Hard sphere model - Revision history

132.230.196.72: Fixed broken links in ref. 5

2023-05-23T11:47:32Z

Fixed broken links in ref. 5

← Older revision		Revision as of 13:47, 23 May 2023
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	<ref>[http://dx.doi.org/10.1063/1.1743957 B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics '''27''' pp. 1208-1209 (1957)]</ref>, much of this work undertaken at the Los Alamos Scientific Laboratory on the world's first electronic digital computer ENIAC <ref>[http://ftp.arl.army.mil/~mike/comphist/eniac-story.html The ENIAC Story]</ref>.		<ref>[http://dx.doi.org/10.1063/1.1743957 B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics '''27''' pp. 1208-1209 (1957)]</ref>, much of this work undertaken at the Los Alamos Scientific Laboratory on the world's first electronic digital computer ENIAC <ref>[http://ftp.arl.army.mil/~mike/comphist/eniac-story.html The ENIAC Story]</ref>.
	==Liquid phase radial distribution function==		==Liquid phase radial distribution function==
	The following are a series of plots of the hard sphere [[radial distribution function]] <ref>The [[total correlation function]] data was produced using the [~~http~~://~~www~~.vscht.cz/fch/software/hsmd/hspline-8-2004.zip computer code] written by [~~http~~://~~www~~.vscht.cz/~~fch~~/~~en/people/Jiri.Kolafa.html~~ Jiří Kolafa]</ref> shown for different values of the number density <math>\rho</math>. The horizontal axis is in units of <math>\sigma</math> where <math>\sigma</math> is set to be 1. Click on image of interest to see a larger view.		The following are a series of plots of the hard sphere [[radial distribution function]] <ref>The [[total correlation function]] data was produced using the [https://old.vscht.cz/fch/software/hsmd/hspline-8-2004.zip computer code] written by [https://web.vscht.cz/~kolafaj/ Jiří Kolafa]</ref> shown for different values of the number density <math>\rho</math>. The horizontal axis is in units of <math>\sigma</math> where <math>\sigma</math> is set to be 1. Click on image of interest to see a larger view.
	:{\| border="1"		:{\| border="1"
	\|-		\|-

Toastedcrumpets: Added the values of Moir et al 2021 for coexistence density, pressure, and chemical potential.

2021-12-22T14:10:10Z

Added the values of Moir et al 2021 for coexistence density, pressure, and chemical potential.

← Older revision		Revision as of 16:10, 22 December 2021
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	\|-		\|-
	\| 1.033(3) \|\| 0.935(2) \|\| <ref name="Miguel"> [http://dx.doi.org/10.1063/1.3023062 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics '''129''' 214112 (2008)]</ref>		\| 1.033(3) \|\| 0.935(2) \|\| <ref name="Miguel"> [http://dx.doi.org/10.1063/1.3023062 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics '''129''' 214112 (2008)]</ref>
			\|-
			\| 1.03715(9) \|\| 0.93890(7) \|\| <ref name="MoirEtAl2021"> [https://doi.org/10.1063/5.0058892 Craig Moir, Leo Lue, and Marcus N. Bannerman "Tethered-particle model: The calculation of free energies for hard-sphere systems", Journal of Chemical Physics '''155''' 064504 (2021)]</ref>
	\|}		\|}
	The coexistence [[pressure]] has been calculated to be		The coexistence [[pressure]] has been calculated to be
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	\|-		\|-
	\| 11.43(17) \|\| <ref>[http://dx.doi.org/10.1063/1.3244562 G. Odriozola "Replica exchange Monte Carlo applied to hard spheres", Journal of Chemical Physics '''131''' 144107 (2009)]</ref>		\| 11.43(17) \|\| <ref>[http://dx.doi.org/10.1063/1.3244562 G. Odriozola "Replica exchange Monte Carlo applied to hard spheres", Journal of Chemical Physics '''131''' 144107 (2009)]</ref>
			\|-
			\| 11.550(4) \|\| <ref name="MoirEtAl2021"></ref>
	\|}		\|}
	The coexistence [[chemical potential]] has been calculated to be		The coexistence [[chemical potential]] has been calculated to be
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	\|-		\|-
	\| 15.980(11) \|\| <ref name="Miguel"></ref>		\| 15.980(11) \|\| <ref name="Miguel"></ref>
			\|-
			\| 16.053(4) \|\| <ref name="MoirEtAl2021"></ref>
	\|}		\|}
	The [[Helmholtz energy function]] (in units of <math>Nk_BT</math>) is given by		The [[Helmholtz energy function]] (in units of <math>Nk_BT</math>) is given by

132.230.195.72: The coexistence densities of Enrique de Miguel were adopted incorrectly, because they refer to the (111) crystal orientation instead of the (100) orientation.

2020-08-26T13:55:00Z

The coexistence densities of Enrique de Miguel were adopted incorrectly, because they refer to the (111) crystal orientation instead of the (100) orientation.

← Older revision		Revision as of 15:55, 26 August 2020
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	\| 1.037 \|\| 0.938 \|\| <ref>[http://dx.doi.org/10.1063/1.476396 Ruslan L. Davidchack and Brian B. Laird "Simulation of the hard-sphere crystal–melt interface", Journal of Chemical Physics '''108''' pp. 9452-9462 (1998)]</ref>		\| 1.037 \|\| 0.938 \|\| <ref>[http://dx.doi.org/10.1063/1.476396 Ruslan L. Davidchack and Brian B. Laird "Simulation of the hard-sphere crystal–melt interface", Journal of Chemical Physics '''108''' pp. 9452-9462 (1998)]</ref>
	\|-		\|-
	\| 1.~~035~~(3) \|\| 0.~~936~~(2) \|\| <ref name="Miguel"> [http://dx.doi.org/10.1063/1.3023062 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics '''129''' 214112 (2008)]</ref>		\| 1.033(3) \|\| 0.935(2) \|\| <ref name="Miguel"> [http://dx.doi.org/10.1063/1.3023062 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics '''129''' 214112 (2008)]</ref>
	\|}		\|}
	The coexistence [[pressure]] has been calculated to be		The coexistence [[pressure]] has been calculated to be

132.230.195.72: Correction of liquid density value. 0.667close_packingdensity=0.943. Uncertainties added.

2020-08-26T13:09:33Z

Correction of liquid density value. 0.667*close_packing*density=0.943. Uncertainties added.

← Older revision		Revision as of 15:09, 26 August 2020
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	\| <math>\rho^_{\mathrm {solid}}</math> \|\| <math>\rho^_{\mathrm {liquid}}</math> \|\| Reference		\| <math>\rho^_{\mathrm {solid}}</math> \|\| <math>\rho^_{\mathrm {liquid}}</math> \|\| Reference
	\|-		\|-
	\| 1.041\|\| 0.~~945~~ \|\| <ref name="HooverRee"></ref>		\| 1.041(4)\|\| 0.943(4) \|\| <ref name="HooverRee"></ref>
	\|-		\|-
	\| 1.0376\|\| 0.9391 \|\| <ref name="FrenkelSmitBook">Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261.</ref>		\| 1.0376\|\| 0.9391 \|\| <ref name="FrenkelSmitBook">Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261.</ref>

65.112.8.207: /* Solid structure */

2019-06-05T14:46:24Z

Solid structure

← Older revision		Revision as of 16:46, 5 June 2019
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	==Solid structure==		==Solid structure==
	The [http://mathworld.wolfram.com/KeplerConjecture.html Kepler conjecture] states that the optimal packing for three dimensional spheres is either cubic or hexagonal close [[Lattice Structures \| packing]], both of which have maximum densities of <math>\pi/(3 \sqrt{2}) \approx 0.~~74048~~%</math><ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>		The [http://mathworld.wolfram.com/KeplerConjecture.html Kepler conjecture] states that the optimal packing for three dimensional spheres is either cubic or hexagonal close [[Lattice Structures \| packing]], both of which have maximum densities of <math>\pi/(3 \sqrt{2}) \approx 74.048%</math><ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>
	<ref>[https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/ Jacob Aron "Proof confirmed of 400-year-old fruit-stacking problem", New Scientist daily news 12 August (2014)]</ref>		<ref>[https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/ Jacob Aron "Proof confirmed of 400-year-old fruit-stacking problem", New Scientist daily news 12 August (2014)]</ref>
	<ref>[http://dx.doi.org/10.1103/PhysRevE.52.3632 C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E '''52''' pp. 3632-3636 (1995)]</ref>. However, for hard spheres at close packing the [[Building up a face centered cubic lattice \|face centred cubic]] phase is the more stable		<ref>[http://dx.doi.org/10.1103/PhysRevE.52.3632 C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E '''52''' pp. 3632-3636 (1995)]</ref>. However, for hard spheres at close packing the [[Building up a face centered cubic lattice \|face centred cubic]] phase is the more stable

132.230.195.76: /* Helmholtz energy function */

2019-02-08T08:02:49Z

Helmholtz energy function

← Older revision		Revision as of 10:02, 8 February 2019
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	\|}		\|}

	In <ref name="Schilling"></ref> the free energies are given without the ideal gas contribution <math>\ln(\rho^*)-1</math> , ~~hence~~ it was added to the free energies in the table.		In <ref name="Schilling"></ref> the free energies are given without the ideal gas contribution <math>\ln(\rho^*)-1</math> . Hence, it was added to the free energies in the table.

	==Interfacial Helmholtz energy function==		==Interfacial Helmholtz energy function==

132.230.195.76: /* Helmholtz energy function */

2018-10-16T11:17:36Z

Helmholtz energy function

← Older revision		Revision as of 13:17, 16 October 2018
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	\| 1.150000 \|\| 6.274 \|\| Table VI <ref name="VegaNoya"></ref>		\| 1.150000 \|\| 6.274 \|\| Table VI <ref name="VegaNoya"></ref>
	\|}		\|}

			In <ref name="Schilling"></ref> the free energies are given without the ideal gas contribution <math>\ln(\rho^*)-1</math> , hence it was added to the free energies in the table.

	==Interfacial Helmholtz energy function==		==Interfacial Helmholtz energy function==

132.230.195.76: /* Helmholtz energy function */ Correction of inconsistency: Schilling and Schmidt give their values without ideal gas contribution ln(rho)-1. So I added this amount. This is now also consistent with the notion of Helmholtz free energy in the link above.

2018-10-11T10:18:59Z

Helmholtz energy function: Correction of inconsistency: Schilling and Schmidt give their values without ideal gas contribution ln(rho)-1. So I added this amount. This is now also consistent with the notion of Helmholtz free energy in the link above.

← Older revision		Revision as of 12:18, 11 October 2018
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	\| <math>\rho^*</math> \|\| <math>A/(Nk_BT)</math>\|\| Reference		\| <math>\rho^*</math> \|\| <math>A/(Nk_BT)</math>\|\| Reference
	\|-		\|-
	\| 0.25 \|\| 0.~~620~~ <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"> [http://dx.doi.org/10.1063/1.3274951 T. Schilling and F. Schmid "Computing absolute free energies of disordered structures by molecular simulation", Journal of Chemical Physics '''131''' 231102 (2009)]</ref>		\| 0.25 \|\| −1.766 <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"> [http://dx.doi.org/10.1063/1.3274951 T. Schilling and F. Schmid "Computing absolute free energies of disordered structures by molecular simulation", Journal of Chemical Physics '''131''' 231102 (2009)]</ref>
	\|-		\|-
	\| 0.50 \|\| 1.~~541~~ <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"></ref>		\| 0.50 \|\| −0.152 <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"></ref>
	\|-		\|-
	\| 0.75 \|\| 3.~~009~~ <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"></ref>		\| 0.75 \|\| 1.721 <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"></ref>
	\|-		\|-
	\| 1.04086 \|\| 4.959 \|\| Table VI <ref name="VegaNoya"></ref>		\| 1.04086 \|\| 4.959 \|\| Table VI <ref name="VegaNoya"></ref>
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	\| 1.150000 \|\| 6.274 \|\| Table VI <ref name="VegaNoya"></ref>		\| 1.150000 \|\| 6.274 \|\| Table VI <ref name="VegaNoya"></ref>
	\|}		\|}

	==Interfacial Helmholtz energy function==		==Interfacial Helmholtz energy function==
	The [[Helmholtz energy function]] of the solid–liquid [[interface]] has been calculated using the [[cleaving method]] giving (Ref. <ref>[http://dx.doi.org/10.1063/1.3514144 Ruslan L. Davidchack "Hard spheres revisited: Accurate calculation of the solid–liquid interfacial free energy", Journal of Chemical Physics '''133''' 234701 (2010)]</ref> Table I):		The [[Helmholtz energy function]] of the solid–liquid [[interface]] has been calculated using the [[cleaving method]] giving (Ref. <ref>[http://dx.doi.org/10.1063/1.3514144 Ruslan L. Davidchack "Hard spheres revisited: Accurate calculation of the solid–liquid interfacial free energy", Journal of Chemical Physics '''133''' 234701 (2010)]</ref> Table I):

Carl McBride: Fixed reference errors

2018-06-18T11:54:07Z

Fixed reference errors

Carl McBride: /* Solid structure */ Added a recent publication

2018-01-15T11:54:29Z

Solid structure: Added a recent publication

← Older revision		Revision as of 13:54, 15 January 2018
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	<ref>[https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/ Jacob Aron "Proof confirmed of 400-year-old fruit-stacking problem", New Scientist daily news 12 August (2014)]</ref>		<ref>[https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/ Jacob Aron "Proof confirmed of 400-year-old fruit-stacking problem", New Scientist daily news 12 August (2014)]</ref>
	<ref>[http://dx.doi.org/10.1103/PhysRevE.52.3632 C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E '''52''' pp. 3632-3636 (1995)]</ref>. However, for hard spheres at close packing the [[Building up a face centered cubic lattice \|face centred cubic]] phase is the more stable		<ref>[http://dx.doi.org/10.1103/PhysRevE.52.3632 C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E '''52''' pp. 3632-3636 (1995)]</ref>. However, for hard spheres at close packing the [[Building up a face centered cubic lattice \|face centred cubic]] phase is the more stable
	<ref>[http://dx.doi.org/10.1039/a701761h Leslie V. Woodcock "Computation of the free energy for alternative crystal structures of hard spheres", Faraday Discussions '''106''' pp. 325-338 (1997)]</ref>, with a [[Helmholtz energy function]] difference in the [[thermodynamic limit]] between the hexagonal close packed and face centered cubic crystals at close packing of 0.001164(8) <math>Nk_BT</math><ref>[http://dx.doi.org/10.1080/00268976.2014.982736 Eva G. Noya and Noé G. Almarza "Entropy of hard spheres in the close-packing limit", Molecular Physics '''113''' pp. 1061-1068 (2015)]</ref>.		<ref>[http://dx.doi.org/10.1039/a701761h Leslie V. Woodcock "Computation of the free energy for alternative crystal structures of hard spheres", Faraday Discussions '''106''' pp. 325-338 (1997)]</ref>, with a [[Helmholtz energy function]] difference in the [[thermodynamic limit]] between the hexagonal close packed and face centered cubic crystals at close packing of 0.001164(8) <math>Nk_BT</math><ref>[http://dx.doi.org/10.1080/00268976.2014.982736 Eva G. Noya and Noé G. Almarza "Entropy of hard spheres in the close-packing limit", Molecular Physics '''113''' pp. 1061-1068 (2015)]</ref>. Recently evidence has been found for a metastable cI16 phase <ref>[https://doi.org/10.1063/1.5009099 Vadim B. Warshavsky, David M. Ford, and Peter A. Monson "On the mechanical stability of the body-centered cubic phase and the emergence of a metastable cI16 phase in classical hard sphere solids", Journal of Chemical Physics '''148''' 024502 (2018)]</ref> indicating the ''"cI16 is a mechanically stable structure that can spontaneously emerge from a bcc starting point but it is thermodynamically metastable relative to fcc or hcp".''
	*See also: [[Equations of state for crystals of hard spheres]]		*See also: [[Equations of state for crystals of hard spheres]]

← Older revision		Revision as of 13:54, 18 June 2018
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	where the [[second virial coefficient]], <math>B_2</math>, is given by		where the [[second virial coefficient]], <math>B_2</math>, is given by
	:<math>B_2 = \frac{2\pi}{3}\sigma^3</math>.		:<math>B_2 = \frac{2\pi}{3}\sigma^3</math>.
	Carnahan and Starling <ref>[http://dx.doi.org/10.1063/1.1672048 N. F.Carnahan and K. E.Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics '''51''' pp. 635-636 (1969)]</ref> provided the following expression for <math>{\mathrm g}(\sigma^+)</math> (Eq. 3 in <ref name="Tao1" > </ref>)		Carnahan and Starling <ref>[http://dx.doi.org/10.1063/1.1672048 N. F.Carnahan and K. E.Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics '''51''' pp. 635-636 (1969)]</ref> provided the following expression for <math>{\mathrm g}(\sigma^+)</math> (Eq. 3 in <ref name="Tao1" ></ref>)
	:<math>{\mathrm g}(\sigma^+)= \frac{1-\eta/2}{(1-\eta)^3}</math>		:<math>{\mathrm g}(\sigma^+)= \frac{1-\eta/2}{(1-\eta)^3}</math>
	where <math>\eta</math> is the [[packing fraction]].		where <math>\eta</math> is the [[packing fraction]].
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	\| <math>\rho^_{\mathrm {solid}}</math> \|\| <math>\rho^_{\mathrm {liquid}}</math> \|\| Reference		\| <math>\rho^_{\mathrm {solid}}</math> \|\| <math>\rho^_{\mathrm {liquid}}</math> \|\| Reference
	\|-		\|-
	\| 1.041\|\| 0.945 \|\| <ref name="HooverRee"> </ref>		\| 1.041\|\| 0.945 \|\| <ref name="HooverRee"></ref>
	\|-		\|-
	\| 1.0376\|\| 0.9391 \|\| <ref name="FrenkelSmitBook">Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261.</ref>		\| 1.0376\|\| 0.9391 \|\| <ref name="FrenkelSmitBook">Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261.</ref>
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	\| 11.5727(10)\|\| <ref name="FernandezUCM">[http://dx.doi.org/10.1103/PhysRevLett.108.165701 L. A. Fernández, V. Martín-Mayor, B. Seoane, and P. Verrocchio "Equilibrium Fluid-Solid Coexistence of Hard Spheres", Physical Review Letters '''108''' 165701 (2012)]</ref>		\| 11.5727(10)\|\| <ref name="FernandezUCM">[http://dx.doi.org/10.1103/PhysRevLett.108.165701 L. A. Fernández, V. Martín-Mayor, B. Seoane, and P. Verrocchio "Equilibrium Fluid-Solid Coexistence of Hard Spheres", Physical Review Letters '''108''' 165701 (2012)]</ref>
	\|-		\|-
	\| 11.57(10) \|\| <ref name="Fortini"> </ref>		\| 11.57(10) \|\| <ref name="Fortini"></ref>
	\|-		\|-
	\| 11.567\|\| <ref name="FrenkelSmitBook"> </ref>		\| 11.567\|\| <ref name="FrenkelSmitBook"></ref>
	\|-		\|-
	\| 11.55(11) \|\| <ref>[http://dx.doi.org/10.1088/0953-8984/9/41/006 Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of Physics: Condensed Matter '''9''' pp. 8591-8599 (1997)]</ref>		\| 11.55(11) \|\| <ref>[http://dx.doi.org/10.1088/0953-8984/9/41/006 Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of Physics: Condensed Matter '''9''' pp. 8591-8599 (1997)]</ref>
	\|-		\|-
	\| 11.54(4) \|\| <ref name="Noya"> </ref>		\| 11.54(4) \|\| <ref name="Noya"></ref>
	\|-		\|-
	\| 11.50(9) \|\| <ref>[http://dx.doi.org/10.1103/PhysRevLett.85.5138 N. B. Wilding and A. D. Bruce "Freezing by Monte Carlo Phase Switch", Physical Review Letters '''85''' pp. 5138-5141 (2000)]</ref>		\| 11.50(9) \|\| <ref>[http://dx.doi.org/10.1103/PhysRevLett.85.5138 N. B. Wilding and A. D. Bruce "Freezing by Monte Carlo Phase Switch", Physical Review Letters '''85''' pp. 5138-5141 (2000)]</ref>
	\|-		\|-
	\| 11.48(11) \|\| <ref name="Miguel"> </ref>		\| 11.48(11) \|\| <ref name="Miguel"></ref>
	\|-		\|-
	\| 11.43(17) \|\| <ref>[http://dx.doi.org/10.1063/1.3244562 G. Odriozola "Replica exchange Monte Carlo applied to hard spheres", Journal of Chemical Physics '''131''' 144107 (2009)]</ref>		\| 11.43(17) \|\| <ref>[http://dx.doi.org/10.1063/1.3244562 G. Odriozola "Replica exchange Monte Carlo applied to hard spheres", Journal of Chemical Physics '''131''' 144107 (2009)]</ref>
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	\| <math>\mu (k_BT) </math> \|\| Reference		\| <math>\mu (k_BT) </math> \|\| Reference
	\|-		\|-
	\| 15.980(11) \|\| <ref name="Miguel"> </ref>		\| 15.980(11) \|\| <ref name="Miguel"></ref>
	\|}		\|}
	The [[Helmholtz energy function]] (in units of <math>Nk_BT</math>) is given by		The [[Helmholtz energy function]] (in units of <math>Nk_BT</math>) is given by
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	\| <math>A_{\mathrm {solid}}</math> \|\| <math>A_{\mathrm {liquid}}</math> \|\| Reference		\| <math>A_{\mathrm {solid}}</math> \|\| <math>A_{\mathrm {liquid}}</math> \|\| Reference
	\|-		\|-
	\| 4.887(3) \|\| 3.719(8) \|\| <ref name="Miguel"> </ref>		\| 4.887(3) \|\| 3.719(8) \|\| <ref name="Miguel"></ref>
	\|}		\|}

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	\| 0.25 \|\| 0.620 <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"> [http://dx.doi.org/10.1063/1.3274951 T. Schilling and F. Schmid "Computing absolute free energies of disordered structures by molecular simulation", Journal of Chemical Physics '''131''' 231102 (2009)]</ref>		\| 0.25 \|\| 0.620 <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"> [http://dx.doi.org/10.1063/1.3274951 T. Schilling and F. Schmid "Computing absolute free energies of disordered structures by molecular simulation", Journal of Chemical Physics '''131''' 231102 (2009)]</ref>
	\|-		\|-
	\| 0.50 \|\| 1.541 <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"> </ref>		\| 0.50 \|\| 1.541 <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"></ref>
	\|-		\|-
	\| 0.75 \|\| 3.009 <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"> </ref>		\| 0.75 \|\| 3.009 <math>\pm</math> 0.002 \|\| Table I <ref name="Schilling"></ref>
	\|-		\|-
	\| 1.04086 \|\| 4.959 \|\| Table VI <ref name="VegaNoya"> </ref>		\| 1.04086 \|\| 4.959 \|\| Table VI <ref name="VegaNoya"></ref>
	\|-		\|-
	\| 1.099975 \|\| 5.631 \|\| Table VI <ref name="VegaNoya"> </ref>		\| 1.099975 \|\| 5.631 \|\| Table VI <ref name="VegaNoya"></ref>
	\|-		\|-
	\| 1.150000 \|\| 6.274 \|\| Table VI <ref name="VegaNoya"> </ref>		\| 1.150000 \|\| 6.274 \|\| Table VI <ref name="VegaNoya"></ref>
	\|}		\|}
	==Interfacial Helmholtz energy function==		==Interfacial Helmholtz energy function==
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	\| <math>\gamma_{\{100\}}</math> \|\| 0.5820(19)		\| <math>\gamma_{\{100\}}</math> \|\| 0.5820(19)
	\|-		\|-
	\| <math>\gamma_{\{100\}}</math> \|\| 0.636(11) <ref name="FernandezUCM"> </ref>		\| <math>\gamma_{\{100\}}</math> \|\| 0.636(11) <ref name="FernandezUCM"></ref>
	\|-		\|-
	\| <math>\gamma_{\{110\}}</math> \|\| 0.5590(20)		\| <math>\gamma_{\{110\}}</math> \|\| 0.5590(20)
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	==Solid structure==		==Solid structure==
	The [http://mathworld.wolfram.com/KeplerConjecture.html Kepler conjecture] states that the optimal packing for three dimensional spheres is either cubic or hexagonal close [[Lattice Structures \| packing]], both of which have maximum densities of <math>\pi/(3 \sqrt{2}) \approx 0.74048%</math> <ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>		The [http://mathworld.wolfram.com/KeplerConjecture.html Kepler conjecture] states that the optimal packing for three dimensional spheres is either cubic or hexagonal close [[Lattice Structures \| packing]], both of which have maximum densities of <math>\pi/(3 \sqrt{2}) \approx 0.74048%</math><ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>
	<ref>[https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/ Jacob Aron "Proof confirmed of 400-year-old fruit-stacking problem", New Scientist daily news 12 August (2014)]</ref>		<ref>[https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/ Jacob Aron "Proof confirmed of 400-year-old fruit-stacking problem", New Scientist daily news 12 August (2014)]</ref>
	<ref>[http://dx.doi.org/10.1103/PhysRevE.52.3632 C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E '''52''' pp. 3632-3636 (1995)]</ref>. However, for hard spheres at close packing the [[Building up a face centered cubic lattice \|face centred cubic]] phase is the more stable		<ref>[http://dx.doi.org/10.1103/PhysRevE.52.3632 C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E '''52''' pp. 3632-3636 (1995)]</ref>. However, for hard spheres at close packing the [[Building up a face centered cubic lattice \|face centred cubic]] phase is the more stable

Hard sphere model - Revision history

132.230.196.72: Fixed broken links in ref. 5

Toastedcrumpets: Added the values of Moir et al 2021 for coexistence density, pressure, and chemical potential.

132.230.195.72: The coexistence densities of Enrique de Miguel were adopted incorrectly, because they refer to the (111) crystal orientation instead of the (100) orientation.

132.230.195.72: Correction of liquid density value. 0.667*close_packing*density=0.943. Uncertainties added.

65.112.8.207: /* Solid structure */

132.230.195.76: /* Helmholtz energy function */

132.230.195.76: /* Helmholtz energy function */

132.230.195.76: /* Helmholtz energy function */ Correction of inconsistency: Schilling and Schmidt give their values without ideal gas contribution ln(rho)-1. So I added this amount. This is now also consistent with the notion of Helmholtz free energy in the link above.

Carl McBride: Fixed reference errors

Carl McBride: /* Solid structure */ Added a recent publication

132.230.195.72: Correction of liquid density value. 0.667close_packingdensity=0.943. Uncertainties added.