Package etomica.statmech

Class LennardJones

java.lang.Object

etomica.statmech.LennardJones

public final class LennardJones
extends java.lang.Object

Implementation of some semi-empirical equations of state for the Lennard-Jones model. Equations for the fcc solid phase and liquid-fcc saturation conditions are based on the empirical formulas proposed by van der Hoef: [1] M.A. van der Hoef, "Free energy of the Lennard-Jones solid", Journal of Chemical Physics, vol 113, page 8142 (2000). Equations for vapor-fcc saturation conditions are based on equations given in: [2] M.S. van der Hoef, "Gas-solid coexistence of the Lennard-Jones system", Journal of Chemical Physics, vol 117, page 5093 (2002). No fluid-phase equations are (yet) implemented here. This is a library class, containing only static methods, and thus is not intended to be instantiated.

Method Summary

Modifier and Type	Method	Description
static double	aResidualFcc(double T, double rho)	Molar Helmholtz free energy for the fcc solid, in excess of an ideal gas at the same temperature and density.
static double	B2(double T)	Returns the 2nd virial coefficient.
static double[]	liquidFccCoexDensities(double T)	Returns the densities of coexisting (saturated) liquid and fcc phases (i.e., melting and freezing densities).
static double	liquidFccCoexPressure(double T)	Returns the saturation pressure for fluid-fcc coexistence (i.e., the melting or freezing pressure).
static void	main(java.lang.String[] args)
static double	uFcc(double T, double rho)	Total ensemble-averaged internal potential energy for the fcc solid.
static double	uStaticFcc(double rho)	Potential energy per atom of a static fcc lattice with a LJ particle at each site.
static double[]	vaporFccCoexDensities(double T)	Returns the densities of coexisting (saturated) vapor and fcc phases (sublimation densities).
static double	vaporFccCoexPressure(double T)	Returns the saturation pressure for vapor-fcc coexistence.
static double	ZFcc(double T, double rho)	Compressibility factor, P/rho/kT for the fcc solid.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

aResidualFcc

public static double aResidualFcc(double T,
                                  double rho)

Molar Helmholtz free energy for the fcc solid, in excess of an ideal gas at the same temperature and density. Total molar free energy is obtained by adding T*(ln[rho] - 1) + 3*T*ln[Lambda], where Lambda is the thermal deBroglie wavelength. More precisely, one should add the term given, plus ln[N!]/N - (ln[N]-1), which goes to zero for N->infinity, where N is the number of atoms (typically specified only for small systems). Note that this detail does not account for all finite-size effects. Based on Eq. 21 of van der Hoef [1].

Parameters:

T - the temperature, in units of epsilon/k (thus input is kT/epsilon)

rho - the number density, in units of 1/sigma^3 (thus input is rho*sigma^3)

Returns:

the excess Helmholtz free energy per atom, in units of epsilon

uFcc

public static double uFcc(double T,
                          double rho)

Total ensemble-averaged internal potential energy for the fcc solid. Based on Eq. 19 of van der Hoef [1].

Parameters:

T - temperature, in units of epsilon/k

rho - number density, in units of 1/sigma^3

Returns:

internal potential energy, in units of epsilon (thus u/epsilon is returned)

ZFcc

public static double ZFcc(double T,
                          double rho)

Compressibility factor, P/rho/kT for the fcc solid. Based on Eq. 20 of van der Hoef [1].

Parameters:

T - temperature, in units of epsilon/k

rho - number density, in units of 1/sigma^3

Returns:

compressibility factor (dimensionless)

uStaticFcc

public static double uStaticFcc(double rho)

Potential energy per atom of a static fcc lattice with a LJ particle at each site.

Parameters:

rho - number density, in units of 1/sigma^3

Returns:

the static-lattice energy, in units of epsilon

liquidFccCoexDensities

public static double[] liquidFccCoexDensities(double T)

Returns the densities of coexisting (saturated) liquid and fcc phases (i.e., melting and freezing densities). Uses Eqs. (25) and (26) of van der Hoef [1].

Parameters:

T - temperature, in units of epsilon/k

Returns:

coexistence densities, in units of 1/sigma^3; liquid is element 0, fcc solid is element 1

liquidFccCoexPressure

public static double liquidFccCoexPressure(double T)

Returns the saturation pressure for fluid-fcc coexistence (i.e., the melting or freezing pressure). Uses Eq. (24) of van der Hoef [1], which is based on a formula proposed by Agrawal and Kofke, referenced therein.

Parameters:

T - temperature, in units of epsilon/k

Returns:

saturation pressure, in units of epsilon/sigma^3 (thus p sigma^3/epsilon is returned)

vaporFccCoexDensities

public static double[] vaporFccCoexDensities(double T)

Returns the densities of coexisting (saturated) vapor and fcc phases (sublimation densities). Uses Eqs. (2) and (3) of van der Hoef [2].

Parameters:

T - temperature, in units of epsilon/k

Returns:

coexistence densities, in units of 1/sigma^3; vapor is element 0, fcc solid is element 1

vaporFccCoexPressure

public static double vaporFccCoexPressure(double T)

Returns the saturation pressure for vapor-fcc coexistence. Uses Eq. (1) of van der Hoef [2].

Parameters:

T - temperature, in units of epsilon/k

Returns:

saturation pressure, in units of epsilon/sigma^3 (thus p sigma^3/epsilon is returned)

B2

public static double B2(double T)

Returns the 2nd virial coefficient. Evaluated using analytic expression for Mayer integral given in terms of the confluent hypergeometric function.

Parameters:

T - temperature, in units of epsilon/k

Returns:

2nd virial coefficient, in units of sigma^3

main

public static void main(java.lang.String[] args)