How dense objects, particles, atoms, and molecules can be packed is intimately related to the properties of the corresponding hosts and macrosystems. We present results from extensive Monte Carlo simulations on maximally compressed packings of linear, freely jointed chains of tangent hard spheres of uniform size in films whose thickness is equal to the monomer diameter. We demonstrate that fully flexible chains of hard spheres can be packed as efficiently as monomeric analogs, within a statistical tolerance of less than 1%. The resulting ordered polymer morphology corresponds to an almost perfect hexagonal triangular (TRI) crystal of the p6m wallpaper group, whose sites are occupied by the chain monomers. The Flory scaling exponent, which corresponds to the maximally dense polymer packing in 2D, has a value of ν = 0.62, which lies between the limits of 0.50 (compact and collapsed state) and 0.75 (self-avoiding random walk).
REFERENCES
1.
W.
Weaire
and T.
Aste
, The Pursuit of Perfect Packing
(Taylor & Francis
, New York
, 2008
).2.
J.
Conway
and N.
Sloane
, Sphere Packings, Lattices and Groups
(Springer Verlag
, New York
, 1998
).3.
S.
Torquato
and F. H.
Stillinger
, “Jammed hard-particle packings: From Kepler to Bernal and beyond
,” Rev. Mod. Phys.
82
, 2633
–2672
(2010
).4.
T. C.
Hales
, J.
Harrison
, S.
McLaughlin
, T.
Nipkow
, S.
Obua
, and R.
Zumkeller
, “A revision of the proof of the Kepler conjecture
,” Discrete Comput. Geom.
44
, 1
–34
(2010
).5.
T.
Hales
, M.
Adams
, G.
Bauer
, T. D.
Dang
, J.
Harrison
, L. T.
Hoang
, C.
Kaliszyk
, V.
Magron
, S.
McLaughlin
, T. T.
Nguyen
, Q. T.
Nguyen
, T.
Nipkow
, S.
Obua
, J.
Pleso
, J.
Rute
, A.
Solovyev
, T. H. A.
Ta
, N. T.
Tran
, T. D.
Trieu
, J.
Urban
, K.
Vu
, and R.
Zumkeller
, “A formal proof of the Kepler conjecture
,” Forum Math. Pi
5
, e2
(2017
).6.
M. S.
Viazovska
, “The sphere packing problem in dimension 8
,” Ann. Math.
185
, 991
–1015
(2017
).7.
H.
Cohn
, A.
Kumar
, S. D.
Miller
, D.
Radchenko
, and M.
Viazovska
, “The sphere packing problem in dimension 24
,” Ann. Math.
185
, 1017
–1033
(2017
).8.
A.
Donev
, S.
Torquato
, F. H.
Stillinger
, and R.
Connelly
, “Jamming in hard sphere and disk packings
,” J. Appl. Phys.
95
, 989
–999
(2004
).9.
S.
Torquato
, T. M.
Truskett
, and P. G.
Debenedetti
, “Is random close packing of spheres well defined?
,” Phys. Rev. Lett.
84
, 2064
–2067
(2000
).10.
S.
Torquato
and F. H.
Stillinger
, “Multiplicity of generation, selection, and classification procedures for jammed hard-particle packings
,” J. Phys. Chem. B
105
, 11849
–11853
(2001
).11.
J. D.
Bernal
, “Geometry of the structure of monatomic liquids
,” Nature
185
, 68
–70
(1960
).12.
J. D.
Bernal
and J. L.
Finney
, “Random close-packed hard-sphere model.II. Geometry of random packing of hard spheres
,” Discuss. Faraday Soc.
43
, 62
–69
(1967
).13.
G. D.
Scott
, K. R.
Knight
, J. D.
Bernal
, and J.
Mason
, “Radial distribution of random close packing of equal spheres
,” Nature
194
, 956
–957
(1962
).14.
C. S.
O’Hern
, L. E.
Silbert
, A. J.
Liu
, and S. R.
Nagel
, “Jamming at zero temperature and zero applied stress: The epitome of disorder
,” Phys. Rev. E
68
, 011306
(2003
).15.
C.
Song
, P.
Wang
, and H. A.
Makse
, “A phase diagram for jammed matter
,” Nature
453
, 629
–632
(2008
).16.
A.
Baule
, F.
Morone
, H. J.
Herrmann
, and H. A.
Makse
, “Edwards statistical mechanics for jammed granular matter
,” Rev. Mod. Phys.
90
, 015006
(2018
).17.
K.
Wang
, C. M.
Song
, P.
Wang
, and H. A.
Makse
, “Edwards thermodynamics of the jamming transition for frictionless packings: Ergodicity test and role of angoricity and compactivity
,” Phys. Rev. E
86
, 011305
(2012
).18.
M.
van Hecke
, “Jamming of soft particles: Geometry, mechanics, scaling and isostaticity
,” J. Phys.: Condens. Matter
22
, 033101
(2010
).19.
G.
Parisi
and F.
Zamponi
, “Mean-field theory of hard sphere glasses and jamming
,” Rev. Mod. Phys.
82
, 789
–845
(2010
).20.
P.
Rissone
, E. I.
Corwin
, and G.
Parisi
, “Long-range anomalous decay of the correlation in jammed packings
,” Phys. Rev. Lett.
127
, 038001
(2021
).21.
A.
Donev
, S.
Torquato
, F. H.
Stillinger
, and R.
Connelly
, “A linear programming algorithm to test for jamming in hard-sphere packings
,” J. Comput. Phys.
197
, 139
–166
(2004
).22.
P.
Ballesta
, A.
Duri
, and L.
Cipelletti
, “Unexpected drop of dynamical heterogeneities in colloidal suspensions approaching the jamming transition
,” Nat. Phys.
4
, 550
–554
(2008
).23.
A.
Donev
, F. H.
Stillinger
, and S.
Torquato
, “Unexpected density fluctuations in jammed disordered sphere packings
,” Phys. Rev. Lett.
95
, 090604
(2005
).24.
Y. L.
Jin
, J. G.
Puckett
, and H. A.
Makse
, “Statistical theory of correlations in random packings of hard particles
,” Phys. Rev. E
89
, 052207
(2014
).25.
A.
Zaccone
, “Explicit analytical solution for random close packing in d=2 and d=3
,” Phys. Rev. Lett.
128
, 028002
(2022
).26.
R.
Blumenfeld
, “Disorder criterion and explicit solution for the disc random packing problem
,” Phys. Rev. Lett.
127
, 118002
(2021
).27.
S.
Atkinson
, F. H.
Stillinger
, and S.
Torquato
, “Existence of isostatic, maximally random jammed monodisperse hard-disk packings
,” Proc. Natl. Acad. Sci. U. S. A.
111
, 18436
–18441
(2014
).28.
S.
Atkinson
, Y.
Jiao
, and S.
Torquato
, “Maximally dense packings of two-dimensional convex and concave noncircular particles
,” Phys. Rev. E
86
, 031302
(2012
).29.
S.
Torquato
and Y.
Jiao
, “Robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming
,” Phys. Rev. E
82
, 061302
(2010
).30.
J. A.
Anderson
, J.
Antonaglia
, J. A.
Millan
, M.
Engel
, and S. C.
Glotzer
, “Shape and symmetry determine two-dimensional melting transitions of hard regular polygons
,” Phys. Rev. X
7
, 021001
(2017
).31.
A.
Donev
, F. H.
Stillinger
, and S.
Torquato
, “Do binary hard disks exhibit an ideal glass transition?
,” Phys. Rev. Lett.
96
, 225502
(2006
).32.
N.
Xu
, J.
Blawzdziewicz
, and C. S.
O’Hern
, “Random close packing revisited: Ways to pack frictionless disks
,” Phys. Rev. E
71
, 061306
(2005
).33.
F. H.
Stillinger
, R. L.
Kornegay
, and E. A.
Dimarzio
, “Systematic approach to explanation of rigid disk phase transition
,” J. Chem. Phys.
40
, 1564
–1576
(1964
).34.
W. M.
Visscher
and M.
Bolsterl
, “Random packing of equal and unequal spheres in two and three dimensions
,” Nature
239
, 504
(1972
).35.
E. L.
Hinrichsen
, J.
Feder
, and T.
Jøssang
, “Random packing of disks in two dimensions
,” Phys. Rev. A
41
, 4199
–4209
(1990
).36.
S.
Meyer
, C.
Song
, Y.
Jin
, K.
Wang
, and H. A.
Makse
, “Jamming in two-dimensional packings
,” Physica A
389
, 5137
–5144
(2010
).37.
T.
Unger
, J.
Kertesz
, and D. E.
Wolf
, “Force indeterminacy in the jammed state of hard disks
,” Phys. Rev. Lett.
94
, 178001
(2005
).38.
B. I.
Halperin
and D. R.
Nelson
, “Theory of two-dimensional melting
,” Phys. Rev. Lett.
41
, 121
–124
(1978
).39.
D. R.
Nelson
and B. I.
Halperin
, “Dislocation-mediated melting in two dimensions
,” Phys. Rev. B
19
, 2457
–2484
(1979
).40.
A. L.
Thorneywork
, J. L.
Abbott
, D. G. A. L.
Aarts
, and R. P. A.
Dullens
, “Two-dimensional melting of colloidal hard spheres
,” Phys. Rev. Lett.
118
, 158001
(2017
).41.
M. A.
Bates
and D.
Frenkel
, “Influence of vacancies on the melting transition of hard disks in two dimensions
,” Phys. Rev. E
61
, 5223
–5227
(2000
).42.
E. P.
Bernard
and W.
Krauth
, “Two-step melting in two dimensions: First-order liquid-hexatic transition
,” Phys. Rev. Lett.
107
, 155704
(2011
).43.
M.
Engel
, J. A.
Anderson
, S. C.
Glotzer
, M.
Isobe
, E. P.
Bernard
, and W.
Krauth
, “Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods
,” Phys. Rev. E
87
, 042134
(2013
).44.
J. A.
Forrest
and K.
Dalnoki-Veress
, “The glass transition in thin polymer films
,” Adv. Colloid Interface Sci.
94
, 167
–196
(2001
).45.
M. D.
Ediger
and J. A.
Forrest
, “Dynamics near free surfaces and the glass transition in thin polymer films: A view to the future
,” Macromolecules
47
, 471
–478
(2014
).46.
B.
Li
, S.
Zhang
, J. S.
Andre
, and Z.
Chen
, “Relaxation behavior of polymer thin films: Effects of free surface, buried interface, and geometrical confinement
,” Prog. Polym. Sci.
120
, 101431
(2021
).47.
Y.-X.
Liu
and E.-Q.
Chen
, “Polymer crystallization of ultrathin films on solid substrates
,” Coord. Chem. Rev.
254
, 1011
–1037
(2010
).48.
N. C.
Karayiannis
and M.
Laso
, “Dense and nearly jammed random packings of freely jointed chains of tangent hard spheres
,” Phys. Rev. Lett.
100
, 050602
(2008
).49.
M.
Laso
, N. C.
Karayiannis
, K.
Foteinopoulou
, M. L.
Mansfield
, and M.
Kröger
, “Random packing of model polymers: Local structure, topological hindrance and universal scaling
,” Soft Matter
5
, 1762
–1770
(2009
).50.
N. C.
Karayiannis
, K.
Foteinopoulou
, and M.
Laso
, “Contact network in nearly jammed disordered packings of hard-sphere chains
,” Phys. Rev. E
80
, 011307
(2009
).51.
N. C.
Karayiannis
, K.
Foteinopoulou
, and M.
Laso
, “The structure of random packings of freely jointed chains of tangent hard spheres
,” J. Chem. Phys.
130
, 164908
(2009
).52.
K.
Foteinopoulou
, N. C.
Karayiannis
, M.
Laso
, M.
Kroger
, and M. L.
Mansfield
, “Universal scaling, entanglements, and knots of model chain molecules
,” Phys. Rev. Lett.
101
, 265702
(2008
).53.
N. C.
Karayiannis
, K.
Foteinopoulou
, and M.
Laso
, “Entropy-driven crystallization in dense systems of athermal chain molecules
,” Phys. Rev. Lett.
103
, 045703
(2009
).54.
N. C.
Karayiannis
, K.
Foteinopoulou
, and M.
Laso
, “The role of bond tangency and bond gap in hard sphere crystallization of chains
,” Soft Matter
11
, 1688
–1700
(2015
).55.
M.
Herranz
, K.
Foteinopoulou
, N. C.
Karayiannis
, and M.
Laso
, “Polymorphism and perfection in crystallization of hard sphere polymers
,” Polymers
14
, 4435
(2022
).56.
M. D.
Rintoul
and S.
Torquato
, “Metastability and crystallization in hard-sphere systems
,” Phys. Rev. Lett.
77
, 4198
–4201
(1996
).57.
N. C.
Karayiannis
, R.
Malshe
, M.
Kröger
, J. J.
de Pablo
, and M.
Laso
, “Evolution of fivefold local symmetry during crystal nucleation and growth in dense hard-sphere packings
,” Soft Matter
8
, 844
–858
(2012
).58.
N. N.
Medvedev
, A.
Bezrukov
, and D.
Shtoyan
, “From amorphous solid to defective crystal. A study of structural peculiarities in close packings of hard spheres
,” J. Struct. Chem.
45
, S23
–S30
(2004
).59.
B.
O’Malley
and I.
Snook
, “Crystal nucleation in the hard sphere system
,” Phys. Rev. Lett.
90
, 085702
(2003
).60.
S.
Auer
and D.
Frenkel
, “Prediction of absolute crystal-nucleation rate in hard-sphere colloids
,” Nature
409
, 1020
–1023
(2001
).61.
I.
Sanchez-Burgos
, E.
Sanz
, C.
Vega
, and J. R.
Espinosa
, “Fcc vs. hcp competition in colloidal hard-sphere nucleation: On their relative stability, interfacial free energy and nucleation rate
,” Phys. Chem. Chem. Phys.
23
, 19611
–19626
(2021
).62.
J. C.
Gaines
, W. W.
Smith
, L.
Regan
, and C. S.
O’Hern
, “Random close packing in protein cores
,” Phys. Rev. E
93
, 032415
(2016
).63.
A. T.
Grigas
, Z.
Mei
, J. D.
Treado
, Z. A.
Levine
, L.
Regan
, and C. S.
O’Hern
, “Using physical features of protein core packing to distinguish real proteins from decoys
,” Protein Sci.
29
, 1931
–1944
(2020
).64.
J. C.
Gaines
, A. H.
Clark
, L.
Regan
, and C. S.
O’Hern
, “Packing in protein cores
,” J. Phys.: Condens. Matter
29
, 293001
(2017
).65.
M.
Herranz
, D.
Martínez-Fernández
, P. M.
Ramos
, K.
Foteinopoulou
, N. C.
Karayiannis
, and M.
Laso
, “Simu-D: A simulator-descriptor suite for polymer-based systems under extreme conditions
,” Int. J. Mol. Sci.
22
, 12464
(2021
).66.
P. M.
Ramos
, N. C.
Karayiannis
, and M.
Laso
, “Off-lattice simulation algorithms for athermal chain molecules under extreme confinement
,” J. Comput. Phys.
375
, 918
–934
(2018
).67.
M.
Herranz
, M.
Santiago
, K.
Foteinopoulou
, N. C.
Karayiannis
, and M.
Laso
, “Crystal, fivefold and glass formation in clusters of polymers interacting with the square well potential
,” Polymers
12
, 1111
(2020
).68.
N. C.
Karayiannis
and M.
Laso
, “Monte Carlo scheme for generation and relaxation of dense and nearly jammed random structures of freely jointed hard-sphere chains
,” Macromolecules
41
, 1537
–1551
(2008
).69.
W.
Humphrey
, A.
Dalke
, and K.
Schulten
, “VMD: Visual molecular dynamics
,” J. Mol. Graphics Modell.
14
, 33
–38
(1996
).70.
N. C.
Karayiannis
, K.
Foteinopoulou
, and M.
Laso
, “The characteristic crystallographic element norm: A descriptor of local structure in atomistic and particulate systems
,” J. Chem. Phys.
130
, 074704
(2009
).71.
P. M.
Ramos
, M.
Herranz
, K.
Foteinopoulou
, N. C.
Karayiannis
, and M.
Laso
, “Identification of local structure in 2-D and 3-D atomic systems through crystallographic analysis
,” Crystals
10
, 1008
(2020
).72.
P. J.
Flory
, Principles of Polymer Chemistry
(Cornell University Press
, Ithaca
, 2010
).73.
A. D.
Schlüter
, P.
Payamyar
, and H. C.
Öttinger
, “How the world changes by going from one- to two-dimensional polymers in solution
,” Macromol. Rapid Commun.
37
, 1638
–1650
(2016
).74.
P.
deGennes
, Scaling Concepts in Polymer Physics
(Cornell University Press
, Ithaca
, 1980
).75.
V.
Baranau
and U.
Tallarek
, “Random-close packing limits for monodisperse and polydisperse hard spheres
,” Soft Matter
10
, 3826
–3841
(2014
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
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