Reversible Aggregation and Colloidal Cluster Morphology: The Importance of the Extended Law of Corresponding States

Continuity of Short-Time Dynamics Crossing the Liquid-Liquid Phase Separation in Charge-Tuned Protein Solutions

Liquid-liquid phase separation (LLPS) constitutes a crucial phenomenon in biological self-organization, not only intervening in the formation of membraneless organelles but also triggering pathological protein aggregation, which is a hallmark in neurodegenerative diseases. Employing incoherent quasi-elastic neutron spectroscopy (QENS), we examine the short-time self-diffusion of a model protein undergoing LLPS as a function of phase splitting and temperature to access information on the nanosecond hydrodynamic response to the cluster formation both within and outside the LLPS regime. We investigate the samples as they dissociate into microdroplets of a dense protein phase dispersed in a dilute phase as well as the separated dense and dilute phases obtained from centrifugation. By interpreting the QENS results in terms of the local concentrations in the two phases determined by UV-vis spectroscopy, we hypothesize that the short-time transient protein cluster size distribution is conserved at the transition point while the local volume fractions separate.

Patchy colloidal gels under the influence of gravity

In this contribution, gravitational effects in gel-forming patchy colloidal systems are studied. We focus on how the gel structure is modified by gravity. Through Monte Carlo computer simulations of gel-like states recently identified by the rigidity percolation criterion [J. A. S. Gallegos et al., Phys. Rev. E 104, 064606 (2021)], the influence of the gravitational field, characterized by the gravitational Péclet number, Pe, on patchy colloids is studied in terms of the patchy coverage, χ. Our findings point out that there exists a threshold Péclet number, Pe_g, that depends on χ above which the gravitational field enhances the particle bonding and, in consequence, promotes the aggregation or clustering of particles; the smaller the χ value, the higher the Pe_g. Interestingly, when χ ∼ 1 (near the isotropic limit), our results are consistent with an experimentally determined threshold Pe value where gravity affects the gel formation in short-range attractive colloids. In addition, our results show that the cluster size distribution and the density profile undergo variations that lead to changes in the percolating cluster, i.e., gravity is able to modify the structure of the gel-like states. These changes have an important impact on the structural rigidity of the patchy colloidal dispersion; the percolating cluster goes from a uniform spatially network to a heterogeneous percolated structure, where an interesting structural scenario emerges, namely, depending on the Pe value, the new heterogeneous gel-like states can coexist with both diluted and dense phases or they simply reach a crystalline-like state. In the isotropic case, the increase in the Pe number can shift the critical temperature to higher temperatures; however, when Pe > 0.01, the binodal disappears and the particles fully sediment at the bottom of the sample cell. Furthermore, gravity moves the rigidity percolation threshold to lower densities. Finally, we also note that within the values of the Péclet number here explored, the cluster morphology is barely altered.

Characterizing Experimental Monoclonal Antibody Interactions and Clustering Using a Coarse-Grained Simulation Library and a Viscosity Model

Attractive protein-protein interactions in concentrated monoclonal antibody (mAb) solutions may lead to the formation of clusters that increase viscosity. Here, we propose an analytical model that relates mAb solution viscosity to clustering by accounting for the contributions of suboptimal mAb packing within a cluster and cluster fractal dimension. The influence of short-range, anisotropic attractions and long-range Coulombic repulsion on cluster properties is investigated by analyzing the cluster-size distributions, cluster fractal dimensions, radial distribution functions, and static structure factors from a library of coarse-grained molecular dynamics simulations. The library spans a vast range of mAb charges and attractive interactions in solutions of varying ionic strength. We present a framework for combining the viscosity model and simulation library to successfully characterize the attraction, repulsion, and clustering of an experimental mAb in three different pH and cosolute conditions by fitting the measured viscosity or structure factor from small-angle X-ray scattering. At low ionic strength, the cluster-size distribution is impacted by strong charges, and both the viscosity and net charge or structure factor and net charge must be considered to deconvolute the effects of short-range attraction and long-range repulsion.

Protein solutions close to liquid-liquid phase separation exhibit a universal osmotic equation of state and dynamical behavior

Liquid-liquid phase separation (LLPS) of protein solutions is governed by highly complex protein-protein interactions. Nevertheless, it has been suggested that based on the extended law of corresponding states (ELCS), as proposed for colloids with short-range attractions, one can rationalize not only the thermodynamics, but also the structure and dynamics of such systems. This claim is systematically and comprehensively tested here by static and dynamic light scattering experiments. Spinodal lines, the isothermal osmotic compressibility κ_T and the relaxation rate of concentration fluctuations Γ are determined for protein solutions in the vicinity of LLPS. All these quantities are found to exhibit a corresponding-states behavior. This means that, for different solution conditions, these quantities are essentially the same if considered at similar reduced temperature or second virial coefficient. For moderately concentrated solutions, the volume fraction ϕ dependence of κ_T and Γ can be consistently described by Baxter's model of adhesive hard spheres. The off-critical, asymptotic T behavior of κ_T and Γ close to LLPS is consistent with the scaling laws predicted by mean-field theory. Thus, the present work aims at a comprehensive experimental test of the applicability of the ELCS to structural and dynamical properties of concentrated protein solutions.

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.

Enhancement in the diffusivity of Brownian spheroids in the presence of spheres

In the present paper, we have extended the simulation technique Brownian cluster dynamics (BCD) to analyze the dynamics of the binary mixture of hard ellipsoids and spheres. The shape dependent diffusional properties have been incorporated into BCD using Perrin's factor and compared with analytical results of a one-component ellipsoidal system. We have investigated pathways to enhance the diffusivity of spheroids in the binary mixture by manipulating the phase behavior of the system through varying the fraction of spheres in the binary mixture. We show that at low volume fraction the spherical particles have a higher diffusion coefficient than the ellipsoids due to the higher friction coefficient. However, at a higher volume fraction, we show that the diffusion coefficient of the ellipsoids increases irrespective of the aspect ratio due to the anisotropic shape.

Universal effective interactions of globular proteins close to liquid–liquid phase separation: Corresponding-states behavior reflected in the structure factor

Intermolecular interactions in protein solutions, in general, contain many contributions. If short-range attractions dominate, the state diagram exhibits liquid–liquid phase separation (LLPS) that is metastable with respect to crystallization. In this case, the extended law of corresponding states (ELCS) suggests that thermodynamic properties are insensitive to details of the underlying interaction potential. Using lysozyme solutions, we investigate the applicability of the ELCS to the static structure factor and how far effective colloidal interaction models can help to rationalize the phase behavior and interactions of protein solutions in the vicinity of the LLPS binodal. The (effective) structure factor has been determined by small-angle x-ray scattering. It can be described by Baxter’s adhesive hard-sphere model, which implies a single fit parameter from which the normalized second virial coefficient b2 is inferred and found to quantitatively agree with previous results from static light scattering. The b2 values are independent of protein concentration but systematically vary with temperature and solution composition, i.e., salt and additive content. If plotted as a function of temperature normalized by the critical temperature, the values of b2 follow a universal behavior. These findings validate the applicability of the ELCS to globular protein solutions and indicate that the ELCS can also be reflected in the structure factor.

Extended law of corresponding states: square-well oblates

The vapour-liquid coexistence collapse in the reduced temperature,T_r=T/T_c, reduced density,ρ_r=ρ/ρ_c, plane is known as a principle of corresponding states, and Noro and Frenkel have extended it for pair potentials of variable range. Here, we provide a theoretical basis supporting this extension, and show that it can also be applied to short-range pair potentials where both repulsive and attractive parts can be anisotropic. We observe that the binodals of oblate hard ellipsoids for a given aspect ratio (κ= 1/3) with varying short-range square-well interactions collapse into a single master curve in theΔB2*-ρ_rplane, whereΔB2*=(B2(T)-B2(Tc))/v0,B₂is the second virial coefficient, andv₀is the volume of the hard body. This finding is confirmed by both REMC simulation and second virial perturbation theory for varying square-well shells, mimicking uniform, equator, and pole attractions. Our simulation results reveal that the extended law of corresponding states is not related to the local structure of the fluid.

Location of the gel-like boundary in patchy colloidal dispersions: Rigidity percolation, structure, and particle dynamics

During the past decade, there has been a hot debate about the physical mechanisms that determine when a colloidal dispersion approaches the gel transition. However, there is still no consensus on a possible unique route that leads to the conditions for the formation of a gel-like state. Based on gel states identified in experiments, Valadez-Pérez et al. [Phys. Rev. E 88, 060302(R) (2013)PLEEE81539-375510.1103/PhysRevE.88.060302] proposed rigidity percolation as the precursor of colloidal gelation in adhesive hard-sphere dispersions with coordination number 〈n_{b}〉 equal to 2.4. Although this criterion was originally established to describe mechanical transitions in network-forming molecular materials with highly directional interactions, it worked well to explain gel formation in colloidal suspensions with isotropic short-range attractive forces. Recently, this idea has also been used to account for the dynamical arrest experimentally observed in attractive spherocylinders. Then, by assuming that rigidity percolation also drives gelation in spherical colloids interacting with short-ranged and highly directional potentials, we locate the thermodynamic states where gelation seems to occur in dispersions made up of patchy colloids. To check whether the criterion 〈n_{b}〉=2.4 also holds in patchy colloidal systems, we apply the so-called bond-bending analysis to determine the fraction of floppy modes at some percolating clusters. This analysis confirms that the condition 〈n_{b}〉=2.4 is a good approximation to determine those percolating clusters that are either mechanically stable or rigid. Furthermore, our results point out that not all combinations of patches and coverages lead to a gel-like state. Additionally, we systematically study the structure and the cluster size distribution along those thermodynamic states identified as gels. We show that for high coverage values, the structure is very similar for systems that have the same coverage regardless the number or the position of the patches on the particle surface. Finally, by using dynamic Monte Carlo computer simulations, we calculate both the mean-square displacement and the intermediate scattering function at and in the neighborhood of the gel-like states.

Potential-invariant network structures in Asakura-Oosawa mixtures with very short attraction range

We systematically investigated the structure and aggregate morphology of gel networks formed by colloid-polymer mixtures with a moderate colloid volume fraction and different values of the polymer-colloid size ratio, always in the limit of short-range attraction. Using the coordinates obtained from confocal microscopy experiments, we determined the radial, angular, and nearest-neighbor distribution functions together with the cluster radius of gyration as a function of size ratio and polymer concentration. The analysis of the structural correlations reveals that the network structure becomes increasingly less sensitive to the potential strength with the decreasing polymer-colloid size ratio. For the larger size ratios, compact clusters are formed at the onset of network formation and become progressively more branched and elongated with increasing polymer concentration/attraction strength. For the smallest size ratios, we observe that the aggregate structures forming the gel network are characterized by similar morphological parameters for different values of the size ratio and the polymer concentration, indicating a limited evolution of the gel structure with variations of the parameters that determine the interaction potential between colloids.

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.

Concentration and size effects on the size-selective particle purification method using the critical Casimir force

Critical Casimir force (CCF) is a solvent fluctuation introduced interaction between particles dispersed in a binary solvent. Recently, it has been demonstrated that the CCF induced attraction between particles can trigger particle size-sensitive aggregation, and has thus been used as an efficient way to purify nanoparticles by size. Here, combining small angle neutron scattering and dynamic light scattering, we investigate the effects of size and concentration on this particle size separation method. Increasing the particle concentration does not significantly affect the purification method, but the solvent composition needs to be adjusted for an optimized efficiency. This purification method is further demonstrated to work also very efficiently for systems with particle size ranging from 15 nm to about 50 nm with a very large size polydispersity. These results indicate that for both short-ranged and long-ranged attraction relative to the particle diameter, the CCF introduced particle aggregation is always size sensitive. This implies that particle aggregation is strongly affected by size polydispersity for many colloidal systems. We further propose a method to use light scattering to help identify the temperature range within which this particle purification method can work efficiently instead of using neutron scattering.

Colloidal systems with a short-range attraction and long-range repulsion: Phase diagrams, structures, and dynamics

Colloidal systems with both a short-range attraction and long-range repulsion (SALR) have rich phases compared with the traditional hard sphere systems or sticky hard sphere systems. The competition between the short-range attraction and long-range repulsion results in the frustrated phase separation, which leads to the formation of intermediate range order (IRO) structures and introduces new phases to both equilibrium and nonequilibrium phase diagrams, such as clustered fluid, cluster percolated fluid, Wigner glass, and cluster glass. One hallmark feature of many SALR systems is the appearance of the IRO peak in the interparticle structure factor, which is associated with different types of IRO structures. The relationship between the IRO peak and the clustered fluid state has been careful investigated. Not surprisingly, the morphology of clusters in solutions can be affected and controlled by the SALR potential. And the effect of the SALR potential on the dynamic properties is also reviewed here. Even though much progress has been made in understanding SALR systems, many future works are still needed to have quantitative comparisons between experiments and simulations/theories and understand the differences from different experimental systems. Owing to the large parameter space available for SALR systems, many exciting features of SALR systems are not fully explored yet. Because proteins in low-salinity solutions have SALR interactions, the understanding of SALR systems can greatly help understand protein behavior in concentrated solutions or crowded conditions.