Coarse-grained depletion potentials for anisotropic colloids: Application to lock-and-key systems.

When colloids are mixed with a depletant such as a non-adsorbing polymer, one observes attractive effective interactions between the colloidal particles. If these particles are anisotropic, analysis of these effective interactions is challenging in general. We present a method for inference of approximate (coarse-grained) effective interaction potentials between such anisotropic particles. Using the example of indented (lock-and-key) colloids, we show how numerical solutions can be used to integrate out the (hard sphere) depletant, leading to a depletion potential that accurately characterises the effective interactions. The accuracy of the method is based on matching of contributions to the second virial coefficient of the colloids. The simplest version of our method yields a piecewise-constant effective potential; we also show how this scheme can be generalised to other functional forms, where appropriate.

Porous Liquid Phases for Indented Colloids with Depletion Interactions.

We study indented spherical colloids, interacting via depletion forces. These systems exhibit liquid-vapor phase transitions whose properties are determined by a combination of strong "lock-and-key" bonds and weaker nonspecific interactions. As the propensity for lock-and-key binding increases, the critical point moves to significantly lower density, and the coexisting phases change their structure. In particular, the liquid phase is porous, exhibiting large percolating voids. The properties of this system depend strongly on the topological structure of an underlying bond network: we comment on the implications of this fact for the assembly of equilibrium states with controlled porous structures.

Self-assembly and crystallisation of indented colloids at a planar wall.

We report experimental and simulation studies of the structure of a monolayer of indented ("lock and key") colloids, on a planar surface. On adding a non-absorbing polymer with prescribed radius and volume fraction, depletion interactions are induced between the colloids, with controlled range and strength. For spherical particles, this leads to crystallisation, but the indented colloids crystallise less easily than spheres, in both simulation and experiment. Nevertheless, simulations show that indented colloids do form plastic (rotator) crystals. We discuss the conditions under which this occurs, and the possibilities of lower-symmetry crystal states. We also comment on the kinetic accessibility of these states.

Shape-sensitive crystallization in colloidal superball fluids.

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 "Λ1" lattice that was recently predicted as the densest packing of superdisks. As the size of the depletant is reduced, however, we observe a transition to a square phase. The differences in these entropically stabilized phases result from an interplay between the size of the depletants and the fine structure of the superball shape. We find qualitative agreement of our experimental results both with a phase diagram computed on the basis of the volume accessible to the depletants and with simulations. By using a mixture of depletants, one of which is thermosensitive, we induce solid-to-solid phase transitions between square and canted structures. The use of depletant size to leverage fine features of the shape of particles in driving their self-assembly demonstrates a general and powerful mechanism for engineering novel materials.

Characterizing the variation of propagation constants in multicore fiber.

We demonstrate a numerical technique that can evaluate the core-to-core variations in propagation constant in multicore fiber. Using a Markov Chain Monte Carlo process, we replicate the interference patterns of light that has coupled between the cores during propagation. We describe the algorithm and verify its operation by successfully reconstructing target propagation constants in a fictional fiber. Then we carry out a reconstruction of the propagation constants in a real fiber containing 37 single-mode cores. We find that the range of fractional propagation constant variation across the cores is approximately ± 2 × 10(-5).

Three-body interactions in complex fluids: virial coefficients from simulation finite-size effects.

A simulation technique is described for quantifying the contribution of three-body interactions to the thermodynamical properties of coarse-grained representations of complex fluids. The method is based on a new approach for determining virial coefficients from the measured volume-dependent asymptote of a certain structural function. By comparing the third virial coefficient B3 for a complex fluid with that of an approximate coarse-grained model described by a pair potential, three body effects can be quantified. The strategy is applicable to both Molecular Dynamics and Monte Carlo simulation. Its utility is illustrated via measurements of three-body effects in models of star polymers and in highly size-asymmetrical colloid-polymer mixtures.

Quantifying the effects of neglecting many-body interactions in coarse-grained models of complex fluids.

We describe a general simulation scheme for assessing the thermodynamic consequences of neglecting many-body effects in coarse-grained models of complex fluids. The method exploits the fact that the asymptote of a simple-to-measure structural function provides direct estimates of virial coefficients. Comparing the virial coefficients of an atomistically detailed system with those of a coarse-grained version described by pair potentials, permits the role of many-body effects to be quantified. The approach is applied to two models: (i) a size-asymmetrical colloid-polymer mixture, and (ii) a solution of star polymers. In the latter case, coarse-graining to an effective fluid described by pair potentials is found to neglect important aspects of the true behavior.

Self-assembly of colloidal polymers via depletion-mediated lock and key binding.

We study the depletion-induced self-assembly of indented colloids. Using state-of-the-art Monte Carlo simulation techniques that treat the depletant particles explicitly, we demonstrate that colloids assemble by a lock-and-key mechanism, leading to colloidal polymerization. The morphology of the chains that are formed depends sensitively on the size of the colloidal indentation, with smaller values additionally permitting chain branching. In contrast to the case of spheres with attractive patches, Wertheim's thermodynamic perturbation theory fails to provide a fully quantitative description of the polymerization transition. We trace this failure to a neglect of packing effects and we introduce a modified theory that accounts better for the shape of the colloids, yielding improved agreement with simulation.

Depletion potentials in highly size-asymmetric binary hard-sphere mixtures: comparison of simulation results with theory.

We report a detailed study, using state-of-the-art simulation and theoretical methods, of the effective (depletion) potential between a pair of big hard spheres immersed in a reservoir of much smaller hard spheres, the size disparity being measured by the ratio of diameters q ≡ σ(s)/σ(b). Small particles are treated grand canonically, their influence being parameterized in terms of their packing fraction in the reservoir η(s)(r). Two Monte Carlo simulation schemes--the geometrical cluster algorithm, and staged particle insertion--are deployed to obtain accurate depletion potentials for a number of combinations of q ≤ 0.1 and η(s)(r). After applying corrections for simulation finite-size effects, the depletion potentials are compared with the prediction of new density functional theory (DFT) calculations based on the insertion trick using the Rosenfeld functional and several subsequent modifications. While agreement between the DFT and simulation is generally good, significant discrepancies are evident at the largest reservoir packing fraction accessible to our simulation methods, namely, η(s)(r) = 0.35. These discrepancies are, however, small compared to those between simulation and the much poorer predictions of the Derjaguin approximation at this η(s)(r). The recently proposed morphometric approximation performs better than Derjaguin but is somewhat poorer than DFT for the size ratios and small-sphere packing fractions that we consider. The effective potentials from simulation, DFT, and the morphometric approximation were used to compute the second virial coefficient B(2) as a function of η(s)(r). Comparison of the results enables an assessment of the extent to which DFT can be expected to correctly predict the propensity toward fluid-fluid phase separation in additive binary hard-sphere mixtures with q ≤ 0.1. In all, the new simulation results provide a fully quantitative benchmark for assessing the relative accuracy of theoretical approaches for calculating depletion potentials in highly size-asymmetric mixtures.

Monte Carlo cluster algorithm for fluid phase transitions in highly size-asymmetrical binary mixtures.

Highly size-asymmetrical fluid mixtures arise in a variety of physical contexts, notably in suspensions of colloidal particles to which much smaller particles have been added in the form of polymers or nanoparticles. Conventional schemes for simulating models of such systems are hamstrung by the difficulty of relaxing the large species in the presence of the small one. Here we describe how the rejection-free geometrical cluster algorithm of Liu and Luijten [J. Liu and E. Luijten, Phys. Rev. Lett. 92, 035504 (2004)] can be embedded within a restricted Gibbs ensemble to facilitate efficient and accurate studies of fluid phase behavior of highly size-asymmetrical mixtures. After providing a detailed description of the algorithm, we summarize the bespoke analysis techniques of [Ashton et al., J. Chem. Phys. 132, 074111 (2010)] that permit accurate estimates of coexisting densities and critical-point parameters. We apply our methods to study the liquid-vapor phase diagram of a particular mixture of Lennard-Jones particles having a 10:1 size ratio. As the reservoir volume fraction of small particles is increased in the range of 0%-5%, the critical temperature decreases by approximately 50%, while the critical density drops by some 30%. These trends imply that in our system, adding small particles decreases the net attraction between large particles, a situation that contrasts with hard-sphere mixtures where an attractive depletion force occurs.

Fluid phase coexistence and critical behavior from simulations in the restricted Gibbs ensemble.

The symmetrical restricted Gibbs ensemble (RGE) is a version of the Gibbs ensemble in which particles are exchanged between two boxes of fixed equal volumes. It has recently come to prominence because--when combined with specialized algorithms--it provides for the study of near-coexistence density fluctuations in highly size-asymmetric binary mixtures. Hitherto, however, a detailed framework for extracting accurate estimates of critical point and coexistence curve parameters from RGE density fluctuations has been lacking. Here we address this problem by exploiting an exact link between the RGE density fluctuations and those of the grand canonical ensemble. In the subcritical region we propose and test a simple method for obtaining accurate estimates of coexistence densities. In the critical region we identify an observable that serves as a finite system size estimator for the critical point parameters, and present a finite-size scaling theory that allows extrapolation to the thermodynamic limit.

Interplay between function and structure in complex networks.

We show that abrupt structural transitions can arise in functionally optimal networks, driven by small changes in the level of transport congestion. Our results offer an explanation as to why so many diverse species of network structure arise in nature (e.g., fungal systems) under essentially the same environmental conditions. Our findings are based on an exactly solvable model system which mimics a variety of biological and social networks. We then extend our analysis by introducing a renormalization scheme involving cost motifs, to describe analytically the average shortest path across multiple-ring-and-hub networks. As a consequence, we uncover a "skin effect" whereby the structure of the inner multi-ring core can cease to play any role in terms of determining the average shortest path across the network.

Effect of congestion costs on shortest paths through complex networks.

We analyze analytically the effect of congestion costs within a physically relevant, yet exactly solvable, network model featuring central hubs. These costs lead to a competition between centralized and decentralized transport pathways. In stark contrast to conventional no-cost networks, there now exists an optimal number of connections to the central hub in order to minimize the shortest path. Our results shed light on an open problem in biology, informatics, and sociology, concerning the extent to which decentralized versus centralized design benefits real-world complex networks.