Molecular Dynamics Simulation - Revision history

Cmditradmin: /* References */

2020-05-12T18:53:18Z

References

← Older revision		Revision as of 11:53, 12 May 2020
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	==References==		==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.		# 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.		# 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.		# 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.		# 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.		# 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.		# 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.		# 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.

Cmditradmin: /* References */

2020-05-12T18:52:46Z

References

← Older revision		Revision as of 11:52, 12 May 2020
Line 122:		Line 122:
	==References==		==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.		# 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.		# 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.		# 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.		# 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.		# 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.		# 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.		# 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.

Cmditradmin: /* Force Field */

2020-05-11T22:25:13Z

Force Field

← Older revision		Revision as of 15:25, 11 May 2020
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	<math>K_{\phi,ijkl} =</math> dihedral energy constant for linearly bonded atoms i, j, k, and l, units of energy		<math>K_{\phi,ijkl} =</math> dihedral energy constant for linearly bonded atoms i, j, k, and l, units of energy

	<math>n = integer		<math>n = </math>integer

	<math>\phi_{ijkl} =</math> dihedral angle for linearly bonded atoms i, j, k, and l, units of degrees		<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>\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>\epsilon_{ij} =</math> Lennard-Jones energy constant for non-bonded atoms i and j, units of energy
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	<math>r_{ij} =</math> distance between 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>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>q_i, q_j = </math>charges of non-bonded atoms i and j, units of elementary\ charge

	<math>\kappa =</math> dielectric constant, unitless~~</math>~~		<math>\kappa =</math> dielectric constant, unitless

Cmditradmin: /* Force Field */

2020-05-11T22:23:25Z

Force Field

← Older revision		Revision as of 15:23, 11 May 2020
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	<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)		<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:		where:

	<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )		<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>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>K_{b,ij} = </math>bond energy constant for bonded atoms i and j, units of \frac{energy}{distance^2}

Cmditradmin: /* Force Field */

2020-05-11T22:22:31Z

Force Field

← Older revision		Revision as of 15:22, 11 May 2020
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	b_ij = distance between bonded atoms i and j, units of distance		b_ij = distance between bonded atoms i and j, units of distance

	~~b_(o~~,ij) = bond ~~distance~~ constant for 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}

	~~K_(θ,ijk)~~ = ~~angle energy constant for~~ bonded atoms i, j~~, and k~~, units of ~~energy/(degrees^2 )~~		<math>b_{ij} =</math> distance between bonded atoms i and j, units of distance

	~~θ_ijk~~ = ~~angle between~~ bonded atoms i, j~~, and k~~, units of ~~degrees~~		<math>b_{o,ij} =</math> bond distance constant for bonded atoms i and j, units of distance

	~~θ_(o~~,ijk) = angle constant for bonded atoms i, j, and k, units of degrees		<math>K_{\theta,ijk} = </math>angle energy constant for bonded atoms i, j, and k, units of \frac{energy}{degrees^2}

	~~K_(ζ,ijkl)~~ = ~~improper dihedral energy constant for~~ bonded atoms i, j, k~~, and l~~, units of ~~energy/(~~degrees~~^2 )~~		<math>\theta_{ijk} = </math>angle between bonded atoms i, j, and k, units of degrees

	~~ζ_ijkl~~ = ~~improper~~ angle for bonded atoms i, j, k~~, and l~~, units of degrees		<math>\theta_{o,ijk} =</math> angle constant for bonded atoms i, j, and k, units of degrees

	~~ζ_(o~~,ijkl) = improper ~~angle~~ constant for bonded atoms i, j, k, and l, 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}

	~~K_(ϕ,~~ijkl) = ~~dihedral energy constant~~ for ~~linearly~~ bonded atoms i, j, k, and l, units of ~~energy~~		<math>\zeta_{ijkl} = </math>improper angle for bonded atoms i, j, k, and l, units of degrees

	n = ~~integer~~		<math>\zeta_{o,ijkl} =</math> improper angle constant for bonded atoms i, j, k, and l, units of degrees

	~~ϕ_ijkl~~ = dihedral ~~angle~~ for linearly 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

	~~δ_n~~ = ~~dihedral angle constant for linearly bonded atoms i, j, k, and l, units of degrees~~		<math>n = integer

	~~ϵ_ij~~ = ~~Lennard-Jones energy constant~~ for ~~non-~~bonded atoms i and j, units of ~~energy~~		<math>\phi_{ijkl} =</math> dihedral angle for linearly bonded atoms i, j, k, and l, units of degrees

	~~σ_ij~~ = ~~Lennard-Jones distant~~ constant for ~~non-~~bonded atoms i and j, units of ~~distance~~		<math>\delta_n = dihedral angle constant for linearly bonded atoms i, j, k, and l, units of degrees

	~~r_ij~~ = ~~distance between~~ non-bonded atoms i and j, units of ~~distance~~		<math>\epsilon_{ij} =</math> Lennard-Jones energy constant for non-bonded atoms i and j, units of energy

	~~K_Coulomb~~ = ~~Coulomb~~ constant, ~~8.988 × 10^9 (Nm^2)/C^2~~		<math>\sigma_{ij} =</math> Lennard-Jones distant constant for non-bonded atoms i and j, units of distance

	~~q_i, q_j~~ = ~~charges of~~ non-bonded atoms i and j, units of ~~elementary charge~~		<math>r_{ij} =</math> distance between non-bonded atoms i and j, units of distance

	κ = ~~dielectric~~ constant, ~~unitless~~		<math>K_{Coulomb} =</math> Coulomb constant,\ 8.988\ \times\ {10}^9\frac{Nm^2}{C^2}

	<math>~~K_{b,ij} = bond energy constant for bonded atoms i and j, units of \frac{energy}{distance^2}~~		<math>q_i, q_j = </math>charges of non-bonded atoms i and j, units of elementary\ charge
	~~b_{ij} = distance between bonded atoms i and j, units of distance~~
	~~b_{o,ij} = bond distance constant for bonded atoms i and j, units of distance~~		<math>\kappa =</math> dielectric constant, unitless</math>
	~~K_{\theta,ijk} = angle energy constant for bonded atoms i, j, and k, units of \frac{energy}{degrees^2}~~
	~~\theta_{ijk} = angle between bonded atoms i, j, and k, units of degrees~~
	~~\theta_{o,ijk} = angle constant for bonded atoms i, j, and k, units of degrees~~
	~~K_{\zeta,ijkl} = improper dihedral energy constant for bonded atoms i, j, k, and l, units of \frac{energy}{degrees^2}~~
	~~\zeta_{ijkl} = improper angle for bonded atoms i, j, k, and l, units of degrees~~
	~~\zeta_{o,ijkl} = improper angle constant for bonded atoms i, j, k, and l, units of degrees~~
	~~K_{\phi,ijkl} = dihedral energy constant for linearly bonded atoms i, j, k, and l, units of energy~~
	~~n = integer~~
	~~\phi_{ijkl} = dihedral angle for linearly bonded atoms i, j, k, and l, units of degrees~~
	~~\delta_n = dihedral angle constant for linearly bonded atoms i, j, k, and l, units of degrees~~
	~~\epsilon_{ij} = Lennard-Jones energy constant for non-bonded atoms i and j, units of energy~~
	~~\sigma_{ij} = Lennard-Jones distant constant for non-bonded atoms i and j, units of distance~~
	~~r_{ij} = distance between non-bonded atoms i and j, units of distance~~
	~~K_{Coulomb} = Coulomb constant,\ 8.988\ \times\ {10}^9\frac{Nm^2}{C^2}~~
	q_i, q_j = charges of non-bonded atoms i and j, units of elementary\ charge
	\kappa = dielectric constant, unitless</math>

Cmditradmin: /* Force Field */

2020-05-11T22:19:43Z

Force Field

← Older revision		Revision as of 15:19, 11 May 2020
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	κ = dielectric constant, unitless		κ = dielectric constant, unitless

			<math>K_{b,ij} = bond energy constant for bonded atoms i and j, units of \frac{energy}{distance^2}
			b_{ij} = distance between bonded atoms i and j, units of distance
			b_{o,ij} = bond distance constant for bonded atoms i and j, units of distance
			K_{\theta,ijk} = angle energy constant for bonded atoms i, j, and k, units of \frac{energy}{degrees^2}
			\theta_{ijk} = angle between bonded atoms i, j, and k, units of degrees
			\theta_{o,ijk} = angle constant for bonded atoms i, j, and k, units of degrees
			K_{\zeta,ijkl} = improper dihedral energy constant for bonded atoms i, j, k, and l, units of \frac{energy}{degrees^2}
			\zeta_{ijkl} = improper angle for bonded atoms i, j, k, and l, units of degrees
			\zeta_{o,ijkl} = improper angle constant for bonded atoms i, j, k, and l, units of degrees
			K_{\phi,ijkl} = dihedral energy constant for linearly bonded atoms i, j, k, and l, units of energy
			n = integer
			\phi_{ijkl} = dihedral angle for linearly bonded atoms i, j, k, and l, units of degrees
			\delta_n = dihedral angle constant for linearly bonded atoms i, j, k, and l, units of degrees
			\epsilon_{ij} = Lennard-Jones energy constant for non-bonded atoms i and j, units of energy
			\sigma_{ij} = Lennard-Jones distant constant for non-bonded atoms i and j, units of distance
			r_{ij} = distance between non-bonded atoms i and j, units of distance
			K_{Coulomb} = Coulomb constant,\ 8.988\ \times\ {10}^9\frac{Nm^2}{C^2}
			q_i, q_j = charges of non-bonded atoms i and j, units of elementary\ charge
			\kappa = dielectric constant, unitless</math>


	For an interactive thorough exploration of this force field, visit https://interactive-md-ff.herokuapp.com/.		For an interactive thorough exploration of this force field, visit https://interactive-md-ff.herokuapp.com/.

Cmditradmin: /* Integrator */

2020-05-11T22:17:29Z

Integrator

← Older revision		Revision as of 15:17, 11 May 2020
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	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).		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).

	r(t+Δt)=r(t)+Δt v(t)+~~(Δt~~^~~2 F~~(t))/2m (3)		<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)

	v(t+Δt)=v(t)+~~(Δt~~(F(t)+ F(t+~~Δt)~~))/2m (4)		<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:		where:

Cmditradmin: /* Force Field */

2020-05-11T22:15:45Z

Force Field

← Older revision		Revision as of 15:15, 11 May 2020
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	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		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
	V_FF=Σ_bonds 1/2 K_(b,ij) (b_ij-b_(o,ij) )^2+Σ_angles 1/2 K_(θ,ijk) (θ_ijk-θ_(o,ijk) )^2+Σ_impropers 1/2 K_(ζ,ijkl) (ζ_ijkl-ζ_(o,ijkl) )^2+ Σ_dihedrals K_(ϕ,ijkl) (1+cos⁡(nϕ_ijkl-δ_n ) )+ Σ_pairs [4ϵ_ij [(σ_ij/r_ij )^12-(σ_ij/r_ij )^6 ]+K_Coulomb (q_i q_j)/(κr_ij )] (1)

	<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>
			<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:		where:
	<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )		<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )

Cmditradmin: /* Force Field */

2020-05-11T22:13:51Z

Force Field

← Older revision		Revision as of 15:13, 11 May 2020
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	V_FF=Σ_bonds 1/2 K_(b,ij) (b_ij-b_(o,ij) )^2+Σ_angles 1/2 K_(θ,ijk) (θ_ijk-θ_(o,ijk) )^2+Σ_impropers 1/2 K_(ζ,ijkl) (ζ_ijkl-ζ_(o,ijkl) )^2+ Σ_dihedrals K_(ϕ,ijkl) (1+cos⁡(nϕ_ijkl-δ_n ) )+ Σ_pairs [4ϵ_ij [(σ_ij/r_ij )^12-(σ_ij/r_ij )^6 ]+K_Coulomb (q_i q_j)/(κr_ij )] (1)		V_FF=Σ_bonds 1/2 K_(b,ij) (b_ij-b_(o,ij) )^2+Σ_angles 1/2 K_(θ,ijk) (θ_ijk-θ_(o,ijk) )^2+Σ_impropers 1/2 K_(ζ,ijkl) (ζ_ijkl-ζ_(o,ijkl) )^2+ Σ_dihedrals K_(ϕ,ijkl) (1+cos⁡(nϕ_ijkl-δ_n ) )+ Σ_pairs [4ϵ_ij [(σ_ij/r_ij )^12-(σ_ij/r_ij )^6 ]+K_Coulomb (q_i q_j)/(κr_ij )] (1)

	<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>		<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>
	where:		where:
	<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )		<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )

Cmditradmin: /* Force Field */

2020-05-11T22:13:10Z

Force Field

← Older revision		Revision as of 15:13, 11 May 2020
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	V_FF=Σ_bonds 1/2 K_(b,ij) (b_ij-b_(o,ij) )^2+Σ_angles 1/2 K_(θ,ijk) (θ_ijk-θ_(o,ijk) )^2+Σ_impropers 1/2 K_(ζ,ijkl) (ζ_ijkl-ζ_(o,ijkl) )^2+ Σ_dihedrals K_(ϕ,ijkl) (1+cos⁡(nϕ_ijkl-δ_n ) )+ Σ_pairs [4ϵ_ij [(σ_ij/r_ij )^12-(σ_ij/r_ij )^6 ]+K_Coulomb (q_i q_j)/(κr_ij )] (1)		V_FF=Σ_bonds 1/2 K_(b,ij) (b_ij-b_(o,ij) )^2+Σ_angles 1/2 K_(θ,ijk) (θ_ijk-θ_(o,ijk) )^2+Σ_impropers 1/2 K_(ζ,ijkl) (ζ_ijkl-ζ_(o,ijkl) )^2+ Σ_dihedrals K_(ϕ,ijkl) (1+cos⁡(nϕ_ijkl-δ_n ) )+ Σ_pairs [4ϵ_ij [(σ_ij/r_ij )^12-(σ_ij/r_ij )^6 ]+K_Coulomb (q_i q_j)/(κr_ij )] (1)

			<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>
	where:		where:
	<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )		<math>K_(b,ij)</math> = bond energy constant for bonded atoms i and j, units of energy/(distance^2 )
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	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:		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>~~		<math>F=ma=-\frac{\delta V_{FF}}{\delta r}</math>(2)
	~~<math>F=ma=- \frac{-(δV_FF)}{δr~~} </math> (2)

	where:		where: