We consider depletion effects of a pear-shaped colloidal particle in a hard-sphere solvent for two different model realizations of the pear-shaped colloidal particle. The two models are the pear hard Gaussian overlap (PHGO) particles and the hard pears of revolution (HPR). The motivation for this study is to provide a microscopic understanding for the substantially different mesoscopic self-assembly properties of these pear-shaped colloids, in dense suspensions, that have been reported in the previous studies. This is done by determining their differing depletion attractions via Monte Carlo simulations of PHGO and HPR particles in a pool of hard spheres and comparing them with excluded volume calculations of numerically obtained ideal configurations on the microscopic level. While the HPR model behaves as predicted by the analysis of excluded volumes, the PHGO model showcases a preference for splay between neighboring particles, which can be attributed to the special non-additive characteristics of the PHGO contact function. Lastly, we propose a potentially experimentally realizable pear-shaped particle model, the non-additive hard pear of revolution model, which is based on the HPR model but also features non-additive traits similar to those of PHGO particles to mimic their depletion behavior.
REFERENCES
1.
P. W. A.
Schönhöfer
, M.
Marechal
, D. J.
Cleaver
, and G. E.
Schröder-Turk
, “Self-assembly and entropic effects in pear-shaped colloid systems: I. Shape sensitivity of bilayer phases in colloidal pear-shaped particle systems
,” J. Chem. Phys.
152
, 034903
(2020
).2.
J. N.
Israelachvili
, D. J.
Mitchell
, and B. W.
Ninham
, “Theory of self-assembly of hydrocarbon amphiphiles into micelles and bilayers
,” J. Chem. Soc., Faraday Trans. 2
72
, 1525
–1568
(1976
).3.
S. T.
Hyde
, Z.
Blum
, T.
Landh
, S.
Lidin
, B. W.
Ninham
, S.
Andersson
, and K.
Larsson
, The Language of Shape: The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology
(Elsevier
, 1996
).4.
J.-W.
Kim
, R. J.
Larsen
, and D. A.
Weitz
, “Synthesis of nonspherical colloidal particles with anisotropic properties
,” J. Am. Chem. Soc.
128
(44
), 14374
–14377
(2006
).5.
A.
Perro
, S.
Reculusa
, E.
Bourgeat-Lami
, E.
Duguet
, and S.
Ravaine
, “Synthesis of hybrid colloidal particles: From snowman-like to raspberry-like morphologies
,” Colloids Surf., A
284-285
, 78
–83
(2006
).6.
T. S.
Ahmadi
, Z. L.
Wang
, T. C.
Green
, A.
Henglein
, and M. A.
El-Sayed
, “Shape-controlled synthesis of colloidal platinum nanoparticles
,” Science
272
(5270
), 1924
–1925
(1996
).7.
A. R.
Tao
, S.
Habas
, and P.
Yang
, “Shape control of colloidal metal nanocrystals
,” Small
4
(3
), 310
–325
(2008
).8.
J. A.
Champion
, Y. K.
Katare
, and S.
Mitragotri
, “Making polymeric micro- and nanoparticles of complex shapes
,” Proc. Natl. Acad. Sci. U. S. A.
104
(29
), 11901
–11904
(2007
).9.
S. C.
Glotzer
and M. J.
Solomon
, “Anisotropy of building blocks and their assembly into complex structures
,” Nat. Mater.
6
(8
), 557
(2007
).10.
J. A. C.
Veerman
and D.
Frenkel
, “Phase diagram of a system of hard spherocylinders by computer simulation
,” Phys. Rev. A
41
(6
), 3237
(1990
).11.
D.
Frenkel
, B. M.
Mulder
, and J. P.
McTague
, “Phase diagram of a system of hard ellipsoids
,” Phys. Rev. Lett.
52
(4
), 287
(1984
).12.
P. F.
Damasceno
, M.
Engel
, and S. C.
Glotzer
, “Predictive self-assembly of polyhedra into complex structures
,” Science
337
(6093
), 453
–457
(2012
).13.
M. Z.
Miskin
, G.
Khaira
, J. J.
de Pablo
, and H. M.
Jaeger
, “Turning statistical physics models into materials design engines
,” Proc. Natl. Acad. Sci. U. S. A.
113
(1
), 34
–39
(2016
).14.
D.
Chen
, G.
Zhang
, and S.
Torquato
, “Inverse design of colloidal crystals via optimized patchy interactions
,” J. Phys. Chem. B
122
(35
), 8462
–8468
(2018
).15.
Q.
Chen
, S. C.
Bae
, and S.
Granick
, “Directed self-assembly of a colloidal kagome lattice
,” Nature
469
(7330
), 381
–384
(2011
).16.
G.
van Anders
, D.
Klotsa
, A. S.
Karas
, P. M.
Dodd
, and S. C.
Glotzer
, “Digital alchemy for materials design: Colloids and beyond
,” ACS Nano
9
(10
), 9542
–9553
(2015
).17.
Y.
Geng
, G.
van Anders
, P. M.
Dodd
, J.
Dshemuchadse
, and S. C.
Glotzer
, “Engineering entropy for the inverse design of colloidal crystals from hard shapes
,” Sci. Adv.
5
(7
), eaaw0514
(2019
).18.
F.
Barmes
, M.
Ricci
, C.
Zannoni
, and D. J.
Cleaver
, “Computer simulations of hard pear-shaped particles
,” Phys. Rev. E
68
, 021708
(2003
).19.
L. J.
Ellison
, D. J.
Michel
, F.
Barmes
, and D. J.
Cleaver
, “Entropy-driven formation of the gyroid cubic phase
,” Phys. Rev. Lett.
97
(23
), 237801
(2006
).20.
P. W. A.
Schönhöfer
, L. J.
Ellison
, M.
Marechal
, D. J.
Cleaver
, and G. E.
Schröder-Turk
, “Purely entropic self-assembly of the bicontinuous Iad gyroid phase in equilibrium hard-pear systems
,” Interface Focus
7
, 20160161
(2017
).21.
P. W. A.
Schönhöfer
, D. J.
Cleaver
, and G. E.
Schröder-Turk
, “Double diamond phase in pear-shaped nanoparticle systems with hard sphere solvent
,” J. Phys. D: Appl. Phys.
51
(46
), 464003
(2018
).22.
S.
Asakura
and F.
Oosawa
, “On interaction between two bodies immersed in a solution of macromolecules
,” J. Chem. Phys.
22
(7
), 1255
–1256
(1954
).23.
S.
Asakura
and F.
Oosawa
, “Interaction between particles suspended in solutions of macromolecules
,” J. Polym. Sci.
33
(126
), 183
–192
(1958
).24.
A.
Vrij
, “Polymers at interfaces and the interactions in colloidal dispersions
,” Pure Appl. Chem.
48
(4
), 471
–483
(1976
).25.
Y.
Mao
, M. E.
Cates
, and H. N. W.
Lekkerkerker
, “Depletion force in colloidal systems
,” Physica A
222
(1-4
), 10
–24
(1995
).26.
R.
Roth
, R.
Evans
, and S.
Dietrich
, “Depletion potential in hard-sphere mixtures: Theory and applications
,” Phys. Rev. E
62
(4
), 5360
(2000
).27.
X. L.
Chu
, A. D.
Nikolov
, and D. T.
Wasan
, “Effects of particle size and polydispersity on the depletion and structural forces in colloidal dispersions
,” Langmuir
12
(21
), 5004
–5010
(1996
).28.
T.
Biben
, P.
Bladon
, and D.
Frenkel
, “Depletion effects in binary hard-sphere fluids
,” J. Phys.: Condens. Matter
8
(50
), 10799
(1996
).29.
R.
Dickman
, P.
Attard
, and V.
Simonian
, “Entropic forces in binary hard sphere mixtures: Theory and simulation
,” J. Chem. Phys.
107
(1
), 205
–213
(1997
).30.
B.
Götzelmann
, R.
Evans
, and S.
Dietrich
, “Depletion forces in fluids
,” Phys. Rev. E
57
(6
), 6785
(1998
).31.
B.
Widom
and J. S.
Rowlinson
, “New model for the study of liquid–vapor phase transitions
,” J. Chem. Phys.
52
(4
), 1670
–1684
(1970
).32.
H. N. W.
Lekkerkerker
and R.
Tuinier
, Colloids and the Depletion Interaction
(Springer
, Dordrecht, The Netherlands
, 2011
), Vol. 833.33.
E.
Eisenriegler
, A.
Hanke
, and S.
Dietrich
, “Polymers interacting with spherical and rodlike particles
,” Phys. Rev. E
54
(2
), 1134
(1996
).34.
D. G. A. L.
Aarts
, R.
Tuinier
, and H. N. W.
Lekkerkerker
, “Phase behaviour of mixtures of colloidal spheres and excluded-volume polymer chains
,” J. Phys.: Condens. Matter
14
(33
), 7551
(2002
).35.
Y.
Mao
, M. E.
Cates
, and H. N. W.
Lekkerkerker
, “Theory of the depletion force due to rodlike polymers
,” J. Chem. Phys.
106
(9
), 3721
–3729
(1997
).36.
R.
Roth
, “Depletion potentials in colloidal mixtures of spheres and rods
,” J. Phys.: Condens. Matter
15
(1
), S277
(2002
).37.
W.
Li
, T.
Yang
, and H.-R.
Ma
, “Depletion potentials in colloidal mixtures of hard spheres and rods
,” J. Chem. Phys.
128
(4
), 044910
(2008
).38.
M.
Piech
and J. Y.
Walz
, “Depletion interactions produced by nonadsorbing charged and uncharged spheroids
,” J. Colloid Interface Sci.
232
(1
), 86
–101
(2000
).39.
L.
Harnau
and S.
Dietrich
, “Depletion potential in colloidal mixtures of hard spheres and platelets
,” Phys. Rev. E
69
(5
), 051501
(2004
).40.
E.
Evans
and D.
Needham
, “Attraction between lipid bilayer membranes in concentrated solutions of nonadsorbing polymers: Comparison of mean-field theory with measurements of adhesion energy
,” Macromolecules
21
(6
), 1822
–1831
(1988
).41.
J. C.
Crocker
, J. A.
Matteo
, A. D.
Dinsmore
, and A. G.
Yodh
, “Entropic attraction and repulsion in binary colloids probed with a line optical tweezer
,” Phys. Rev. Lett.
82
(21
), 4352
(1999
).42.
P. D.
Kaplan
, L. P.
Faucheux
, and A. J.
Libchaber
, “Direct observation of the entropic potential in a binary suspension
,” Phys. Rev. Lett.
73
(21
), 2793
(1994
).43.
A. D.
Dinsmore
, P. B.
Warren
, W. C. K.
Poon
, and A. G.
Yodh
, “Fluid-solid transitions on walls in binary hard-sphere mixtures
,” Europhys. Lett.
40
(3
), 337
(1997
).44.
W.
Knoben
, N. A. M.
Besseling
, and M. A.
Cohen Stuart
, “Direct measurement of depletion and hydrodynamic forces in solutions of a reversible supramolecular polymer
,” Langmuir
23
(11
), 6095
–6105
(2007
).45.
T.
Biben
and J.-P.
Hansen
, “Phase separation of asymmetric binary hard-sphere fluids
,” Phys. Rev. Lett.
66
(17
), 2215
(1991
).46.
H. N. W.
Lekkerkerker
and A.
Stroobants
, “On the spinodal instability of highly asymmetric hard sphere suspensions
,” Physica A
195
(3-4
), 387
–397
(1993
).47.
Y.
Rosenfeld
, “Phase separation of asymmetric binary hard-sphere fluids: Self-consistent density functional theory
,” Phys. Rev. Lett.
72
(24
), 3831
(1994
).48.
Y.
Rosenfeld
, “Phase separation of asymmetric binary hard-sphere fluids: Self-consistent density functional theory
,” J. Phys. Chem.
99
(9
), 2857
–2864
(1995
).49.
A.
Imhof
and J. K. G.
Dhont
, “Experimental phase diagram of a binary colloidal hard-sphere mixture with a large size ratio
,” Phys. Rev. Lett.
75
(8
), 1662
(1995
).50.
M.
Dijkstra
, R.
van Roij
, and R.
Evans
, “Phase diagram of highly asymmetric binary hard-sphere mixtures
,” Phys. Rev. E
59
(5
), 5744
(1999
).51.
R.
Tuinier
, J.
Rieger
, and C. G.
De Kruif
, “Depletion-induced phase separation in colloid–polymer mixtures
,” Adv. Colloid Interface Sci.
103
(1
), 1
–31
(2003
).52.
S.
Sacanna
, W. T. M.
Irvine
, P. M.
Chaikin
, and D. J.
Pine
, “Lock and key colloids
,” Nature
464
(7288
), 575
(2010
).53.
Y.
Wang
, Y.
Wang
, X.
Zheng
, G.-R.
Yi
, S.
Sacanna
, D. J.
Pine
, and M.
Weck
, “Three-dimensional lock and key colloids
,” J. Am. Chem. Soc.
136
(19
), 6866
–6869
(2014
).54.
E.
Eisenriegler
, A.
Bringer
, and R.
Maassen
, “Polymer depletion interaction of small mesoscopic particles: Effects beyond leading order and anisotropic particles
,” J. Chem. Phys.
118
(17
), 8093
–8105
(2003
).55.
E. A.
Lord
and A. L.
Mackay
, “Periodic minimal surfaces of cubic symmetry
,” Curr. Sci.
85
, 346
–362
(2003
).56.
R.
Roth
, R. H. H. G.
van Roij
, D.
Andrienko
, K. R.
Mecke
, and S.
Dietrich
, “Entropic torque
,” Phys. Rev. Lett.
89
(8
), 088301
(2002
).57.
M.
Adams
, Z.
Dogic
, S. L.
Keller
, and S.
Fraden
, “Entropically driven microphase transitions in mixtures of colloidal rods and spheres
,” Nature
393
(6683
), 349
(1998
).58.
G. H.
Koenderink
, G. A.
Vliegenthart
, S. G. J. M.
Kluijtmans
, A.
van Blaaderen
, A. P.
Philipse
, and H. N. W.
Lekkerkerker
, “Depletion-induced crystallization in colloidal rod-sphere mixtures
,” Langmuir
15
(14
), 4693
–4696
(1999
).59.
A.
Suzuki
, M.
Yamazaki
, and T.
Ito
, “Osmoelastic coupling in biological structures: Formation of parallel bundles of actin filaments in a crystalline-like structure caused by osmotic stress
,” Biochemistry
28
(15
), 6513
–6518
(1989
).60.
M.
Adams
and S.
Fraden
, “Phase behavior of mixtures of rods (tobacco mosaic virus) and spheres (polyethylene oxide, bovine serum albumin)
,” Biophys. J.
74
(1
), 669
–677
(1998
).61.
M. A.
Bates
and D.
Frenkel
, “Phase behavior of model mixtures of colloidal disks and polymers
,” Phys. Rev. E
62
(4
), 5225
(2000
).62.
Z.
Dogic
, K. R.
Purdy
, E.
Grelet
, M.
Adams
, and S.
Fraden
, “Isotropic-nematic phase transition in suspensions of filamentous virus and the neutral polymer dextran
,” Phys. Rev. E
69
(5
), 051702
(2004
).63.
S.
Belli
, M.
Dijkstra
, and R.
van Roij
, “Depletion-induced biaxial nematic states of boardlike particles
,” J. Phys.: Condens. Matter
24
(28
), 284128
(2012
).64.
R.
Aliabadi
, M.
Moradi
, and S.
Varga
, “Tracking three-phase coexistences in binary mixtures of hard plates and spheres
,” J. Chem. Phys.
144
(7
), 074902
(2016
).65.
Á. G.
García
, J.
Opdam
, and R.
Tuinier
, “Phase behaviour of colloidal superballs mixed with non-adsorbing polymers
,” Euro. Phys. J. E
41
(9
), 110
(2018
).66.
Á. G.
García
, R.
Tuinier
, J. V.
Maring
, J.
Opdam
, H. H.
Wensink
, and H. N. W.
Lekkerkerker
, “Depletion-driven four-phase coexistences in discotic systems
,” Mol. Phys.
116
, 2757
(2018
).67.
S. S.
Cohen
, “The isolation and crystallization of plant viruses and other protein macro molecules by means of hydrophilic colloids
,” J. Biol. Chem.
144
(2
), 353
–362
(1942
); ; available at https://www.jbc.org/content/144/2/353.full.pdf.68.
T. G.
Mason
, “Osmotically driven shape-dependent colloidal separations
,” Phys. Rev. E
66
(6
), 060402
(2002
).69.
D.
Baranov
, A.
Fiore
, M.
van Huis
, C.
Giannini
, A.
Falqui
, U.
Lafont
, H.
Zandbergen
, M.
Zanella
, R.
Cingolani
, and L.
Manna
, “Assembly of colloidal semiconductor nanorods in solution by depletion attraction
,” Nano Lett.
10
(2
), 743
–749
(2010
).70.
K.
Park
, H.
Koerner
, and R. A.
Vaia
, “Depletion-induced shape and size selection of gold nanoparticles
,” Nano Lett.
10
(4
), 1433
–1439
(2010
).71.
C. H.
Bennett
, “Efficient estimation of free energy differences from Monte Carlo data
,” J. Comput. Phys.
22
(2
), 245
–268
(1976
).72.
M. P.
Allen
and D. J.
Tildesley
, Computer Simulation of Liquids
(Oxford University Press
, 1991
).73.
W.
Li
and H. R.
Ma
, “Depletion potential near curved surfaces
,” Phys. Rev. E
66
(6
), 061407
(2002
).74.
F.
Wang
and D. P.
Landau
, “Efficient, multiple-range random walk algorithm to calculate the density of states
,” Phys. Rev. Lett.
86
(10
), 2050
(2001
).75.
M. S.
Shell
, P. G.
Debenedetti
, and A. Z.
Panagiotopoulos
, “Generalization of the Wang-Landau method for off-lattice simulations
,” Phys. Rev. E
66
(5
), 056703
(2002
).76.
H.
Miao
, Y.
Li
, and H.
Ma
, “Depletion interaction between two ellipsoids
,” J. Chem. Phys.
140
(15
), 154904
(2014
).77.
H. W.
Hatch
, W. P.
Krekelberg
, S. D.
Hudson
, and V. K.
Shen
, “Depletion-driven crystallization of cubic colloids sedimented on a surface
,” J. Chem. Phys.
144
(19
), 194902
(2016
).78.
Z.
Jin
and J.
Wu
, “Hybrid MC–DFT method for studying multidimensional entropic forces
,” J. Phys. Chem. B
115
(6
), 1450
–1460
(2011
).79.
P.-M.
König
, R.
Roth
, and S.
Dietrich
, “Depletion forces between nonspherical objects
,” Phys. Rev. E
74
(4
), 041404
(2006
).80.
M.
Kinoshita
, “Interaction between big bodies with high asphericity immersed in small spheres
,” Chem. Phys. Lett.
387
(1-3
), 47
–53
(2004
).81.
P. W. A.
Schönhöfer
, G. E.
Schröder-Turk
, and M.
Marechal
, “Density functional theory for hard uniaxial particles: Complex ordering of pear-shaped and spheroidal particles near a substrate
,” J. Chem. Phys.
148
(12
), 124104
(2018
).82.
T. C.
Hudson
, M. C.
Lin
, J.
Cohen
, S.
Gottschalk
, and D.
Manocha
, “V-COLLIDE: Accelerated collision detection for VRML
,” in Proceedings of the Second Symposium on Virtual Reality Modeling Language–VRML’97
(ACM
, 1997
), p. 117–ff
.83.
P.
Jiménez
, F.
Thomas
, and C.
Torras
, “3D collision detection: A survey
,” Comput. Graphics
25
(2
), 269
–285
(2001
).84.
A.
Perera
, “Fluids of hard natural and Gaussian ellipsoids: A comparative study by integral equation theories
,” J. Chem. Phys.
129
(19
), 194504
(2008
).85.
P.
Ballone
, G.
Pastore
, G.
Galli
, and D.
Gazzillo
, “Additive and non-additive hard sphere mixtures: Monte Carlo simulation and integral equation results
,” Mol. Phys.
59
(2
), 275
–290
(1986
).86.
E.
Lomba
, M.
Alvarez
, L. L.
Lee
, and N. G.
Almarza
, “Phase stability of binary non-additive hard-sphere mixtures: A self-consistent integral equation study
,” J. Chem. Phys.
104
(11
), 4180
–4188
(1996
).87.
R.
Roth
and R.
Evans
, “The depletion potential in non-additive hard-sphere mixtures
,” Europhys. Lett.
53
(2
), 271
(2001
).88.
P.
Hopkins
and M.
Schmidt
, “Binary non-additive hard sphere mixtures: Fluid demixing, asymptotic decay of correlations and free fluid interfaces
,” J. Phys.: Condens. Matter
22
(32
), 325108
(2010
).89.
K.
Zhang
, M.
Fan
, Y.
Liu
, J.
Schroers
, M. D.
Shattuck
, and C. S.
O’Hern
, “Beyond packing of hard spheres: The effects of core softness, non-additivity, intermediate-range repulsion, and many-body interactions on the glass-forming ability of bulk metallic glasses
,” J. Chem. Phys.
143
(18
), 184502
(2015
).90.
W.
Kob
and H. C.
Andersen
, “Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture I: The van Hove correlation function
,” Phys. Rev. E
51
(5
), 4626
(1995
).91.
H. W.
Sheng
, M. J.
Kramer
, A.
Cadien
, T.
Fujita
, and M. W.
Chen
, “Highly optimized embedded-atom-method potentials for fourteen fcc metals
,” Phys. Rev. B
83
(13
), 134118
(2011
).92.
M.
Matsuoka
, “Solid liquid equilibria of binary organic mixtures
,” Bunri Gijutsu (Separation Process Engineering)
7
, 245
–249
(1977
).93.
S.
Punnathanam
and P. A.
Monson
, “Crystal nucleation in binary hard sphere mixtures: A Monte Carlo simulation study
,” J. Chem. Phys.
125
(2
), 024508
(2006
).94.
K.
Zhao
and T. G.
Mason
, “Directing colloidal self-assembly through roughness-controlled depletion attractions
,” Phys. Rev. Lett.
99
(26
), 268301
(2007
).95.
D. J.
Kraft
, R.
Ni
, F.
Smallenburg
, M.
Hermes
, K.
Yoon
, D. A.
Weitz
, A.
van Blaaderen
, J.
Groenewold
, M.
Dijkstra
, and W. K.
Kegel
, “Surface roughness directed self-assembly of patchy particles into colloidal micelles
,” Proc. Natl. Acad. Sci. U. S. A.
109
(27
), 10787
–10792
(2012
).96.
See https://www.sidefx.com for SideFX, Houdini 17: Procedural content creation tools for film, TV and gamedev; accessed
2018
.97.
Note that these “overlaps” do not enable the pear-shaped particles to invade the space occupied by other pears according to the PHGO potential. The interactions are governed by a hard-core potential.
98.
In these theories, every single configuration has to be treated individually to calculate depletion interactions.
99.
Note that the MC dynamics described here do, of course, not represent true particle dynamics or trajectories.
100.
Here, the term “overlap” might be misleading as the particles do not technically overlap in terms of their PHGO contact function but according to the best possible illustration using the Bézier representation. However, it also has to be mentioned that the spheres interact with the pear according to this Bézier shape. Thus, the solvent particles interact with the PHGO particles in terms of a different effective shape than two PHGO particles with each other.
© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.