impedance, Z, the previously stated conditions for Ohm’s law are relaxed, and the time-dependent expressions for V and I are substituted into Eq. 1: 

(4) <math>Z= \frac{V\left (t)}{I\left (t\right)}= \frac{V_{o} \cos⁡\left (\omega t\right)}{I_{o} \cos ⁡\left ( \omega t + \varphi\right)}=Z_{o} \frac{ \cos\left⁡(\omega t\right)}{\cos⁡\left(\omega t+\varphi\right)}</math> 

Again, we see that if the system exhibits no phase shift (φ=0), Eq. 4 reduces to Ohm’s Law3.

==== Equivalent Electrical Circuit Elements ====
Interpretation of EIS data has traditionally relied on models containing Equivalent Electrical Circuit (EEC) elements, where spectra are represented as combinations of circuit elements, such as resistors, capacitors, and inductors. These elements are then attributed to physical processes in the system (e.g. double layer capacitance, charge transfer resistance, etc.). Although this method may be appropriate for simple systems with well-defined physics, it may not be able to discern the differences between proposed local processes, such as reaction mechanisms. In any case, the impedance relations for these elements are given below. 

(5) Z_resistor=R 

(6) <math>Z_capacitor=\frac{1}{j\omega C}</math> 

(7) Z_inductor=jωL 

Where C is capacitance in F and L is inductance in H. Note that j=√-1, contrary to IUPAC convention so as to not be confused with current. Other elements without analogy to electrical circuits have also been used to represent EIS behavior not captured by the three included here (e.g. Warburg Impedance and constant phase element). For details regarding the derivation of Eqs. 5-7 and explanation of other EEC elements please see 3.
In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. 

〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
(b)
Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Z_eq=Z_1+Z_2+⋯+Z_n

(b)
Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

==== EEC Model Example ====
As an example, let us examine a circuit composed of a resistor element, Rs, followed by a resistor, Rr, and a capacitor, Cdl, in parallel. Graphical and mathematical representations of this circuit are given in Figure 4.

Z_eq=R_s+1/(1/R_r +jωC_dl )=R_s+R_r/(1+jωR_r C_dl )

(b)
Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

==== Nyquist Plots ====
At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
[[File:Example.jpg|thumb]]
Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
(9) C=1/(R_r ω_c )
Furthermore, the shape of features, such as multiple semicircles or a 45˚ low-frequency tail, gives possible insight into the governing kinetic or transport phenomena. Further details on analyzing Nyquist plots can be found in Ref. 2 or Chapter 16 Ref. 3.

==== Bode Plots ====
Another common representation is the Bode Plot, in which the impedance magnitude and phase angle are plotted against frequency. The magnitude and phase angle are given by Eqs. 10 and 11, respectively.
(10) |Z|=√(〖Z_Re〗^2+〖Z_Im〗^2 ) 

(11) φ=〖tan〗^(-1) (Z_Im/Z_Re ) 

Due to the large range of values often encountered in |Z| and f, they are plotted on log scales for easier examination of small values. Again, the high-frequency limit of |Z| yields Rs, and the low-frequency limit yields Rs+Rr. The slope of the transition region between the two asymptotic limits reveals the power of the frequency dependence in the imaginary part (-1 in our example). The frequency at which φ=-45˚ should give the characteristic frequency of the feature; however, in our example we see it crosses this line at both f=500 Hz and f=3,300 Hz. This error is due to the dominance of Rs at high frequencies, which obscures the behavior of the process responsible for the EIS feature. As such, Rs, or an estimate of Rs, should be subtracted from the real and imaginary parts of the entire data set.
Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area. 
==== Nonlinear EIS ====
NLEIS is essentially an extension of EIS which operates on many of the same principles. The key difference is the use of perturbation amplitudes which produce responses larger than appropriate for linear treatment. This allows for direct investigation of nonlinear system behavior. These nonlinearities are analyzed by collecting response signals at integer multiples of the input frequency, ω. Additionally, it is performed using current perturbations rather than voltage perturbations often used in EIS.
Higher Harmonic Analysis
Before addressing how NLEIS spectra are reported and interpreted, it is useful to understand how higher order harmonics are extracted from a complex signal. First, recall that a steady periodic function can be expressed as a Fourier series4. 

(12) f(t)=∑_(n=1)^∞▒〖(a_n 〗 cos⁡(nωt)+b_n sin⁡(nωt)) 

Or, using trigonometric identities for complex numbers: 

(13) f(t)=∑_(n=1)^∞▒〖(c ̃_n 〗 exp⁡(jnωt)+c ̃_(-n) exp⁡(-jnωt)) 

Where c ̃_n=(a_n+b_n)/2 and c ̃_(-n)=(a_n-b_n)/2. Now, when this is applied to the voltage response from a current perturbation with frequency ω ̃ we find: 

(14) V(t;ω ̃,i ̃ )=1/2 ∑_(k=1)^∞▒〖(V ̂_k (ω ̃,i ̃)〗 exp⁡(kjω ̃t)+V ̂_(-k) (ω ̃,i ̃)exp⁡(-kjω ̃t)) 

Where V ̂_(∓k)=V ̂'_k±jV ̂_k'' are complex Fourier coefficients for the kth harmonic. As with impedance data, a prime indicates the real part and a double prime indicates the imaginary part. The nonlinear dependence of the Fourier coefficients is then expressed as a power series: 

(15) V ̂_k (ω ̃,i ̃ )=∑_(r=1)^∞▒〖α^(k+2r-2) V ̂_(k+2r-2) (ω ̃)〗 

Where r is the order of nonlinear contribution to the kth order harmonic. For example, with k=r=1 we have V ̂_1,1, which is simply the linear response. These coefficients are found by fitting data from multiple amplitude perturbations at each desired frequency5. These amplitudes should be chosen to elicit purely linear behavior at the lover limit, and behavior that is one term higher than the desired power series order (e.g. 5th harmonic if truncating the power series at k=3).
NLEIS Representation
Once these Fourier coefficients are fit, they can be presented in a Complex-Plane representation, similar to Nyquist plots. An NLEIS spectrum of La0.6Sr0.4Co0.2Fe0.8O3-δ (LSFC-6428) at 600 ˚C under various PO2 environments is provided as an example in Figure 76. In this case, all harmonic data has been nondimensionalized and normalized with respect to the maximum absolute value of the imaginary component of the linear response. Other workers have chosen to report harmonic data in dimensional, non-normalized form6. Unfortunately, at this time there is no way to interpret higher harmonic data without support from a physical model like there is in the linear EIS community (e.g. low-frequency 45˚ tail indicates semi-infinite diffusion).

Fig. 7 2nd harmonic spectra of a porous LSCF-6428 electrode at 600 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

==== NLEIS Interpretation ====
Interpretation of NLEIS spectra instead relies on proposing physical models for phenomena in an electrochemical system and calculating harmonic spectra thereof. In fact, it is through this framework that NLEIS can verify, or at least eliminate, proposed mechanisms which may be indistinguishable in linear EIS.
To extend current knowledge of oxygen reduction behavior on porous LSCF-6428 electrodes, Tim Geary performed extensive NLEIS studies. He performed experiments under several temperatures and PO2 relevant for intermediate temperature SOFC operation and evaluated several modeling scenarios. Details regarding the modeling scenarios can be found in Ref. 6. As an example, Figure 8 compares the results between models assuming oxygen chemisorption and dissociative adsorption limited rate laws. Both models assume the surface behaves identically to the bulk, and bulk transport is the only pathway for oxygen vacancies. Although neither scenario accurately captured the electrode behavior, it did indicate that a more complex framework of surface thermodynamics was merited.

Fig. 8 2nd Harmonic spectra of a porous LSCF-6428 electrode at 650 ˚C. Lines extending from origin are phasor lines intersecting Uk,k at 1/k ω ̃_c; 1/2 ω ̃_c in this case6.

Conducting Tip Atomic Force Microscopy

2020-05-28T17:44:48Z

Cmditradmin: /* Links */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: center; width: 33%">[[Main_Page#Research Equipment, Devices and Techniques|Return to Research Tool Menu]]</td>

</tr>
</table>

=== Overview ===
Atomic Force Microscopy (AFM)is a well established process for visualizing ultrafine surface characteristics. In normal AFM scanning mode a fine needle is drawn very near a surface and is gently bent by the various atomic forces. The conducting tip gives you the chance to measure electrical conductivity at discrete locations and then correlate these measurement with the surface scan that reveals the shape.

see [[wikipedia:Atomic_force_microscope]]

=== Operation ===
In C-AFM a metal-coated cantilever is moved back and forth across a sample’s surface. The vertical deflection of the cantilever is measured by monitoring the deflection of a laser beam reflected off the back of the cantilever, giving a topographic map of the surface. By applying a voltage to the tip and measuring the current flow we generate corresponding maps of sample topography and electrical properties. It is also possible to obtain current-voltage curves at a single point with an area of ~20 nm2.

<swf width="640" height="452">http://depts.washington.edu/cmditr/media/afm.swf</swf>

This video is also available on YouTube:

AFM Video, Part 1]

{{#ev:youtube|H48w-aTE9pg}}

Part 2

{{#ev:youtube|RLardiQXruY}}

=== Significance ===
This is of particular interest to the field of photonics research because the structure of thin coatings has a huge effect on the performance of devices.

Application example from Alex Veneman at U of A.:

Indium Tin Oxide (ITO) is the most commonly used anode in Organic Photovoltaics (OPVs) due to its optical transparency and relatively high electrical conductivity. ITO is an imperfect electrode, and electron transfer between the ITO and adjacent organic layers is hampered by heterogeneous coverage of surface contaminants and the fact that the oxide itself is most likely a heterogeneous mixture of phases with varying electrical properties.

ITO for organic Light-Emitting Diodes (OLEDs) and OPVs is often treated by methods such as detergent or solvent cleaning, oxygen plasma or ozone cleaning and/or coating with poly(3,4-ethylenedioxythiophene) heavily doped with poly(styrenesulfonic acid) (PEDOT:PSS). The effect of these modifications at the nanoscopic level is still not fully understood, and although the effect of current-voltage properties of the devices has been studied, a working model describing their physical effects at the relevant length scales are lacking. In this work we use Conducting-Probe Atomic Force Microscopy (C-AFM) to study these surface modifications at the nanometer length scale, and compare these results to current-voltage data for macroscopic OPVs.

Our results indicate that PEDOT:PSS is a ‘band-aid’ fix for the deeper problem of heterogeneity of the ITO surface. PEDOT:PSS electrically wires over ‘dead’ spots on the ITO, making an electrically uniform electrode, but it also introduces another energy barrier to the device that increases the diode quality factor and thus decreases fill factor. We also find that aggressive acid etching of the ITO surface results in increased homogeneity, and much improved repeatability in the manufacture of devices.

[[Image:Itopics.png|thumb|500px|center|These C-AFM images demonstrate the electrical heterogeneity of the ITO surface. This lack of uniformity is due to carbonaceous impurities and hydroxide species contaminating the surface. Additionally it is unclear whether the ITO is composed of a single or multiple phases of varying electrical activity. Modification of the ITO surface can increase the electrical activity of the film by removing contaminant species and possible changing the relative ratio of phases present on the surface.
]]

[[Image:ITO-IVCURVE.png|thumb|600px|center|Each semi-log plot on the left shows current-voltage curves at several different ~20 nm2 areas on the same organic film. The data indicate that the electrical heterogeneity of the ITO affects the current flowing through the above Copper Phthalocyanine layer (top row). The addition of a PEDOT:PSS mediator layer (bottom row) makes the electrode electrically uniform by allowing current to pass over any insulating regions on the ITO surface. The PEDOT:PSS/CuPc interface is also rectifying in such a manner as to collect photocurrent and suppress dark current.
It should also be noted that the acid etch produces a surface that is already very uniform and is actually hindered by the addition of the PEDOT:PSS layer (right column).
]]
[[category:Research equipment]]
=== Links===
[http://cei.washington.edu/educationspace/presentations/Piffy%20the%20Photoinduced%20Force%20Microscope.pdf Photoinduced Force Microscope]

[https://cleanenergy.shinyapps.io/bundle/ Conducting tip AFM applet]

[http://depts.washington.edu/nanolab/NUE_UNIQUE/NUE_UNIQUE_Workshop.htm Lab activities using AFM]

Conducting Tip Atomic Force Microscopy

2020-05-28T17:39:11Z

Cmditradmin: /* Links */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: center; width: 33%">[[Main_Page#Research Equipment, Devices and Techniques|Return to Research Tool Menu]]</td>

</tr>
</table>

=== Overview ===
Atomic Force Microscopy (AFM)is a well established process for visualizing ultrafine surface characteristics. In normal AFM scanning mode a fine needle is drawn very near a surface and is gently bent by the various atomic forces. The conducting tip gives you the chance to measure electrical conductivity at discrete locations and then correlate these measurement with the surface scan that reveals the shape.

see [[wikipedia:Atomic_force_microscope]]

=== Operation ===
In C-AFM a metal-coated cantilever is moved back and forth across a sample’s surface. The vertical deflection of the cantilever is measured by monitoring the deflection of a laser beam reflected off the back of the cantilever, giving a topographic map of the surface. By applying a voltage to the tip and measuring the current flow we generate corresponding maps of sample topography and electrical properties. It is also possible to obtain current-voltage curves at a single point with an area of ~20 nm2.

<swf width="640" height="452">http://depts.washington.edu/cmditr/media/afm.swf</swf>

This video is also available on YouTube:

AFM Video, Part 1]

{{#ev:youtube|H48w-aTE9pg}}

Part 2

{{#ev:youtube|RLardiQXruY}}

=== Significance ===
This is of particular interest to the field of photonics research because the structure of thin coatings has a huge effect on the performance of devices.

Application example from Alex Veneman at U of A.:

Indium Tin Oxide (ITO) is the most commonly used anode in Organic Photovoltaics (OPVs) due to its optical transparency and relatively high electrical conductivity. ITO is an imperfect electrode, and electron transfer between the ITO and adjacent organic layers is hampered by heterogeneous coverage of surface contaminants and the fact that the oxide itself is most likely a heterogeneous mixture of phases with varying electrical properties.

ITO for organic Light-Emitting Diodes (OLEDs) and OPVs is often treated by methods such as detergent or solvent cleaning, oxygen plasma or ozone cleaning and/or coating with poly(3,4-ethylenedioxythiophene) heavily doped with poly(styrenesulfonic acid) (PEDOT:PSS). The effect of these modifications at the nanoscopic level is still not fully understood, and although the effect of current-voltage properties of the devices has been studied, a working model describing their physical effects at the relevant length scales are lacking. In this work we use Conducting-Probe Atomic Force Microscopy (C-AFM) to study these surface modifications at the nanometer length scale, and compare these results to current-voltage data for macroscopic OPVs.

Our results indicate that PEDOT:PSS is a ‘band-aid’ fix for the deeper problem of heterogeneity of the ITO surface. PEDOT:PSS electrically wires over ‘dead’ spots on the ITO, making an electrically uniform electrode, but it also introduces another energy barrier to the device that increases the diode quality factor and thus decreases fill factor. We also find that aggressive acid etching of the ITO surface results in increased homogeneity, and much improved repeatability in the manufacture of devices.

[[Image:Itopics.png|thumb|500px|center|These C-AFM images demonstrate the electrical heterogeneity of the ITO surface. This lack of uniformity is due to carbonaceous impurities and hydroxide species contaminating the surface. Additionally it is unclear whether the ITO is composed of a single or multiple phases of varying electrical activity. Modification of the ITO surface can increase the electrical activity of the film by removing contaminant species and possible changing the relative ratio of phases present on the surface.
]]

[[Image:ITO-IVCURVE.png|thumb|600px|center|Each semi-log plot on the left shows current-voltage curves at several different ~20 nm2 areas on the same organic film. The data indicate that the electrical heterogeneity of the ITO affects the current flowing through the above Copper Phthalocyanine layer (top row). The addition of a PEDOT:PSS mediator layer (bottom row) makes the electrode electrically uniform by allowing current to pass over any insulating regions on the ITO surface. The PEDOT:PSS/CuPc interface is also rectifying in such a manner as to collect photocurrent and suppress dark current.
It should also be noted that the acid etch produces a surface that is already very uniform and is actually hindered by the addition of the PEDOT:PSS layer (right column).
]]
[[category:Research equipment]]
=== Links===

[https://cleanenergy.shinyapps.io/bundle/ Conducting tip AFM applet]
[http://depts.washington.edu/nanolab/NUE_UNIQUE/NUE_UNIQUE_Workshop.htm Lab activities using AFM]

Radiometer measurements

2020-05-20T17:43:19Z

Cmditradmin: Created page with "<object style="border: 2px solid black;" data="/radiometer/widget.html" width="500" height="240"></object>"

Main Page

2020-05-20T17:28:38Z

Cmditradmin: /* High School */

[[File:CEI_logo_tag_color.360x105.jpg]] 
[http://cei.washington.edu Link to the Clean Energy Institute]

== Clean Energy ==
===Energy Basics===
*[[Energy Basics]]
*[[Survey of Renewables]]
*[[Clean Energy Policy]]

===Solar Basics===
[[Image:Solar_tracker_in_Lixouri.jpg|thumb|200px|Solar tracker]]
*[[Electromagnetic Radiation]]
*[[Solar Technologies]]
*[[Solar Cell Introduction|Solar Cell Introduction]]
*[[Physics of Solar Cells]]
*[[Band-gap]]
 

===Solar Materials===
[[Image:Perovskite_ABO3.jpg|thumb|200px|Perovskite]]
*[[Silicon Solar]]
*[[Perovskites]]
*[[CIGS, CIS, CdTe]]
*[[Quantum Dots]]
*[[Luminescent Solar Concentrator]]
*[[Graetzel or Dye Sensitized Solar Cell]]
*[[Photonic Crystal]]
*[[2D Materials]]
 

'''Organic Solar Cells'''
[[Image:Opvtestcells.png|thumb|200px|OPV Test Cells]]
*[[Major Processes in Organic Solar Cells]]
*[[Organic Heterojunctions in Solar Cells]]
*[[Energy vs Charge Transfer at Heterojunctions]]
*[[Current OPV Research Directions]]
 

== Energy Storage ==
=== Introduction ===
[[Image:UET.jpg|thumb|200px|Unienergy Flow Battery]]
*[[Storage Basics]]
*[[Battery Basics]]
*[[MACCOR Battery Tester]]
*[[Hydrogen Policy]]
*[[Data Center Energy Management]]

=== Battery Types ===
*[[Coin Cell Battery]]
*[[Lithium Ion Battery]]
*[[Redox Flow Battery]]
*[[Supercapacitor]]
=== Battery Materials ===
*[[Solid State Electrolyte]]
 

== Grid Integration ==
[[Image:EGRIDSubregionmap.jpg|thumb|200px|US Power Grids]]
*[[Smart Grid]]
*[[Grid Integration of Renewables]]
*[[MatPower Simulation of Submodular Optimization for Voltage Control in Power Systems]]
 

== Photonics Core Concepts and Applications ==
[http://depts.washington.edu/cmditr/media/Photonics.html Concept Map CMDITR]

[[Image:Wordle3.png|thumb|center|600px|This graphic was created by processing the CMDITR 2009 annual report in the Wordle program. The larger the word the more times it appeared in the text.]]
 

===Basics of Light ===
[[Image:Snells_law_wavefronts.gif|thumb|150px|]]
*[[Propagation, Reflection and Refraction]]
*[[Dispersion and Scattering of Light]]
*[[Diffraction of Light]]

*[[Luminescence Phenomena]]
*[[Color and Chromaticity]]

 

=== Optical Fibers, Waveguides, and Lasers ===
[[Image:Military_laser_experiment.jpg|thumb|200px|]]

*[[Optical Fibers]]
*[[Total Internal Reflection]]
*[[Planar Dielectric Waveguides]]
*[[Optical Fiber Waveguides]]
*[[Dispersion and Attenuation Phenomena]]
*[[Lasers]]

===Molecular Orbitals===
[[Image:HAtomOrbitals.png|thumb|200px]]
*[[Atomic Orbitals and Nodes]]
*[[Electronegativity and Bonding Between Atoms]]
*[[Sigma and pi Orbitals|Sigma and Pi Orbitals]]
*[[Polarization and Polarizability]]
*[[Electronic Coupling Between Orbitals]]
*[[Donors and Acceptors]]
 

===Electronic Band Structure of Organic Materials===
[[Image:Ethylene.JPG|thumb|200px|]]
*[[Introduction to Band Structure]]
*[[Electronic Structure of Hydrogen]]
*[[The Polyene Series]]
*[[Bloch's Theorem]]
*[[Electrical Properties]]
*[[Electronic States vs Molecular Levels]]
 

===Absorption and Emission of Light===
[[Image:Abs Emis stokes.png|thumb|200px|]]
*[[Introduction to Absorption]]
*[[Changes in Absorption Spectra]]
*[[Jablonksi Diagram]]
*[[Fluorescence Process]]
*[[Transition Dipole Moment]]
*[[Absorption and Emission]]
*[[Photochromism]]
*[[Interchain Interactions]]

 

===Transport Properties===
[[Image:rubrene.png|thumb|150px|]]
*[[Charge Carrier Mobility]]
*[[Band Regime versus Hopping Regime]]
*[[Electronic Coupling]]
*[[Model Calculations of Electronic Coupling]]
*[[Marcus Theory and Reorganization Energy]]
*[[Electron-Phonon Coupling]]

 

===Liquid Crystals and Displays===
[[Image:Fächertextur.jpg|thumb|200px|]]
*[[Liquid Crystals]]
*[[Double Refraction and Birefringence]]
*[[Director – Degrees of Order in Liquid Crystals]]
*[[Classification and Examples of Liquid Crystals]]
*[[Alignment]]
*[[Freederickz Transition and Dielectric Anisotropy]]
*[[Liquid Crystal Displays]]

 

===Organic Light Emitting Diodes===
[[Image:PNNL_Light_Lab_041.jpg|thumb|200px|Blue phosphorescent OLED developed by Pacific Northwest National Laboratory.]]
*[[OLED Device Applications]]
*[[Light Emitting Electrochemical Processes]]
*[[The OLED Test Cell]]
*[[What is a Light Emitting Diode?]]
*[[The First OLEDs]]
*[[Organic/Organic Heterojunctions in OLEDs]]
*[[OLED Charge Mobilities]]
*[[Organic Heterojunctions]]
*[[Fluorescent/Phosphorescent Dopants]]
*[[Metal Complex Dopants]]
 

===Organic Electronics===
*[[Organic Electronics Overview]]
*[[Synthesis of Organic Semiconductors]](In progress)
*[[Organic Field Effect Transistors]]

==Non linear Optics and Devices==

===Quantum Mechanical and Perturbation Theory of Polarizability===
*[[Quantum-Mechanical Theory of Molecular Polarizabilities]]
*[[Mathematical Expansion of the Dipole Moment]]
*[[Perturbation Theory]]

===Second-order Processes, Materials & Characterization ===
[[Image:MachZehnder.gif|thumb|200px]]
*[[Second-order Processes]]
*[[Structure-Property Relationships]]
*[[Second-order NLO Materials]]
*[[Second-order Material Design]]
*[[Terahertz Radiation]]
*[[Second-order Material Characterization]]

 

===Third-order Processes, Materials & Characterization ===
[[Image:Tpa_concentrated.png|thumb|100px|]]
*[[Introduction to Third-order Processes and Materials]]
*[[Two Photon Absorption]]
*[[Characterization of Third-order Materials]]
 

===Organic Photonics Applications in Information Technology ===
[[Image:Dualmz packaged.png|thumb|200px|]]
*[[Optical Networks]]
*[[Passive Optical Polymers]]
*[[Electro-optic Polymers and Devices]]
*[[Materials Processing and Fabrication]]
 

===Photonics Integration===
[[Image:Si_waveguide_em.jpg‎|thumb|200px|]]
*[[The Need for Photonic Integration]]
*[[Photonics Integration]]
 

== Research Equipment, Devices and Techniques ==
These wiki pages include training videos produced at University of Washington, University of Arizona, Georgia Tech, and Norfolk State University. These videos can also be accessed through the [http://www.youtube.com/cmditr | CMDITR YouTube channel]
 
[[Image:PES.jpg|right|200px|link=Photoelectron Spectrometer XPS and UPS]]

=== Characterization ===
'''Composition'''
*[[Photoelectron Spectrometer XPS and UPS]]
*[[UV/VIS/NIR spectrometer]]
*[[Fourier transform infrared spectroscopy (FTIR)/Raman spectroscopy]]
*[[NMR spectrometer]]
*[[Electron Spin Resonance (ESR)]]
*[[Energy Dispersive X-Ray Spectroscopy(EDX)]]
*[[Real Time Gas Analyzer]]

 
'''Structure'''
[[Image:Sirion sem.png|thumb|200px| ]]
*[[Scanning Electron Microscope]]
*[[Transmission Electron Microscope]]
*[[Conducting Tip Atomic Force Microscopy]]
*[[Photoconducting Tip Atomic Force Microscopy pcAFM]]
*[[Profilometer]]
*[[X-ray Diffraction]]
*[[Raman microscope]]

 
'''Properties'''
[[Image:Zscan.png|thumb|200px| ]]
*[[Two-Photon Spectroscopy]]
*[[Hyper Rayleigh Scattering]]
*[[Teng-Man Method]]
*[[Attenuated Total Reflectance]]
*[[External quantum efficiency]]
*[[Metricon Prism Coupler]]
*[[Femtosecond Z-Scan Spectrometer]]
*[[Femtosecond Pump-Probe Spectroscopy]]
*[[Electron Loss Spectroscopy and Plasmonics]]
*[[Confocal microscope]]
*[[Superconducting Quantum Interference Device- SQUID]]
*[[Fluorometer]]
*[[Ellipsometer]]
*[[Surface Analyzer]]
*[[Xray Absorption Near Edge Spectroscopy- XANES]]
*[[Grazing-Incidence Small-Angle X-ray Scattering (GISAXS)]]
*[[Electrochemical Impedence Spectroscopy EIS]]
*[[Pulse Pick Laser for measuring excitons]]
*[[Automation of current / voltage measurements]]
*[[Measuring Capacitance]]
*[[MACCOR Battery Tester]]

=== Synthesis===

*[[Lead Sulfide Quantum Dot Synthesis]]
*[[Silver Nano Prism Synthesis]]
*[[Cadmium Selenide Nanocrystal Synthesis]]
*[[General Synthesis Techniques]]
*[[Ionic Liquid Synthesis]]
*[[Molecular Dynamics Simulation]]
*[[Supercritical fluids]]

===Device Fabrication===
[[Image:glovebox.png|thumb|200px| ]]
*[[Spin coater]]
*[[Make a Coin Cell Battery]]
*[[Make an Electrode for Coin Cell]]
*[[Make a Perovskite Solar Cell]]
*[[Graphene Synthesis and Characterization]]

*[[Organic Photovoltaic Fabrication and Test Apparatus]]
*[[E-beam Lithography]]
*[[Physical Vapor Deposition PVD - Vacuum/thermal coater]]
*[[Physical Vapor Transport]]
*[[Magnetron Sputtering Coater]]
*[[Chemical Vapor Deposition]]
*[[Spark Plasma Sintering (SPS)]]
*[[Laser Mask Generator]]
*[[OFET fabrication and characterization]]
*[[Mach-Zehnder Device]]

'''In Development'''
*Reactive ion etcher
*Plasma etcher
*[[Molecular Beam Epitaxy]]
*[[Atomic Layer Deposition ALD]]
*[[Liquid Phase Deposition]]

*SPM

== Professional Development for Research and Career Planning ==
*[[Writing a Successful Proposal]]
*[[How to Keep a Lab Notebook]]
*[[Writing a Scientific Paper]]
*[[How to Give a Research Presentation]]
*[[Poster Session]]
*[[Basic Illustration Techniques]]
*[[Mentoring]]
*[[Teaching]]
*[[Outreach | Communicating Science to the Public]]
*[[Responsible Conduct of Research- RCR]]
*[[Career Planning]]
*[[Green Chemistry]]
*[[Laboratory Safety]]
*[[Keys to Success in Graduate School]]

==Acronyms and Unit Abbreviations==
*[[Acronyms]]
*[[Variables and Constants]]
*[[Units]]

==[[External Education Links]]==

==K-12 Outreach Kits and Labs==
[[Image:AssembledCell_small.JPG|thumb|200px|]]
=== Middle School ===
*[[K-12 Outreach Introduction]]
*[[Basic Optics - Outreach Kit]]
*[[Photovoltaics- Outreach Kit]]
*[[Solar Car Derby]]
*[[Solar Cell Photon Toss Exhibit]]
*[[Cuprous Oxide Solar Cell]]

=== High School ===
*[[Lasers and Telecommunication- Outreach Kit]]
*[[Nanocrystalline - Dye Solar Cell Lab]]
*[[Phosphorescent Decay Lab]]
*[[PV Characterization Lab]]
*[[SunDawg Exhibit]]
*[[Radiometer measurements]]
 

=== Introductory College ===
*[[Organic Photovoltaic Device Lab]]
*[[Nanoparticle Lab]]
 

==[[Suggested Wiki Sequence By Audience]]==

== [[Photonics Wiki Showcase]] ==

== [[Concept Map]] ==

==[[Credits and Reviewers]]==

Supercritical fluids

2020-05-12T19:02:58Z

Cmditradmin: Created page with "Supercritical fluids are used in certain synthesis processes. {{#ev:youtube|rBdFXudHz6M}}"

Supercritical fluids are used in certain synthesis processes.
{{#ev:youtube|rBdFXudHz6M}}

Main Page

2020-05-12T19:00:00Z

Cmditradmin: /* Synthesis */

[[File:CEI_logo_tag_color.360x105.jpg]] 
[http://cei.washington.edu Link to the Clean Energy Institute]

== Clean Energy ==
===Energy Basics===
*[[Energy Basics]]
*[[Survey of Renewables]]
*[[Clean Energy Policy]]

===Solar Basics===
[[Image:Solar_tracker_in_Lixouri.jpg|thumb|200px|Solar tracker]]
*[[Electromagnetic Radiation]]
*[[Solar Technologies]]
*[[Solar Cell Introduction|Solar Cell Introduction]]
*[[Physics of Solar Cells]]
*[[Band-gap]]
 

===Solar Materials===
[[Image:Perovskite_ABO3.jpg|thumb|200px|Perovskite]]
*[[Silicon Solar]]
*[[Perovskites]]
*[[CIGS, CIS, CdTe]]
*[[Quantum Dots]]
*[[Luminescent Solar Concentrator]]
*[[Graetzel or Dye Sensitized Solar Cell]]
*[[Photonic Crystal]]
*[[2D Materials]]
 

'''Organic Solar Cells'''
[[Image:Opvtestcells.png|thumb|200px|OPV Test Cells]]
*[[Major Processes in Organic Solar Cells]]
*[[Organic Heterojunctions in Solar Cells]]
*[[Energy vs Charge Transfer at Heterojunctions]]
*[[Current OPV Research Directions]]
 

== Energy Storage ==
=== Introduction ===
[[Image:UET.jpg|thumb|200px|Unienergy Flow Battery]]
*[[Storage Basics]]
*[[Battery Basics]]
*[[MACCOR Battery Tester]]
*[[Hydrogen Policy]]
*[[Data Center Energy Management]]

=== Battery Types ===
*[[Coin Cell Battery]]
*[[Lithium Ion Battery]]
*[[Redox Flow Battery]]
*[[Supercapacitor]]
=== Battery Materials ===
*[[Solid State Electrolyte]]
 

== Grid Integration ==
[[Image:EGRIDSubregionmap.jpg|thumb|200px|US Power Grids]]
*[[Smart Grid]]
*[[Grid Integration of Renewables]]
*[[MatPower Simulation of Submodular Optimization for Voltage Control in Power Systems]]
 

== Photonics Core Concepts and Applications ==
[http://depts.washington.edu/cmditr/media/Photonics.html Concept Map CMDITR]

[[Image:Wordle3.png|thumb|center|600px|This graphic was created by processing the CMDITR 2009 annual report in the Wordle program. The larger the word the more times it appeared in the text.]]
 

===Basics of Light ===
[[Image:Snells_law_wavefronts.gif|thumb|150px|]]
*[[Propagation, Reflection and Refraction]]
*[[Dispersion and Scattering of Light]]
*[[Diffraction of Light]]

*[[Luminescence Phenomena]]
*[[Color and Chromaticity]]

 

=== Optical Fibers, Waveguides, and Lasers ===
[[Image:Military_laser_experiment.jpg|thumb|200px|]]

*[[Optical Fibers]]
*[[Total Internal Reflection]]
*[[Planar Dielectric Waveguides]]
*[[Optical Fiber Waveguides]]
*[[Dispersion and Attenuation Phenomena]]
*[[Lasers]]

===Molecular Orbitals===
[[Image:HAtomOrbitals.png|thumb|200px]]
*[[Atomic Orbitals and Nodes]]
*[[Electronegativity and Bonding Between Atoms]]
*[[Sigma and pi Orbitals|Sigma and Pi Orbitals]]
*[[Polarization and Polarizability]]
*[[Electronic Coupling Between Orbitals]]
*[[Donors and Acceptors]]
 

===Electronic Band Structure of Organic Materials===
[[Image:Ethylene.JPG|thumb|200px|]]
*[[Introduction to Band Structure]]
*[[Electronic Structure of Hydrogen]]
*[[The Polyene Series]]
*[[Bloch's Theorem]]
*[[Electrical Properties]]
*[[Electronic States vs Molecular Levels]]
 

===Absorption and Emission of Light===
[[Image:Abs Emis stokes.png|thumb|200px|]]
*[[Introduction to Absorption]]
*[[Changes in Absorption Spectra]]
*[[Jablonksi Diagram]]
*[[Fluorescence Process]]
*[[Transition Dipole Moment]]
*[[Absorption and Emission]]
*[[Photochromism]]
*[[Interchain Interactions]]

 

===Transport Properties===
[[Image:rubrene.png|thumb|150px|]]
*[[Charge Carrier Mobility]]
*[[Band Regime versus Hopping Regime]]
*[[Electronic Coupling]]
*[[Model Calculations of Electronic Coupling]]
*[[Marcus Theory and Reorganization Energy]]
*[[Electron-Phonon Coupling]]

 

===Liquid Crystals and Displays===
[[Image:Fächertextur.jpg|thumb|200px|]]
*[[Liquid Crystals]]
*[[Double Refraction and Birefringence]]
*[[Director – Degrees of Order in Liquid Crystals]]
*[[Classification and Examples of Liquid Crystals]]
*[[Alignment]]
*[[Freederickz Transition and Dielectric Anisotropy]]
*[[Liquid Crystal Displays]]

 

===Organic Light Emitting Diodes===
[[Image:PNNL_Light_Lab_041.jpg|thumb|200px|Blue phosphorescent OLED developed by Pacific Northwest National Laboratory.]]
*[[OLED Device Applications]]
*[[Light Emitting Electrochemical Processes]]
*[[The OLED Test Cell]]
*[[What is a Light Emitting Diode?]]
*[[The First OLEDs]]
*[[Organic/Organic Heterojunctions in OLEDs]]
*[[OLED Charge Mobilities]]
*[[Organic Heterojunctions]]
*[[Fluorescent/Phosphorescent Dopants]]
*[[Metal Complex Dopants]]
 

===Organic Electronics===
*[[Organic Electronics Overview]]
*[[Synthesis of Organic Semiconductors]](In progress)
*[[Organic Field Effect Transistors]]

==Non linear Optics and Devices==

===Quantum Mechanical and Perturbation Theory of Polarizability===
*[[Quantum-Mechanical Theory of Molecular Polarizabilities]]
*[[Mathematical Expansion of the Dipole Moment]]
*[[Perturbation Theory]]

===Second-order Processes, Materials & Characterization ===
[[Image:MachZehnder.gif|thumb|200px]]
*[[Second-order Processes]]
*[[Structure-Property Relationships]]
*[[Second-order NLO Materials]]
*[[Second-order Material Design]]
*[[Terahertz Radiation]]
*[[Second-order Material Characterization]]

 

===Third-order Processes, Materials & Characterization ===
[[Image:Tpa_concentrated.png|thumb|100px|]]
*[[Introduction to Third-order Processes and Materials]]
*[[Two Photon Absorption]]
*[[Characterization of Third-order Materials]]
 

===Organic Photonics Applications in Information Technology ===
[[Image:Dualmz packaged.png|thumb|200px|]]
*[[Optical Networks]]
*[[Passive Optical Polymers]]
*[[Electro-optic Polymers and Devices]]
*[[Materials Processing and Fabrication]]
 

===Photonics Integration===
[[Image:Si_waveguide_em.jpg‎|thumb|200px|]]
*[[The Need for Photonic Integration]]
*[[Photonics Integration]]
 

== Research Equipment, Devices and Techniques ==
These wiki pages include training videos produced at University of Washington, University of Arizona, Georgia Tech, and Norfolk State University. These videos can also be accessed through the [http://www.youtube.com/cmditr | CMDITR YouTube channel]
 
[[Image:PES.jpg|right|200px|link=Photoelectron Spectrometer XPS and UPS]]

=== Characterization ===
'''Composition'''
*[[Photoelectron Spectrometer XPS and UPS]]
*[[UV/VIS/NIR spectrometer]]
*[[Fourier transform infrared spectroscopy (FTIR)/Raman spectroscopy]]
*[[NMR spectrometer]]
*[[Electron Spin Resonance (ESR)]]
*[[Energy Dispersive X-Ray Spectroscopy(EDX)]]
*[[Real Time Gas Analyzer]]

 
'''Structure'''
[[Image:Sirion sem.png|thumb|200px| ]]
*[[Scanning Electron Microscope]]
*[[Transmission Electron Microscope]]
*[[Conducting Tip Atomic Force Microscopy]]
*[[Photoconducting Tip Atomic Force Microscopy pcAFM]]
*[[Profilometer]]
*[[X-ray Diffraction]]
*[[Raman microscope]]

 
'''Properties'''
[[Image:Zscan.png|thumb|200px| ]]
*[[Two-Photon Spectroscopy]]
*[[Hyper Rayleigh Scattering]]
*[[Teng-Man Method]]
*[[Attenuated Total Reflectance]]
*[[External quantum efficiency]]
*[[Metricon Prism Coupler]]
*[[Femtosecond Z-Scan Spectrometer]]
*[[Femtosecond Pump-Probe Spectroscopy]]
*[[Electron Loss Spectroscopy and Plasmonics]]
*[[Confocal microscope]]
*[[Superconducting Quantum Interference Device- SQUID]]
*[[Fluorometer]]
*[[Ellipsometer]]
*[[Surface Analyzer]]
*[[Xray Absorption Near Edge Spectroscopy- XANES]]
*[[Grazing-Incidence Small-Angle X-ray Scattering (GISAXS)]]
*[[Electrochemical Impedence Spectroscopy EIS]]
*[[Pulse Pick Laser for measuring excitons]]
*[[Automation of current / voltage measurements]]
*[[Measuring Capacitance]]
*[[MACCOR Battery Tester]]

=== Synthesis===

*[[Lead Sulfide Quantum Dot Synthesis]]
*[[Silver Nano Prism Synthesis]]
*[[Cadmium Selenide Nanocrystal Synthesis]]
*[[General Synthesis Techniques]]
*[[Ionic Liquid Synthesis]]
*[[Molecular Dynamics Simulation]]
*[[Supercritical fluids]]

===Device Fabrication===
[[Image:glovebox.png|thumb|200px| ]]
*[[Spin coater]]
*[[Make a Coin Cell Battery]]
*[[Make an Electrode for Coin Cell]]
*[[Make a Perovskite Solar Cell]]
*[[Graphene Synthesis and Characterization]]

*[[Organic Photovoltaic Fabrication and Test Apparatus]]
*[[E-beam Lithography]]
*[[Physical Vapor Deposition PVD - Vacuum/thermal coater]]
*[[Physical Vapor Transport]]
*[[Magnetron Sputtering Coater]]
*[[Chemical Vapor Deposition]]
*[[Spark Plasma Sintering (SPS)]]
*[[Laser Mask Generator]]
*[[OFET fabrication and characterization]]
*[[Mach-Zehnder Device]]

'''In Development'''
*Reactive ion etcher
*Plasma etcher
*[[Molecular Beam Epitaxy]]
*[[Atomic Layer Deposition ALD]]
*[[Liquid Phase Deposition]]

*SPM

== Professional Development for Research and Career Planning ==
*[[Writing a Successful Proposal]]
*[[How to Keep a Lab Notebook]]
*[[Writing a Scientific Paper]]
*[[How to Give a Research Presentation]]
*[[Poster Session]]
*[[Basic Illustration Techniques]]
*[[Mentoring]]
*[[Teaching]]
*[[Outreach | Communicating Science to the Public]]
*[[Responsible Conduct of Research- RCR]]
*[[Career Planning]]
*[[Green Chemistry]]
*[[Laboratory Safety]]
*[[Keys to Success in Graduate School]]

==Acronyms and Unit Abbreviations==
*[[Acronyms]]
*[[Variables and Constants]]
*[[Units]]

==[[External Education Links]]==

==K-12 Outreach Kits and Labs==
[[Image:AssembledCell_small.JPG|thumb|200px|]]
=== Middle School ===
*[[K-12 Outreach Introduction]]
*[[Basic Optics - Outreach Kit]]
*[[Photovoltaics- Outreach Kit]]
*[[Solar Car Derby]]
*[[Solar Cell Photon Toss Exhibit]]
*[[Cuprous Oxide Solar Cell]]

=== High School ===
*[[Lasers and Telecommunication- Outreach Kit]]
*[[Nanocrystalline - Dye Solar Cell Lab]]
*[[Phosphorescent Decay Lab]]
*[[PV Characterization Lab]]
*[[SunDawg Exhibit]]
 

=== Introductory College ===
*[[Organic Photovoltaic Device Lab]]
*[[Nanoparticle Lab]]
 

==[[Suggested Wiki Sequence By Audience]]==

== [[Photonics Wiki Showcase]] ==

== [[Concept Map]] ==

==[[Credits and Reviewers]]==

Cadmium Selenide Nanocrystal Synthesis

2020-05-12T18:58:01Z

Cmditradmin: /* Technique */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: center; width: 33%">[[Main_Page#Synthesis_and_Fabrication|Return to Research Tool Menu]]</td>

</tr>
</table>

=== Overview ===
[[Image:CdSeqdots.jpg|thumb|300px|Cadmium Selenide Quantum Dots]]
Cadmium selenide nanocrystals or quantum dots are used to create absorbers whose bandgap can be fine tuned using the size of the crystals. The particles can be incorporated into LEDs or photovoltaic devices.
 

=== Significance ===
{{#ev:youtube|JJ8QADJnpc0}}

=== Technique ===
{{#ev:youtube|6oDHZtxTEoo}}
{{#ev:youtube|Fi055GPBofo}}

=== Links ===
[[Silver Nano Prism Synthesis]]

[[Lead Sulfide Quantum Dot Synthesis]]

[[wikipedia:Cadmium selenide]]

Cadmium Selenide Nanocrystal Synthesis

2020-05-12T18:56:37Z

Cmditradmin: /* Technique */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: center; width: 33%">[[Main_Page#Synthesis_and_Fabrication|Return to Research Tool Menu]]</td>

</tr>
</table>

=== Overview ===
[[Image:CdSeqdots.jpg|thumb|300px|Cadmium Selenide Quantum Dots]]
Cadmium selenide nanocrystals or quantum dots are used to create absorbers whose bandgap can be fine tuned using the size of the crystals. The particles can be incorporated into LEDs or photovoltaic devices.
 

=== Significance ===
{{#ev:youtube|JJ8QADJnpc0}}

=== Technique ===
{{#ev:youtube|6oDHZtxTEoo}}
{{#ev:youtube|rBdFXudHz6M}}

=== Links ===
[[Silver Nano Prism Synthesis]]

[[Lead Sulfide Quantum Dot Synthesis]]

[[wikipedia:Cadmium selenide]]

Molecular Dynamics Simulation

2020-05-12T18:53:18Z

Cmditradmin: /* References */

Molecular Dynamics Simulations

==Introduction==

Molecular dynamics (MD) simulations are a computational tool for exploring molecular scale structure and dynamics in a wide variety of materials. They have been used to study solutions, bulk materials, interfaces, crystal structures, proteins, and polymers. Researchers can use this computational approach to explore properties and behaviors of these materials at small length and time scales that are not always directly accessible by experimental methods. It can also provide new perspectives or additional information about complex phenomena observed in experiments.1 MD can be used to study diffusion, polymer self-assembly, preferred molecular conformations, and more.

There is a diverse and complex set of bonded and non-bonded interactions between atoms and molecules that arise from electromagnetic forces.2 The most accurate method for modeling these interactions is a quantum mechanical approach. However, these ab initio calculations are computationally expensive and are therefore limited in the time and length scales at which they can model molecular systems. Molecular dynamics simulations instead approximate these interactions using classical mechanics. While some accuracy is lost, these systems are still able to capture larger-scale behaviors on the order of picoseconds to milliseconds (ps-ms) and Angstroms to millimeters (Å-mm), and provide researchers with useful information.

[[File:P3HT.png|thumb]]

Figure 1. Snapshot from a molecular dynamics simulation of poly(3-hexylthiophene) (P3HT). Alkyl side chains (carbon and hydrogen atoms) are shown in blue, backbone hydrogen atoms are shown in white, backbone carbon atoms are shown in red, and sulfur atoms are shown in yellow.

==Applications in Clean Energy Research==

Molecular dynamics simulations have been used to study materials in many areas of clean energy research, such as solar cell technologies or batteries. These tools can give researchers a deeper understanding of the materials and can also help researchers design improved materials for new technologies. For example, molecular dynamics simulations have been critical in understanding charge transport mechanisms in conjugated polymers (CPs), semiconducting materials that are found in the active layers of organic photovoltaics (OPVs). Their structure is comprised of a conjugated backbone with alternating double and single bonds that enable charge transport. Side chains are also included in the polymer structure to improve their solubility. While not yet as efficient, these polymers enable flexible, lightweight, and solution-processable alternatives to more prevalent inorganic equivalents.

Poly(3-hexylthiophene) (P3HT) is a common, well-studied conjugated polymer in the field. It provides a model system for studying the molecular behavior of CPs and how it relates to their macroscopic performance. In the work of McMahon, et al., researchers used atomistic molecular dynamics simulations of P3HT to understand the effects of structural defects in the polymer chain on the charge delocalization in the material.3 MD simulations were used to capture large-scale molecular structures at various time points along the trajectory. These ‘frozen’ structures were then fed to more accurate quantum mechanic calculations to understand the charge movement. Another group, Tapping and coworkers, used coarse-grained molecular dynamics to study P3HT on an even larger scale.4 They paired the coarse-grained structures of self-assembled P3HT nanofibers in solution with ab-initio approaches to track exciton movement through the fiber.

==Fundamentals of Molecular Dynamics Simulations==

==Force Field==

Atoms in a molecular dynamics simulation will move in response to the sum of bonded and non-bonded interactions with neighboring atoms. These include bonds, van der Waals forces, steric repulsion and Coulomb forces. There are also contributions from the angles and dihedrals (planarity) formed by three or more bonded atoms. All bonded and non-bonded interactions are approximated with equations of various functional forms and constants derived from ab initio calculations or empirical fitting approaches. For some parameters, energy as a function of a conformational change (e.g. distance between two atoms) is fit to determine the relevant parameters (e.g. bond energy and distance constants). Other parameters are modified to recreate macroscopic properties of the system (e.g. density) during a simulation.

The molecular dynamics force field can be defined as an equation and set of constants that describe the potential energy from all bonded and non-bonded interactions in an atomic system. Different force fields can be found in the literature for a wide range of molecular system. They vary in both the functional form and the parameter values, all with varying degrees of specificity in the atoms, molecules and materials to which they can be applied. A simple functional form for a force field (Class 1) is defined in equation (1) below.5

<math>V_{FF}=\mathrm{\Sigma}_{bonds}\frac{1}{2}K_{b,ij}\left(b_{ij}-b_{o,ij}\right)^2+\mathrm{\Sigma}_{angles}\frac{1}{2}K_{\theta,ijk}\left(\theta_{ijk}-\theta_{o,ijk}\right)^2+\mathrm{\Sigma}_{impropers}\frac{1}{2}K_{\zeta,ijkl}\left(\zeta_{ijkl}-\zeta_{o,ijkl}\right)^2+\ \mathrm{\Sigma}_{dihedrals}K_{\phi,ijkl}\left(1+cos{\left(n\phi_{ijkl}-\delta_n\right)}\right)+\ \mathrm{\Sigma}_{pairs}\left[4\epsilon_{ij}\left[\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^6\right]+K_{Coulomb}\frac{q_iq_j}{\kappa r_{ij}}\right]</math>(1)

where:

<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )

<math>b_ij = </math>distance between bonded atoms i and j, units of distance

<math>K_{b,ij} = </math>bond energy constant for bonded atoms i and j, units of \frac{energy}{distance^2}

<math>b_{ij} =</math> distance between bonded atoms i and j, units of distance

<math>b_{o,ij} =</math> bond distance constant for bonded atoms i and j, units of distance

<math>K_{\theta,ijk} = </math>angle energy constant for bonded atoms i, j, and k, units of \frac{energy}{degrees^2}

<math>\theta_{ijk} = </math>angle between bonded atoms i, j, and k, units of degrees

<math>\theta_{o,ijk} =</math> angle constant for bonded atoms i, j, and k, units of degrees

<math>K_{\zeta,ijkl} =</math> improper dihedral energy constant for bonded atoms i, j, k, and l, units of \frac{energy}{degrees^2}

<math>\zeta_{ijkl} = </math>improper angle for bonded atoms i, j, k, and l, units of degrees

<math>\zeta_{o,ijkl} =</math> improper angle constant for bonded atoms i, j, k, and l, units of degrees

<math>K_{\phi,ijkl} =</math> dihedral energy constant for linearly bonded atoms i, j, k, and l, units of energy

<math>n = </math>integer

<math>\phi_{ijkl} =</math> dihedral angle for linearly bonded atoms i, j, k, and l, units of degrees

<math>\delta_n = </math>dihedral angle constant for linearly bonded atoms i, j, k, and l, units of degrees

<math>\epsilon_{ij} =</math> Lennard-Jones energy constant for non-bonded atoms i and j, units of energy

<math>\sigma_{ij} =</math> Lennard-Jones distant constant for non-bonded atoms i and j, units of distance

<math>r_{ij} =</math> distance between non-bonded atoms i and j, units of distance

<math>K_{Coulomb} =</math> Coulomb constant,\<math> 8.988\ \times\ {10}^9\frac{Nm^2}{C^2}</math>

<math>q_i, q_j = </math>charges of non-bonded atoms i and j, units of elementary\ charge

<math>\kappa =</math> dielectric constant, unitless

For an interactive thorough exploration of this force field, visit https://interactive-md-ff.herokuapp.com/.

Classical mechanics are used to determine the forces each atom experiences due to these bonded and non-bonded interactions as well as the direction and speed at which they move in response to those forces. Following Newton’s second law of motion and utilizing the force field in equation (1), the force can be defined as1,6:
<math>F=ma=-\frac{\delta V_{FF}}{\delta r}</math>(2)

where:
F = force

m = mass

a = acceleration

This equation can then be solved for the position and velocity of all atoms at every time step during the simulation. This algorithm is known as the integrator.

==Integrator==

The full length of the simulation in time (e.g. 1 nanosecond) is broken down into discrete time steps (e.g. 1 femtosecond). At each time step, the bonded and non-bonded forces on each atom are summed using equations (1) and (2), and the atoms ‘step’ forward in position and time in response to those forces. One method to approximate an atom’s next position is the Velocity-Verlet algorithm shown in equations (3) and (4).6 This integrator computes an atom’s position and velocity at every time step. By stringing this information together at every time step over the duration of the simulation (e.g. 1,000,000 steps of 1 fs each for a 1 ns simulation), a trajectory of the atomic motions can be created. This data can then be analyzed to uncover information about the molecular-level behavior in the system (e.g. diffusion, mean squared displacement).

<math>r\left(t+\mathrm{\Delta t}\right)=r\left(t\right)+\mathrm{\Delta\ t}\ v\left(t\right)+\frac{\mathrm{\Delta}t^2F(t)}{2m}</math> (3)

<math>v\left(t+\mathrm{\Delta t}\right)=v\left(t\right)+\frac{\mathrm{\Delta t}(F\left(t\right)+\ F\left(t+\mathrm{\Delta t}\right))}{2m}\ </math>(4)

where:
r = atomic position
v = velocity
F = force
m = atomic mass
t = current time in simulation
Δt = time step

At the first time step, when t+Δt = 0, there is no information about the about the forces acting on the atoms or their velocities at time t. Therefore, the simulation is initialized with random set of velocities taken from a distribution based around the simulation’s temperature. The system must then be allowed to reach equilibration before the data can be used for analysis.

==Ensembles==

In a molecular dynamics simulation, the behavior of a fixed number of atoms is explored (constant N). Additional simulation conditions are also applied with respect to volume, pressure, and energy. There are three common ensembles that define these simulation conditions: NVE (constant number of atoms, volume, and energy), NVT (constant number of atoms, volume, and temperature) and NPT (constant number of atoms, pressure, and temperature). A variety of algorithms have been developed to enforce these conditions, such as the Andersen or Nosé-Hoover thermostats for controlling temperature. A researcher will choose an ensemble based on the questions they are trying to answer.6

==Atomistic and Coarse-Grained Simulations==

Depending on the time and length scales a researcher would like to explore, they may choose to use a coarse-grained force field. Instead of modeling each atom discretely, small clusters of atoms are grouped together into a single functional group and the interactions are approximated at a larger scale7, saving time in calculating interactions at every atomic site and allowing the simulation to reach larger lengths and times with reasonable computational expense. An example of coarse-graining a molecule is shown in Figure 2.

[[File:coarse_grain_P3HT.png|thumb]]

Figure 2. Poly(3-hexylthiophene) (P3HT) monomer for a coarse-grained molecular dynamics simulation. Clusters of atoms are turned into larger ‘beads’ with force field parameters representative of the large-scale collective behavior of the atoms or functional groups.

==References==

# A. Arbe, F. Alvarez and J. Colmenero, Neutron scattering and molecular dynamics simulations: synergetic tools to unravel structure and dynamics in polymers, Soft Matter, 2012, 8, 8257.
# J. N. Israelachvili, Intermolecular and Surface Forces, Elsevier Inc., 3rd ed., 2011.
# D. P. McMahon, D. L. Cheung, L. Goris, J. Dacuña, A. Salleo and A. Troisi, Relation between microstructure and charge transport in polymers of different regioregularity, J. Phys. Chem. C, 2011, 115, 19386–19393.
# P. C. Tapping, S. N. Clafton, K. N. Schwarz, T. W. Kee and D. M. Huang, Molecular-Level Details of Morphology-Dependent Exciton Migration in Poly(3-hexylthiophene) Nanostructures, J. Phys. Chem. C, 2015, 119, 7047–7059.
# M. Moreno, M. Casalegno, G. Raos, S. V. Meille and R. Po, Molecular Modeling of Crystalline Alkylthiophene Oligomers and Polymers, J. Phys. Chem. B, 2010, 114, 1591–1602.
# D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, San Diego, 2nd edn., 2002.
# K. N. Schwarz, T. W. Kee and D. M. Huang, Coarse-grained simulations of the solution-phase self-assembly of poly(3-hexylthiophene) nanostructures, Nanoscale, 2013, 5, 2017.

Molecular Dynamics Simulation

2020-05-12T18:52:46Z

Cmditradmin: /* References */

Molecular Dynamics Simulations

==Introduction==

Molecular dynamics (MD) simulations are a computational tool for exploring molecular scale structure and dynamics in a wide variety of materials. They have been used to study solutions, bulk materials, interfaces, crystal structures, proteins, and polymers. Researchers can use this computational approach to explore properties and behaviors of these materials at small length and time scales that are not always directly accessible by experimental methods. It can also provide new perspectives or additional information about complex phenomena observed in experiments.1 MD can be used to study diffusion, polymer self-assembly, preferred molecular conformations, and more.

There is a diverse and complex set of bonded and non-bonded interactions between atoms and molecules that arise from electromagnetic forces.2 The most accurate method for modeling these interactions is a quantum mechanical approach. However, these ab initio calculations are computationally expensive and are therefore limited in the time and length scales at which they can model molecular systems. Molecular dynamics simulations instead approximate these interactions using classical mechanics. While some accuracy is lost, these systems are still able to capture larger-scale behaviors on the order of picoseconds to milliseconds (ps-ms) and Angstroms to millimeters (Å-mm), and provide researchers with useful information.

[[File:P3HT.png|thumb]]

Figure 1. Snapshot from a molecular dynamics simulation of poly(3-hexylthiophene) (P3HT). Alkyl side chains (carbon and hydrogen atoms) are shown in blue, backbone hydrogen atoms are shown in white, backbone carbon atoms are shown in red, and sulfur atoms are shown in yellow.

==Applications in Clean Energy Research==

Molecular dynamics simulations have been used to study materials in many areas of clean energy research, such as solar cell technologies or batteries. These tools can give researchers a deeper understanding of the materials and can also help researchers design improved materials for new technologies. For example, molecular dynamics simulations have been critical in understanding charge transport mechanisms in conjugated polymers (CPs), semiconducting materials that are found in the active layers of organic photovoltaics (OPVs). Their structure is comprised of a conjugated backbone with alternating double and single bonds that enable charge transport. Side chains are also included in the polymer structure to improve their solubility. While not yet as efficient, these polymers enable flexible, lightweight, and solution-processable alternatives to more prevalent inorganic equivalents.

Poly(3-hexylthiophene) (P3HT) is a common, well-studied conjugated polymer in the field. It provides a model system for studying the molecular behavior of CPs and how it relates to their macroscopic performance. In the work of McMahon, et al., researchers used atomistic molecular dynamics simulations of P3HT to understand the effects of structural defects in the polymer chain on the charge delocalization in the material.3 MD simulations were used to capture large-scale molecular structures at various time points along the trajectory. These ‘frozen’ structures were then fed to more accurate quantum mechanic calculations to understand the charge movement. Another group, Tapping and coworkers, used coarse-grained molecular dynamics to study P3HT on an even larger scale.4 They paired the coarse-grained structures of self-assembled P3HT nanofibers in solution with ab-initio approaches to track exciton movement through the fiber.

==Fundamentals of Molecular Dynamics Simulations==

==Force Field==

Atoms in a molecular dynamics simulation will move in response to the sum of bonded and non-bonded interactions with neighboring atoms. These include bonds, van der Waals forces, steric repulsion and Coulomb forces. There are also contributions from the angles and dihedrals (planarity) formed by three or more bonded atoms. All bonded and non-bonded interactions are approximated with equations of various functional forms and constants derived from ab initio calculations or empirical fitting approaches. For some parameters, energy as a function of a conformational change (e.g. distance between two atoms) is fit to determine the relevant parameters (e.g. bond energy and distance constants). Other parameters are modified to recreate macroscopic properties of the system (e.g. density) during a simulation.

The molecular dynamics force field can be defined as an equation and set of constants that describe the potential energy from all bonded and non-bonded interactions in an atomic system. Different force fields can be found in the literature for a wide range of molecular system. They vary in both the functional form and the parameter values, all with varying degrees of specificity in the atoms, molecules and materials to which they can be applied. A simple functional form for a force field (Class 1) is defined in equation (1) below.5

<math>V_{FF}=\mathrm{\Sigma}_{bonds}\frac{1}{2}K_{b,ij}\left(b_{ij}-b_{o,ij}\right)^2+\mathrm{\Sigma}_{angles}\frac{1}{2}K_{\theta,ijk}\left(\theta_{ijk}-\theta_{o,ijk}\right)^2+\mathrm{\Sigma}_{impropers}\frac{1}{2}K_{\zeta,ijkl}\left(\zeta_{ijkl}-\zeta_{o,ijkl}\right)^2+\ \mathrm{\Sigma}_{dihedrals}K_{\phi,ijkl}\left(1+cos{\left(n\phi_{ijkl}-\delta_n\right)}\right)+\ \mathrm{\Sigma}_{pairs}\left[4\epsilon_{ij}\left[\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^6\right]+K_{Coulomb}\frac{q_iq_j}{\kappa r_{ij}}\right]</math>(1)

where:

<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )

<math>b_ij = </math>distance between bonded atoms i and j, units of distance

<math>K_{b,ij} = </math>bond energy constant for bonded atoms i and j, units of \frac{energy}{distance^2}

<math>b_{ij} =</math> distance between bonded atoms i and j, units of distance

<math>b_{o,ij} =</math> bond distance constant for bonded atoms i and j, units of distance

<math>K_{\theta,ijk} = </math>angle energy constant for bonded atoms i, j, and k, units of \frac{energy}{degrees^2}

<math>\theta_{ijk} = </math>angle between bonded atoms i, j, and k, units of degrees

<math>\theta_{o,ijk} =</math> angle constant for bonded atoms i, j, and k, units of degrees

<math>K_{\zeta,ijkl} =</math> improper dihedral energy constant for bonded atoms i, j, k, and l, units of \frac{energy}{degrees^2}

<math>\zeta_{ijkl} = </math>improper angle for bonded atoms i, j, k, and l, units of degrees

<math>\zeta_{o,ijkl} =</math> improper angle constant for bonded atoms i, j, k, and l, units of degrees

<math>K_{\phi,ijkl} =</math> dihedral energy constant for linearly bonded atoms i, j, k, and l, units of energy

<math>n = </math>integer

<math>\phi_{ijkl} =</math> dihedral angle for linearly bonded atoms i, j, k, and l, units of degrees

<math>\delta_n = </math>dihedral angle constant for linearly bonded atoms i, j, k, and l, units of degrees

<math>\epsilon_{ij} =</math> Lennard-Jones energy constant for non-bonded atoms i and j, units of energy

<math>\sigma_{ij} =</math> Lennard-Jones distant constant for non-bonded atoms i and j, units of distance

<math>r_{ij} =</math> distance between non-bonded atoms i and j, units of distance

<math>K_{Coulomb} =</math> Coulomb constant,\<math> 8.988\ \times\ {10}^9\frac{Nm^2}{C^2}</math>

<math>q_i, q_j = </math>charges of non-bonded atoms i and j, units of elementary\ charge

<math>\kappa =</math> dielectric constant, unitless

For an interactive thorough exploration of this force field, visit https://interactive-md-ff.herokuapp.com/.

Classical mechanics are used to determine the forces each atom experiences due to these bonded and non-bonded interactions as well as the direction and speed at which they move in response to those forces. Following Newton’s second law of motion and utilizing the force field in equation (1), the force can be defined as1,6:
<math>F=ma=-\frac{\delta V_{FF}}{\delta r}</math>(2)

where:
F = force

m = mass

a = acceleration

This equation can then be solved for the position and velocity of all atoms at every time step during the simulation. This algorithm is known as the integrator.

==Integrator==

The full length of the simulation in time (e.g. 1 nanosecond) is broken down into discrete time steps (e.g. 1 femtosecond). At each time step, the bonded and non-bonded forces on each atom are summed using equations (1) and (2), and the atoms ‘step’ forward in position and time in response to those forces. One method to approximate an atom’s next position is the Velocity-Verlet algorithm shown in equations (3) and (4).6 This integrator computes an atom’s position and velocity at every time step. By stringing this information together at every time step over the duration of the simulation (e.g. 1,000,000 steps of 1 fs each for a 1 ns simulation), a trajectory of the atomic motions can be created. This data can then be analyzed to uncover information about the molecular-level behavior in the system (e.g. diffusion, mean squared displacement).

<math>r\left(t+\mathrm{\Delta t}\right)=r\left(t\right)+\mathrm{\Delta\ t}\ v\left(t\right)+\frac{\mathrm{\Delta}t^2F(t)}{2m}</math> (3)

<math>v\left(t+\mathrm{\Delta t}\right)=v\left(t\right)+\frac{\mathrm{\Delta t}(F\left(t\right)+\ F\left(t+\mathrm{\Delta t}\right))}{2m}\ </math>(4)

where:
r = atomic position
v = velocity
F = force
m = atomic mass
t = current time in simulation
Δt = time step

At the first time step, when t+Δt = 0, there is no information about the about the forces acting on the atoms or their velocities at time t. Therefore, the simulation is initialized with random set of velocities taken from a distribution based around the simulation’s temperature. The system must then be allowed to reach equilibration before the data can be used for analysis.

==Ensembles==

In a molecular dynamics simulation, the behavior of a fixed number of atoms is explored (constant N). Additional simulation conditions are also applied with respect to volume, pressure, and energy. There are three common ensembles that define these simulation conditions: NVE (constant number of atoms, volume, and energy), NVT (constant number of atoms, volume, and temperature) and NPT (constant number of atoms, pressure, and temperature). A variety of algorithms have been developed to enforce these conditions, such as the Andersen or Nosé-Hoover thermostats for controlling temperature. A researcher will choose an ensemble based on the questions they are trying to answer.6

==Atomistic and Coarse-Grained Simulations==

Depending on the time and length scales a researcher would like to explore, they may choose to use a coarse-grained force field. Instead of modeling each atom discretely, small clusters of atoms are grouped together into a single functional group and the interactions are approximated at a larger scale7, saving time in calculating interactions at every atomic site and allowing the simulation to reach larger lengths and times with reasonable computational expense. An example of coarse-graining a molecule is shown in Figure 2.

[[File:coarse_grain_P3HT.png|thumb]]

Figure 2. Poly(3-hexylthiophene) (P3HT) monomer for a coarse-grained molecular dynamics simulation. Clusters of atoms are turned into larger ‘beads’ with force field parameters representative of the large-scale collective behavior of the atoms or functional groups.

==References==

# 1 A. Arbe, F. Alvarez and J. Colmenero, Neutron scattering and molecular dynamics simulations: synergetic tools to unravel structure and dynamics in polymers, Soft Matter, 2012, 8, 8257.
# 2 J. N. Israelachvili, Intermolecular and Surface Forces, Elsevier Inc., 3rd ed., 2011.
# 3 D. P. McMahon, D. L. Cheung, L. Goris, J. Dacuña, A. Salleo and A. Troisi, Relation between microstructure and charge transport in polymers of different regioregularity, J. Phys. Chem. C, 2011, 115, 19386–19393.
# 4 P. C. Tapping, S. N. Clafton, K. N. Schwarz, T. W. Kee and D. M. Huang, Molecular-Level Details of Morphology-Dependent Exciton Migration in Poly(3-hexylthiophene) Nanostructures, J. Phys. Chem. C, 2015, 119, 7047–7059.
# 5 M. Moreno, M. Casalegno, G. Raos, S. V. Meille and R. Po, Molecular Modeling of Crystalline Alkylthiophene Oligomers and Polymers, J. Phys. Chem. B, 2010, 114, 1591–1602.
# 6 D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, San Diego, 2nd edn., 2002.
# 7 K. N. Schwarz, T. W. Kee and D. M. Huang, Coarse-grained simulations of the solution-phase self-assembly of poly(3-hexylthiophene) nanostructures, Nanoscale, 2013, 5, 2017.

Molecular Dynamics Simulation

2020-05-11T22:25:13Z

Cmditradmin: /* Force Field */

Molecular Dynamics Simulations

==Introduction==

Molecular dynamics (MD) simulations are a computational tool for exploring molecular scale structure and dynamics in a wide variety of materials. They have been used to study solutions, bulk materials, interfaces, crystal structures, proteins, and polymers. Researchers can use this computational approach to explore properties and behaviors of these materials at small length and time scales that are not always directly accessible by experimental methods. It can also provide new perspectives or additional information about complex phenomena observed in experiments.1 MD can be used to study diffusion, polymer self-assembly, preferred molecular conformations, and more.

There is a diverse and complex set of bonded and non-bonded interactions between atoms and molecules that arise from electromagnetic forces.2 The most accurate method for modeling these interactions is a quantum mechanical approach. However, these ab initio calculations are computationally expensive and are therefore limited in the time and length scales at which they can model molecular systems. Molecular dynamics simulations instead approximate these interactions using classical mechanics. While some accuracy is lost, these systems are still able to capture larger-scale behaviors on the order of picoseconds to milliseconds (ps-ms) and Angstroms to millimeters (Å-mm), and provide researchers with useful information.

[[File:P3HT.png|thumb]]

Figure 1. Snapshot from a molecular dynamics simulation of poly(3-hexylthiophene) (P3HT). Alkyl side chains (carbon and hydrogen atoms) are shown in blue, backbone hydrogen atoms are shown in white, backbone carbon atoms are shown in red, and sulfur atoms are shown in yellow.

==Applications in Clean Energy Research==

Molecular dynamics simulations have been used to study materials in many areas of clean energy research, such as solar cell technologies or batteries. These tools can give researchers a deeper understanding of the materials and can also help researchers design improved materials for new technologies. For example, molecular dynamics simulations have been critical in understanding charge transport mechanisms in conjugated polymers (CPs), semiconducting materials that are found in the active layers of organic photovoltaics (OPVs). Their structure is comprised of a conjugated backbone with alternating double and single bonds that enable charge transport. Side chains are also included in the polymer structure to improve their solubility. While not yet as efficient, these polymers enable flexible, lightweight, and solution-processable alternatives to more prevalent inorganic equivalents.

Poly(3-hexylthiophene) (P3HT) is a common, well-studied conjugated polymer in the field. It provides a model system for studying the molecular behavior of CPs and how it relates to their macroscopic performance. In the work of McMahon, et al., researchers used atomistic molecular dynamics simulations of P3HT to understand the effects of structural defects in the polymer chain on the charge delocalization in the material.3 MD simulations were used to capture large-scale molecular structures at various time points along the trajectory. These ‘frozen’ structures were then fed to more accurate quantum mechanic calculations to understand the charge movement. Another group, Tapping and coworkers, used coarse-grained molecular dynamics to study P3HT on an even larger scale.4 They paired the coarse-grained structures of self-assembled P3HT nanofibers in solution with ab-initio approaches to track exciton movement through the fiber.

==Fundamentals of Molecular Dynamics Simulations==

==Force Field==

Atoms in a molecular dynamics simulation will move in response to the sum of bonded and non-bonded interactions with neighboring atoms. These include bonds, van der Waals forces, steric repulsion and Coulomb forces. There are also contributions from the angles and dihedrals (planarity) formed by three or more bonded atoms. All bonded and non-bonded interactions are approximated with equations of various functional forms and constants derived from ab initio calculations or empirical fitting approaches. For some parameters, energy as a function of a conformational change (e.g. distance between two atoms) is fit to determine the relevant parameters (e.g. bond energy and distance constants). Other parameters are modified to recreate macroscopic properties of the system (e.g. density) during a simulation.

The molecular dynamics force field can be defined as an equation and set of constants that describe the potential energy from all bonded and non-bonded interactions in an atomic system. Different force fields can be found in the literature for a wide range of molecular system. They vary in both the functional form and the parameter values, all with varying degrees of specificity in the atoms, molecules and materials to which they can be applied. A simple functional form for a force field (Class 1) is defined in equation (1) below.5

<math>V_{FF}=\mathrm{\Sigma}_{bonds}\frac{1}{2}K_{b,ij}\left(b_{ij}-b_{o,ij}\right)^2+\mathrm{\Sigma}_{angles}\frac{1}{2}K_{\theta,ijk}\left(\theta_{ijk}-\theta_{o,ijk}\right)^2+\mathrm{\Sigma}_{impropers}\frac{1}{2}K_{\zeta,ijkl}\left(\zeta_{ijkl}-\zeta_{o,ijkl}\right)^2+\ \mathrm{\Sigma}_{dihedrals}K_{\phi,ijkl}\left(1+cos{\left(n\phi_{ijkl}-\delta_n\right)}\right)+\ \mathrm{\Sigma}_{pairs}\left[4\epsilon_{ij}\left[\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^6\right]+K_{Coulomb}\frac{q_iq_j}{\kappa r_{ij}}\right]</math>(1)

where:

<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )

<math>b_ij = </math>distance between bonded atoms i and j, units of distance

<math>K_{b,ij} = </math>bond energy constant for bonded atoms i and j, units of \frac{energy}{distance^2}

<math>b_{ij} =</math> distance between bonded atoms i and j, units of distance

<math>b_{o,ij} =</math> bond distance constant for bonded atoms i and j, units of distance

<math>K_{\theta,ijk} = </math>angle energy constant for bonded atoms i, j, and k, units of \frac{energy}{degrees^2}

<math>\theta_{ijk} = </math>angle between bonded atoms i, j, and k, units of degrees

<math>\theta_{o,ijk} =</math> angle constant for bonded atoms i, j, and k, units of degrees

<math>K_{\zeta,ijkl} =</math> improper dihedral energy constant for bonded atoms i, j, k, and l, units of \frac{energy}{degrees^2}

<math>\zeta_{ijkl} = </math>improper angle for bonded atoms i, j, k, and l, units of degrees

<math>\zeta_{o,ijkl} =</math> improper angle constant for bonded atoms i, j, k, and l, units of degrees

<math>K_{\phi,ijkl} =</math> dihedral energy constant for linearly bonded atoms i, j, k, and l, units of energy

<math>n = </math>integer

<math>\phi_{ijkl} =</math> dihedral angle for linearly bonded atoms i, j, k, and l, units of degrees

<math>\delta_n = </math>dihedral angle constant for linearly bonded atoms i, j, k, and l, units of degrees

<math>\epsilon_{ij} =</math> Lennard-Jones energy constant for non-bonded atoms i and j, units of energy

<math>\sigma_{ij} =</math> Lennard-Jones distant constant for non-bonded atoms i and j, units of distance

<math>r_{ij} =</math> distance between non-bonded atoms i and j, units of distance

<math>K_{Coulomb} =</math> Coulomb constant,\<math> 8.988\ \times\ {10}^9\frac{Nm^2}{C^2}</math>

<math>q_i, q_j = </math>charges of non-bonded atoms i and j, units of elementary\ charge

<math>\kappa =</math> dielectric constant, unitless

For an interactive thorough exploration of this force field, visit https://interactive-md-ff.herokuapp.com/.

Classical mechanics are used to determine the forces each atom experiences due to these bonded and non-bonded interactions as well as the direction and speed at which they move in response to those forces. Following Newton’s second law of motion and utilizing the force field in equation (1), the force can be defined as1,6:
<math>F=ma=-\frac{\delta V_{FF}}{\delta r}</math>(2)

where:
F = force

m = mass

a = acceleration

This equation can then be solved for the position and velocity of all atoms at every time step during the simulation. This algorithm is known as the integrator.

==Integrator==

The full length of the simulation in time (e.g. 1 nanosecond) is broken down into discrete time steps (e.g. 1 femtosecond). At each time step, the bonded and non-bonded forces on each atom are summed using equations (1) and (2), and the atoms ‘step’ forward in position and time in response to those forces. One method to approximate an atom’s next position is the Velocity-Verlet algorithm shown in equations (3) and (4).6 This integrator computes an atom’s position and velocity at every time step. By stringing this information together at every time step over the duration of the simulation (e.g. 1,000,000 steps of 1 fs each for a 1 ns simulation), a trajectory of the atomic motions can be created. This data can then be analyzed to uncover information about the molecular-level behavior in the system (e.g. diffusion, mean squared displacement).

<math>r\left(t+\mathrm{\Delta t}\right)=r\left(t\right)+\mathrm{\Delta\ t}\ v\left(t\right)+\frac{\mathrm{\Delta}t^2F(t)}{2m}</math> (3)

<math>v\left(t+\mathrm{\Delta t}\right)=v\left(t\right)+\frac{\mathrm{\Delta t}(F\left(t\right)+\ F\left(t+\mathrm{\Delta t}\right))}{2m}\ </math>(4)

where:
r = atomic position
v = velocity
F = force
m = atomic mass
t = current time in simulation
Δt = time step

At the first time step, when t+Δt = 0, there is no information about the about the forces acting on the atoms or their velocities at time t. Therefore, the simulation is initialized with random set of velocities taken from a distribution based around the simulation’s temperature. The system must then be allowed to reach equilibration before the data can be used for analysis.

==Ensembles==

In a molecular dynamics simulation, the behavior of a fixed number of atoms is explored (constant N). Additional simulation conditions are also applied with respect to volume, pressure, and energy. There are three common ensembles that define these simulation conditions: NVE (constant number of atoms, volume, and energy), NVT (constant number of atoms, volume, and temperature) and NPT (constant number of atoms, pressure, and temperature). A variety of algorithms have been developed to enforce these conditions, such as the Andersen or Nosé-Hoover thermostats for controlling temperature. A researcher will choose an ensemble based on the questions they are trying to answer.6

==Atomistic and Coarse-Grained Simulations==

Depending on the time and length scales a researcher would like to explore, they may choose to use a coarse-grained force field. Instead of modeling each atom discretely, small clusters of atoms are grouped together into a single functional group and the interactions are approximated at a larger scale7, saving time in calculating interactions at every atomic site and allowing the simulation to reach larger lengths and times with reasonable computational expense. An example of coarse-graining a molecule is shown in Figure 2.

[[File:coarse_grain_P3HT.png|thumb]]

Figure 2. Poly(3-hexylthiophene) (P3HT) monomer for a coarse-grained molecular dynamics simulation. Clusters of atoms are turned into larger ‘beads’ with force field parameters representative of the large-scale collective behavior of the atoms or functional groups.

==References==

1 A. Arbe, F. Alvarez and J. Colmenero, Neutron scattering and molecular dynamics simulations: synergetic tools to unravel structure and dynamics in polymers, Soft Matter, 2012, 8, 8257.
2 J. N. Israelachvili, Intermolecular and Surface Forces, Elsevier Inc., 3rd ed., 2011.
3 D. P. McMahon, D. L. Cheung, L. Goris, J. Dacuña, A. Salleo and A. Troisi, Relation between microstructure and charge transport in polymers of different regioregularity, J. Phys. Chem. C, 2011, 115, 19386–19393.
4 P. C. Tapping, S. N. Clafton, K. N. Schwarz, T. W. Kee and D. M. Huang, Molecular-Level Details of Morphology-Dependent Exciton Migration in Poly(3-hexylthiophene) Nanostructures, J. Phys. Chem. C, 2015, 119, 7047–7059.
5 M. Moreno, M. Casalegno, G. Raos, S. V. Meille and R. Po, Molecular Modeling of Crystalline Alkylthiophene Oligomers and Polymers, J. Phys. Chem. B, 2010, 114, 1591–1602.
6 D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, San Diego, 2nd edn., 2002.
7 K. N. Schwarz, T. W. Kee and D. M. Huang, Coarse-grained simulations of the solution-phase self-assembly of poly(3-hexylthiophene) nanostructures, Nanoscale, 2013, 5, 2017.

Molecular Dynamics Simulation

2020-05-11T22:23:25Z

Cmditradmin: /* Force Field */

Molecular Dynamics Simulations

==Introduction==

Molecular dynamics (MD) simulations are a computational tool for exploring molecular scale structure and dynamics in a wide variety of materials. They have been used to study solutions, bulk materials, interfaces, crystal structures, proteins, and polymers. Researchers can use this computational approach to explore properties and behaviors of these materials at small length and time scales that are not always directly accessible by experimental methods. It can also provide new perspectives or additional information about complex phenomena observed in experiments.1 MD can be used to study diffusion, polymer self-assembly, preferred molecular conformations, and more.

There is a diverse and complex set of bonded and non-bonded interactions between atoms and molecules that arise from electromagnetic forces.2 The most accurate method for modeling these interactions is a quantum mechanical approach. However, these ab initio calculations are computationally expensive and are therefore limited in the time and length scales at which they can model molecular systems. Molecular dynamics simulations instead approximate these interactions using classical mechanics. While some accuracy is lost, these systems are still able to capture larger-scale behaviors on the order of picoseconds to milliseconds (ps-ms) and Angstroms to millimeters (Å-mm), and provide researchers with useful information.

[[File:P3HT.png|thumb]]

Figure 1. Snapshot from a molecular dynamics simulation of poly(3-hexylthiophene) (P3HT). Alkyl side chains (carbon and hydrogen atoms) are shown in blue, backbone hydrogen atoms are shown in white, backbone carbon atoms are shown in red, and sulfur atoms are shown in yellow.

==Applications in Clean Energy Research==

Molecular dynamics simulations have been used to study materials in many areas of clean energy research, such as solar cell technologies or batteries. These tools can give researchers a deeper understanding of the materials and can also help researchers design improved materials for new technologies. For example, molecular dynamics simulations have been critical in understanding charge transport mechanisms in conjugated polymers (CPs), semiconducting materials that are found in the active layers of organic photovoltaics (OPVs). Their structure is comprised of a conjugated backbone with alternating double and single bonds that enable charge transport. Side chains are also included in the polymer structure to improve their solubility. While not yet as efficient, these polymers enable flexible, lightweight, and solution-processable alternatives to more prevalent inorganic equivalents.

Poly(3-hexylthiophene) (P3HT) is a common, well-studied conjugated polymer in the field. It provides a model system for studying the molecular behavior of CPs and how it relates to their macroscopic performance. In the work of McMahon, et al., researchers used atomistic molecular dynamics simulations of P3HT to understand the effects of structural defects in the polymer chain on the charge delocalization in the material.3 MD simulations were used to capture large-scale molecular structures at various time points along the trajectory. These ‘frozen’ structures were then fed to more accurate quantum mechanic calculations to understand the charge movement. Another group, Tapping and coworkers, used coarse-grained molecular dynamics to study P3HT on an even larger scale.4 They paired the coarse-grained structures of self-assembled P3HT nanofibers in solution with ab-initio approaches to track exciton movement through the fiber.

==Fundamentals of Molecular Dynamics Simulations==

==Force Field==

Atoms in a molecular dynamics simulation will move in response to the sum of bonded and non-bonded interactions with neighboring atoms. These include bonds, van der Waals forces, steric repulsion and Coulomb forces. There are also contributions from the angles and dihedrals (planarity) formed by three or more bonded atoms. All bonded and non-bonded interactions are approximated with equations of various functional forms and constants derived from ab initio calculations or empirical fitting approaches. For some parameters, energy as a function of a conformational change (e.g. distance between two atoms) is fit to determine the relevant parameters (e.g. bond energy and distance constants). Other parameters are modified to recreate macroscopic properties of the system (e.g. density) during a simulation.

The molecular dynamics force field can be defined as an equation and set of constants that describe the potential energy from all bonded and non-bonded interactions in an atomic system. Different force fields can be found in the literature for a wide range of molecular system. They vary in both the functional form and the parameter values, all with varying degrees of specificity in the atoms, molecules and materials to which they can be applied. A simple functional form for a force field (Class 1) is defined in equation (1) below.5

<math>V_{FF}=\mathrm{\Sigma}_{bonds}\frac{1}{2}K_{b,ij}\left(b_{ij}-b_{o,ij}\right)^2+\mathrm{\Sigma}_{angles}\frac{1}{2}K_{\theta,ijk}\left(\theta_{ijk}-\theta_{o,ijk}\right)^2+\mathrm{\Sigma}_{impropers}\frac{1}{2}K_{\zeta,ijkl}\left(\zeta_{ijkl}-\zeta_{o,ijkl}\right)^2+\ \mathrm{\Sigma}_{dihedrals}K_{\phi,ijkl}\left(1+cos{\left(n\phi_{ijkl}-\delta_n\right)}\right)+\ \mathrm{\Sigma}_{pairs}\left[4\epsilon_{ij}\left[\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^6\right]+K_{Coulomb}\frac{q_iq_j}{\kappa r_{ij}}\right]</math>(1)

where:

<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )

<math>b_ij = </math>distance between bonded atoms i and j, units of distance

<math>K_{b,ij} = </math>bond energy constant for bonded atoms i and j, units of \frac{energy}{distance^2}

<math>b_{ij} =</math> distance between bonded atoms i and j, units of distance

<math>b_{o,ij} =</math> bond distance constant for bonded atoms i and j, units of distance

<math>K_{\theta,ijk} = </math>angle energy constant for bonded atoms i, j, and k, units of \frac{energy}{degrees^2}

<math>\theta_{ijk} = </math>angle between bonded atoms i, j, and k, units of degrees

<math>\theta_{o,ijk} =</math> angle constant for bonded atoms i, j, and k, units of degrees

<math>K_{\zeta,ijkl} =</math> improper dihedral energy constant for bonded atoms i, j, k, and l, units of \frac{energy}{degrees^2}

<math>\zeta_{ijkl} = </math>improper angle for bonded atoms i, j, k, and l, units of degrees

<math>\zeta_{o,ijkl} =</math> improper angle constant for bonded atoms i, j, k, and l, units of degrees

<math>K_{\phi,ijkl} =</math> dihedral energy constant for linearly bonded atoms i, j, k, and l, units of energy

<math>n = integer

<math>\phi_{ijkl} =</math> dihedral angle for linearly bonded atoms i, j, k, and l, units of degrees

<math>\delta_n = dihedral angle constant for linearly bonded atoms i, j, k, and l, units of degrees

<math>\epsilon_{ij} =</math> Lennard-Jones energy constant for non-bonded atoms i and j, units of energy

<math>\sigma_{ij} =</math> Lennard-Jones distant constant for non-bonded atoms i and j, units of distance

<math>r_{ij} =</math> distance between non-bonded atoms i and j, units of distance

<math>K_{Coulomb} =</math> Coulomb constant,\ 8.988\ \times\ {10}^9\frac{Nm^2}{C^2}

<math>q_i, q_j = </math>charges of non-bonded atoms i and j, units of elementary\ charge

<math>\kappa =</math> dielectric constant, unitless</math>

For an interactive thorough exploration of this force field, visit https://interactive-md-ff.herokuapp.com/.

Classical mechanics are used to determine the forces each atom experiences due to these bonded and non-bonded interactions as well as the direction and speed at which they move in response to those forces. Following Newton’s second law of motion and utilizing the force field in equation (1), the force can be defined as1,6:
<math>F=ma=-\frac{\delta V_{FF}}{\delta r}</math>(2)

where:
F = force

m = mass

a = acceleration

This equation can then be solved for the position and velocity of all atoms at every time step during the simulation. This algorithm is known as the integrator.

==Integrator==

The full length of the simulation in time (e.g. 1 nanosecond) is broken down into discrete time steps (e.g. 1 femtosecond). At each time step, the bonded and non-bonded forces on each atom are summed using equations (1) and (2), and the atoms ‘step’ forward in position and time in response to those forces. One method to approximate an atom’s next position is the Velocity-Verlet algorithm shown in equations (3) and (4).6 This integrator computes an atom’s position and velocity at every time step. By stringing this information together at every time step over the duration of the simulation (e.g. 1,000,000 steps of 1 fs each for a 1 ns simulation), a trajectory of the atomic motions can be created. This data can then be analyzed to uncover information about the molecular-level behavior in the system (e.g. diffusion, mean squared displacement).

<math>r\left(t+\mathrm{\Delta t}\right)=r\left(t\right)+\mathrm{\Delta\ t}\ v\left(t\right)+\frac{\mathrm{\Delta}t^2F(t)}{2m}</math> (3)

<math>v\left(t+\mathrm{\Delta t}\right)=v\left(t\right)+\frac{\mathrm{\Delta t}(F\left(t\right)+\ F\left(t+\mathrm{\Delta t}\right))}{2m}\ </math>(4)

where:
r = atomic position
v = velocity
F = force
m = atomic mass
t = current time in simulation
Δt = time step

At the first time step, when t+Δt = 0, there is no information about the about the forces acting on the atoms or their velocities at time t. Therefore, the simulation is initialized with random set of velocities taken from a distribution based around the simulation’s temperature. The system must then be allowed to reach equilibration before the data can be used for analysis.

==Ensembles==

In a molecular dynamics simulation, the behavior of a fixed number of atoms is explored (constant N). Additional simulation conditions are also applied with respect to volume, pressure, and energy. There are three common ensembles that define these simulation conditions: NVE (constant number of atoms, volume, and energy), NVT (constant number of atoms, volume, and temperature) and NPT (constant number of atoms, pressure, and temperature). A variety of algorithms have been developed to enforce these conditions, such as the Andersen or Nosé-Hoover thermostats for controlling temperature. A researcher will choose an ensemble based on the questions they are trying to answer.6

==Atomistic and Coarse-Grained Simulations==

Depending on the time and length scales a researcher would like to explore, they may choose to use a coarse-grained force field. Instead of modeling each atom discretely, small clusters of atoms are grouped together into a single functional group and the interactions are approximated at a larger scale7, saving time in calculating interactions at every atomic site and allowing the simulation to reach larger lengths and times with reasonable computational expense. An example of coarse-graining a molecule is shown in Figure 2.

[[File:coarse_grain_P3HT.png|thumb]]

Figure 2. Poly(3-hexylthiophene) (P3HT) monomer for a coarse-grained molecular dynamics simulation. Clusters of atoms are turned into larger ‘beads’ with force field parameters representative of the large-scale collective behavior of the atoms or functional groups.

==References==

1 A. Arbe, F. Alvarez and J. Colmenero, Neutron scattering and molecular dynamics simulations: synergetic tools to unravel structure and dynamics in polymers, Soft Matter, 2012, 8, 8257.
2 J. N. Israelachvili, Intermolecular and Surface Forces, Elsevier Inc., 3rd ed., 2011.
3 D. P. McMahon, D. L. Cheung, L. Goris, J. Dacuña, A. Salleo and A. Troisi, Relation between microstructure and charge transport in polymers of different regioregularity, J. Phys. Chem. C, 2011, 115, 19386–19393.
4 P. C. Tapping, S. N. Clafton, K. N. Schwarz, T. W. Kee and D. M. Huang, Molecular-Level Details of Morphology-Dependent Exciton Migration in Poly(3-hexylthiophene) Nanostructures, J. Phys. Chem. C, 2015, 119, 7047–7059.
5 M. Moreno, M. Casalegno, G. Raos, S. V. Meille and R. Po, Molecular Modeling of Crystalline Alkylthiophene Oligomers and Polymers, J. Phys. Chem. B, 2010, 114, 1591–1602.
6 D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, San Diego, 2nd edn., 2002.
7 K. N. Schwarz, T. W. Kee and D. M. Huang, Coarse-grained simulations of the solution-phase self-assembly of poly(3-hexylthiophene) nanostructures, Nanoscale, 2013, 5, 2017.

Molecular Dynamics Simulation

2020-05-11T22:22:31Z

Cmditradmin: /* Force Field */

Molecular Dynamics Simulations

==Introduction==

Molecular dynamics (MD) simulations are a computational tool for exploring molecular scale structure and dynamics in a wide variety of materials. They have been used to study solutions, bulk materials, interfaces, crystal structures, proteins, and polymers. Researchers can use this computational approach to explore properties and behaviors of these materials at small length and time scales that are not always directly accessible by experimental methods. It can also provide new perspectives or additional information about complex phenomena observed in experiments.1 MD can be used to study diffusion, polymer self-assembly, preferred molecular conformations, and more.

There is a diverse and complex set of bonded and non-bonded interactions between atoms and molecules that arise from electromagnetic forces.2 The most accurate method for modeling these interactions is a quantum mechanical approach. However, these ab initio calculations are computationally expensive and are therefore limited in the time and length scales at which they can model molecular systems. Molecular dynamics simulations instead approximate these interactions using classical mechanics. While some accuracy is lost, these systems are still able to capture larger-scale behaviors on the order of picoseconds to milliseconds (ps-ms) and Angstroms to millimeters (Å-mm), and provide researchers with useful information.

[[File:P3HT.png|thumb]]

Figure 1. Snapshot from a molecular dynamics simulation of poly(3-hexylthiophene) (P3HT). Alkyl side chains (carbon and hydrogen atoms) are shown in blue, backbone hydrogen atoms are shown in white, backbone carbon atoms are shown in red, and sulfur atoms are shown in yellow.

==Applications in Clean Energy Research==

Molecular dynamics simulations have been used to study materials in many areas of clean energy research, such as solar cell technologies or batteries. These tools can give researchers a deeper understanding of the materials and can also help researchers design improved materials for new technologies. For example, molecular dynamics simulations have been critical in understanding charge transport mechanisms in conjugated polymers (CPs), semiconducting materials that are found in the active layers of organic photovoltaics (OPVs). Their structure is comprised of a conjugated backbone with alternating double and single bonds that enable charge transport. Side chains are also included in the polymer structure to improve their solubility. While not yet as efficient, these polymers enable flexible, lightweight, and solution-processable alternatives to more prevalent inorganic equivalents.

Poly(3-hexylthiophene) (P3HT) is a common, well-studied conjugated polymer in the field. It provides a model system for studying the molecular behavior of CPs and how it relates to their macroscopic performance. In the work of McMahon, et al., researchers used atomistic molecular dynamics simulations of P3HT to understand the effects of structural defects in the polymer chain on the charge delocalization in the material.3 MD simulations were used to capture large-scale molecular structures at various time points along the trajectory. These ‘frozen’ structures were then fed to more accurate quantum mechanic calculations to understand the charge movement. Another group, Tapping and coworkers, used coarse-grained molecular dynamics to study P3HT on an even larger scale.4 They paired the coarse-grained structures of self-assembled P3HT nanofibers in solution with ab-initio approaches to track exciton movement through the fiber.

==Fundamentals of Molecular Dynamics Simulations==

==Force Field==

Atoms in a molecular dynamics simulation will move in response to the sum of bonded and non-bonded interactions with neighboring atoms. These include bonds, van der Waals forces, steric repulsion and Coulomb forces. There are also contributions from the angles and dihedrals (planarity) formed by three or more bonded atoms. All bonded and non-bonded interactions are approximated with equations of various functional forms and constants derived from ab initio calculations or empirical fitting approaches. For some parameters, energy as a function of a conformational change (e.g. distance between two atoms) is fit to determine the relevant parameters (e.g. bond energy and distance constants). Other parameters are modified to recreate macroscopic properties of the system (e.g. density) during a simulation.

The molecular dynamics force field can be defined as an equation and set of constants that describe the potential energy from all bonded and non-bonded interactions in an atomic system. Different force fields can be found in the literature for a wide range of molecular system. They vary in both the functional form and the parameter values, all with varying degrees of specificity in the atoms, molecules and materials to which they can be applied. A simple functional form for a force field (Class 1) is defined in equation (1) below.5

<math>V_{FF}=\mathrm{\Sigma}_{bonds}\frac{1}{2}K_{b,ij}\left(b_{ij}-b_{o,ij}\right)^2+\mathrm{\Sigma}_{angles}\frac{1}{2}K_{\theta,ijk}\left(\theta_{ijk}-\theta_{o,ijk}\right)^2+\mathrm{\Sigma}_{impropers}\frac{1}{2}K_{\zeta,ijkl}\left(\zeta_{ijkl}-\zeta_{o,ijkl}\right)^2+\ \mathrm{\Sigma}_{dihedrals}K_{\phi,ijkl}\left(1+cos{\left(n\phi_{ijkl}-\delta_n\right)}\right)+\ \mathrm{\Sigma}_{pairs}\left[4\epsilon_{ij}\left[\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^6\right]+K_{Coulomb}\frac{q_iq_j}{\kappa r_{ij}}\right]</math>(1)
where:
<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )

b_ij = distance between bonded atoms i and j, units of distance

<math>K_{b,ij} = </math>bond energy constant for bonded atoms i and j, units of \frac{energy}{distance^2}

<math>b_{ij} =</math> distance between bonded atoms i and j, units of distance

<math>b_{o,ij} =</math> bond distance constant for bonded atoms i and j, units of distance

<math>K_{\theta,ijk} = </math>angle energy constant for bonded atoms i, j, and k, units of \frac{energy}{degrees^2}

<math>\theta_{ijk} = </math>angle between bonded atoms i, j, and k, units of degrees

<math>\theta_{o,ijk} =</math> angle constant for bonded atoms i, j, and k, units of degrees

<math>K_{\zeta,ijkl} =</math> improper dihedral energy constant for bonded atoms i, j, k, and l, units of \frac{energy}{degrees^2}

<math>\zeta_{ijkl} = </math>improper angle for bonded atoms i, j, k, and l, units of degrees

<math>\zeta_{o,ijkl} =</math> improper angle constant for bonded atoms i, j, k, and l, units of degrees

<math>K_{\phi,ijkl} =</math> dihedral energy constant for linearly bonded atoms i, j, k, and l, units of energy

<math>n = integer

<math>\phi_{ijkl} =</math> dihedral angle for linearly bonded atoms i, j, k, and l, units of degrees

<math>\delta_n = dihedral angle constant for linearly bonded atoms i, j, k, and l, units of degrees

<math>\epsilon_{ij} =</math> Lennard-Jones energy constant for non-bonded atoms i and j, units of energy

<math>\sigma_{ij} =</math> Lennard-Jones distant constant for non-bonded atoms i and j, units of distance

<math>r_{ij} =</math> distance between non-bonded atoms i and j, units of distance

<math>K_{Coulomb} =</math> Coulomb constant,\ 8.988\ \times\ {10}^9\frac{Nm^2}{C^2}

<math>q_i, q_j = </math>charges of non-bonded atoms i and j, units of elementary\ charge

<math>\kappa =</math> dielectric constant, unitless</math>

For an interactive thorough exploration of this force field, visit https://interactive-md-ff.herokuapp.com/.

Classical mechanics are used to determine the forces each atom experiences due to these bonded and non-bonded interactions as well as the direction and speed at which they move in response to those forces. Following Newton’s second law of motion and utilizing the force field in equation (1), the force can be defined as1,6:
<math>F=ma=-\frac{\delta V_{FF}}{\delta r}</math>(2)

where:
F = force

m = mass

a = acceleration

This equation can then be solved for the position and velocity of all atoms at every time step during the simulation. This algorithm is known as the integrator.

==Integrator==

The full length of the simulation in time (e.g. 1 nanosecond) is broken down into discrete time steps (e.g. 1 femtosecond). At each time step, the bonded and non-bonded forces on each atom are summed using equations (1) and (2), and the atoms ‘step’ forward in position and time in response to those forces. One method to approximate an atom’s next position is the Velocity-Verlet algorithm shown in equations (3) and (4).6 This integrator computes an atom’s position and velocity at every time step. By stringing this information together at every time step over the duration of the simulation (e.g. 1,000,000 steps of 1 fs each for a 1 ns simulation), a trajectory of the atomic motions can be created. This data can then be analyzed to uncover information about the molecular-level behavior in the system (e.g. diffusion, mean squared displacement).

<math>r\left(t+\mathrm{\Delta t}\right)=r\left(t\right)+\mathrm{\Delta\ t}\ v\left(t\right)+\frac{\mathrm{\Delta}t^2F(t)}{2m}</math> (3)

<math>v\left(t+\mathrm{\Delta t}\right)=v\left(t\right)+\frac{\mathrm{\Delta t}(F\left(t\right)+\ F\left(t+\mathrm{\Delta t}\right))}{2m}\ </math>(4)

where:
r = atomic position
v = velocity
F = force
m = atomic mass
t = current time in simulation
Δt = time step

At the first time step, when t+Δt = 0, there is no information about the about the forces acting on the atoms or their velocities at time t. Therefore, the simulation is initialized with random set of velocities taken from a distribution based around the simulation’s temperature. The system must then be allowed to reach equilibration before the data can be used for analysis.

==Ensembles==

In a molecular dynamics simulation, the behavior of a fixed number of atoms is explored (constant N). Additional simulation conditions are also applied with respect to volume, pressure, and energy. There are three common ensembles that define these simulation conditions: NVE (constant number of atoms, volume, and energy), NVT (constant number of atoms, volume, and temperature) and NPT (constant number of atoms, pressure, and temperature). A variety of algorithms have been developed to enforce these conditions, such as the Andersen or Nosé-Hoover thermostats for controlling temperature. A researcher will choose an ensemble based on the questions they are trying to answer.6

==Atomistic and Coarse-Grained Simulations==

Depending on the time and length scales a researcher would like to explore, they may choose to use a coarse-grained force field. Instead of modeling each atom discretely, small clusters of atoms are grouped together into a single functional group and the interactions are approximated at a larger scale7, saving time in calculating interactions at every atomic site and allowing the simulation to reach larger lengths and times with reasonable computational expense. An example of coarse-graining a molecule is shown in Figure 2.

[[File:coarse_grain_P3HT.png|thumb]]

Figure 2. Poly(3-hexylthiophene) (P3HT) monomer for a coarse-grained molecular dynamics simulation. Clusters of atoms are turned into larger ‘beads’ with force field parameters representative of the large-scale collective behavior of the atoms or functional groups.

==References==

1 A. Arbe, F. Alvarez and J. Colmenero, Neutron scattering and molecular dynamics simulations: synergetic tools to unravel structure and dynamics in polymers, Soft Matter, 2012, 8, 8257.
2 J. N. Israelachvili, Intermolecular and Surface Forces, Elsevier Inc., 3rd ed., 2011.
3 D. P. McMahon, D. L. Cheung, L. Goris, J. Dacuña, A. Salleo and A. Troisi, Relation between microstructure and charge transport in polymers of different regioregularity, J. Phys. Chem. C, 2011, 115, 19386–19393.
4 P. C. Tapping, S. N. Clafton, K. N. Schwarz, T. W. Kee and D. M. Huang, Molecular-Level Details of Morphology-Dependent Exciton Migration in Poly(3-hexylthiophene) Nanostructures, J. Phys. Chem. C, 2015, 119, 7047–7059.
5 M. Moreno, M. Casalegno, G. Raos, S. V. Meille and R. Po, Molecular Modeling of Crystalline Alkylthiophene Oligomers and Polymers, J. Phys. Chem. B, 2010, 114, 1591–1602.
6 D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, San Diego, 2nd edn., 2002.
7 K. N. Schwarz, T. W. Kee and D. M. Huang, Coarse-grained simulations of the solution-phase self-assembly of poly(3-hexylthiophene) nanostructures, Nanoscale, 2013, 5, 2017.