Greg van Anders

Understanding shape entropy through local dense packing

Significance Many natural systems are structured by the ordering of repeated, distinct shapes. Understanding how this happens is difficult because shape affects structure in two ways. One is how the shape of a cell or nanoparticle, for example, affects its surface, chemical, or other intrinsic properties. The other is an emergent, entropic effect that arises from the geometry of the shape itself, which we term “shape entropy,” and is not well understood. In this paper, we determine how shape entropy affects structure. We quantify the mechanism and determine when shape entropy competes with intrinsic shape effects. Our results show that in a wide class of systems, shape affects bulk structure because crowded particles optimize their local packing. Entropy drives the phase behavior of colloids ranging from dense suspensions of hard spheres or rods to dilute suspensions of hard spheres and depletants. Entropic ordering of anisotropic shapes into complex crystals, liquid crystals, and even quasicrystals was demonstrated recently in computer simulations and experiments. The ordering of shapes appears to arise from the emergence of directional entropic forces (DEFs) that align neighboring particles, but these forces have been neither rigorously defined nor quantified in generic systems. Here, we show quantitatively that shape drives the phase behavior of systems of anisotropic particles upon crowding through DEFs. We define DEFs in generic systems and compute them for several hard particle systems. We show they are on the order of a few times the thermal energy (kBT) at the onset of ordering, placing DEFs on par with traditional depletion, van der Waals, and other intrinsic interactions. In experimental systems with these other interactions, we provide direct quantitative evidence that entropic effects of shape also contribute to self-assembly. We use DEFs to draw a distinction between self-assembly and packing behavior. We show that the mechanism that generates directional entropic forces is the maximization of entropy by optimizing local particle packing. We show that this mechanism occurs in a wide class of systems and we treat, in a unified way, the entropy-driven phase behavior of arbitrary shapes, incorporating the well-known works of Kirkwood, Onsager, and Asakura and Oosawa.

Biomimetic Hierarchical Assembly of Helical Supraparticles from Chiral Nanoparticles

Chiroptical materials found in butterflies, beetles, stomatopod crustaceans, and other creatures are attributed to biocomposites with helical motifs and multiscale hierarchical organization. These structurally sophisticated materials self-assemble from primitive nanoscale building blocks, a process that is simpler and more energy efficient than many top-down methods currently used to produce similarly sized three-dimensional materials. Here, we report that molecular-scale chirality of a CdTe nanoparticle surface can be translated to nanoscale helical assemblies, leading to chiroptical activity in the visible electromagnetic range. Chiral CdTe nanoparticles coated with cysteine self-organize around Te cores to produce helical supraparticles. D-/L-Form of the amino acid determines the dominant left/right helicity of the supraparticles. Coarse-grained molecular dynamics simulations with a helical pair-potential confirm the assembly mechanism and the origin of its enantioselectivity, providing a framework for engineering three-dimensional chiral materials by self-assembly. The helical supraparticles further self-organize into lamellar crystals with liquid crystalline order, demonstrating the possibility of hierarchical organization and with multiple structural motifs and length scales determined by molecular-scale asymmetry of nanoparticle interactions.

Self-assembly of archimedean tilings with enthalpically and entropically patchy polygons

Considerable progress in the synthesis of anisotropic patchy nanoplates (nanoplatelets) promises a rich variety of highly ordered two-dimensional superlattices. Recent experiments of superlattices assembled from nanoplates confirm the accessibility of exotic phases and motivate the need for a better understanding of the underlying self-assembly mechanisms. Here, we present experimentally accessible, rational design rules for the self-assembly of the Archimedean tilings from polygonal nanoplates. The Archimedean tilings represent a model set of target patterns that (i) contain both simple and complex patterns, (ii) are comprised of simple regular shapes, and (iii) contain patterns with potentially interesting materials properties. Via Monte Carlo simulations, we propose a set of design rules with general applicability to one- and two-component systems of polygons. These design rules, specified by increasing levels of patchiness, correspond to a reduced set of anisotropy dimensions for robust self-assembly of the Archimedean tilings. We show for which tilings entropic patches alone are sufficient for assembly and when short-range enthalpic interactions are required. For the latter, we show how patchy these interactions should be for optimal yield. This study provides a minimal set of guidelines for the design of anisostropic patchy particles that can self-assemble all 11 Archimedean tilings.

Binding kinetics of lock and key colloids

Using confocal microscopy and first passage time analysis, we measure and predict the rates of formation and breakage of polymer-depletion-induced bonds between lock-and-key colloidal particles and find that an indirect route to bond formation is accessed at a rate comparable to that of the direct formation of these bonds. In the indirect route, the pocket of the lock particle is accessed by nonspecific bonding of the key particle with the lock surface, followed by surface diffusion leading to specific binding in the pocket of the lock. The surprisingly high rate of indirect binding is facilitated by its high entropy relative to that of the pocket. Rate constants for forward and reverse transitions among free, nonspecific, and specific bonds are reported, compared to theoretical values, and used to determine the free energy difference between the nonspecific and specific binding states.

First results from lattice simulation of the PWMM

We present results of lattice simulations of the Plane Wave Matrix Model (PWMM). The PWMM is a theory of supersymmetric quantum mechanics that has a well-defined canonical ensemble. We simulate this theory by applying rational hybrid Monte Carlo techniques to a naïve lattice action. We examine the strong coupling behaviour of the model focussing on the deconfinement transition.

Clusters of polyhedra in spherical confinement

Significance What is the best way to pack objects into a container? This simple question, one that is relevant to everyday life, biology, and nanoscience, is easy to state but surprisingly difficult to answer. Here, we use computational methods to determine dense packings of a set of polyhedra inside a sphere, for up to 60 constituent packers. Our dense packings display a wide variety of symmetries and structures, and indicate that the presence of the spherical container suppresses packing effects due to polyhedral shape. Our results have implications for a range of biological phenomena and experimental applications, including blood clotting, cell aggregation, drug delivery, colloidal engineering, and the creation of metamaterials. Dense particle packing in a confining volume remains a rich, largely unexplored problem, despite applications in blood clotting, plasmonics, industrial packaging and transport, colloidal molecule design, and information storage. Here, we report densest found clusters of the Platonic solids in spherical confinement, for up to N=60 constituent polyhedral particles. We examine the interplay between anisotropic particle shape and isotropic 3D confinement. Densest clusters exhibit a wide variety of symmetry point groups and form in up to three layers at higher N. For many N values, icosahedra and dodecahedra form clusters that resemble sphere clusters. These common structures are layers of optimal spherical codes in most cases, a surprising fact given the significant faceting of the icosahedron and dodecahedron. We also investigate cluster density as a function of N for each particle shape. We find that, in contrast to what happens in bulk, polyhedra often pack less densely than spheres. We also find especially dense clusters at so-called magic numbers of constituent particles. Our results showcase the structural diversity and experimental utility of families of solutions to the packing in confinement problem.

Shape-driven solid–solid transitions in colloids

Significance Despite the fundamental importance of solid–solid transitions for metallurgy, ceramics, earth science, reconfigurable materials, and colloidal matter, the details of how materials transform between two solid structures are poorly understood. We introduce a class of simple model systems in which the direct control of local order, via colloid shape change, induces solid–solid phase transitions and characterize how the transitions happen thermodynamically. We find that, within a single shape family, there are solid–solid transitions that can occur with or without a thermal activation barrier. Our results provide means for the study of solid–solid phase transitions and have implications for designing reconfigurable materials. Solid–solid phase transitions are the most ubiquitous in nature, and many technologies rely on them. However, studying them in detail is difficult because of the extreme conditions (high pressure/temperature) under which many such transitions occur and the high-resolution equipment needed to capture the intermediate states of the transformations. These difficulties mean that basic questions remain unanswered, such as whether so-called diffusionless solid–solid transitions, which have only local particle rearrangement, require thermal activation. Here, we introduce a family of minimal model systems that exhibits solid–solid phase transitions that are driven by changes in the shape of colloidal particles. By using particle shape as the control variable, we entropically reshape the coordination polyhedra of the particles in the system, a change that occurs indirectly in atomic solid–solid phase transitions via changes in temperature, pressure, or density. We carry out a detailed investigation of the thermodynamics of a series of isochoric, diffusionless solid–solid phase transitions within a single shape family and find both transitions that require thermal activation or are “discontinuous” and transitions that occur without thermal activation or are “continuous.” In the discontinuous case, we find that sufficiently large shape changes can drive reconfiguration on timescales comparable with those for self-assembly and without an intermediate fluid phase, and in the continuous case, solid–solid reconfiguration happens on shorter timescales than self-assembly, providing guidance for developing means of generating reconfigurable colloidal materials.

Comments on holographic Fermi surfaces

Recently, a mechanism for the development of a fermi surface in a holographic model of large N QCD with a baryon chemical potential was proposed. We examine similar constructions to determine when this mechanism persists. We find a class of models in which it does.

Relevance of packing to colloidal self-assembly

Significance Understanding how structural order forms in matter is a key challenge in designing materials. In the 1920s, Pauling proposed packing as a mechanism for driving structural order based on observed correlations between the structure of crystals and the mathematical packing of hard spheres. We study the ordering of several systems of hard colloids in which structural order correlates with mathematical packing and find, surprisingly, that structural order cannot arise from packing. Our approach provides statistical mechanics approaches for investigating the mathematics of packing and raises questions about the role of packing in determining the structural order of matter. Since the 1920s, packing arguments have been used to rationalize crystal structures in systems ranging from atomic mixtures to colloidal crystals. Packing arguments have recently been applied to complex nanoparticle structures, where they often, but not always, work. We examine when, if ever, packing is a causal mechanism in hard particle approximations of colloidal crystals. We investigate three crystal structures composed of their ideal packing shapes. We show that, contrary to expectations, the ordering mechanism cannot be packing, even when the thermodynamically self-assembled structure is the same as that of the densest packing. We also show that the best particle shapes for hard particle colloidal crystals at any finite pressure are imperfect versions of the ideal packing shape.