Equations of state for crystals of hard spheres: Difference between revisions
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Carl McBride (talk | contribs) m (→Speedy equation of state: Added table number.) |
Carl McBride (talk | contribs) (Added the Almarza equation of state for the fcc solid.) |
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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== | ||
( | <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== | ==Hall 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: | ||
:<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>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> | ||
where | where | ||
:<math>\xi = \pi \sqrt{2}/6-y</math> | :<math>\xi = \pi \sqrt{2}/6-y</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) | ||
:<math>\frac{pV}{Nk_BT} = \frac{3}{1-z} -\frac{a(z-b)}{(z-c)}</math> | :<math>\frac{pV}{Nk_BT} = \frac{3}{1-z} -\frac{a(z-b)}{(z-c)}</math> | ||
where | where | ||
:<math>z= (N/V)\sigma^3/\sqrt{2}</math> | :<math>z= (N/V)\sigma^3/\sqrt{2}</math> | ||
and ( | and (Table 1) | ||
:{| border="1" | :{| border="1" | ||
|- | |- | ||
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| face-centred cubic || 0.5921 || 0.7072 || 0.601 | | face-centred cubic || 0.5921 || 0.7072 || 0.601 | ||
|} | |} | ||
==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>: | |||
:<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== | ||
<references/> | |||
{{Numeric}} | |||
[[category: equations of state]] | [[category: equations of state]] |
Revision as of 12:23, 13 May 2009
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
where where is the volume at close packing, is the pressure, is the temperature and is the Boltzmann constant.
Hall equation of state
[2] Eq. 12:
where
Speedy equation of state
([3], 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
Almarza equation of state
For the face-centred cubic solid phase [4]:
References
- ↑ 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)
- ↑ Kenneth R. Hall "Another Hard-Sphere Equation of State", Journal of Chemical Physics 57 pp. 2252-2254 (1972)
- ↑ Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of Physics: Condensed Matter 10 pp. 4387-4391 (1998)
- ↑ N. G. Almarza "A cluster algorithm for Monte Carlo simulation at constant pressure", Journal of Chemical Physics 130 184106 (2009)