Hard sphere: virial coefficients

2020-12-26T16:34:55Z

Urrutia: New related reading by Lyberg

The [[virial equation of state]] of the [[hard sphere model]], in various dimensions, has long been of interest.
In 3-dimensions analytical results were derived for <math>B_2</math> by [[Johannes Diderik van der Waals]]<ref>[http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publDetail&pId=PU00014537 J. D. van der Waals "Simple deduction of the characteristic equation for substances with extended and composite molecules", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''1''' pp. 138-143 (1899)]</ref>, <math>B_3</math> by Jäger <ref>G. Jäger "", Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) '''105''' pp. 15- (1896)</ref>
and [[Ludwig Eduard Boltzmann]] <ref>L. Boltzmann "",Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) '''105''' pp. 695- (1896)</ref>
<ref>L. Boltzmann "On the characteristic equation of v.d.Waals", Versl. Gewone Vergad. Afd. Natuurkd., K. Ned. Akad. Wet. '''7''' pp. 484- (1899)</ref>, and <math>B_4</math> by [[Johannis Jacobus van Laar]]
<ref>[http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?page_id=&pagetype=publDetail&pId=PU00014563 J. J. Van Laar "Calculation of the second correction to the quantity b of the equation of condition of Van der Waals", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''1''' pp. 273-287 (1899)]</ref>
as well as Boltzmann <ref>[http://dx.doi.org/10.1119/1.1986605 John H. Nairn and John E. Kilpatrick "van der Waals, Boltzmann, and the Fourth Virial Coefficient of Hard Spheres", American Journal of Physics '''40''' pp. 503-515 (1972)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRev.85.777 B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review '''85''' pp. 777-783 (1952)]</ref>.
The calculation of <math>B_5</math> had to wait for the Rosenbluths
<ref>[http://dx.doi.org/10.1063/1.1740207 Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics '''22''' pp. 881- (1954)]</ref> in 1954. Thus far no analytical expressions for <math>B_5</math> and beyond have been derived. One has:

:<math>\frac{B_2}{V(\mathbb{R}^3)}=4</math>

:<math>\frac{B_3}{V(\mathbb{R}^3)^2}=10</math>

:<math>\frac{B_4}{V(\mathbb{R}^3)^3}= \frac{2707\pi+[438\sqrt{2}-4131 \arccos(1/3)]}{70\pi}= 18.3647684</math>

where <math>V(\mathbb{R}^3)</math> is the volume of a sphere in three dimensions. For [[hard disks]] (ie. 2-dimensional hard spheres) one has<ref>[http://dx.doi.org/10.1103/PhysRevE.71.021105 Stanislav Labík, Jirí Kolafa, and Anatol Malijevský, "Virial coefficients of hard spheres and hard disks up to the ninth", Physical Review E '''71''' pp. 021105 (2005)]</ref>

:<math>\frac{B_2}{V(\mathbb{R}^2)}=2</math>

:<math>\frac{B_3}{V(\mathbb{R}^2)^2}=\frac{16}{3}- \frac{4 \sqrt{3}}{\pi}</math>

:<math>\frac{B_4}{V(\mathbb{R}^2)^3}= 16-\frac{36\sqrt{3}}{\pi}+\frac{80}{\pi^2}</math>

where <math>V(\mathbb{R}^2)</math> is the area of a circle.
{| style="width:100%; height:250px; text-align:center" border="1"
|-
| Virial / Dimension || 2 || 3 || 4 || 5 || 6 || 7 || 8
|-
| <math>B_3/B_2^2</math> || 0.782004... || 0.625 || 0.506340... || 0.414063... || 0.340941... || 0.282227... || 0.234614...
|-
| <math>B_4/B_2^3</math> || 0.53223180...|| 0.2869495... || 0.15184606... || 0.0759724807... || 0.03336314... || 0.00986494662... || -0.00255768...
|-
| <math>B_5/B_2^4</math> || 0.33355604(1) || 0.110252(1) || 0.0357041(17)|| 0.0129551(13) || 0.0075231(11) || 0.0070724(10) || 0.00743092(93)
|-
| <math>B_6/B_2^5</math> || 0.1988425(42)|| 0.03888198(91)|| 0.0077359(16) || 0.0009815(14) || -0.0017385(13)|| -0.0035121(11) || -0.0045164(11)
|-
| <math>B_7/B_2^6</math> || 0.1148728(43)||0.01302354(91) || 0.0014303(19) || 0.0004162(19)|| 0.0013066(18)|| 0.0025386(16) || 0.0034149(15)
|-
| <math>B_8/B_2^7</math> || 0.0649930(34)|| 0.0041832(11)|| 0.0002888(18) || -0.0001120(20) || -0.0008950(30) || -0.0019937(28)|| -0.0028624(26)
|-
| <math>B_9/B_2^8</math> ||0.0362193(35) || 0.0013094(13)|| 0.0000441(22) || 0.0000747(26) || 0.0006673(45) || 0.0016869(41)|| 0.0025969(38)
|-
| <math>B_{10}/B_2^9</math> || 0.0199537(80)|| 0.0004035(15)|| 0.0000113(31)|| -0.0000492(48) || -0.000525(16) || -0.001514(14) || -0.002511(13)
|-
| <math>B_{11}/B_2^{10}</math> || || 0.000122 (4)|| || || || ||
|-
| <math>B_{12}/B_2^{11}</math> || || 0.000027 (7)|| || || || ||
|}
This table is taken directly from Table 1 in Ref.<ref>[http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)]</ref>. The values of <math>B_{11}</math> and <math>B_{12}</math> for three dimensional hard spheres are taken from <ref>[http://link.aps.org/doi/10.1103/PhysRevLett.110.200601 Richard J. Wheatley "Calculation of High-Order Virial Coefficients with Applications to Hard and Soft Spheres", Physical review Letters, '''110''' 200601 (2013)]]</ref>.
==See also==
*[[Equations of state for hard disks]]
*[[Equations of state for hard spheres]]
== References ==
<references/>
'''Related reading'''
*[https://doi.org/10.1007/s10955-005-3020-6 I. Lyberg "The fourth virial coefficient of a fluid of hard spheres in odd dimensions", Journal of Statistical Physics '''119''' pp. 747-764 (2005)]
*[http://dx.doi.org/10.1023/B:JOSS.0000013959.30878.d2 N. Clisby and B.M. McCoy "Analytic Calculation of B4 for Hard Spheres in Even Dimensions", Journal of Statistical Physics '''114''' pp. 1343-1361 (2004)]
*[http://dx.doi.org/10.1063/1.2821962 Marvin Bishop, Nathan Clisby and Paula A. Whitlock "The equation of state of hard hyperspheres in nine dimensions for low to moderate densities", Journal of Chemical Physics '''128''' 034506 (2008)]
*[http://dx.doi.org/10.1063/1.2951456 René D. Rohrmann, Miguel Robles, Mariano López de Haro, and Andrés Santos "Virial series for fluids of hard hyperspheres in odd dimensions", Journal of Chemical Physics '''129''' 014510 (2008)]
*[http://dx.doi.org/10.1063/1.2958914 André O. Guerrero and Adalberto B. M. S. Bassi "On Padé approximants to virial series", Journal of Chemical Physics '''129''' 044509 (2008)]
*[http://dx.doi.org/10.1063/1.3558779 Miguel Ángel G. Maestre, Andrés Santos, Miguel Robles, and Mariano López de Haro "On the relation between virial coefficients and the close-packing of hard disks and hard spheres", Journal of Chemical Physics '''134''' 084502 (2011)]
*[http://dx.doi.org/10.1080/00268976.2014.904945 Cheng Zhang and B. Montgomery Pettitt "Computation of high-order virial coefficients in high-dimensional hard-sphere fluids by Mayer sampling", Molecular Physics '''112''' pp. 1427-1447 (2014)]

[[category:virial coefficients]]
[[category: hard sphere]]
{{numeric}}

Exact solution of the Percus Yevick integral equation for hard spheres

2018-01-22T15:25:28Z

Urrutia: I replaced x by \eta in all the eqs. refered to Thiele's work. This is correct because the "dimensionless number density" (eq.6 of Thiele) divided by 4 is the packing fraction. I also modify the cross ref to Wertheim1 ref. to solve a problem with it.

The exact solution for the [[Percus Yevick]] [[Integral equations |integral equation]] for the [[hard sphere model]]
was derived by M. S. Wertheim in 1963 <ref name="wertheim1" >[http://dx.doi.org/10.1103/PhysRevLett.10.321 M. S. Wertheim "Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres", Physical Review Letters '''10''' 321 - 323 (1963)]</ref> (see also <ref>[http://dx.doi.org/10.1063/1.1704158 M. S. Wertheim "Analytic Solution of the Percus-Yevick Equation", Journal of Mathematical Physics, '''5''' pp. 643-651 (1964)]</ref>), and for [[mixtures]] by Joel Lebowitz in 1964 <ref>[http://dx.doi.org/10.1103/PhysRev.133.A895 J. L. Lebowitz, "Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres", Physical Review '''133''' pp. A895 - A899 (1964)]</ref>.

The [[direct correlation function]] is given by (Eq. 6 of <ref name="wertheim1" /> )

:<math>C(r/R) = - \frac{(1+2\eta)^2 - 6\eta(1+ \frac{1}{2} \eta)^2(r/R) + \eta(1+2\eta)^2\frac{(r/R)^3}{2}}{(1-\eta)^4}</math>

where

:<math>\eta = \frac{1}{6} \pi R^3 \rho</math>

and <math>R</math> is the hard sphere diameter.
The [[Equations of state | equation of state]] is given by (Eq. 7 of <ref name="wertheim1" />)

:<math>\frac{\beta P}{\rho} = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math>

where <math>\beta</math> is the [[inverse temperature]]. Everett Thiele also studied this system <ref>[http://dx.doi.org/10.1063/1.1734272 Everett Thiele "Equation of State for Hard Spheres", Journal of Chemical Physics, '''39''' pp. 474-479 (1963)]</ref>,
resulting in (Eq. 23)

:<math>\left.h_0(r)\right. = ar+ br^2 + cr^4</math>

where (Eq. 24)

:<math>a = \frac{(2\eta+1)^2}{(\eta-1)^4}</math>

and

:<math>b= - \frac{12\eta + 12\eta^2 + 3\eta^3}{2(\eta-1)^4}</math>

and

:<math>c= \frac{\eta(2\eta+1)^2}{2(\eta-1)^4}</math>

The [[pressure]] via the pressure route (Eq.s 32 and 33) is

:<math>P=nk_BT\frac{(1+2\eta+3\eta^2)}{(1-\eta)^2}</math>

and the [[Compressibility equation |compressibility]] route is

:<math>P=nk_BT\frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math>

==A derivation of the Carnahan-Starling equation of state==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048 G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics '''54''' pp. 1523-1525 (1971)] </ref> Eq. 6) that one can arrive at the [[Carnahan-Starling equation of state]] by adding two thirds of the exact solution via the compressibility route, to one third via the pressure route, i.e.

:<math>Z = \frac{ p V}{N k_B T} = \frac{2}{3} \left[ \frac{(1+\eta+\eta^2)}{(1-\eta)^3} \right] + \frac{1}{3} \left[ \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2} \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }</math>

The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry '''93''' pp. 6916-6919 (1989)]</ref> ).

==References==
<references/>

[[Category: Integral equations]]

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Hard sphere: virial coefficients

Exact solution of the Percus Yevick integral equation for hard spheres