We show that an analogy between crowding in fluid and jammed phases of hard spheres captures the density dependence of the kissing number for a family of numerically generated jammed states. We extend this analogy to jams of mixtures of hard spheres in d = 3 dimensions and, thus, obtain an estimate of the random close packing volume fraction, ϕRCP, as a function of size polydispersity. We first consider mixtures of particle sizes with discrete distributions. For binary systems, we show agreement between our predictions and simulations using both our own results and results reported in previous studies, as well as agreement with recent experiments from the literature. We then apply our approach to systems with continuous polydispersity using three different particle size distributions, namely, the log-normal, Gamma, and truncated power-law distributions. In all cases, we observe agreement between our theoretical findings and numerical results up to rather large polydispersities for all particle size distributions when using as reference our own simulations and results from the literature. In particular, we find ϕRCP to increase monotonically with the relative standard deviation, sσ, of the distribution and to saturate at a value that always remains below 1. A perturbative expansion yields a closed-form expression for ϕRCP that quantitatively captures a distribution-independent regime for sσ < 0.5. Beyond that regime, we show that the gradual loss in agreement is tied to the growth of the skewness of size distributions.
REFERENCES
1.
B. J.
Alder
and T. E.
Wainwright
, Phys. Rev.
127
, 359
(1962
).2.
3.
J. G.
Kirkwood
and E. M.
Boggs
, J. Chem. Phys.
10
, 394
(1942
).4.
J.
de Boer
, Rep. Prog. Phys.
12
, 305
(1949
).5.
P. N.
Pusey
and W.
van Megen
, Nature
320
, 340
(1986
).6.
A.
Vrij
, J. W.
Jansen
, J. K. G.
Dhont
, C.
Pathmamanoharan
, M. M.
Kops-Werkhoven
, and H. M.
Fijnaut
, Faraday Discuss. Chem. Soc.
76
, 19
(1983
).7.
T. H.
Besseling
, M.
Hermes
, A.
Fortini
, M.
Dijkstra
, A.
Imhof
, and A.
van Blaaderen
, Soft Matter
8
, 6931
(2012
).8.
A.
Mulero
, Theory and Simulation of Hard-Sphere Fluids and Related Systems
, Lecture Notes in Physics Vol. 753 (Springer-Verlag
, Berlin
, 2008
).9.
J.
Hansen
and I.
McDonald
, Theory of Simple Liquids
(Elsevier Science
, New York
, 2006
).10.
B. J.
Alder
and T. E.
Wainwright
, J. Chem. Phys.
27
, 1208
(1957
).11.
W. W.
Wood
and J. D.
Jacobson
, J. Chem. Phys.
27
, 1207
(1957
).12.
W. G.
Hoover
and F. H.
Ree
, J. Chem. Phys.
49
, 3609
(1968
).13.
S.
Torquato
and F. H.
Stillinger
, Rev. Mod. Phys.
82
, 2633
(2010
).14.
T. C.
Hales
, Ann. Math.
162
, 1065
(2005
).15.
T. C.
Hales
, J.
Harrison
, S.
McLaughlin
, T.
Nipkow
, S.
Obua
, and R.
Zumkeller
, Discrete Comput. Geom.
44
, 1
(2010
).16.
T.
Hales
, M.
Adams
, G.
Bauer
, T. D.
Dang
, J.
Harrison
, L. T.
Hoang
, C.
Kaliszyk
, V.
Magron
, S.
Mclaughlin
, T. T.
Nguyen
et al, Forum Math., Pi
5
, e2
(2017
).17.
P. N.
Pusey
, E.
Zaccarelli
, C.
Valeriani
, E.
Sanz
, W. C. K.
Poon
, and M. E.
Cates
, Philos. Trans. R. Soc., A
367
, 4993
(2009
).18.
E.
Zaccarelli
, C.
Valeriani
, E.
Sanz
, W. C. K.
Poon
, M. E.
Cates
, and P. N.
Pusey
, Phys. Rev. Lett.
103
, 135704
(2009
).19.
E.
Sanz
, C.
Valeriani
, E.
Zaccarelli
, W. C. K.
Poon
, P. N.
Pusey
, and M. E.
Cates
, Phys. Rev. Lett.
106
, 215701
(2011
).20.
A.
van Blaaderen
and P.
Wiltzius
, Science
270
, 1177
(1995
).21.
M.
van Hecke
, J. Phys.: Condens. Matter
22
, 033101
(2009
).22.
23.
J. D.
Bernal
and J.
Mason
, Nature
188
, 910
(1960
).24.
R. D.
Kamien
and A. J.
Liu
, Phys. Rev. Lett.
99
, 155501
(2007
).25.
G.
Parisi
and F.
Zamponi
, J. Chem. Phys.
123
, 144501
(2005
).26.
S.
Wilken
, R. E.
Guerra
, D.
Levine
, and P. M.
Chaikin
, Phys. Rev. Lett.
127
, 38002
(2021
).27.
F.
Krzakala
and J.
Kurchan
, Phys. Rev. E
76
, 021122
(2007
).28.
S.
Torquato
and F. H.
Stillinger
, Phys. Rev. E
73
, 031106
(2006
).29.
G.
Parisi
and F.
Zamponi
, Rev. Mod. Phys.
82
, 789
(2010
).30.
M.
Hermes
and M.
Dijkstra
, Europhys. Lett.
89
, 38005
(2010
).31.
R.
Mari
, F.
Krzakala
, and J.
Kurchan
, Phys. Rev. Lett.
103
, 025701
(2009
).32.
L.
Berthier
and T. A.
Witten
, Phys. Rev. E
80
, 021502
(2009
).33.
R. J.
Speedy
, Mol. Phys.
95
, 169
(1998
).34.
I.
Biazzo
, F.
Caltagirone
, G.
Parisi
, and F.
Zamponi
, Phys. Rev. Lett.
102
, 195701
(2009
).35.
36.
P.
Charbonneau
, J.
Kurchan
, G.
Parisi
, P.
Urbani
, and F.
Zamponi
, Annu. Rev. Condens. Matter Phys.
8
, 265
(2017
).37.
S.
Torquato
, T. M.
Truskett
, and P. G.
Debenedetti
, Phys. Rev. Lett.
84
, 2064
(2000
).38.
S.
Torquato
and M. D.
Rintoul
, Phys. Rev. Lett.
75
, 4067
(1995
).39.
M. D.
Rintoul
and S.
Torquato
, J. Chem. Phys.
105
, 9258
(1996
).40.
T. M.
Truskett
, S.
Torquato
, and P. G.
Debenedetti
, Phys. Rev. E
62
, 993
(2000
).41.
A. R.
Kansal
, S.
Torquato
, and F. H.
Stillinger
, Phys. Rev. E
66
, 041109
(2002
).42.
P. J.
Steinhardt
, D. R.
Nelson
, and M.
Ronchetti
, Phys. Rev. B
28
, 784
(1983
).43.
E. J.
Le Fevre
, Nat. Phys. Sci.
235
, 20
(1972
).44.
45.
T.
Aste
and A.
Coniglio
, Europhys. Lett.
67
, 165
(2004
).46.
E.
Katzav
, R.
Berdichevsky
, and M.
Schwartz
, Phys. Rev. E
99
, 012146
(2019
).47.
S.
Torquato
and F. H.
Stillinger
, J. Appl. Phys.
102
, 093511
(2007
).48.
49.
S.
Atkinson
, F. H.
Stillinger
, and S.
Torquato
, Proc. Natl. Acad. Sci. U. S. A.
111
, 18436
(2014
).50.
A.
Zaccone
and E.
Scossa-Romano
, Phys. Rev. B
83
, 184205
(2011
).51.
A.
Zaccone
, Phys. Rev. Lett.
128
, 028002
(2022
).52.
S.-E.
Phan
, W. B.
Russel
, J.
Zhu
, and P. M.
Chaikin
, J. Chem. Phys.
108
, 9789
(1998
).53.
L.
Meng
, P.
Lu
, and S.
Li
, Particuology
16
, 155
(2014
).54.
R. S.
Farr
and R. D.
Groot
, J. Chem. Phys.
131
, 244104
(2009
).55.
A. V.
Kyrylyuk
, A.
Wouterse
, and A. P.
Philipse
, Trends in Colloid and Interface Science XXIII
(Springer
, Berlin, Heidelberg
, 2010
), pp. 29
–33
.56.
Y.
Yuan
, L.
Liu
, Y.
Zhuang
, W.
Jin
, and S.
Li
, Phys. Rev. E
98
, 042903
(2018
).57.
R. S.
Farr
, Powder Technol.
245
, 28
(2013
).58.
L.
Berthier
, D.
Coslovich
, A.
Ninarello
, and M.
Ozawa
, Phys. Rev. Lett.
116
, 238002
(2016
).59.
A.
Ninarello
, L.
Berthier
, and D.
Coslovich
, Phys. Rev. X
7
, 021039
(2017
).60.
H.
Cramer
, Mathematical Methods of Statistics
(Princeton University Press
, Princeton
, 1954
).61.
M.
Kotlarchyk
, R. B.
Stephens
, and J. S.
Huang
, J. Phys. Chem.
92
, 1533
(1988
).62.
63.
D.
Hexner
, A. J.
Liu
, and S. R.
Nagel
, Phys. Rev. Lett.
121
, 115501
(2018
).64.
J. J.
Shynk
, Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
(John Wiley and Sons
, New York
, 2012
).65.
H.
Pishro-Nik
, Introduction to Probability, Statistics and Random Processes
(Kappa Research, LCC
, Amherst, MA
, 2014
).66.
A.
Donev
, S.
Torquato
, and F. H.
Stillinger
, Phys. Rev. E
71
, 011105
(2005
).67.
E.
Lerner
, G.
Düring
, and M.
Wyart
, Soft Matter
9
, 8252
(2013
).68.
P.
Charbonneau
, J.
Kurchan
, G.
Parisi
, P.
Urbani
, and F.
Zamponi
, J. Stat. Mech.: Theory Exp.
2014
, P10009
.69.
P.
Dimon
, S. K.
Sinha
, D. A.
Weitz
, C. R.
Safinya
, G. S.
Smith
, W. A.
Varady
, and H. M.
Lindsay
, Phys. Rev. Lett.
57
, 595
(1986
).70.
M.
Lattuada
, H.
Wu
, and M.
Morbidelli
, J. Colloid Interface Sci.
268
, 106
(2003
).71.
S.
Torquato
, J. Chem. Phys.
149
, 020901
(2018
).72.
M. P.
Allen
and D. J.
Tildesley
, Computer Simulation of Liquids
(Oxford University Press
, 2017
).73.
S.
Torquato
, Random Heterogeneous Materials: Microstructure and Macroscopic Properties
(Springer-Verlag
, New York
, 2002
).74.
C.
Likos
,“Maximizing space efficiency without order, analytically
,” J. Club Condens. Matter Phys.
(published online, 2022)75.
J. L.
Lebowitz
, Phys. Rev.
133
, A895
(1964
).76.
G. A.
Mansoori
, N. F.
Carnahan
, K. E.
Starling
, and T. W.
Leland
, J. Chem. Phys.
54
, 1523
(1971
).77.
A.
Santos
, S. B.
Yuste
, and M. L.
de Haro
, Mol. Phys.
96
, 1
(1999
).78.
79.
V.
Baranau
and U.
Tallarek
, Soft Matter
10
, 3826
(2014
).80.
B. D.
Lubachevsky
and F. H.
Stillinger
, J. Stat. Phys.
60
, 561
(1990
).81.
B. D.
Lubachevsky
, J. Comput. Phys.
94
, 255
(1991
).82.
H.
Yuan
, Z.
Zhang
, W.
Kob
, and Y.
Wang
, Phys. Rev. Lett.
127
, 278001
(2021
).83.
S.
Suo
, C.
Zhai
, M.
Xu
, M.
Kamlah
, and Y.
Gan
, “An unexplored valley of binary packing: The loose jamming state
,” arXiv:2205.01934 (2022
).84.
V.
Ogarko
and S.
Luding
, Soft Matter
9
, 9530
(2013
).85.
A.
Santos
, S. B.
Yuste
, M. L.
de Haro
, M.
Alawneh
, and D.
Henderson
, Mol. Phys.
107
, 685
(2009
).86.
C.-H.
Sow
, K.
Harada
, A.
Tonomura
, G.
Crabtree
, and D. G.
Grier
, Phys. Rev. Lett.
80
, 2693
(1998
).87.
M.
Mungan
, C.-H.
Sow
, S. N.
Coppersmith
, and D. G.
Grier
, Phys. Rev. B
58
, 14588
(1998
).88.
J. C.
Petit
, N.
Kumar
, S.
Luding
, and M.
Sperl
, Phys. Rev. Lett.
125
, 215501
(2020
).89.
Y.
Hara
, H.
Mizuno
, and A.
Ikeda
, Phys. Rev. Res.
3
, 023091
(2021
).90.
91.
M. S.
Wertheim
, Phys. Rev. Lett.
10
, 321
(1963
).92.
N. F.
Carnahan
and K. E.
Starling
, J. Chem. Phys.
51
, 635
(1969
).93.
Y.
Song
, E. A.
Mason
, and R. M.
Stratt
, J. Phys. Chem.
93
, 6916
(1989
).94.
D. A.
Young
and B. J.
Alder
, J. Chem. Phys.
70
, 473
(1979
).95.
96.
M.
Abramovitz
and I. A.
Stegun
, Handbook of Mathematical Functions
(Dover
, New York
, 1972
).97.
S.
Atkinson
, F. H.
Stillinger
, and S.
Torquato
, Phys. Rev. E
88
, 062208
(2013
).98.
W. S.
Jodrey
and E. M.
Tory
, Phys. Rev. A
32
, 2347
(1985
).99.
R.
Jullien
, J.-F.
Sadoc
, and R.
Mosseri
, J. Phys. I
7
, 1677
(1997
).100.
J.
Tobochnik
and P. M.
Chapin
, J. Chem. Phys.
88
, 5824
(1988
).101.
A. Z.
Zinchenko
, J. Comput. Phys.
114
, 298
(1994
).102.
W. M.
Visscher
and M.
Bolsterli
, Nature
239
, 504
(1972
).104.
A.
Santos
, and M.
López de Haro
, “A heuristic approach for the densest packing fraction of hard-sphere mixtures
,” Phys. A: Stat. Mech. Appl.
(published online, 2023).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
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