# Difference between revisions of "Uhlenbeck-Ford model"

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− | 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 | + | 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. |

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* <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 | + | * <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. | ||

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where | where | ||

− | * <math>b \equiv (\pi\sigma^2)^{3/2} | + | * <math>P_{\rm UF}</math> is the system pressure; |

+ | * <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 \equiv B_n/b^{n-1}</math> are reduced virial coefficients. | + | * <math>\tilde{B}_n(p) \equiv B_n(p)/b^{n-1}</math> are reduced virial coefficients. |

− | == Helmholtz free-energy | + | == Helmholtz free-energy == |

The Helmholtz free-energy expression for the UFM is given by | The Helmholtz free-energy expression for the UFM is given by | ||

:<math>\frac{\beta F_{\rm UF}(x,p)}{N} =\sum_{n=1}^{\infty} \frac{\tilde{B}_{n+1}(p)}{n} \,x^n</math> | :<math>\frac{\beta F_{\rm UF}(x,p)}{N} =\sum_{n=1}^{\infty} \frac{\tilde{B}_{n+1}(p)}{n} \,x^n</math> | ||

+ | |||

+ | where | ||

+ | * <math>F_{\rm UF}</math> is the Helmholtz free energy; | ||

+ | * <math>N</math> is the number of particles; | ||

+ | |||

+ | == Virial coefficients == | ||

+ | |||

+ | :{| border="1" | ||

+ | |- | ||

+ | | || p=1 || p=2 || p=10 | ||

+ | |- | ||

+ | | <math>\tilde{B}_2</math> || 1 || 1.646 446 609 40 || 4.014 383 975 40 | ||

+ | |- | ||

+ | | <math>\tilde{B}_3</math> ||0.256 600 119 639 833 673 11 || | ||

+ | 0.943 195 827 15 || 8.031 625 170 36 | ||

+ | |- | ||

+ | | <math>\tilde{B}_4</math> || -0.125 459 957 055 044 678 34 || -0.334 985 180 88 || 5.791 113 868 79 | ||

+ | |- | ||

+ | | <math>\tilde{B}_5</math> || 0.013 325 655 173 205 441 00 || -0.303 688 574 00 || | ||

+ | -7.658 371 037 27 | ||

+ | |- | ||

+ | | <math>\tilde{B}_6</math> || 0.038 460 935 830 869 671 55 || 0.395 857 781 68 || | ||

+ | |- | ||

+ | | <math>\tilde{B}_7</math> || -0.033 083 442 903 149 717 39 || 0.053 655 130 98 || | ||

+ | |- | ||

+ | | <math>\tilde{B}_8</math> || 0.004 182 418 769 682 387 35 || -0.408 481 406 00 || | ||

+ | |- | ||

+ | | <math>\tilde{B}_9</math> || 0.015 197 607 195 500 874 66 || || | ||

+ | |- | ||

+ | | <math>\tilde{B}_{10}</math> || -0.013 849 654 134 575 142 93 || || | ||

+ | |- | ||

+ | | <math>\tilde{B}_{11}</math> || 0.001 334 757 917 110 966 11 || || | ||

+ | |- | ||

+ | | <math>\tilde{B}_{12}</math> || 0.007 604 327 248 812 125 87 || || | ||

+ | |- | ||

+ | | <math>\tilde{B}_{13}</math> || -0.006 726 441 408 781 588 25 || || | ||

+ | |} | ||

+ | |||

+ | == Phase diagram == | ||

==References== | ==References== |

## Revision as of 15:40, 12 October 2017

The **Uhlenbeck-Ford model** (UFM) was originally proposed by G. Uhlenbeck and G. Ford ^{[1]} 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 where is the Boltzmman constant and 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 ^{[2]} showed that this model can be used as a reference system for fluid-phase free-energy calculations.

## Contents

## Functional form

The **Uhlenbeck-Ford model** is given by :

where

- is a scaling factor;
- is the well depth (energy);
- is the interparticle distance;
- is a length-scale parameter.

## Equation of state

The UFM's equation of state using virial expansion has recently been studied by Paula Leite, Freitas, Azevedo and de Koning ^{[2]}. This equation of state is given by

where

- is the system pressure;
- is a constant;
- is an adimensional variable;
- are reduced virial coefficients.

## Helmholtz free-energy

The Helmholtz free-energy expression for the UFM is given by

where

- is the Helmholtz free energy;
- is the number of particles;

## Virial coefficients

p=1 p=2 p=10 1 1.646 446 609 40 4.014 383 975 40 0.256 600 119 639 833 673 11 0.943 195 827 15 || 8.031 625 170 36

-0.125 459 957 055 044 678 34 -0.334 985 180 88 5.791 113 868 79 0.013 325 655 173 205 441 00 -0.303 688 574 00 -7.658 371 037 27

0.038 460 935 830 869 671 55 0.395 857 781 68 -0.033 083 442 903 149 717 39 0.053 655 130 98 0.004 182 418 769 682 387 35 -0.408 481 406 00 0.015 197 607 195 500 874 66 -0.013 849 654 134 575 142 93 0.001 334 757 917 110 966 11 0.007 604 327 248 812 125 87 -0.006 726 441 408 781 588 25

## Phase diagram

## References

- ↑ [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.]
- ↑
^{2.0}^{2.1}>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)

- Related reading

- 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) - 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) - 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)