Editing Hard sphere model

Jump to navigation Jump to search
Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.

The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.

Latest revision Your text
Line 18: Line 18:
<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 [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.
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.
:{| border="1"
:{| border="1"
|-  
|-  
Line 31: Line 31:
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]].
Line 50: Line 50:
The hard sphere system undergoes a [[Solid-liquid phase transitions |liquid-solid]] [[First-order transitions |first order transition]] <ref name="HooverRee">[http://dx.doi.org/10.1063/1.1670641    William G. Hoover and Francis H. Ree "Melting Transition and Communal Entropy for Hard Spheres", Journal of Chemical Physics '''49''' pp. 3609-3617  (1968)]</ref>
The hard sphere system undergoes a [[Solid-liquid phase transitions |liquid-solid]] [[First-order transitions |first order transition]] <ref name="HooverRee">[http://dx.doi.org/10.1063/1.1670641    William G. Hoover and Francis H. Ree "Melting Transition and Communal Entropy for Hard Spheres", Journal of Chemical Physics '''49''' pp. 3609-3617  (1968)]</ref>
<ref>[http://dx.doi.org/10.1063/1.4870524 Miguel Robles, Mariano López de Haro and Andrés Santos "Note: Equation of state and the freezing point in the hard-sphere model", Journal of Chemical Physics '''140''' 136101 (2014)]</ref>, sometimes referred to as the Kirkwood-Alder transition <ref name="GastRussel">[http://dx.doi.org/10.1063/1.882495 Alice P. Gast and William B. Russel "Simple Ordering in Complex Fluids", Physics Today '''51''' (12) pp. 24-30  (1998)]</ref>.
<ref>[http://dx.doi.org/10.1063/1.4870524 Miguel Robles, Mariano López de Haro and Andrés Santos "Note: Equation of state and the freezing point in the hard-sphere model", Journal of Chemical Physics '''140''' 136101 (2014)]</ref>, sometimes referred to as the Kirkwood-Alder transition <ref name="GastRussel">[http://dx.doi.org/10.1063/1.882495 Alice P. Gast and William B. Russel "Simple Ordering in Complex Fluids", Physics Today '''51''' (12) pp. 24-30  (1998)]</ref>.
The liquid-solid coexistence densities (<math>\rho^* = \rho \sigma^3=6\eta/\pi</math>) has been calculated to be
The liquid-solid coexistence densities (<math>\rho^* = \rho \sigma^3</math>) has been calculated to be
:{| border="1"
:{| border="1"
|-  
|-  
| <math>\rho^*_{\mathrm {solid}}</math> || <math>\rho^*_{\mathrm {liquid}}</math> || Reference
| <math>\rho^*_{\mathrm {solid}}</math> || <math>\rho^*_{\mathrm {liquid}}</math> || Reference
|-  
|-  
| 1.041(4)|| 0.943(4) || <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>
Line 67: Line 67:
| 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>
|-  
|-  
| 1.033(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>
| 1.035(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>
|-
| 1.03715(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
Line 78: Line 76:
| 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>
|-
| 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
Line 99: Line 95:
| <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>
|-
| 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  
Line 108: Line 102:
| <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>
|}
|}
The melting and crystallization process has been studied by Isobe and Krauth <ref>[http://dx.doi.org/10.1063/1.4929529  Masaharu Isobe and Werner Krauth "Hard-sphere melting and crystallization with event-chain Monte Carlo", Journal of Chemical Physics '''143''' 084509 (2015)]</ref>.


==Helmholtz energy function==
==Helmholtz energy function==
Line 119: Line 111:
| <math>\rho^*</math> || <math>A/(Nk_BT)</math>|| Reference
| <math>\rho^*</math> || <math>A/(Nk_BT)</math>|| Reference
|-  
|-  
| 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>
| 0.25 || 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 || −0.152 <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 || 1.721 <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>
|}
|}
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==
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):
Line 142: Line 131:
| <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)
Line 152: Line 141:


==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 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>
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>[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>. 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".''
<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>
*See also: [[Equations of state for crystals of hard spheres]]
*See also: [[Equations of state for crystals of hard spheres]]
==Direct correlation function==
==Direct correlation function==
For the [[direct correlation function]] see:
For the [[direct correlation function]] see:
Please note that all contributions to SklogWiki are considered to be released under the Creative Commons Attribution Non-Commercial Share Alike (see SklogWiki:Copyrights for details). If you do not want your writing to be edited mercilessly and redistributed at will, then do not submit it here.
You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource. Do not submit copyrighted work without permission!

To edit this page, please answer the question that appears below (more info):

Cancel Editing help (opens in new window)