Uhlenbeck-Ford model - Revision history

Rleite: /* Reference system */

2019-07-11T00:44:06Z

Reference system

← Older revision		Revision as of 02:44, 11 July 2019
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	== Reference system ==		== Reference system ==

	The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using [[thermodynamic integration]] techniques for different natures of fluid description ([[Lennard-Jones ~~model \|Lennard~~-~~Jones]] and [[Stillinger~~-~~Weber potential \| Stillinger~~-~~Weber]]~~) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. Recently, water-like models using the UF model as an interacting potential were reported and added on LAMMPS source-code. The respective publication <ref name="CMS">[https://doi.org/10.1016/j.commatsci.2018.12.029 R. Paula Leite, and M. de Koning "Nonequilibrium free-energy calculations of fluids using LAMMPS", Computational Material Science '''159''', 316-326 (2019).]</ref> and a complete guide <ref name="FFE">[https://github.com/plrodolfo/FluidFreeEnergyforLAMMPS]</ref> are available on reference section. The typical values for <math>p</math> are <math>50 \sim 100.</math>.		The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using [[thermodynamic integration]] techniques for different natures of fluid description (Lennard-Jones, Silicon-SW, CuZr-EAM-metallic glass, rigid and flexible water models) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. Recently, water-like models using the UF model as an interacting potential were reported and added on LAMMPS source-code. The respective publication <ref name="CMS">[https://doi.org/10.1016/j.commatsci.2018.12.029 R. Paula Leite, and M. de Koning "Nonequilibrium free-energy calculations of fluids using LAMMPS", Computational Material Science '''159''', 316-326 (2019).]</ref> and a complete guide <ref name="FFE">[https://github.com/plrodolfo/FluidFreeEnergyforLAMMPS]</ref> are available on reference section. The typical values for <math>p</math> are <math>50 \sim 100.</math>.

	==References==		==References==

Rleite at 00:40, 11 July 2019

2019-07-11T00:40:01Z

← Older revision		Revision as of 02:40, 11 July 2019
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	== Reference system ==		== Reference system ==

	The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using [[thermodynamic integration]] techniques for different natures of fluid description ([[Lennard-Jones model \|Lennard-Jones]] and [[Stillinger-Weber potential \| Stillinger-Weber]]) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. Recently, water-like models using the UF model as an interacting potential were reported and added on LAMMPS source-code. The respective publication <ref name="CMS"></ref> and a complete guide <ref name="FFE"></ref> are available on reference section. The typical values for <math>p</math> are <math>50 \sim 100.</math>.		The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using [[thermodynamic integration]] techniques for different natures of fluid description ([[Lennard-Jones model \|Lennard-Jones]] and [[Stillinger-Weber potential \| Stillinger-Weber]]) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. Recently, water-like models using the UF model as an interacting potential were reported and added on LAMMPS source-code. The respective publication <ref name="CMS">[https://doi.org/10.1016/j.commatsci.2018.12.029 R. Paula Leite, and M. de Koning "Nonequilibrium free-energy calculations of fluids using LAMMPS", Computational Material Science '''159''', 316-326 (2019).]</ref> and a complete guide <ref name="FFE">[https://github.com/plrodolfo/FluidFreeEnergyforLAMMPS]</ref> are available on reference section. The typical values for <math>p</math> are <math>50 \sim 100.</math>.

	==References==		==References==

Rleite: /* Reference system */

2019-07-11T00:36:03Z

Reference system

← Older revision		Revision as of 02:36, 11 July 2019
Line 83:		Line 83:
	== Reference system ==		== Reference system ==

	The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using [[thermodynamic integration]] techniques for different natures of fluid description ([[Lennard-Jones model \|Lennard-Jones]] and [[Stillinger-Weber potential \| Stillinger-Weber]]) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. Recently, water-like models using the UF model as an interacting potential were reported and added on LAMMPS source-code. A complete guide ~~can be found on these~~ <ref name="~~CMS~~"></ref> ~~links~~. The typical values for <math>p</math> are <math>50 \sim 100.</math>.		The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using [[thermodynamic integration]] techniques for different natures of fluid description ([[Lennard-Jones model \|Lennard-Jones]] and [[Stillinger-Weber potential \| Stillinger-Weber]]) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. Recently, water-like models using the UF model as an interacting potential were reported and added on LAMMPS source-code. The respective publication <ref name="CMS"></ref> and a complete guide <ref name="FFE"></ref> are available on reference section. The typical values for <math>p</math> are <math>50 \sim 100.</math>.

	==References==		==References==

Rleite: /* Reference system */

2019-07-11T00:34:37Z

Reference system

← Older revision		Revision as of 02:34, 11 July 2019
Line 83:		Line 83:
	== Reference system ==		== Reference system ==

	The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using [[thermodynamic integration]] techniques for different natures of fluid description ([[Lennard-Jones model \|Lennard-Jones]] and [[Stillinger-Weber potential \| Stillinger-Weber]]) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. The typical values for <math>p</math> are <math>50 \sim 100.</math>.		The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using [[thermodynamic integration]] techniques for different natures of fluid description ([[Lennard-Jones model \|Lennard-Jones]] and [[Stillinger-Weber potential \| Stillinger-Weber]]) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. Recently, water-like models using the UF model as an interacting potential were reported and added on LAMMPS source-code. A complete guide can be found on these <ref name="CMS"></ref> links. The typical values for <math>p</math> are <math>50 \sim 100.</math>.

	==References==		==References==

Carl McBride: Added some internal links

2017-10-17T10:39:26Z

Added some internal links

← Older revision		Revision as of 12:39, 17 October 2017
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	The '''Uhlenbeck-Ford model''' (UFM), originally named Gaussian gas <ref>D. McQuarrie, Statistical Mechanics (University Science Books, 2000).</ref>, was proposed by G. Uhlenbeck and G. Ford <ref name="UF">G. Uhlenbeck and G. Ford, in Studies in Statistical Mechanics— The Theory of Linear Graphs with Application to the Theory of the Virial Development of the Properties of Gases, edited by G. E. Uhlenbeck and J. de Boer (North-Holland, Amsterdam, 1962), Vol. 2. </ref> for the theoretical study of imperfect gases. This model is characterized by an ultrasoft, purely repulsive pairwise interaction potential that diverges logarithmically at the origin and features an energy scale that coincides with the thermal energy unit <math>k_B T</math> where <math>k_B</math> is the ~~Boltzmman~~ constant and <math>T</math> the absolute temperature. The particular functional form of the potential permits, in principle, that the virial coefficients and, therefore, the equation of state and excess free energies for the fluid phase be evaluated analytically. A recent study <ref name="JCP">[http://dx.doi.org/10.1063/1.4967775 R. Paula Leite, R. Freitas, R. Azevedo and M. de Koning "The Uhlenbeck-Ford model: Exact virial coefficients and application as a reference system in fluid-phase free-energy calculations", Journal of Chemical Physics '''145''', 194101 (2016).]</ref> showed that this model can be used as a reference system for fluid-phase free-energy calculations.		The '''Uhlenbeck-Ford model''' (UFM), originally named Gaussian gas <ref>D. McQuarrie, Statistical Mechanics (University Science Books, 2000).</ref>, was proposed by G. Uhlenbeck and G. Ford <ref name="UF">G. Uhlenbeck and G. Ford, in Studies in Statistical Mechanics— The Theory of Linear Graphs with Application to the Theory of the Virial Development of the Properties of Gases, edited by G. E. Uhlenbeck and J. de Boer (North-Holland, Amsterdam, 1962), Vol. 2. </ref> for the theoretical study of imperfect gases. This model is characterized by an ultrasoft, purely repulsive pairwise interaction potential that diverges logarithmically at the origin and features an energy scale that coincides with the thermal energy unit <math>k_B T</math> where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> the absolute [[temperature]]. The particular functional form of the potential permits, in principle, that the virial coefficients and, therefore, the equation of state and excess free energies for the fluid phase be evaluated analytically. A recent study <ref name="JCP">[http://dx.doi.org/10.1063/1.4967775 R. Paula Leite, R. Freitas, R. Azevedo and M. de Koning "The Uhlenbeck-Ford model: Exact virial coefficients and application as a reference system in fluid-phase free-energy calculations", Journal of Chemical Physics '''145''', 194101 (2016).]</ref> showed that this model can be used as a reference system for fluid-phase free-energy calculations.


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	* <math> \sigma </math> is a length-scale parameter.		* <math> \sigma </math> is a length-scale parameter.

	Note that increasing the value of p gives rise to a stronger repulsion.		Note that increasing the value of <math>p</math> gives rise to a stronger repulsion.

	== Equation of state ==		== Equation of state ==

	The equation of state for the UFM fluid using virial expansion has recently been studied by Paula Leite, Freitas, Azevedo and de Koning <ref name="JCP"></ref> and is given by		The [[Equations of state \|equation of state]] for the UFM fluid using [[Virial equation of state \| virial expansion]] has recently been studied by Paula Leite, Freitas, Azevedo and de Koning <ref name="JCP"></ref> and is given by

	:<math>\beta bP_{\rm UF}(x,p) = x + \sum_{n=2}^{\infty} \tilde{B}_n(p) \,x^n</math>,		:<math>\beta bP_{\rm UF}(x,p) = x + \sum_{n=2}^{\infty} \tilde{B}_n(p) \,x^n</math>,

	where		where
	*<math>P_{\rm UF}(x,p)</math> is the system pressure.		*<math>P_{\rm UF}(x,p)</math> is the system [[pressure]].
	*<math>b \equiv \frac{1}{2}(\pi\sigma^2)^{3/2} </math> is a constant.		*<math>b \equiv \frac{1}{2}(\pi\sigma^2)^{3/2} </math> is a constant.
	*<math>x \equiv b\rho </math> is an adimensional variable.		*<math>x \equiv b\rho </math> is an adimensional variable.
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	== Excess Helmholtz free-energy ==		== Excess Helmholtz free-energy ==

	The excess Helmholtz free-energy expression for the UFM, which can be obtained integrating the equation of state with respect to volume <ref name="JCP"></ref>, is given by		The excess [[Helmholtz energy function \| Helmholtz free-energy]] expression for the UFM, which can be obtained integrating the equation of state with respect to volume <ref name="JCP"></ref>, is given by

	:<math>\frac{\beta F^{\rm (exc)}_{\rm UF}(x,p)}{N} =\sum_{n=1}^{\infty} \frac{\tilde{B}_{n+1}(p)}{n} \,x^n</math>,		:<math>\frac{\beta F^{\rm (exc)}_{\rm UF}(x,p)}{N} =\sum_{n=1}^{\infty} \frac{\tilde{B}_{n+1}(p)}{n} \,x^n</math>,
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	== Virial coefficients ==		== Virial coefficients ==
	As the virial coefficients for the UFM can be calculated analytically, in principle, Paula Leite, Freitas, Azevedo and de Koning <ref name="JCP"></ref> recalculated the first eight coefficients previously known <math>(\tilde{B}_2</math> to <math>\tilde{B}_9)</math> and obtained values for four new coefficients <math>(\tilde{B}_{10}</math> to <math>\tilde{B}_{13})</math> for <math>p=1</math>. Some coefficients for <math>p=2</math> and <math>p=10</math> are also shown in the table below.		As the [[Virial equation of state \| virial coefficients]] for the UFM can be calculated analytically, in principle, Paula Leite, Freitas, Azevedo and de Koning <ref name="JCP"></ref> recalculated the first eight coefficients previously known <math>(\tilde{B}_2</math> to <math>\tilde{B}_9)</math> and obtained values for four new coefficients <math>(\tilde{B}_{10}</math> to <math>\tilde{B}_{13})</math> for <math>p=1</math>. Some coefficients for <math>p=2</math> and <math>p=10</math> are also shown in the table below.

	Table 1: Numerical representation of the exact virial coefficients for the UFM.		Table 1: Numerical representation of the exact virial coefficients for the UFM.
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	== Numerical representations ==		== Numerical representations ==
	Due to the number of integrals needed to compute the virial coefficients, a numerical representation using molecular dynamics data is usually adopted. In this sense, extensive molecular dynamics simulations have been performed and using a piece-wise fitting procedure, called cubic splines, numerical expressions for the UFM's equation of state were constructed. Moreover, this representation also leads to an excellent description for the excess Helmholtz free energy <math>F^{\rm (exc)}_{\rm UF}(x,p)</math>.		Due to the number of integrals needed to compute the virial coefficients, a numerical representation using molecular dynamics data is usually adopted. In this sense, extensive [[molecular dynamics]] simulations have been performed and using a piece-wise fitting procedure, called cubic splines, numerical expressions for the UFM's equation of state were constructed. Moreover, this representation also leads to an excellent description for the excess Helmholtz free energy <math>F^{\rm (exc)}_{\rm UF}(x,p)</math>.

	All spline representations of equations of state and excess Helmholtz free energies are available by means of the ~~'''''~~python~~'''''~~ script ''ufGenerator.py'' supplied in the supplementary material of Paula Leite ''et al''. paper <ref name="JCP"></ref>.		All spline representations of equations of state and excess Helmholtz free energies are available by means of the python script ''ufGenerator.py'' supplied in the supplementary material of Paula Leite ''et al''. paper <ref name="JCP"></ref>.

	== Phase diagram ==		== Phase diagram ==
	The phase diagram of the UFM in the <math>(p^{-1},P_{\rm UF}^*)</math> has been constructed by Paula Leite, Santos-Flórez and de Koning <ref name="PRE">[https://doi.org/10.1103/PhysRevE.96.032115 R. Paula Leite, P. A. Santos-Flórez and M. de Koning "Uhlenbeck-Ford model: Phase diagram and corresponding-states analysis", Physical Review E '''96''', 032115 (2017).]</ref>. Beyond the fluid phase, body-centered-cubic (bcc) and face-centered-cubic (fcc) are the only thermodynamically stable crystalline phases. These three phases meet each other at a triple point. Furthermore, they reported the existence of two reentrant transition sequences as a function of the number density, one featuring a fluid-bcc-fluid succession and another displaying a bcc-fcc-bcc sequence near the triple point.		The [[Phase diagrams \| phase diagram]] of the UFM in the <math>(p^{-1},P_{\rm UF}^*)</math> has been constructed by Paula Leite, Santos-Flórez and de Koning <ref name="PRE">[https://doi.org/10.1103/PhysRevE.96.032115 R. Paula Leite, P. A. Santos-Flórez and M. de Koning "Uhlenbeck-Ford model: Phase diagram and corresponding-states analysis", Physical Review E '''96''', 032115 (2017).]</ref>. Beyond the fluid phase, [[Building up a body centered cubic lattice \|body-centered-cubic]] (bcc) and [[Building up a face centered cubic lattice \| face-centered-cubic]] (fcc) are the only thermodynamically stable crystalline phases. These three phases meet each other at a [[triple point]]. Furthermore, they reported the existence of two reentrant transition sequences as a function of the number density, one featuring a fluid-bcc-fluid succession and another displaying a bcc-fcc-bcc sequence near the triple point.

	== Reference system ==		== Reference system ==

	The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using thermodynamic integration ~~technique~~ for different natures of fluid description (Lennard-Jones and Stillinger-Weber) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. The typical values for <math>p</math> are <math>50 \sim 100.</math>.		The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using [[thermodynamic integration]] techniques for different natures of fluid description ([[Lennard-Jones model \|Lennard-Jones]] and [[Stillinger-Weber potential \| Stillinger-Weber]]) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. The typical values for <math>p</math> are <math>50 \sim 100.</math>.

	==References==		==References==

Rleite: /* Reference system */

2017-10-14T14:06:31Z

Reference system

← Older revision		Revision as of 16:06, 14 October 2017
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	== Reference system ==		== Reference system ==

	The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using thermodynamic integration technique for different natures of fluid description (Lennard-Jones and Stillinger-Weber) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. The typical values for <math>p</math> are <math>50 \~~approx~~ 100.</math>.		The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using thermodynamic integration technique for different natures of fluid description (Lennard-Jones and Stillinger-Weber) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. The typical values for <math>p</math> are <math>50 \sim 100.</math>.


	==References==		==References==

Rleite at 14:03, 14 October 2017

2017-10-14T14:03:27Z

← Older revision		Revision as of 16:03, 14 October 2017
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	== Phase diagram ==		== Phase diagram ==
	The phase diagram of the UFM in the <math>(p^{-1},P_{\rm UF}^*)</math> has been constructed by Paula Leite, Santos-Flórez and de Koning <ref name="PRE">[https://doi.org/10.1103/PhysRevE.96.032115 R. Paula Leite, P. A. Santos-Flórez and M. de Koning "Uhlenbeck-Ford model: Phase diagram and corresponding-states analysis", Physical Review E '''96''', 032115 (2017).]</ref>. Beyond the fluid phase, body-centered-cubic (bcc) and face-centered-cubic (fcc) are the only thermodynamically stable crystalline phases. These three phases meet each other at a triple point. Furthermore, they reported the existence of two reentrant transition sequences as a function of the number density, one featuring a fluid-bcc-fluid succession and another displaying a bcc-fcc-bcc sequence near the triple point.		The phase diagram of the UFM in the <math>(p^{-1},P_{\rm UF}^*)</math> has been constructed by Paula Leite, Santos-Flórez and de Koning <ref name="PRE">[https://doi.org/10.1103/PhysRevE.96.032115 R. Paula Leite, P. A. Santos-Flórez and M. de Koning "Uhlenbeck-Ford model: Phase diagram and corresponding-states analysis", Physical Review E '''96''', 032115 (2017).]</ref>. Beyond the fluid phase, body-centered-cubic (bcc) and face-centered-cubic (fcc) are the only thermodynamically stable crystalline phases. These three phases meet each other at a triple point. Furthermore, they reported the existence of two reentrant transition sequences as a function of the number density, one featuring a fluid-bcc-fluid succession and another displaying a bcc-fcc-bcc sequence near the triple point.

			== Reference system ==

			The UFM can be used as a reference system for fluid-phase free-energy calculations. Examples of these kind of calculations using thermodynamic integration technique for different natures of fluid description (Lennard-Jones and Stillinger-Weber) can be found in Paula Leite ''et al''. paper <ref name="JCP"></ref>. The typical values for <math>p</math> are <math>50 \approx 100.</math>.


	==References==		==References==

Rleite: /* Numerical representations */

2017-10-14T13:55:01Z

Numerical representations

← Older revision		Revision as of 15:55, 14 October 2017
Line 74:		Line 74:

	== Numerical representations ==		== Numerical representations ==
	Due to the number of integrals needed to compute the virial coefficients, a numerical representation using molecular dynamics data is usually adopted. In this sense, ~~Paula Leite, Freitas, Azevedo and de Koning <ref name="JCP"></ref> have been performed~~ extensive molecular dynamics simulations and ~~constructed~~ numerical expressions for the UFM's equation of state ~~using a piece-wise fitting procedure called cubic splines~~. Moreover, this representation also leads to an excellent description for the excess Helmholtz free energy <math>F^{\rm (exc)}_{\rm UF}(x,p)</math>.		Due to the number of integrals needed to compute the virial coefficients, a numerical representation using molecular dynamics data is usually adopted. In this sense, extensive molecular dynamics simulations have been performed and using a piece-wise fitting procedure, called cubic splines, numerical expressions for the UFM's equation of state were constructed. Moreover, this representation also leads to an excellent description for the excess Helmholtz free energy <math>F^{\rm (exc)}_{\rm UF}(x,p)</math>.

	All spline representations of equations of state and excess Helmholtz free energies are available by means of the '''''python''''' script ''ufGenerator.py'' supplied in the supplementary material of Paula Leite ''et al''. paper <ref name="JCP"></ref>.		All spline representations of equations of state and excess Helmholtz free energies are available by means of the '''''python''''' script ''ufGenerator.py'' supplied in the supplementary material of Paula Leite ''et al''. paper <ref name="JCP"></ref>.

Rleite: /* Numerical representations */

2017-10-14T10:06:35Z

Numerical representations

← Older revision		Revision as of 12:06, 14 October 2017
Line 76:		Line 76:
	Due to the number of integrals needed to compute the virial coefficients, a numerical representation using molecular dynamics data is usually adopted. In this sense, Paula Leite, Freitas, Azevedo and de Koning <ref name="JCP"></ref> have been performed extensive molecular dynamics simulations and constructed numerical expressions for the UFM's equation of state using a piece-wise fitting procedure called cubic splines. Moreover, this representation also leads to an excellent description for the excess Helmholtz free energy <math>F^{\rm (exc)}_{\rm UF}(x,p)</math>.		Due to the number of integrals needed to compute the virial coefficients, a numerical representation using molecular dynamics data is usually adopted. In this sense, Paula Leite, Freitas, Azevedo and de Koning <ref name="JCP"></ref> have been performed extensive molecular dynamics simulations and constructed numerical expressions for the UFM's equation of state using a piece-wise fitting procedure called cubic splines. Moreover, this representation also leads to an excellent description for the excess Helmholtz free energy <math>F^{\rm (exc)}_{\rm UF}(x,p)</math>.

	All spline representations of equations of state and excess Helmholtz free energies are available by means of the '''''python''''' script ''ufGenerator.py'' supplied in the supplementary material of Paula Leite ''et. al''. paper <ref name="JCP"></ref>.		All spline representations of equations of state and excess Helmholtz free energies are available by means of the '''''python''''' script ''ufGenerator.py'' supplied in the supplementary material of Paula Leite ''et al''. paper <ref name="JCP"></ref>.

	== Phase diagram ==		== Phase diagram ==

Rleite: /* Numerical representations */

2017-10-14T10:05:48Z

Numerical representations

← Older revision		Revision as of 12:05, 14 October 2017
Line 76:		Line 76:
	Due to the number of integrals needed to compute the virial coefficients, a numerical representation using molecular dynamics data is usually adopted. In this sense, Paula Leite, Freitas, Azevedo and de Koning <ref name="JCP"></ref> have been performed extensive molecular dynamics simulations and constructed numerical expressions for the UFM's equation of state using a piece-wise fitting procedure called cubic splines. Moreover, this representation also leads to an excellent description for the excess Helmholtz free energy <math>F^{\rm (exc)}_{\rm UF}(x,p)</math>.		Due to the number of integrals needed to compute the virial coefficients, a numerical representation using molecular dynamics data is usually adopted. In this sense, Paula Leite, Freitas, Azevedo and de Koning <ref name="JCP"></ref> have been performed extensive molecular dynamics simulations and constructed numerical expressions for the UFM's equation of state using a piece-wise fitting procedure called cubic splines. Moreover, this representation also leads to an excellent description for the excess Helmholtz free energy <math>F^{\rm (exc)}_{\rm UF}(x,p)</math>.

	All spline representations of equations of state and excess Helmholtz free energies are available by means of the '''''python''''' script ''ufGenerator.py'' supplied in the supplementary material of Paula Leite, et. al. paper <ref name="JCP"></ref>.		All spline representations of equations of state and excess Helmholtz free energies are available by means of the '''''python''''' script ''ufGenerator.py'' supplied in the supplementary material of Paula Leite ''et. al''. paper <ref name="JCP"></ref>.

	== Phase diagram ==		== Phase diagram ==