The role of particle shape in self-assembly processes is a double-edged sword. On the one hand, particle shape and particle elongation are often considered the most fundamental determinants of soft matter structure formation. On the other hand, structure formation is often highly sensitive to details of shape. Here, we address the question of particle shape sensitivity for the self-assembly of hard pear-shaped particles by studying two models for this system: (a) the pear hard Gaussian overlap (PHGO) and (b) the hard pears of revolution (HPR) model. Hard pear-shaped particles, given by the PHGO model, are known to form a bicontinuous gyroid phase spontaneously. However, this model does not replicate an additive object perfectly and, hence, varies slightly in shape from a “true” pear-shape. Therefore, we investigate in the first part of this series the stability of the gyroid phase in pear-shaped particle systems. We show, based on the HPR phase diagram, that the gyroid phase does not form in pears with such a “true” hard pear-shaped potential. Moreover, we acquire first indications from the HPR and PHGO pair-correlation functions that the formation of the gyroid is probably attributed to the small non-additive properties of the PHGO potential.
REFERENCES
1.
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
).2.
D.
Frenkel
, B. M.
Mulder
, and J. P.
McTague
, “Phase diagram of a system of hard ellipsoids
,” Phys. Rev. Lett.
52
(4
), 287
(1984
).3.
A.
Haji-Akbari
, M.
Engel
, and S. C.
Glotzer
, “Phase diagram of hard tetrahedra
,” J. Chem. Phys.
135
(19
), 194101
(2011
).4.
R.
Ni
, A. P.
Gantapara
, J.
de Graaf
, R.
van Roij
, and M.
Dijkstra
, “Phase diagram of colloidal hard superballs: From cubes via spheres to octahedra
,” Soft Matter
8
(34
), 8826
–8834
(2012
).5.
P. F.
Damasceno
, M.
Engel
, and S. C.
Glotzer
, “Crystalline assemblies and densest packings of a family of truncated tetrahedra and the role of directional entropic forces
,” ACS Nano
6
(1
), 609
–614
(2011
).6.
P. F.
Damasceno
, M.
Engel
, and S. C.
Glotzer
, “Predictive self-assembly of polyhedra into complex structures
,” Science
337
(6093
), 453
–457
(2012
).7.
A. P.
Gantapara
, J.
de Graaf
, R.
van Roij
, and M.
Dijkstra
, “Phase diagram and structural diversity of a family of truncated cubes: Degenerate close-packed structures and vacancy-rich states
,” Phys. Rev. Lett.
111
(1
), 015501
(2013
).8.
C. X.
Du
, G.
van Anders
, R. S.
Newman
, and S. C.
Glotzer
, “Shape-driven solid–solid transitions in colloids
,” Proc. Natl. Acad. Sci. U. S. A.
114
(20
), E3892
–E3899
(2017
).9.
D.
Klotsa
, E. R.
Chen
, M.
Engel
, and S. C.
Glotzer
, “Intermediate crystalline structures of colloids in shape space
,” Soft Matter
14
(43
), 8692
–8697
(2018
).10.
P.
Bolhuis
and D.
Frenkel
, “Tracing the phase boundaries of hard spherocylinders
,” J. Chem. Phys.
106
(2
), 666
–687
(1997
).11.
B. S.
John
, C.
Juhlin
, and F. A.
Escobedo
, “Phase behavior of colloidal hard perfect tetragonal parallelepipeds
,” J. Chem. Phys.
128
(4
), 044909
(2008
).12.
M.
Marechal
, S.
Dussi
, and M.
Dijkstra
, “Density functional theory and simulations of colloidal triangular prisms
,” J. Chem. Phys.
146
(12
), 124905
(2017
).13.
The standard deviation of the diameter distribution has to be with the mean diameter .
14.
P.
Bartlett
and P. B.
Warren
, “Reentrant melting in polydispersed hard spheres
,” Phys. Rev. Lett.
82
(9
), 1979
(1999
).15.
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
).16.
M. A.
Bates
and D.
Frenkel
, “Phase behavior of model mixtures of colloidal disks and polymers
,” Phys. Rev. E
62
(4
), 5225
(2000
).17.
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
).18.
S.
Belli
, M.
Dijkstra
, and R.
van Roij
, “Depletion-induced biaxial nematic states of boardlike particles
,” J. Phys.: Condens. Matter
24
(28
), 284128
(2012
).19.
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
).20.
Á. G.
García
, J.
Opdam
, and R.
Tuinier
, “Phase behaviour of colloidal superballs mixed with non-adsorbing polymers
,” Eur. Phys. J. E
41
(9
), 110
(2018
).21.
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
(21
-22
), 2757
–2772
(2018
).22.
P. W. A.
Schönhöfer
, M.
Marechal
, D. J.
Cleaver
, and G. E.
Schröder-Turk
, “Self-assembly and entropic effects in pear-shaped colloid systems: II. Depletion attraction of pear-shaped particles in a hard sphere solvent
,” J. Chem. Phys.
153
, 034904
(2020
).23.
R. D.
Batten
, F. H.
Stillinger
, and S.
Torquato
, “Phase behavior of colloidal superballs: Shape interpolation from spheres to cubes
,” Phys. Rev. E
81
(6
), 061105
(2010
).24.
Y.
Zhang
, F.
Lu
, D.
van der Lelie
, and O.
Gang
, “Continuous phase transformation in nanocube assemblies
,” Phys. Rev. Lett.
107
(13
), 135701
(2011
).25.
L.
Rossi
, V.
Soni
, D. J.
Ashton
, D. J.
Pine
, A. P.
Philipse
, P. M.
Chaikin
, M.
Dijkstra
, S.
Sacanna
, and W. T. M.
Irvine
, “Shape-sensitive crystallization in colloidal superball fluids
,” Proc. Natl. Acad. Sci. U. S. A.
112
(17
), 5286
–5290
(2015
).26.
S.
Dussi
and M.
Dijkstra
, “Entropy-driven formation of chiral nematic phases by computer simulations
,” Nat. Commun.
7
, 11175
(2016
).27.
M.
Marechal
and M.
Dijkstra
, “Phase behavior and structure of colloidal bowl-shaped particles: Simulations
,” Phys. Rev. E
82
(3
), 031405
(2010
).28.
D.
Wan
, C. X.
Du
, G.
van Anders
, and S. C.
Glotzer
, “FCC-to-BCC phase transitions in convex and concave hard particle systems
,” J. Phys. Chem. B
123
(42
), 9038
–9043
(2019
).29.
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
).30.
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
).31.
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
).32.
F.
Barmes
, M.
Ricci
, C.
Zannoni
, and D. J.
Cleaver
, “Computer simulations of hard pear-shaped particles
,” Phys. Rev. E
68
, 021708
(2003
).33.
A.
Perera
, “Fluids of hard natural and Gaussian ellipsoids: A comparative study by integral equation theories
,” J. Chem. Phys.
129
(19
), 194504
(2008
).34.
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
).35.
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
).36.
R.
Roth
and R.
Evans
, “The depletion potential in non-additive hard-sphere mixtures
,” Europhys. Lett.
53
(2
), 271
(2001
).37.
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
).38.
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
).39.
Additional overlap rules (like adding non-additive features to the blunt ends) are required to imitate the interactions between PHGO particles with physical hard shapes.
40.
The parameter σw indicates the width of the pear-shaped particles.
41.
G.
Odriozola
, “Revisiting the phase diagram of hard ellipsoids
,” J. Chem. Phys.
136
(13
), 134505
(2012
).42.
Note that the “normalization” factor in this case indicates that g(r) converges toward 1 for large distances: limr→∞g(r) = 1.
43.
This only applies to the smectic-A phase. For other smectic phases, it is still more convenient to use the director as a reference.
44.
Note here, that r∥ can become negative. For pear-shaped particles, positive longitudinal distances correspond to a distance in the direction of the thin narrow end, while negative distances have to be assigned to particles, which are placed in the direction of the thick blunt end.
45.
Note that the functions do not contain information about the likeliness of such configurations occurring.
46.
M.
Dennison
, K.
Milinković
, and M.
Dijkstra
, “Phase diagram of hard snowman-shaped particles
,” J. Chem. Phys.
137
(4
), 044507
(2012
).47.
K.
Milinković
, M.
Dennison
, and M.
Dijkstra
, “Phase diagram of hard asymmetric dumbbell particles
,” Phys. Rev. E
87
(3
), 032128
(2013
).48.
J.
van den Hoven
, “Phase behaviour of conical colloids
,” M.Sc. thesis, Universiteit Utrecht
, 2016
.49.
D.
Anderson
, H.
Wennerstroem
, and U.
Olsson
, “Isotropic bicontinuous solutions in surfactant-solvent systems: The L3 phase
,” J. Phys. Chem.
93
(10
), 4243
–4253
(1989
).50.
S.
Engström
, K.
Alfons
, M.
Rasmusson
, and H.
Ljusberg-Wahren
, “Solvent-induced sponge (L3) phases in the solvent-monoolein-water system
,” Prog. Colloid Polym. Sci.
108
, 93
–98
(1998
).51.
H.
Evertsson
, P.
Stilbs
, G.
Lindblom
, and S.
Engström
, “NMR self diffusion measurements of the monooleoylglycerol/poly ethylene glycol/water L3 phase
,” Colloids Surf., B
26
(1-2
), 21
–29
(2002
).52.
V.
Cherezov
, J.
Clogston
, M. Z.
Papiz
, and M.
Caffrey
, “Room to move: Crystallizing membrane proteins in swollen lipidic mesophases
,” J. Mol. Biol.
357
(5
), 1605
–1618
(2006
).53.
S.
Abe
and H.
Takahashi
, “A comparative study of the effects of dimethylsulfoxide and glycerol on the bicontinuous cubic structure of hydrated monoolein and its phase behavior
,” Chem. Phys. Lipids
147
(2
), 59
–68
(2007
).54.
A. B.
Wöhri
, L. C.
Johansson
, P.
Wadsten-Hindrichsen
, W. Y.
Wahlgren
, G.
Fischer
, R.
Horsefield
, G.
Katona
, M.
Nyblom
, F.
Öberg
, G.
Young
, R. J.
Cogdell
, N. J.
Fraser
, S.
Engström
, and R.
Neutze
, “A lipidic-sponge phase screen for membrane protein crystallization
,” Structure
16
(7
), 1003
–1009
(2008
).55.
T.
Landh
, “Phase behavior in the system pine needle oil monoglycerides-poloxamer 407-water at 20 degree
,” J. Phys. Chem.
98
(34
), 8453
–8467
(1994
).56.
J.
Barauskas
, A.
Misiunas
, T.
Gunnarsson
, F.
Tiberg
, and M.
Johnsson
, ““sponge” nanoparticle dispersions in aqueous mixtures of diglycerol monooleate, glycerol dioleate, and polysorbate 80
,” Langmuir
22
(14
), 6328
–6334
(2006
).57.
R.
Iñiguez-Palomares
, H.
Acuña-Campa
, and A.
Maldonado
, “Effect of polymer on the elasticity of surfactant membranes: A light scattering study
,” Phys. Rev. E
84
(1
), 011604
(2011
).58.
M.
Valldeperas
, M.
Wiśniewska
, M.
Ram-On
, E.
Kesselman
, D.
Danino
, T.
Nylander
, and J.
Barauskas
, “Sponge phases and nanoparticle dispersions in aqueous mixtures of mono- and diglycerides
,” Langmuir
32
(34
), 8650
–8659
(2016
).59.
P. N.
Pusey
and W.
Van Megen
, “Phase behaviour of concentrated suspensions of nearly hard colloidal spheres
,” Nature
320
(6060
), 340
(1986
).60.
P. N.
Pusey
and W.
van Megen
, in Physics of Complex and Supramolecular Fluids
, edited by S. A.
Safran
and N. A.
Clark
(Wiley
, New York
, 1987
), pp. 673
–698
.61.
W. C. K.
Poon
, E. R.
Weeks
, and C. P.
Royall
, “On measuring colloidal volume fractions
,” Soft Matter
8
(1
), 21
–30
(2012
).62.
C. P.
Royall
, W. C. K.
Poon
, and E. R.
Weeks
, “In search of colloidal hard spheres
,” Soft Matter
9
(1
), 17
–27
(2013
).63.
R. A.
LaCour
, C. S.
Adorf
, J.
Dshemuchadse
, and S. C.
Glotzer
, “Influence of softness on the stability of binary colloidal crystals
,” ACS Nano
13
(12
), 13829
–13842
(2019
).64.
J. P.
Rolland
, B. W.
Maynor
, L. E.
Euliss
, A. E.
Exner
, G. M.
Denison
, and J. M.
DeSimone
, “Direct fabrication and harvesting of monodisperse, shape-specific nanobiomaterials
,” J. Am. Chem. Soc.
127
(28
), 10096
–10100
(2005
).65.
S. H.
Lee
, S. J.
Gerbode
, B. S.
John
, A. K.
Wolfgang
, F. A.
Escobedo
, I.
Cohen
, and C. M.
Liddell
, “Synthesis and assembly of nonspherical hollow silica colloids under confinement
,” J. Mater. Chem.
18
(41
), 4912
–4916
(2008
).66.
L.
Chen
, H. Z.
An
, and P. S.
Doyle
, “Synthesis of nonspherical microcapsules through controlled polyelectrolyte coating of hydrogel templates
,” Langmuir
31
(33
), 9228
–9235
(2015
).67.
J.
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
).68.
W. K.
Kegel
, D.
Breed
, M.
Elsesser
, and D. J.
Pine
, “Formation of anisotropic polymer colloids by disparate relaxation times
,” Langmuir
22
(17
), 7135
–7136
(2006
).69.
J.-G.
Park
, J. D.
Forster
, and E. R.
Dufresne
, “Synthesis of colloidal particles with the symmetry of water molecules
,” Langmuir
25
(16
), 8903
–8906
(2009
).70.
A.
Perro
, E.
Duguet
, O.
Lambert
, J.-C.
Taveau
, E.
Bourgeat-Lami
, and S.
Ravaine
, “A chemical synthetic route towards ‘colloidal molecules
,’” Angew. Chem.
121
(2
), 367
–371
(2009
).71.
B.
Peng
and A.
Imhof
, “Surface morphology control of cross-linked polymer particles via dispersion polymerization
,” Soft Matter
11
(18
), 3589
–3598
(2015
).72.
J.-B.
Fan
, H.
Liu
, Y.
Song
, Z.
Luo
, Z.
Lu
, and S.
Wang
, “Janus particles synthesis by emulsion interfacial polymerization: Polystyrene as seed or beyond?
,” Macromolecules
51
(5
), 1591
–1597
(2018
).73.
I.
Lesov
, Z.
Valkova
, E.
Vassileva
, G. S.
Georgiev
, K.
Ruseva
, M.
Simeonov
, S.
Tcholakova
, N. D.
Denkov
, and S. K.
Smoukov
, “Bottom-up synthesis of polymeric micro-and nanoparticles with regular anisotropic shapes
,” Macromolecules
51
(19
), 7456
–7462
(2018
).74.
C. I.
Zoldesi
, C. A.
van Walree
, and A.
Imhof
, “Deformable hollow hybrid silica/siloxane colloids by emulsion templating
,” Langmuir
22
(9
), 4343
–4352
(2006
).75.
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
).76.
H.
Shin
and C.
Kim
, “Preparation of spheroidal and ellipsoidal particles from spherical polymer particles by extension of polymer film
,” Colloid Polym. Sci.
290
(13
), 1309
–1315
(2012
).77.
Y.
Yin
, Y.
Lu
, B.
Gates
, and Y.
Xia
, “Template-assisted self-assembly: A practical route to complex aggregates of monodispersed colloids with well-defined sizes, shapes, and structures
,” J. Am. Chem. Soc.
123
(36
), 8718
–8729
(2001
).78.
D.
Dendukuri
, D. C.
Pregibon
, J.
Collins
, T. A.
Hatton
, and P. S.
Doyle
, “Continuous-flow lithography for high-throughput microparticle synthesis
,” Nat. Mater.
5
, 365
–367
(2006
).79.
J.
Guan
, N.
Ferrell
, L. J.
Lee
, and D. J.
Hansford
, “Fabrication of polymeric microparticles for drug delivery by soft lithography
,” Biomaterials
27
(21
), 4034
–4041
(2006
).80.
G. C.
Le Goff
, J.
Lee
, A.
Gupta
, W. A.
Hill
, and P. S.
Doyle
, “High-throughput contact flow lithography
,” Adv. Sci.
2
(10
), 1500149
(2015
).81.
H.
Kruggel-Emden
, S.
Rickelt
, S.
Wirtz
, and V.
Scherer
, “A study on the validity of the multi-sphere discrete element method
,” Powder Technol.
188
(2
), 153
–165
(2008
).82.
D.
Markauskas
, R.
Kačianauskas
, A.
Džiugys
, and R.
Navakas
, “Investigation of adequacy of multi-sphere approximation of elliptical particles for DEM simulations
,” Granular Matter
12
(1
), 107
–123
(2010
).83.
D.
Höhner
, S.
Wirtz
, H.
Kruggel-Emden
, and V.
Scherer
, “Comparison of the multi-sphere and polyhedral approach to simulate non-spherical particles within the discrete element method: Influence on temporal force evolution for multiple contacts
,” Powder Technol.
208
(3
), 643
–656
(2011
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.