ufjc.potential

A module for handling potentials.

This module contains several different classes representing potentials, each having methods to compute relevant nondimensional quantities as functions of nondimensional force or stretch. This module also contains the parent class Potential that is used to assign a potential to a given model using keyword arguments.

Examples

Create a Lennard-Jones potential model with a nondimensional potential energy scale of 8 and evaluate the nondimensional potential energy at a stretch of 1.23:

>>> from ufjc.potential import LennardJonesPotential
>>> model = LennardJonesPotential(varepsilon=8)
>>> model.beta_u(1.23)
4.046654314368616

Do the same with the Lenard-Jones-FENE potential:

>>> from ufjc.potential import LJFENEPotential
>>> model = LJFENEPotential(varepsilon=(8, 8))
>>> model.beta_u(1.23)
8.510502022381505

Create a single-link model in one dimension, instantiate it with the Morse potential, and compute the incremental link stretch under a nondimensional force of 8:

>>> from ufjc.potential import Potential
>>> class Link1D(Potential):
...     def __init__(self, **kwargs):
...         Potential.__init__(self, **kwargs)
>>> _ = Link1D(potential='morse').delta_lambda(8)
>>> Link1D(potential='lj-fene').eta_link(1)
184.0

class CustomPotential(**kwargs)[source]

Bases: object

A custom user-defined potential.

potential

The potential name.

Type:: str

varepsilon

The nondimensional energy scale.

Type:: float

phi

The scaled nondimensional energy function.

Type:: function

eta_link

The nondimensional force as a function of stretch.

Type:: function

delta_lambda

The incremental stretch as a function of the nondimensional force (optional).

Type:: function

kappa

The nondimensional stiffness.

Type:: float

c

The correction parameter.

Type:: float

beta_u(lambda_)[source]

The nondimensional potential energy function,

\[\beta u(\lambda) = \varepsilon\phi(\lambda) .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional potential energy(s).
Return type:: numpy.ndarray

class HarmonicPotential(**kwargs)[source]

Bases: object

The harmonic potential.

varepsilon

The nondimensional energy scale.

Type:: float

kappa

The nondimensional stiffness \(\kappa=\varepsilon\).

Type:: float

c

The correction parameter \(c=1\).

Type:: float

beta_u(lambda_)[source]

The nondimensional potential energy function,

\[\beta u(\lambda) = \varepsilon\phi(\lambda) .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional potential energy(s).
Return type:: numpy.ndarray

delta_lambda(eta)[source]

The incremental stretch as a function of nondimensional force,

\[\Delta\lambda(\eta) = \frac{\eta}{\varepsilon} .\]

Parameters:: eta (array_like) – The nondimensional force(s).
Returns:: The incremental stretch(s).
Return type:: numpy.ndarray

eta_link(lambda_)[source]

The nondimensional force as a function of stretch,

\[\eta(\lambda) = \varepsilon(\lambda - 1) .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional force(s).
Return type:: numpy.ndarray

Example

Compute the nondimensional force at a sample stretch:

>>> from ufjc.potential import HarmonicPotential
>>> HarmonicPotential().eta_link(1.8)
70.4

phi(lambda_)[source]

The scaled nondimensional potential energy function,

\[\phi(\lambda) = \frac{1}{2}(\lambda-1)^2 .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The scaled nondimensional potential energy(s).
Return type:: numpy.ndarray

class LJFENEPotential(**kwargs)[source]

Bases: object

The Lennard-Jones-FENE potential [2].

varepsilon

The nondimensional energy scale.

Type:: float

kappa

The nondimensional stiffness.

Type:: float

c

The correction parameter.

Type:: float

eta_max

The maximum nondimensional force \(\eta_\mathrm{max} = \eta(\lambda_\mathrm{max})\).

Type:: float

lambda_max

The stretch at the maximum nondimensional force.

Type:: float

beta_u(lambda_)[source]

The nondimensional potential energy function,

\[\beta u(\lambda) = \varepsilon\phi(\lambda) = \varepsilon_1\left( \frac{1}{\lambda^{12}} - \frac{2}{\lambda^6} + 1 \right) - \frac{\varepsilon_2}{2}\,\ln\left[ 1 - \left(\frac{\lambda}{\lambda_\mathrm{max}}\right)^2 \right] .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional potential energy(s).
Return type:: numpy.ndarray

eta_link(lambda_)[source]

The nondimensional force as a function of stretch,

\[\eta(\lambda) = 12\varepsilon_1\left( \frac{1}{\lambda^7} - \frac{1}{\lambda^{13}} \right) + \frac{\varepsilon_2\lambda} {\lambda_\mathrm{max}^2 - \lambda^2} .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional force(s).
Return type:: numpy.ndarray

phi(lambda_)[source]

The scaled nondimensional potential energy function,

\[\phi(\lambda) = \frac{\varepsilon_1}{\varepsilon_2}\left( \frac{1}{\lambda^{12}} - \frac{2}{\lambda^6} + 1 \right) - \frac{1}{2}\,\ln\left[ 1 - \left(\frac{\lambda}{\lambda_\mathrm{max}}\right)^2 \right] .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The scaled nondimensional potential energy(s).
Return type:: numpy.ndarray

class LennardJonesPotential(**kwargs)[source]

Bases: object

The Lennard-Jones potential [1].

varepsilon

The nondimensional energy scale.

Type:: float

kappa

The nondimensional stiffness \(\kappa=72\varepsilon\).

Type:: float

c

The correction parameter \(c=2/23\).

Type:: float

eta_max

The maximum nondimensional force \(\eta_\mathrm{max} = \eta(\lambda_\mathrm{max})\).

Type:: float

lambda_max

The stretch at the maximum nondimensional force, \(\lambda_\mathrm{max} = (13/7)^{1/6}\).

Type:: float

beta_u(lambda_)[source]

The nondimensional potential energy function,

\[\beta u(\lambda) = \varepsilon\phi(\lambda) .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional potential energy(s).
Return type:: numpy.ndarray

eta_link(lambda_)[source]

The nondimensional force as a function of stretch,

\[\eta(\lambda) = 12\varepsilon\left( \frac{1}{\lambda^7} - \frac{1}{\lambda^{13}}\right) .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional force(s).
Return type:: numpy.ndarray

phi(lambda_)[source]

The scaled nondimensional potential energy function,

\[\phi(\lambda) = \frac{1}{\lambda^{12}} - \frac{2}{\lambda^6} + 1 .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The scaled nondimensional potential energy(s).
Return type:: numpy.ndarray

class LogSquaredPotential(**kwargs)[source]

Bases: object

The log-squared potential [3].

varepsilon

The nondimensional energy scale.

Type:: float

kappa

The nondimensional stiffness \(\kappa=\varepsilon\).

Type:: float

c

The correction parameter \(c=2/5\).

Type:: float

eta_max

The maximum nondimensional force \(\eta_\mathrm{max} = e^{-1}\varepsilon\).

Type:: float

lambda_max

The stretch at the maximum nondimensional force, \(\lambda_\mathrm{max} = e^{1}\).

Type:: float

beta_u(lambda_)[source]

The nondimensional potential energy function,

\[\beta u(\lambda) = \varepsilon\phi(\lambda) .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional potential energy(s).
Return type:: numpy.ndarray

delta_lambda(eta)[source]

The incremental stretch as a function of nondimensional force,

\[\Delta\lambda(\eta) = e^{-\mathcal{W}(-\eta/\varepsilon)} ,\qquad \eta\in[0,\eta_\mathrm{max}] .\]

Parameters:: eta (array_like) – The nondimensional force(s).
Returns:: The incremental stretch(s).
Return type:: numpy.ndarray

eta_link(lambda_)[source]

The nondimensional force as a function of stretch,

\[\eta(\lambda) = \varepsilon\,\frac{\ln(\lambda)}{\lambda} .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional force(s).
Return type:: numpy.ndarray

Example

Compute the nondimensional force at a sample stretch:

>>> from ufjc.potential import LogSquaredPotential
>>> LogSquaredPotential().eta_link(1.8)
28.736236950770266

phi(lambda_)[source]

The scaled nondimensional potential energy function,

\[\phi(\lambda) = \frac{1}{2}\big[\ln(\lambda)\big]^2 .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The scaled nondimensional potential energy(s).
Return type:: numpy.ndarray

class MiePotential(**kwargs)[source]

Bases: object

The Mie potential [4].

varepsilon

The nondimensional energy scale.

Type:: float

n

The repulsive exponent.

Type:: float

m

The attractive exponent.

Type:: float

kappa

The nondimensional stiffness \(\kappa=nm\varepsilon\).

Type:: float

c

The correction parameter \(c=\frac{4m(m+1)-2n(n+1)}{2m(m^2+5m+4)-n(n^2+5n+4)}\).

Type:: float

eta_max

The maximum nondimensional force \(\eta_\mathrm{max} = \eta(\lambda_\mathrm{max})\).

Type:: float

lambda_max

The stretch at the maximum nondimensional force, \(\lambda_\mathrm{max}=\left[\frac{n+1}{m+1}\right]^{1/(n-m)}\) .

Type:: float

beta_u(lambda_)[source]

The nondimensional potential energy function,

\[\beta u(\lambda) = \varepsilon\phi(\lambda) .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional potential energy(s).
Return type:: numpy.ndarray

eta_link(lambda_)[source]

The nondimensional force as a function of stretch,

\[\eta(\lambda) = \varepsilon\, \frac{nm}{(n-m)} \left(\frac{1}{\lambda^{m+1}} - \frac{1}{\lambda^{n+1}}\right) .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional force(s).
Return type:: numpy.ndarray

phi(lambda_)[source]

The scaled nondimensional potential energy function,

\[\phi(\lambda) = \frac{1}{(n-m)} \left(\frac{m}{\lambda^n} - \frac{n}{\lambda^m}\right) .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The scaled nondimensional potential energy(s).
Return type:: numpy.ndarray

class MorsePotential(**kwargs)[source]

Bases: object

The Morse potential [5].

varepsilon

The nondimensional energy scale.

Type:: float

alpha

The Morse parameter.

Type:: float

kappa

The nondimensional stiffness \(\kappa=2\varepsilon\alpha^2\).

Type:: float

c

The correction parameter \(c=1/(1+3\alpha/2)\).

Type:: float

eta_max

The maximum nondimensional force \(\eta_\mathrm{max} = \sqrt{\kappa\varepsilon/8}\).

Type:: float

lambda_max

The stretch at the maximum nondimensional force, \(\lambda_\mathrm{max} = 1+\ln(2)/\alpha\).

Type:: float

beta_u(lambda_)[source]

The nondimensional potential energy function,

\[\beta u(\lambda) = \varepsilon\phi(\lambda) .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional potential energy(s).
Return type:: numpy.ndarray

delta_lambda(eta)[source]

The incremental stretch as a function of nondimensional force,

\[\Delta\lambda(\eta) = \ln\left(\frac{2}{1 + \sqrt{1 - \eta/\eta_\mathrm{max}}}\right)^{1/\alpha} ,\qquad \eta\in[0,\eta_\mathrm{max}] .\]

Parameters:: eta (array_like) – The nondimensional force(s).
Returns:: The incremental stretch(s).
Return type:: numpy.ndarray

eta_link(lambda_)[source]

The nondimensional force as a function of stretch,

\[\eta(\lambda) = 2\alpha\varepsilon e^{-\alpha(\lambda - 1)} \left[1 - e^{-\alpha(\lambda - 1)}\right]^2 .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional force(s).
Return type:: numpy.ndarray

Example

Compute the nondimensional force at a sample stretch:

>>> from ufjc.potential import MorsePotential
>>> _ = MorsePotential().eta_link(1.23)

phi(lambda_)[source]

The scaled nondimensional potential energy function,

\[\phi(\lambda) = \left[1 - e^{-\alpha(\lambda - 1)}\right]^2 .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The scaled nondimensional potential energy(s).
Return type:: numpy.ndarray

class PolynomialPotential(**kwargs)[source]

Bases: object

A polynomial potential.

varepsilon

The nondimensional energy scale.

Type:: float

coefficients

The coefficients \(a_k\).

Type:: array_like

kappa

The nondimensional stiffness \(\kappa=a_1\varepsilon\).

Type:: float

c

The correction parameter \(c=(1 - \frac{a_2}{2a_1})^{-1}\).

Type:: float

beta_u(lambda_)[source]

The nondimensional potential energy function,

\[\beta u(\lambda) = \varepsilon\phi(\lambda) .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional potential energy(s).
Return type:: numpy.ndarray

eta_link(lambda_)[source]

The nondimensional force as a function of stretch,

\[\eta(\lambda) = \varepsilon\sum_{k=1} a_k\, \lambda^k .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The nondimensional force(s).
Return type:: numpy.ndarray

phi(lambda_)[source]

The scaled nondimensional potential energy function,

\[\phi(\lambda) = \sum_{k=2} \frac{a_{k-1}}{k}\, \lambda^k .\]

Parameters:: lambda (array_like) – The stretch(s).
Returns:: The scaled nondimensional potential energy(s).
Return type:: numpy.ndarray

class Potential(**kwargs)[source]

Bases: object

A class to assign a potential to a given model through inheritance.

potential

The potential type.

Type:: str

pot

The potential model instance.

Type:: object

References

[1]

John Edward Jones. On the determination of molecular fields. II. From the equation of state of a gas. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 106(738):463–477, 1924. doi:10.1098/rspa.1924.0082.

[2]

Kurt Kremer and Gary S Grest. Dynamics of entangled linear polymer melts: A molecular-dynamics simulation. The Journal of Chemical Physics, 92(8):5057–5086, 1990. doi:10.1063/1.458541.

[3]

Yunwei Mao, Brandon Talamini, and Lallit Anand. Rupture of polymers by chain scission. Extreme Mechanics Letters, 13:17–24, 2017. doi:10.1016/j.eml.2017.01.003.

[4]

Gustav Mie. Zur kinetischen theorie der einatomigen körper. Annalen der Physik, 316(8):657–697, 1903. doi:10.1002/andp.19033160802.

[5]

Philip M Morse. Diatomic molecules according to the wave mechanics. II. Vibrational levels. Physical Review, 34(1):57, 1929. doi:10.1103/PhysRev.34.57.