Structure and dynamics of model colloidal clusters with short-range attractions

Baseline Comparisons of Complementary Sampling Methods for Assembly Driven by Short-Ranged Pair Potentials toward Fast and Flexible Hybridization

This work measures baseline sampling characteristics that highlight fundamental differences between sampling methods for assembly driven by short-ranged pair potentials. Such granular comparison is essential for fast, flexible, and accurate hybridization of complementary methods. Besides sampling speed, efficiency, and accuracy of uniform grid coverage, other sampling characteristics measured are (i) accuracy of covering narrow low energy regions that have low effective dimension (ii) ability to localize sampling to specific basins, and (iii) flexibility in sampling distributions. As a proof of concept, we compare a recently developed geometric methodology EASAL (Efficient Atlasing and Search of Assembly Landscapes) and the traditional Monte Carlo (MC) method for sampling the energy landscape of two assembling trans-membrane helices, driven by short-range pair potentials. By measuring the above-mentioned sampling characteristics, we demonstrate that EASAL provides localized and accurate coverage of crucial regions of the energy landscape of low effective dimension, under flexible sampling distributions, with much fewer samples and computational resources than MC sampling. EASAL's empirically validated theoretical guarantees permit credible extrapolation of these measurements and comparisons to arbitrary number and size of assembling units. Promising avenues for hybridizing the complementary advantages of the two methods are discussed.

Atlasing of Assembly Landscapes using Distance Geometry and Graph Rigidity

This Article describes a novel geometric methodology for analyzing free energy and kinetics of assembly driven by short-range pair-potentials in an implicit solvent and provides a proof-of-concept illustration of its unique capabilities. An atlas is a labeled partition of the assembly landscape into a roadmap of maximal, contiguous, nearly-equipotential-energy conformational regions or macrostates, together with their neighborhood relationships. The new methodology decouples the roadmap generation from sampling and produces: (1) a queryable atlas of local potential energy minima, their basin structure, energy barriers, and neighboring basins; (2) paths between a specified pair of basins, each path being a sequence of conformational regions or macrostates below a desired energy threshold; and (3) approximations of relative path lengths, basin volumes (configurational entropy), and path probabilities. Results demonstrating the core algorithm's capabilities and high computational efficiency have been generated by a resource-light, curated open source software implementation EASAL (Efficient Atlasing and Search of Assembly Landscapes, ACM Trans. Math. Softw. 2018 44, 1-48. 10.1145/3204472; see software, Efficient Atlasing and Search of Assembly Landscapes, 2016. https://bitbucket.org/geoplexity/easal; video, Video Illustrating the opensource software EASAL, 2016. https://cise.ufl.edu/~sitharam/EASALvideo.mpeg; and user guide, EASAL software user guide, 2016. https://bitbucket.org/geoplexity/easal/src/master/CompleteUserGuide.pdf). Running on a laptop with Intel(R) Core(TM) i7-7700@3.60 GHz CPU with 16GB of RAM, EASAL atlases several hundred thousand conformational regions or macrostates in minutes using a single compute core. Subsequent path and basin computations each take seconds. A parallelized EASAL version running on the same laptop with 4 cores gives a 3× speedup for atlas generation. The core algorithm's correctness, time complexity, and efficiency-accuracy trade-offs are formally guaranteed using modern distance geometry, geometric constraint systems and combinatorial rigidity. The methodology further links the shape of the input assembling units to a type of intuitive and queryable bar-code of the output atlas, which in turn determine stable assembled structures and kinetics. This succinct input-output relationship facilitates reverse analysis and control toward design. A novel feature that is crucial to both the high sampling efficiency and decoupling of roadmap generation from sampling is a recently developed theory of convex Cayley (distance-based) custom parametrizations specific to assembly, as opposed to folding. Representing microstates with macrostate-specific Cayley parameters, to generate microstate samples, avoids gradient-descent search used by all prevailing methods. Further, these parametrizations convexify conformational regions or macrostates. This ratchets up sampling efficiency, significantly reducing number of repeated and discarded samples. These features of the new stand-alone methodology can also be used to complement the strengths of prevailing methodologies including Molecular Dynamics, Monte Carlo, and Fast Fourier Transform based methods.

From canyons to valleys: Numerically continuing sticky-hard-sphere clusters to the landscapes of smoother potentials

We study the energy landscapes of particles with short-range attractive interactions as the range of the interactions increases. Starting with the set of local minima for 6≤N≤12 hard spheres that are "sticky," i.e., they interact only when their surfaces are exactly in contact, we use numerical continuation to evolve the local minima (clusters) as the range of the potential increases, using both the Lennard-Jones and Morse families of interaction potentials. As the range increases, clusters merge, until at long ranges only one or two clusters are left. We compare clusters obtained by continuation with different potentials and find that for short and medium ranges, up to about 30% of particle diameter, the continued clusters are nearly identical, both within and across families of potentials. For longer ranges, the clusters vary significantly, with more variation between families of potentials than within a family. We analyze the mechanisms behind the merge events and find that most rearrangements occur when a pair of nonbonded particles comes within the range of the potential. An exception occurs for nonharmonic clusters, i.e., those that have a zero eigenvalue in their Hessian, which undergo a more global rearrangement.

From sticky-hard-sphere to Lennard-Jones-type clusters

A relation M_{SHS→LJ} between the set of nonisomorphic sticky-hard-sphere clusters M_{SHS} and the sets of local energy minima M_{LJ} of the (m,n)-Lennard-Jones potential V_{mn}^{LJ}(r)=ɛ/n-m[mr^{-n}-nr^{-m}] is established. The number of nonisomorphic stable clusters depends strongly and nontrivially on both m and n and increases exponentially with increasing cluster size N for N≳10. While the map from M_{SHS}→M_{SHS→LJ} is noninjective and nonsurjective, the number of Lennard-Jones structures missing from the map is relatively small for cluster sizes up to N=13, and most of the missing structures correspond to energetically unfavorable minima even for fairly low (m,n). Furthermore, even the softest Lennard-Jones potential predicts that the coordination of 13 spheres around a central sphere is problematic (the Gregory-Newton problem). A more realistic extended Lennard-Jones potential chosen from coupled-cluster calculations for a rare gas dimer leads to a substantial increase in the number of nonisomorphic clusters, even though the potential curve is very similar to a (6,12)-Lennard-Jones potential.

Free energy of singular sticky-sphere clusters

Networks of particles connected by springs model many condensed-matter systems, from colloids interacting with a short-range potential and complex fluids near jamming, to self-assembled lattices and various metamaterials. Under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero-frequency vibrational modes, yet such singular modes are at the epicenter of many interesting behaviors in the systems above. We consider a system of N spherical particles, and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged. Although the contribution to the partition function from singular clusters diverges in the limit, its asymptotic value can be calculated and depends on only two parameters, characterizing the depth and range of the potential. The result holds for systems that are second-order rigid, a geometric characterization that describes all known ground-state (rigid) sticky clusters. To illustrate the applications of our theory we address the question of emergence: how does crystalline order arise in large systems when it is strongly disfavored in small ones? We calculate the partition functions of all known rigid clusters up to N≤21 and show the cluster landscape is dominated by hyperstatic clusters (those with more than 3N-6 contacts); singular and isostatic clusters are far less frequent, despite their extra vibrational and configurational entropies. Since the most hyperstatic clusters are close to fragments of a close-packed lattice, this underlies the emergence of order in sticky-sphere systems, even those as small as N=10.

A Detailed Study on the Low-Energy Structures of Charged Colloidal Clusters

The target of this investigation is the systematic characterization of the low-energy structures of charged colloidal clusters that may be important to understand the self-assembling process of biomolecules. The aggregation of charged colloidal particles is governed by the attractive short-ranged Morse potential and the Yukawa repulsive tail to describe the long-range charge effect. A global optimization strategy, based on our own evolutionary algorithm, was adopted to discover the low-energy structures of colloidal clusters composed of up to 20 particles. A detailed analysis of the low-energy structures involving charged particles shows that the appearance of the Bernal spiral as the most stable motif occurs, first, at N = 6, but it is favored for larger clusters (N ≥ 13); for 6 ≤ N ≤ 12, there is a competition between the spiral (which is favored for higher charges) and more spherical-like structures. Finally, we study binary clusters composed by two sets of differently charged colloidal particles. Although a great diversity of low-energy structures is observed (especially for aggregates with one of the components in excess), the global minimum is disputed by three structural motifs depending on the composition of the cluster and, in some cases, on the range of the Morse potential.

Segregation of "isotope" particles within colloidal molecules

Clusters of spherical particles are called "colloidal molecules" because they adopt structures that resemble those of true molecules. In this analogy, the particles are the atoms, the attractive interactions between them are bonds, and the different structures that appear in equilibrium are isomers. We take this analogy a step further by doping colloidal molecules with colloidal "isotopes," particles that have the same size but different bonding energies from the other particles in the system. Our molecules are two-dimensional clusters consisting of polystyrene and silica microspheres held together by depletion interactions. Using a combination of optical microscopy and particle tracking, we examine an ensemble of 4- and 5-particle molecules at different isotope ratios. We find that the isotopes tend to segregate to particular positions in the various isomers. We explain these findings using a statistical mechanical model that accounts for the rotational entropy of the isomers and the different interaction potentials between the different types of particles. The model shows how to optimize the yield of any particular isomer, so as to put the isotopes in desired locations. Our experiments and models show that even in systems of particles with isotropic interactions, the structures of self-assembled molecules can in principle be controlled to a surprisingly high extent.

Crystallization and arrest mechanisms of model colloids

We performed dynamic simulations of spheres with short-range attractive interactions for many values of interaction strength and range. Fast crystallization occurs in a localized region of this parameter space, but the character of crystallization pathways is not uniform within this region. Pathways range from one-step, in which a crystal nucleates directly from a gas, to two-step, in which substantial liquid-like clusters form and only subsequently become crystalline. Crystallization can fail because of slow nucleation from either gas or liquid, or because of dynamic arrest caused by strong interactions. Arrested states are characterized by the formation of networks of face-sharing tetrahedra that can be detected by a local common neighbor analysis.

Equilibrium Shape of Colloidal Crystals

Assembling colloidal particles into highly ordered configurations, such as photonic crystals, has significant potential for enabling a broad range of new technologies. Facilitating the nucleation of colloidal crystals and developing successful crystal growth strategies require a fundamental understanding of the equilibrium structure and morphology of small colloidal assemblies. Here, we report the results of a novel computational approach to determine the equilibrium shape of assemblies of colloidal particles that interact via an experimentally validated pair potential. While the well-known Wulff construction can accurately capture the equilibrium shape of large colloidal assemblies, containing O(10(4)) or more particles, determining the equilibrium shape of small colloidal assemblies of O(10) particles requires a generalized Wulff construction technique which we have developed for a proper description of equilibrium structure and morphology of small crystals. We identify and characterize fully several "magic" clusters which are significantly more stable than other similarly sized clusters.