Demixing transition, structure, and depletion forces in binary mixtures of hard-spheres: The role of bridge functions

A mean-field theory for predicting single polymer collapse induced by neutral crowders

Macromolecular crowding can induce the collapse of a single long polymer into a globular form due to depletion forces of entropic nature. This phenomenon has been shown to play a significant role in compacting the genome within the bacterium Escherichia coli into a well-defined region of the cell known as the nucleoid. Motivated by the biological significance of this process, numerous theoretical and computational studies have searched for the primary determinants of the behavior of polymer-crowder phases. However, our understanding of this process remains incomplete and there is debate on a quantitatively unified description. In particular, different simulation studies with explicit crowders have proposed different order parameters as potential predictors for the collapse transition. In this work, we present a comprehensive analysis of published simulation data obtained from different sources. Based on the common behavior we find in this data, we develop a unified phenomenological model that we show to be predictive. Finally, to further validate the accuracy of the model, we conduct new simulations on polymers of various sizes, and investigate the role of jamming of the crowders.

Modeling the structure and thermodynamics of multicomponent and polydisperse hard-sphere dispersions with continuous potentials

A model system of identical particles interacting via a hard-sphere potential is essential in condensed matter physics; it helps to understand in and out of equilibrium phenomena in complex fluids, such as colloidal dispersions. Yet, most of the fixed time-step algorithms to study the transport properties of those systems have drawbacks due to the mathematical nature of the interparticle potential. Because of this, mapping a hard-sphere potential onto a soft potential has been recently proposed [Báez et al., J. Chem. Phys. 149, 164907 (2018)]. More specifically, using the second virial coefficient criterion, one can set a route to estimate the parameters of the soft potential that accurately reproduces the thermodynamic properties of a monocomponent hard-sphere system. However, real colloidal dispersions are multicomponent or polydisperse, making it important to find an efficient way to extend the potential model for dealing with such kind of many-body systems. In this paper, we report on the extension and applicability of the second virial coefficient criterion to build a description that correctly captures the phenomenology of both multicomponent and polydisperse hard-sphere dispersions. To assess the accuracy of the continuous potentials, we compare the structure of soft polydisperse systems with their hard-core counterpart. We also contrast the structural and thermodynamic properties of soft binary mixtures with those obtained through mean-field approximations and the Ornstein-Zernike equation for the two-component hard-sphere dispersion.

Structure ordering and glass transition in size-asymmetric ternary mixtures of hard spheres: Variation from fragile to strong glasses

We investigate the structure and activated dynamics of a binary mixture of colloidal particles dispersed in a solvent of much smaller-sized particles. The solvent degrees of freedom are traced out from the grand partition function of the colloid-solvent mixture which reduces the system from ternary to effective binary mixture of colloidal particles. In the effective binary mixture colloidal particles interact via effective potential that consists of bare potential plus the solvent-induced interaction. Expressions for the effective potentials and pair correlation functions are derived. We used the result of pair correlation functions to determine the number of particles in a cooperatively reorganizing cluster (CRC) in which localized particles form "long-lived" nonchemical bonds with the central particle. For an event of relaxation to take place these bonds have to reorganize irreversibly, the energy involved in the processes is the effective activation energy of relaxation. Results are reported for hard sphere colloidal particles dispersed in a solvent of hard sphere particles. Our results show that the concentration of solvent can be used as a control parameter to fine-tune the microscopic structural ordering and the size of CRC that governs the glassy dynamics. We show that a small variation in the concentration of solvent creates a bigger change in the kinetic fragility which highlights a wide variation in behavior, ranging from fragile to strong glasses. We conclude that the CRC which is determined from the static pair correlation function and the fluctuations embedded in the system is probably the sole player in the physics of glass transition.

Determining depletion interactions by contracting forces

Depletion forces are fundamental for determining the phase behavior of a vast number of materials and colloidal dispersions, and have been used for the manipulation of in- and out-of-equilibrium thermodynamic states. The entropic nature of depletion forces is well understood; however, most theoretical approaches, and also molecular simulations, work quantitatively at moderate size ratios in very diluted systems, since large size asymmetries and high particle concentrations are difficult to deal with. The existing approaches for integrating out the degrees of freedom of the depletant species may fail under these extreme physical conditions. Thus, the main goal of this contribution is to introduce a general physical formulation for obtaining the depletion forces even in those cases where the concentration of all species is relevant. We show that the contraction of the bare forces uniquely determines depletion interactions. Our formulation is tested by studying depletion forces in binary and ternary colloidal mixtures. We report here results for dense systems, with total packing fractions of 45\% and 55\%. Our results open up the possibility of finding an efficient route to determine effective interactions at finite concentration, even at non-equilibrium thermodynamic conditions.

Soft representation of the square-well and square-shoulder potentials to be used in Brownian and molecular dynamics simulations

The discrete hard-sphere (HS), square-well (SW), and square-shoulder (SS) potentials have become the battle horse of molecular and complex fluids because they contain the basic elements to describe the thermodynamic, structural, and transport properties of both types of fluids. The mathematical simplicity of these discrete potentials allows us to obtain some analytical results despite the nature and complexity of the modeled systems. However, the divergent forces arising at the potential discontinuities may lead to severe issues when discrete potentials are used in computer simulations with uniform time steps. One of the few routes to avoid these technical problems is to replace the discrete potentials with continuous and differentiable forms built under strict physical criteria to capture the correct phenomenology. The match of the second virial coefficient between the discrete and the soft potentials has recently been successfully used to construct a continuous representation that mimics some physical properties of HSs (Báezet al2018J. Chem. Phys.149164907). In this paper, we report an extension of this idea to construct soft representations of the discrete SW and SS potentials. We assess the accuracy of the resulting soft potential by studying structural and thermodynamic properties of the modeled systems by using extensive Brownian and molecular dynamics computer simulations. Besides, Monte Carlo results for the original discrete potentials are used as benchmark. We have also implemented the discrete interaction models and their soft counterparts within the integral equations theory of liquids, finding that the most widely used approximations predict almost identical results for both potentials.

Effective potentials induced by mixtures of patchy and hard co-solutes

The addition of co-solutes to colloidal suspensions is often employed to induce tunable depletion interactions. In this work, we investigate effective colloidal interactions arising from binary co-solute mixtures of hard spheres and patchy particles. By changing the relative concentration of the two species, we show that the resulting effective potential V_eff continuously changes from the one obtained for a single-component hard sphere co-solute to that mediated by the single-component patchy particle co-solute. Interestingly, we find that, independent of the relative concentration of the two components, the resulting V_eff is additive, i.e., it is well-described by the linear combination of the effective interactions mediated by respective pure co-solutes. However, a breakdown of the additivity occurs when the co-solute mixture is close to the onset of a demixing transition. These results represent a step forward in understanding and predicting colloidal behavior in complex and crowded environments and for exploiting this knowledge to design targeted colloidal superstructures.

Competing interactions in the depletion forces of ternary colloidal mixtures

Depletion interactions between colloidal particles surrounded by smaller depletants are typically characterized by a strong attraction at contact and a moderately repulsive barrier in front of it that extends at distances similar to the size of the depletants; the appearance and height of the barrier basically depend on the concentration and, therefore, the correlation between depletants. From a thermodynamic point of view, the former can drive the system to phase separation or toward non-equilibrium states, such as gel-like states, but its effects on both local and global properties may be controlled by the latter, which acts as a kind of entropic gate. However, the latter has not been entirely analyzed and understood within the context of colloidal mixtures mainly driven by entropy. In this contribution, we present a systematic study of depletion forces in ternary mixtures of hard spherical particles with two species of depletants, in two and three dimensions. We focus the discussion on how the composition of the depletants becomes the main physical parameter that drives the competition between the attractive well and the repulsive barrier. Our results are obtained by means of the integral equation theory of depletion forces and techniques of contraction of the description adapted to molecular dynamics computer simulations.

Lateral depletion effect on two-dimensional ordering of bacteriorhodopsins in a lipid bilayer: A theoretical study based on a binary hard-disk model

The 2D ordering of bacteriorhodopsins in a lipid bilayer was studied using a binary hard-disk model. The phase diagrams were calculated taking into account the lateral depletion effects. The critical concentrations of the protein ordering for monomers and trimers were obtained from the phase diagrams. The critical concentration ratio agreed well with the experiment when the repulsive core interaction between the depletants, namely, lipids, was taken into account. The results suggest that the depletion effect plays an important role in the association behaviors of transmembrane proteins.

Solvation effects on diffusion processes of a macromolecule: Accuracy required for radial distribution function to calculate diffusion coefficient

We investigate the dependence of the diffusion coefficient of a large solute particle on the solvation structure around a solute. The diffusion coefficient of a hard-sphere system is calculated by using a perturbation theory of large-particle diffusion with radial distribution functions around the solute. To obtain the radial distribution function, some integral equation theories are examined, such as the Percus-Yevick (PY), hypernetted-chain (HNC), and modified HNC theories using a bridge function proposed by Kinoshita (MHNC) closures. In one-component solvent systems, the diffusion coefficient depends on the first-minimum value of the radial distribution function. The results of the MHNC closure are in good agreement with those of calculation using the radial distribution functions of Monte Carlo simulations since the MHNC closure very closely reproduces the radial distribution function of Monte Carlo simulations. In binary-solvent mixtures, the diffusion coefficient is affected by the larger solvent density distribution in the short-range part, particularly the height and sharpness of the first peak and the depth of the first minimum. Since the HNC closure gives the first peak that is higher and sharper than that of the MHNC closure, the calculated diffusion coefficient is smaller than the MHNC closure result. In contrast, the results of the PY closure are qualitatively and quantitatively different from those of the MHNC and HNC closures.

Bidisperse Nanospheres Jammed on a Liquid Surface

Jammed packings of bidisperse nanospheres were assembled on a nonvolatile liquid surface and visualized to the single-particle scale by using an in situ scanning electron microscopy method. The PEGylated silica nanospheres, mixed at different number fractions and size ratios, had large enough in-plane mobilities prior to jamming to form uniform monolayers reproducibly. From the collected nanometer-resolution images, local order and degree of mixing were assessed by standard metrics. For equimolar mixtures, a large-to-small size ratio of about 1.5 minimized correlated metrics for local orientational and positional order, as previously predicted in simulations of bidisperse disk jamming. Despite monolayer uniformity, structural and depletion interactions caused spheres of a similar size to cluster, a feature evident at size ratios above 2. Uniform nanoparticle monolayers of high packing disorder are sought in many liquid interface technologies, and these experiments outlined key design principles, buttressing extensive theory/simulation literature on the topic.

Small-scale demixing in confluent biological tissues

Surface tension governed by differential adhesion can drive fluid particle mixtures to sort into separate regions, i.e., demix. Does the same phenomenon occur in confluent biological tissues? We begin to answer this question for epithelial monolayers with a combination of theory via a vertex model and experiments on keratinocyte monolayers. Vertex models are distinct from particle models in that the interactions between the cells are shape-based, as opposed to distance-dependent. We investigate whether a disparity in cell shape or size alone is sufficient to drive demixing in bidisperse vertex model fluid mixtures. Surprisingly, we observe that both types of bidisperse systems robustly mix on large lengthscales. On the other hand, shape disparity generates slight demixing over a few cell diameters, a phenomenon we term micro-demixing. This result can be understood by examining the differential energy barriers for neighbor exchanges (T1 transitions). Experiments with mixtures of wild-type and E-cadherin-deficient keratinocytes on a substrate are consistent with the predicted phenomenon of micro-demixing, which biology may exploit to create subtle patterning. The robustness of mixing at large scales, however, suggests that despite some differences in cell shape and size, progenitor cells can readily mix throughout a developing tissue until acquiring means of recognizing cells of different types.

Reduced density profile of small particles near a large particle: Results of an integral equation theory with an accurate bridge function and a Monte Carlo simulation

Solute-solvent reduced density profiles of hard-sphere fluids were calculated by using several integral equation theories for liquids. The traditional closures, Percus-Yevick (PY) and the hypernetted-chain (HNC) closures, as well as the theories with bridge functions, Verlet, Duh-Henderson, and Kinoshita (named MHNC), were used for the calculation. In this paper, a one-solute hard-sphere was immersed in a one-component hard-sphere solvent and various size ratios were examined. The profiles between the solute and solvent particles were compared with those calculated by Monte Carlo simulations. The profiles given by the integral equations with the bridge functions were much more accurate than those calculated by conventional integral equation theories, such as the Ornstein-Zernike (OZ) equation with the PY closure. The accuracy of the MHNC-OZ theory was maintained even when the particle size ratio of solute to solvent was 50. For example, the contact values were 5.7 (Monte Carlo), 5.6 (MHNC), 7.8 (HNC), and 4.5 (PY), and the first minimum values were 0.48 (Monte Carlo), 0.46 (MHNC), 0.54 (HNC), and 0.40 (PY) when the packing fraction of the hard-sphere solvent was 0.38 and the size ratio was 50. The asymptotic decay and the oscillation period for MHNC-OZ were also very accurate, although those given by the HNC-OZ theory were somewhat faster than those obtained by Monte Carlo simulations.

Evidence of electrostatic-enhanced depletion attraction in the structural properties and phase behavior of binary charged colloidal suspensions

In this paper we study the structure and phase behavior of binary mixtures of charged particles at low ionic strength. Due to the large size asymmetry between both species, light scattering measurements give us access only to the partial static structure factor that corresponds to the big particles. We observe that the addition of small charged colloids produces a decrease of the main peak of the measured static structure factor and a shift to larger scattering vector values. This finding is in agreement with theory based on integral equations with the Hypernetted-Chain Closure (HNC) relation. The effective interaction between two big particles due to the presence of small particles is obtained by a HNC inversion scheme and used in numerical simulations that adequately reproduce the experiments. We find that the presence of small particles induces an electrostatic depletion screening among the big colloids, creating around them an exclusion zone for the small charged colloids greater than that caused in the case of neutral small colloids, which in turn augments the depletion effect.

Structure of colloidal gels at intermediate concentrations: the role of competing interactions

Colloidal gels formed by colloid-polymer mixtures with an intermediate volume fraction (ϕ_c ≈ 0.4) are investigated by confocal microscopy. In addition, we have performed Monte Carlo simulations based on a simple effective pair potential that includes a short-range attractive contribution representing depletion interactions, and a longer-range repulsive contribution describing the electrostatic interactions due to the presence of residual charges. Despite neglecting non-equilibrium effects, experiments and simulations yield similar gel structures, characterised by, e.g., the pair, angular and bond distribution functions. We find that the structure hardly depends on the strength of the attraction if the electrostatic contribution is fixed, but changes significantly if the electrostatic screening is changed. This delicate balance between attractions and repulsions, which we quantify by the second virial coefficient, also determines the location of the gelation boundary.

Transient dynamics during stress overshoots in binary colloidal glasses

We investigate, using simultaneous rheology and confocal microscopy, the time-dependent stress response and transient single-particle dynamics following a step change in shear rate in binary colloidal glasses with large dynamical asymmetry and different mixing ratios. The transition from solid-like response to flow is characterised by a stress overshoot, whose magnitude is linked to transient superdiffusive dynamics as well as cage compression effects. These and the yield strain at which the overshoot occurs vary with the mixing ratio, and hence the prevailing caging mechanism. The yielding and stress storage are dominated by dynamics on different time and length scales, the short-time in-cage dynamics and the long-time structural relaxation respectively. These time scales and their relation to the characteristic time associated with the applied shear, namely the inverse shear rate, result in two different and distinct regimes of the shear rate dependencies of the yield strain and the magnitude of the stress overshoot.