Equations of state for crystals of hard spheres: Difference between revisions

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A number of [[equations of state]] have been proposed for this system. The usual procedure to obtain precise equations of
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.  
state is to fit [[Computer simulation techniques | computer simulation]] results.  
==Alder, Hoover and Young equation of state==
==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>
<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>
:<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]].
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]].
==Hall equation of state==
==Almarza equation of state==
<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. 12:
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>Z_{\mathrm{solid}}= \frac{1+y+y^2-0.67825y^3-y^4-0.5y^5-6.028e^{\xi(7.9-3.9\xi)}y^6}{1-3y+3y^2-1.04305y^3}</math>
 
:<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
where
:<math>\xi = \pi \sqrt{2}/6-y</math>
 
:<math>\beta = 4(1-v_0/v)</math>
:<math>z(solid)=\frac{pV}{Nk_BT}</math>
 
==Speedy equation of state==
==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)
(<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)
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|-  
|-  
| face-centred cubic || 0.5921 || 0.7072 || 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
|}
|}
==Almarza equation of state==
For the 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>\beta p (v-v_0) = 3 - 1.807846y + 11.56350 y^2 + 141.6000y^3 - 2609.260y^4 + 19328.09 y^5</math>


==References==
==References==

Latest revision as of 08:03, 23 February 2017

The stable phase of the hard sphere model at high densities is thought to have a 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 results.

Alder, Hoover and Young equation of state (face-centred cubic solid)[edit]

[1]

where where is the volume at close packing, is the pressure, is the temperature and is the Boltzmann constant.

Almarza equation of state[edit]

For the face-centred cubic solid phase [2] (Eq. 19):

,

where is the volume per particle, is the volume per particle at close packing, and ; with being the hard sphere diameter.

Hall equation of state (face-centred cubic)[edit]

[3] Eq. 13:

where

Speedy equation of state[edit]

([4], Eq. 2)

where

and (Table 1)

Crystal structure
hexagonal close packed 0.5935 0.7080 0.601
face-centred cubic 0.5921 0.7072 0.601
face-centred cubic [5] 0.620735 0.708194 0.591663

References[edit]

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