Equations of state for crystals of hard spheres

2017-02-23T07:03:10Z

Cabb99: /* Hall equation of state (face-centred cubic) */

The stable phase of the [[hard sphere model]] at high densities is thought to have a [[Building up a face centered cubic lattice |face-centered cubic]] structure.
A number of [[equations of state]] have been proposed for this system. The usual procedure to obtain precise equations of
state is to fit [[Computer simulation techniques | computer simulation]] results.
==Alder, Hoover and Young equation of state (face-centred cubic solid) ==
<ref>[http://dx.doi.org/10.1063/1.1670653 B. J. Alder, W. G. Hoover, and D. A. Young "Studies in Molecular Dynamics. V. High-Density Equation of State and Entropy for Hard Disks and Spheres", Journal of Chemical Physics '''49''' pp 3688-3696 (1968)]</ref>
:<math>\frac{pV}{Nk_BT} = \frac{3}{\alpha} + 2.56 + 0.56 \alpha + O(\alpha^2).</math>
where <math>\alpha = (V-V_0)/V_0</math> where <math>V_0</math> is the volume at close packing, <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>k_B</math> is the [[Boltzmann constant]].
==Almarza equation of state==
For the [[Building up a face centered cubic lattice |face-centred cubic]] solid phase <ref>[http://dx.doi.org/10.1063/1.3133328 N. G. Almarza "A cluster algorithm for Monte Carlo simulation at constant pressure", Journal of Chemical Physics '''130''' 184106 (2009)]</ref> (Eq. 19):

:<math> p \left(v-v_0\right)/k_B T = 3 - 1.807846y + 11.56350 y^2 + 141.6000y^3 - 2609.260y^4 + 19328.09 y^5</math>,

where <math> \left. v \right. </math> is the volume per particle, <math> v_0 \equiv \sigma^3/\sqrt{2} </math> is the volume per particle at close packing,
and <math> y \equiv ( p \sigma^3/k_B T)^{-1} </math>; with <math> \left. \sigma \right. </math> being the hard sphere diameter.

==Hall equation of state (face-centred cubic)==
<ref>[http://dx.doi.org/10.1063/1.1678576 Kenneth R. Hall "Another Hard-Sphere Equation of State", Journal of Chemical Physics '''57''' pp. 2252-2254 (1972)]</ref> Eq. 13:
:<math>z ({\mathrm {solid}}) - \left[ (12-3\beta)/\beta \right]= 2.557696 + 0.1253077 \beta + 0.1762393 \beta^2 -
1.053308 \beta^3 + 2.818621 \beta^4 - 2.921934 \beta^5 + 1.118413 \beta^6</math>

where

:<math>\beta = 4(1-v_0/v)</math>
:<math>z(solid)=\frac{pV}{Nk_BT}</math>

==Speedy equation of state==
(<ref>[http://dx.doi.org/10.1088/0953-8984/10/20/006 Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of Physics: Condensed Matter '''10''' pp. 4387-4391 (1998)]</ref>, Eq. 2)
:<math>\frac{pV}{Nk_BT} = \frac{3}{1-z} -\frac{a(z-b)}{(z-c)}</math>
where
:<math>z= (N/V)\sigma^3/\sqrt{2}</math>
and (Table 1)
:{| border="1"
|-
| Crystal structure || <math>a</math> || <math>b</math> || <math>c</math>
|-
| hexagonal close packed || 0.5935 || 0.7080 || 0.601
|-
| face-centred cubic || 0.5921 || 0.7072 || 0.601
|-
| face-centred cubic <ref>[http://dx.doi.org/10.1063/1.3328823 Marcus N. Bannerman, Leo Lue, and Leslie V. Woodcock "Thermodynamic pressures for hard spheres and closed-virial equation-of-state", Journal of Chemical Physics '''132''' 084507 (2010)]</ref> || 0.620735 || 0.708194 || 0.591663
|}

==References==
<references/>
{{Numeric}}
[[category: equations of state]]

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Equations of state for crystals of hard spheres