Shape-sensitive crystallization in colloidal superball fluids
- aJames Franck institute, Department of Physics, University of Chicago, Chicago, IL 60637;
- bVan’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute for Nano-materials Science, Utrecht University, 3584 CH Utrecht, The Netherlands;
- cSoft Condensed Matter, Debye Institute for Nano-materials Science, Utrecht University, 3584 CC Utrecht, The Netherlands; and
- dCenter for Soft Matter Research, Department of Physics, and
- eMolecular Design Institute, Department of Chemistry, New York University, New York, NY 10003
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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved March 16, 2015 (received for review August 11, 2014)

Significance
Since antiquity it has been known that particle shape plays an essential role in the symmetry and structure of matter. A familiar example comes from dense packings, such as spheres arranged in a face-centered cubic lattice. For colloidal superballs, we observe the transition from hexagonal to rhombic crystals consistent with the densest packings. In addition, we see the existence of square structures promoted by the presence of depletion attractions in the colloidal system. By using a mixture of depletants, one of which is size tunable, we induce solid-to-solid phase transitions between these phases. Our results introduce a general scenario where particle building blocks are designed to assemble not only into their maximum density states, but also into depletion-tunable interaction-dependent structures.
Abstract
Guiding the self-assembly of materials by controlling the shape of the individual particle constituents is a powerful approach to material design. We show that colloidal silica superballs crystallize into canted phases in the presence of depletants. Some of these phases are consistent with the so-called “
Determining the relationship between the macroscopic structure of a material and the properties of its microscopic constituents is a fundamental problem in condensed matter science. A particularly interesting aspect of this problem is to understand how the self-assembly of a collection of particles is determined by their shape. These so-called “packing problems” have long interested physicists, mathematicians, and chemists alike and have been used to understand the structures of many condensed phases of matter (1⇓–3). Computational and experimental advances continue to enable new explorations into fundamental aspects of these problems today (4⇓⇓⇓⇓⇓⇓⇓⇓–13). Recent discoveries include dense packings of tetrahedra into disordered, crystalline, and quasi-crystalline structures (14, 15), as well as the singular dense packings of ellipsoids (16).
Technologically speaking, these discoveries are becoming increasingly crucial as new synthesis techniques are allowing for the creation of more and more complex shaped nanoscopic and microscopic particles (17, 18). The self-assembly of these particles into ordered structures creates new possibilities for the fabrication of novel materials (19⇓⇓⇓–23). Moreover, advances in synthesis techniques have created new capabilities for experimentally investigating how the shapes of particles can be exploited in their self-assembly (24⇓–26).
Here, we experimentally and computationally explore the self-assembly of colloidal superballs interacting with depletion forces. We find that monolayers of superballs can be tuned to equilibrate into both their densest known packings—so-called “
(A) SEM images of sample m = 3.9. The particles have a cubic shape with rounded edges. (B–D) TEM micrographs of samples with m = 3.5, m = 3.0, and m = 2.0, respectively. All samples are uniform with a size polydispersity as low as 3%. (Scale bars: 1 μm.) In B the particles are shown with their corresponding superball fit highlighted in red (see also Fig. S1). (E, Top) Computer-generated models of colloidal superballs with different shape parameters m. A gradual increase of the absolute value of the shape parameter from
Fig. 1 shows SEM and TEM images of the silica superballs used for the experiments. Although the particles still possess a distinct cubic symmetry, they have rounded edges whose curvatures are consistent with superballs of shape parameters
To perform the experiments, silica superballs were dispersed in slightly alkaline water (pH = 9) and were stabilized against aggregation by surface charges. Sodium chloride (10 mM, final concentration) was added to the dispersion to screen the charges and lower the Debye length down to a thickness of about 3 nm, small enough to allow the particles to fully experience their anisotropic shape. Attractive forces between superballs arise by addition of depletion agents with gyration radii of
At low particle concentration, the superballs first sediment to the bottom of the capillary where they are attracted to the glass wall by depletion forces. While diffusing in the plane, the particles cluster together into monolayers. Once clusters are formed, time-lapsed images are collected and analyzed. The images show the appearance of several qualitatively distinct phases (Fig. 2). The particles are found to arrange into crystallite islands, often possessing grain boundaries, which we separate by orientation and analyze independently. We do not exclude a priori the possibility that a cluster does not have a coherent crystal structure.
Representative optical microscope images showing three different ordered structures found in superball samples. (A–C, Right) Histogram of the relative positions of nearest neighbors for each particle in a crystallite (Top) and a histogram of the interparticle bond angles (Bottom) (see also Fig. S2). The structures of the crystallites are characterized by bond angels of
To characterize the structure of each cluster, the positions of the constituent particles are identified for every time-lapsed image. The relative positions of nearest neighbors are then computed for each particle. For spherical superballs the distribution of these positions are found to be consistent with triangular lattices (Fig. 2C). For superballs with intermediate shape parameters (
To understand this interplay, we look at the depletion interactions between the superballs. Each superball is surrounded by an exclusion zone of thickness
Two-dimensional predicted diagram for depletion-stabilized superball phases. The favorability of each lattice type is determined by calculating the bound state energy of a particle. (A) Operationally, the bound state energy is found by computing the difference in the excluded volume for a particular lattice (A, i) and the excluded volume of that lattice when a particle is removed from the interior (A, ii). (B) Change in excluded volume for each lattice type with varying m but fixed q = 2Rg/L, where Rg is the radius of gyration of the depletant and L is the diameter of the superball. To illustrate the behavior of
Comparison between experimental observations, bulk crystal simulations, and calculated phase diagram for superballs at different m and q values. Circles indicate the experimental results, open circles indicate simulation results, and the background color indicates the predicted phase. The approximated phase diagram qualitatively agrees with our experimental and simulation results.
Indeed, the calculations agree with the experimental result that for sufficiently small depletants and sufficiently large m, square lattices, although they are not the densest packings for any finite value of m, are preferred. Square lattices occur when m is large enough such that the overlap in exclusion zones resulting from face-to-face contact is considerable and for q small enough such that depletants are able to fit into the interparticle pores made where the rounded edges of the superballs meet. When the osmotic pressure exerted by a depletant within an interparticle pore is substantial, the cubic phase is stabilized. However, when intermediate-sized depletants, which can no longer fit into the spaces within the lattice, are dispersed with superballs possessing these larger values of m (3.5 and 3.9), the densely packed
As mentioned previously, spherical superballs form triangular lattices, which are equivalent to the
To more carefully probe the stability of our observed lattices, we perform idealized simulations of superballs and depletants (SI Text and Fig. S4). We first simulate finite crystallites and find the results qualitatively agree with experiments (Figs. S5 and S6). A particular choice of initial conditions, however, may influence the vulnerability of the resulting assembly to fall into kinetic traps. To probe the true stability of our candidate lattices, and to remove surface effects that exist in finite crystallites, we perform bulk crystal simulations, using periodic boundary conditions of each candidate lattice (Fig. S7). Fig. 4 shows the resulting stable lattices determined from these simulations. The results qualitatively agree with our excluded volume calculations. It is interesting to note that near phase boundaries both
It is particularly interesting to note that for both experiment and simulation, we identify different crystalline structures as q is varied for m
To drive this transition, we use superballs with shape parameter
Using a mixture of the two depletants, however, allows us to reversibly switch between the two lattice types by varying the temperature. At room temperature, the interactions induced by the pNIPAM are activated, and the superballs once again favor a square lattice. As the temperature is increased, the relative energetic contribution of the pNIPAM depletant decreases, while the contribution of the PEO remains the same. Because the PEO dominates the overall energy at high temperatures, the
Demonstration of reversible solid–solid phase transition of superballs. (A) Colloidal superballs with shape parameter m = 3.9 dispersed in depletant mixture of PEO and pNIPAM. At 27.5 °C, superballs assemble into a square lattice. At 31 °C, energetic contribution of pNIPAM becomes negligible, while that of PEO stays fixed, resulting in the transition into a
By performing simulations of bidepletant superball dispersions we provide further evidence of the simple entropic nature of the geometric mechanism that induces this solid–solid transition. Again we perform periodic simulations of a bulk crystal as well as simulations of finite crystallites. Superballs are dispersed with two species of depletant, one with fixed size ratio
In this article we have demonstrated the reversible assembly of the same superball-shaped colloidal particles into both a square phase and the recently predicted
Acknowledgments
Prof. Nigel B. Wilding, Dr. Krassimir P. Velikov, Dr. Andrei Petukhov, Prof. Willem Kegel, Dr. Ra Ni, Dr. Frank Smallenburg, Theodore Hueckel, Prof. S. R. Nagel, and Prof. T. Witten are thanked for many useful discussions. Prof. Dirk Aarts is thanked for kindly providing the Xanthan polymer. We acknowledge the Materials Research and Engineering Centers (MRSEC) Shared Facilities at The University of Chicago for the use of their instruments, and NSF MRI 1229456. This work was supported by the National Science Foundation MRSEC Program at The University of Chicago (NSF DMR-MRSEC 1420709). W.T.M.I. further acknowledges support from the A. P. Sloan Foundation through a Sloan fellowship, and the Packard Foundation through a Packard fellowship. L.R. and A.P.P. acknowledge Agentschap NL for financial support through Grant FND07002. Engineering and Physical Sciences Research Council (EPSRC) is acknowledged for support to D.J.A. through Grant EP/I036192/1. P.M.C. acknowledges support from NASA (NNX08AK04G).
Footnotes
↵1Present address: Institute of Physics, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands.
↵2L.R., V.S., and D.J.A. contributed equally to this work.
- ↵3To whom correspondence may be addressed. Email: L.Rossi{at}uva.nl, soni{at}uchicago.edu, or wtmirvine{at}uchicago.edu.
Author contributions: L.R., V.S., A.P.P., P.M.C., S.S., and W.T.M.I. designed research; L.R., V.S., D.J.A., M.D., S.S., and W.T.M.I. performed research; L.R., V.S., D.J.A., P.M.C., M.D., and W.T.M.I. analyzed data; and L.R., V.S., D.J.A., D.J.P., A.P.P., P.M.C., M.D., S.S., and W.T.M.I. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1415467112/-/DCSupplemental.
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