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The '''Uhlenbeck-Ford model''' (UFM) | The '''Uhlenbeck-Ford model''' (UFM) was originally proposed by G. Uhlenbeck and G. Ford <ref>[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 coeffcients 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. | ||
== Functional form == | == Functional form == | ||
The '''Uhlenbeck-Ford model''' is given by : | |||
The | |||
:<math>U_{\rm UF}(r) = - \frac{p}{\beta} \ln \left(1-e^{-(r/\sigma)^2} \right)</math> | :<math>U_{\rm UF}(r) = - \frac{p}{\beta} \ln \left(1-e^{-(r/\sigma)^2} \right)</math> | ||
where | where | ||
* <math>p > 0 </math> is a scaling factor | * <math>p > 0 </math> is a scaling factor; | ||
* <math> \beta \equiv (k_B T)^{-1} </math> is the well depth (energy) | * <math> \beta \equiv (k_B T)^{-1} </math> is the well depth (energy); | ||
* <math>r =|\mathbf{r}_1 - \mathbf{r}_2|</math> is the | * <math>r =|\mathbf{r}_1 - \mathbf{r}_2|</math> is the interparticle distance; | ||
* <math> \sigma </math> is a length-scale parameter. | * <math> \sigma </math> is a length-scale parameter. | ||
== Equation of state == | == Equation of state == | ||
The | 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 | ||
:<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 | * <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; | ||
*<math>\tilde{B}_n(p) \equiv B_n(p)/b^{n-1}</math> are the reduced virial coefficients. | * <math>\tilde{B}_n(p) \equiv B_n(p)/b^{n-1}</math> are the reduced virial coefficients. | ||
Note that, due to the functional form of the potential, the equation of state for the UFM fluid can be specified in terms of a function of a single adimensional variable <math>x</math>, regardless of the length-scale <math>\sigma</math> and absolute temperature <math>T</math>, i.é., the virial coefficients are independent of the absolute temperature. | Note that, due to the functional form of the potential, the equation of state for the UFM fluid can be specified in terms of a function of a single adimensional variable <math>x</math>, regardless of the length-scale <math>\sigma</math> and absolute temperature <math>T</math>, i.é., the virial coefficients are independent of the absolute temperature. | ||
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== Excess Helmholtz free-energy == | == Excess Helmholtz free-energy == | ||
The excess | 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 | ||
:<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> | ||
where | where | ||
* <math>F^{\rm (exc)}_{\rm UF}(x,p)</math> is the | * <math>F^{\rm (exc)}_{\rm UF}(x,p)</math> is the Helmholtz free energy; | ||
* <math>N</math> is the number of particles | * <math>N</math> is the number of particles; | ||
== Virial coefficients == | == Virial coefficients == | ||
:{| border="1" | :{| border="1" | ||
|- | |- | ||
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| <math>\tilde{B}_{13}</math> || -0.006 726 441 408 781 588 25 || || | | <math>\tilde{B}_{13}</math> || -0.006 726 441 408 781 588 25 || || | ||
|} | |} | ||
== Phase diagram == | == Phase diagram == | ||
==References== | ==References== | ||
<references/> | <references/> | ||
;Related reading | |||
*[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)] | |||
*[http://aip.scitation.org/doi/suppl/10.1063/1.4967775 R. Paula Leite, R. Freitas, R. Azevedo and M. de Koning "Supplemental Material: 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)] | |||
*[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)] | |||
{{numeric}} | {{numeric}} | ||
[[category: models]] | [[category: models]] |