Generalized phase behavior of cluster formation in colloidal dispersions with competing interactions

A colloidal model for the equilibrium assembly and liquid-liquid phase separation of the reflectin A1 protein

Reflectin is an intrinsically disordered protein known for its ability to modulate the biophotonic camouflage of cephalopods based on its assembly-induced osmotic properties. Its reversible self-assembly into discrete, size-controlled clusters and condensed droplets are known to depend sensitively on the net protein charge, making reflectin stimuli-responsive to pH, phosphorylation, and electric fields. Despite considerable efforts to characterize this behavior, the detailed physical mechanisms of reflectin's assembly are not yet fully understood. Here, we pursue a coarse-grained molecular understanding of reflectin assembly using a combination of experiments and simulations. We hypothesize that reflectin assembly and phase behavior can be explained from a remarkably simple colloidal model whereby individual protein monomers effectively interact via a short-range attractive and long-range repulsive (SA-LR) pair potential. We parameterize a coarse-grained SA-LR interaction potential for reflectin A1 from small-angle x-ray scattering measurements, and then extend it to a range of pH values using Gouy-Chapman theory to model monomer-monomer electrostatic interactions. The pH-dependent SA-LR interaction is then used in molecular dynamics simulations of reflectin assembly, which successfully capture a number of qualitative features of reflectin, including pH-dependent formation of discrete-sized nanoclusters and liquid-liquid phase separation at high pH, resulting in a putative phase diagram for reflectin. Importantly, we find that at low pH size-controlled reflectin clusters are equilibrium assemblies, which dynamically exchange protein monomers to maintain an equilibrium size distribution. These findings provide a mechanistic understanding of the equilibrium assembly of reflectin, and suggest that colloidal-scale models capture key driving forces and interactions to explain thermodynamic aspects of native reflectin behavior. Furthermore, the success of SA-LR interactions presented in this study demonstrates the potential of a colloidal interpretation of interactions and phenomena in a range of intrinsically disordered proteins.

Interactions and pattern formation in a macroscopic magnetocapillary SALR system of mermaid cereal

When particles are deposited at a fluid interface they tend to aggregate by capillary attraction to minimize the overall potential energy of the system. In this work, we embed floating millimetric disks with permanent magnets to introduce a competing repulsion effect and study their pattern formation in equilibrium. The pairwise energy landscape of two disks is described by a short-range attraction and long-range repulsion (SALR) interaction potential, previously documented in a number of microscopic condensed matter systems. Such competing interactions enable a variety of pairwise equilibrium states, including the possibility of a local minimum energy corresponding to a finite disk spacing. Two-dimensional (2D) experiments and simulations in confined geometries demonstrate that as the areal packing fraction is increased, the dilute repulsion-dominated lattice state becomes unstable to the spontaneous formation of localized clusters, which eventually merge into a system-spanning striped pattern. Finally, we demonstrate that the equilibrium pattern can be externally manipulated by the application of a supplemental vertical magnetic force that remotely enhances the effective capillary attraction.

Secondary nucleation in symmetric binary SALR mixtures

Monte Carlo simulation is used to study secondary nucleation, fissioning, or 'reproduction', of giant clusters in a symmetric binary model fluid with competing short-range (SA) and long-range (LR) interactions. Previous work [M. B. Sweatman, Mol. Phys., 116(15-16), 1945-1952] suggests that a pure SALR fluid can exhibit secondary nucleation if the solute concentration is slowly increased. We show this is also true for a binary symmetric SALR mixture where the cross-interactions can be tuned to generate clusters with three different kinds of structure; (i) independent clusters of each component, (ii) contact clusters of different components, and (iii) mixed clusters. In each case, the overall concentration of each component is identical. This binary model is an initial step towards using SALR fluids to model the intra-cellular space of biological cells that contain a wide range of membraneless organelles and the chemical 'soup' at the origin of life.

Arrested states in colloidal fluids with competing interactions: A static replica study

We present the first systematic application of the integral equation implementation of the replica method to the study of arrested states in fluids with microscopic competing interactions (short-range attractive and long-range repulsive, SALR), as exemplified by the prototype Lennard-Jones-Yukawa model. Using a wide set of potential parameters, we provide as many as 11 different phase diagrams on the density (ρ)-temperature (T) plane, embodying both the cluster-phase boundary, TC(ρ), and the locus below which arrest takes place, TD(ρ). We describe how the interplay between TC and TD-with the former falling on top of the other, or the other way around, depending on thermodynamic conditions and potential parameters-gives rise to a rich variety of non-ergodic states interspersed with ergodic ones, of which both the building blocks are clusters or single particles. In a few cases, we find that the TD locus does not extend all over the density range subtended by the TC envelope; under these conditions, the λ-line is within reach of the cluster fluid, with the ensuing possibility to develop ordered microphases. Whenever a comparison is possible, our predictions favorably agree with previous numerical results. Thereby, we demonstrate the reliability and effectiveness of our scheme to provide a unified theoretical framework for the study of arrested states in SALR fluids, irrespective of their nature.

Dilute gel networks vs. clumpy gels in colloidal systems with a competition between repulsive and attractive interactions

Using Brownian dynamics simulations we study gel-forming colloidal systems. The focus of this article lies on the differences of dense and dilute gel networks in terms of structure formation both on a local and a global level. We apply reduction algorithms and observe that dilute networks and dense gels differ in the way structural properties like the thickness of strands emerge. We also analyze the percolation behavior and find that two different regimes of percolation exist which might be responsible for structural differences. In dilute networks we confirm that solidity is mainly a consequence of pentagonal bipyramids forming in the network. In dense gels, tetrahedral structures also influence solidity.

A density functional theory and simulation study of stripe phases in symmetric colloidal mixtures

In a binary mixture, stripes refer to a one-dimensional periodicity of the composition, namely, a regular alternation of layers filled with particles of mostly one species. We have recently introduced [Munaò et al., Phys. Chem. Chem. Phys. 25, 16227 (2023)] a model that possibly provides the simplest binary mixture endowed with stripe order. The model consists of two species of identical hard spheres with equal concentration, which mutually interact through a square-well potential. In that paper, we have numerically shown that stripes are present in both liquid and solid phases when the attraction range is rather long. Here, we study the phase behavior of the model in terms of a density functional theory capable to account for the existence of stripes in the dense mixture. Our theory is accurate in reproducing the phases of the model, at least insofar as the composition inhomogeneities occur on length scales quite larger than the particle size. Then, using Monte Carlo simulations, we prove the existence of solid stripes even when the square well is much thinner than the particle diameter, making our model more similar to a real colloidal mixture. Finally, when the width of the attractive well is equal to the particle diameter, we observe a different and more complex form of compositional order in the solid, where each species of particle forms a regular porous matrix holding in its holes the other species, witnessing a surprising variety of emergent behaviors for a very basic model of interaction.

Microphase versus macrophase separation in the square-well-linear fluid: A theoretical and computational study

Due to the presence of competing interactions, the square-well-linear fluid can exhibit either liquid-vapor equilibrium (macrophase separation) or clustering (microphase separation). Here we address the issue of determining the boundary between these two regimes, i.e., the Lifshitz point, expressed in terms of a relationship between the parameters of the model. To this aim, we carry out Monte Carlo simulations to compute the structure factor of the fluid, whose behavior at low wave vectors accurately captures the tendency of the fluid to form aggregates or, alternatively, to phase separate. Specifically, for a number of different combinations of attraction and repulsion ranges, we make the system go across the Lifshitz point by increasing the strength of the repulsion. We use simulation results to benchmark the performance of two theories of fluids, namely, the hypernetted chain (HNC) equation and the analytically solvable random phase approximation (RPA); in particular, the RPA theory is applied with two different prescriptions as for the direct correlation function inside the core. Overall, the HNC theory proves to be an appropriate tool to characterize the fluid structure and the low-wave-vector behavior of the structure factor is consistent with the threshold between microphase and macrophase separation established through simulation. The structural predictions of the RPA theory turn out to be less accurate, but this theory offers the advantage of providing an analytical expression of the Lifshitz point. Compared to simulation, both RPA schemes predict a Lifshitz point that falls within the macrophase-separation region of parameters: in the best case, barriers roughly twice higher than predicted are required to attain clustering conditions.

Non-equilibrium view of the amorphous solidification of liquids with competing interactions

The interplay between short-range attractions and long-range repulsions (SALR) characterizes the so-called liquids with competing interactions, which are known to exhibit a variety of equilibrium and non-equilibrium phases. The theoretical description of the phenomenology associated with glassy or gel states in these systems has to take into account both the presence of thermodynamic instabilities (such as those defining the spinodal line and the so called λ line) and the limited capability to describe genuine non-equilibrium processes from first principles. Here, we report the first application of the non-equilibrium self-consistent generalized Langevin equation theory to the description of the dynamical arrest processes that occur in SALR systems after being instantaneously quenched into a state point in the regions of thermodynamic instability. The physical scenario predicted by this theory reveals an amazing interplay between the thermodynamically driven instabilities, favoring equilibrium macro- and micro-phase separation, and the kinetic arrest mechanisms, favoring non-equilibrium amorphous solidification of the liquid into an unexpected variety of glass and gel states.

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.

Diffusive Dynamics of Bacterial Proteome as a Proxy of Cell Death

Temperature variations have a big impact on bacterial metabolism and death, yet an exhaustive molecular picture of these processes is still missing. For instance, whether thermal death is determined by the deterioration of the whole or a specific part of the proteome is hotly debated. Here, by monitoring the proteome dynamics of E. coli, we clearly show that only a minor fraction of the proteome unfolds at the cell death. First, we prove that the dynamical state of the E. coli proteome is an excellent proxy for temperature-dependent bacterial metabolism and death. The proteome diffusive dynamics peaks at about the bacterial optimal growth temperature, then a dramatic dynamical slowdown is observed that starts just below the cell's death temperature. Next, we show that this slowdown is caused by the unfolding of just a small fraction of proteins that establish an entangling interprotein network, dominated by hydrophobic interactions, across the cytoplasm. Finally, the deduced progress of the proteome unfolding and its diffusive dynamics are both key to correctly reproduce the E. coli growth rate.

Aggregation behavior of nanoparticles: Revisiting the phase diagram of colloids

Surface functionalization of metal nanoparticles (NPs), e.g., using peptides and proteins, has recently attracted a considerable attention in the field of design of therapeutics and diagnostics. The possibility of diverse functionalization allows them to selectively interact with proteins, while the metal core ensures solubility, making them tunable therapeutic agents against diseases due to mis-folding or aggregation. On the other hand, their action is limited by possible self-aggregation, which could be, however, prevented based on the full understanding of their phase diagram as a function of the environmental variables (temperature, ionic strength of the solution, concentration) and intrinsic characteristics (size, charge, amount, and type of functional groups). A common modeling strategy to study the phase behavior is to represent the NPs as spheres interacting via effective potentials implicitly accounting for the solvation effects. Their size put the NPs into the class of colloids, albeit with particularly complex interactions including both attractive and repulsive features, and a consequently complex phase diagram. In this work, we review the studies exploring the phases of these systems starting from those with only attractive or repulsive interactions, displaying a simpler disperse-clustered-aggregated transitions. The phase diagram is here interpreted focusing on the universal aspects, i.e., those dependent on the general feature of the potentials, and available data are organized in a parametric phase diagram. We then consider the potentials with competing attractive short range well and average-long-range repulsive tail, better representing the NPs. Through the proper combination of the attractive only and repulsive only potentials, we are able to interpret the appearance of novel phases, characterized by aggregates with different structural characteristics. We identify the essential parameters that stabilize the disperse phase potentially useful to optimize NP therapeutic activity and indicate how to tune the phase behavior by changing environmental conditions or the NP chemical–physical properties.

Competition between clustering and phase separation in binary mixtures containing SALR particles

We investigate by Monte Carlo simulations a mixture of particles with competing interactions (hard-sphere two-Yukawa, HSTY) and hard spheres (HS), with same diameters σ and a square-well (SW) cross attraction. In a recent study [G. Munaò et al., J. Phys. Chem. B, 2022, 126, 2027-2039], we have analysed situations-in terms of relative concentration and attraction strength-where HS promote the formation of clusters involving particles of both species under thermodynamic conditions that would not allow for clustering of the pure HSTY fluid. Here, we focus on the role played by the range of cross attraction in determining the equilibrium structure of the mixture, starting from a homogeneous low-density state. When the width of the well exceeds approximately σ, clustering takes place in the system, with aggregates characterised by various sizes and shapes. Only for low HSTY concentrations (less than 10%) a single big cluster appears, anticipating the behaviour observed for a wider well, around 1.2σ. In the latter case, a spherical cluster encompassing almost all particles is the stable structure at equilibrium. We interpret this outcome as a macrophase, liquid-vapour separation where the spherical cluster is just the form taken at low density by the liquid phase inside the vapour phase: indeed, when the density takes larger values, periodic boundary conditions select liquid-vapour interfaces with other non-spherical shapes, similarly as found for a finite sample of simple fluid going through the liquid-vapour coexistence region. For still higher densities we document the existence of a solid phase characterized by the alternation of bilayers filled with particles of one species and bilayers of the other species, giving the solid a peculiar wafer structure.

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.

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.

On the dynamically arrested states of equilibrium and non-equilibrium gels: a comprehensive Brownian dynamics study

In this work a systematic study over a wide number of final thermodynamic states for two gel-forming liquids was performed. Such two kind of gel formers are distinguished by their specific interparticle interaction potential. We explored several thermodynamic states determining the thermodynamic, structural and dynamic properties of both liquids after a sudden temperature change. The thermodynamic analysis allows to identify that the liquid with short range attraction and long range repulsion lacks of a stable gas-liquid phase separation liquid, in contrast with the liquid with short range attractions. Thus, although for some thermodynamic states the structural behavior, measured by the static structure factor, is similar to and characteristic of the gel phase, for the short range attractive fluid the gel phase is a consequence of a spinodal decomposition process. In contrast, gelation in the short range attraction and long range repulsion liquid is not due to a phase separation. We also analyze the similarities and differences of the dynamic behavior of both systems through the analysis of the mean square displacement, the self part of the intermediate scattering function, the diffusion coefficient and theαrelaxation time. Finally, using one of the main results of the non-equilibrium self-consistent generalized Langevin equation theory (NE-SCGLE), we determine the dynamic arrest phase diagram in the volume fraction and temperature (φvsT) plane.

The Role of Structure in Polymer Rheology: Review

The review is devoted to the analysis of the current state of understanding relationships among the deformation-induced structure transformations, observed rheological properties, and the occurrence of non-linear effects for polymer liquids (melts, solutions, and composites). Three levels of non-linearity are the base for consideration. The first one concerns changes in the relaxation spectra of viscoelastic liquids, which are responsible for weak non-linear phenomena. The second one refers to the strong non-linearity corresponding to such changes in the structure of a medium that leads to the emergence of a new relaxation state of a matter. Finally, the third one describes the deformation-induced changes in the phase state and/or the occurring of bifurcations and instability in flow and reflects the thermodynamic non-linear behavior. From a structure point of view, a common cause of the non-linear effects is the orientation of macromolecules and changes in intermolecular interaction, while a dominant factor in describing fluid dynamics of polymer liquids is their elasticity. The modern understanding of thixotropic effects, yielding viscoplastic materials, deformation-induced phase transition, and the experimental observations, demonstrating direct correlations between the structure and rheology of polymer liquids, are the main objects for discussion. All these topics are reviewed and discussed mainly on the basis of the latest five-year publications.

Clustering in Mixtures of SALR Particles and Hard Spheres with Cross Attraction

Self-assembling complex fluids are often modeled as particles with effective competing isotropic interactions, combining a short-range attraction (SA) followed by a longer-range repulsion (LR). For moderately low temperatures and densities, SALR particles form clusters in equilibrium, at least provided that the potential parameters are appropriate. Here we inquire into the possibility that cluster formation in SALR fluids might be pushed by a foreign species even under thermodynamic conditions that would not allow for clusterization of the pure system. To this aim, we study by Monte Carlo simulations a mixture of hard-sphere two-Yukawa particles and hard spheres, with a cross interaction modeled by a square-well attraction, and we investigate the conditions of clustering in terms of strength of attraction and relative concentration of the two species. We find that clusters can occur in the mixture for the same temperature and density where the pure SALR fluid is almost structureless. In particular, we single out a cross attraction such that clusters are formed with a SALR concentration as low as 5%. We also find a situation where nearly pure droplets of hard spheres are held together by a shell of SALR particles. Conversely, we show that clustering can be undermined in the mixture under conditions for which this process takes place in the parent SALR fluid. Using a simple criterion, based on the second virial coefficients of the attractive part of interaction potentials (the so-called "reference attractive fluids"), we are able to predict accurately whether clustering is favored (or hindered) in the mixture, as compared to the pure SALR fluid.

Thermodynamics, static properties and transport behaviour of fluids with competing interactions

Competing interaction fluids have become ideal model systems to study a large number of phenomena, for example, the formation of intermediate range order structures, condensed phases not seen in fluids driven by purely attractive or repulsive forces, the onset of particle aggregation under in- and out-of-equilibrium conditions, which results in the birth of reversible and irreversible aggregates or clusters whose topology and morphology depend additionally on the thermodynamic constrictions, and a particle dynamics that has a strong influence on the transport behaviour and rheological properties of the fluid. In this contribution, we study a system of particles interacting through a potential composed by a continuous succession of a short-ranged square-well (SW), an intermediate-ranged square-shoulder and a long-ranged SW. This potential model is chosen to systematically analyse the contribution of every component of the interaction potential on the phase behaviour, the microstructure, the morphology of the resulting aggregates and the transport phenomena of fluids described by competing interactions. Our results indicate that the inclusion of a barrier and a second well leads to new and interesting effects, which in addition result in variations of the physical properties associated to the competition among interactions.

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.

Early stages of aggregation in fluid mixtures of dimers and spheres: a theoretical and simulation study

We use Monte Carlo simulation and the Reference Interaction Site Model (RISM) theory of molecular fluids to investigate a simple model of colloidal mixture consisting of dimers, made up of two tangent hard monomers of different size, and hard spheres. In addition to steric repulsion, the two species interact via a square-well attraction only between small monomers and spheres. Recently, we have characterized the low-temperature regime of this mixture by Monte Carlo, reporting on the spontaneous formation of a wide spectrum of supramolecular aggregates [Prestipino et al., J. Phys. Chem. B, 2019, 123, 9272]. Here we focus on a regime of temperatures where, on cooling, the appearance of local inhomogeneties first, and the early stages of aggregation thereafter, are observed. In particular, we find signatures of aggregation in the onset of a low-wavevector peak in the structure factors of the mixture, as computed by both theory and simulation. Then, we link the structural information to the microscopic arrangement through a detailed cluster analysis of Monte Carlo configurations. In this regard, we devise a novel method to compute the maximum distance for which two spheres can be regarded as bonded together, a crucial issue in the proper identification of fluid aggregates. The RISM theory provides relatively accurate structural and thermodynamic predictions in comparison with Monte Carlo, but with slightly degrading performances as the fluid progresses inside the locally inhomogeneous phase. Our study certifies the efficacy of the RISM approach as a useful complement to numerical simulation for a reasoned analysis of aggregation properties in colloidal mixtures.

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.

Solution of disordered microphases in the Bethe approximation

The periodic microphases that self-assemble in systems with competing short-range attractive and long-range repulsive (SALR) interactions are structurally both rich and elegant. Significant theoretical and computational efforts have thus been dedicated to untangling their properties. By contrast, disordered microphases, which are structurally just as rich but nowhere near as elegant, have not been as carefully considered. Part of the difficulty is that simple mean-field descriptions make a homogeneity assumption that washes away all of their structural features. Here, we study disordered microphases by exactly solving a SALR model on the Bethe lattice. By sidestepping the homogenization assumption, this treatment recapitulates many of the key structural regimes of disordered microphases, including particle and void cluster fluids as well as gelation. This analysis also provides physical insight into the relationship between various structural and thermal observables, between criticality and physical percolation, and between glassiness and microphase ordering.

Formation and internal ordering of periodic microphases in colloidal models with competing interactions

Theory and simulations predict that colloidal particles with short-range attractive and long-range repulsive interactions form periodic microphases if there is a proper balance between the attractive and repulsive contributions. However, the experimental identification of such structures has remained elusive to date. Using molecular dynamics simulations, we investigate the phase behaviour of a model system that stabilizes a cluster-crystal, a cylindrical and a lamellar phase at low temperatures. Besides the transition from the fluid to the periodic microphases, we also observe the internal freezing of the clusters at a lower temperature. Finally, our study indicates that, for the chosen model parameters, the three periodic microphases are kinetically accessible from the fluid phase.

Inverse design of equilibrium cluster fluids applied to a physically informed model

Inverse design strategies have proven highly useful for the discovery of interaction potentials that prompt self-assembly of a variety of interesting structures. However, often the optimized particle interactions do not have a direct relationship to experimental systems. In this work, we show that Relative Entropy minimization is able to discover physically meaningful parameter sets for a model interaction built from depletion attraction and electrostatic repulsion that yield self-assembly of size-specific clusters. We then explore the sensitivity of the optimized interaction potentials with respect to deviations in the underlying physical quantities, showing that clustering behavior is largely preserved even as the optimized parameters are perturbed.

Smoothed profile method for direct numerical simulations of hydrodynamically interacting particles

A general method is presented for computing the motions of hydrodynamically interacting particles in various kinds of host fluids for arbitrary Reynolds numbers. The method follows the standard procedure for performing direct numerical simulations (DNS) of particulate systems, where the Navier-Stokes equation must be solved consistently with the motion of the rigid particles, which defines the temporal boundary conditions to be satisfied by the Navier-Stokes equation. The smoothed profile (SP) method provides an efficient numerical scheme for coupling the continuum fluid mechanics with the dispersed moving particles, which are allowed to have arbitrary shapes. In this method, the sharp boundaries between solid particles and the host fluid are replaced with a smeared out thin shell (interfacial) region, which can be accurately resolved on a fixed Cartesian grid utilizing a SP function with a finite thickness. The accuracy of the SP method is illustrated by comparison with known exact results. In the present paper, the high degree of versatility of the SP method is demonstrated by considering several types of active and passive particle suspensions.

Generalized equation of state for fluids: From molecular liquids to colloidal dispersions

In this work, a new parameterization for the Statistical Association Fluid Theory for potentials of Variable Range (SAFT-VR) is coupled to the discrete potential theory to represent the thermodynamic properties of several fluids, ranging from molecular liquids to colloidal-like dispersions. In this way, this version of the SAFT-VR approach can be straightforwardly applied to any kind of either simple or complex fluid. In particular, two interaction potentials, namely, the Lennard-Jones and the hard-core attractive Yukawa potentials, are discretized to study the vapor-liquid equilibrium properties of both molecular and complex liquids, respectively. Our results are assessed with Monte Carlo computer simulations and available and accurate theoretical results based on the self-consistent Ornstein-Zernike approximation.

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.

Cluster formation in symmetric binary SALR mixtures

The equilibrium cluster fluid state of a symmetric binary mixture of particles interacting through short-ranged attractive and long-ranged repulsive interactions is investigated through Monte Carlo simulations. We find that the clustering behavior of this system is controlled by the cross-interaction between the two types of particles. For a weak cross-attraction, the system displays a behavior that is a composite of the behavior of individual components, i.e., the two components can both form giant clusters independently and the clusters distribute evenly in the system. For a strong cross-attraction, we instead find that the resulting clusters are mixtures of both components. Between these limits, both components can form relatively pure clusters, but unlike clusters can join at their surfaces to form composite clusters. These insights should help to understand the mechanisms for clustering in experimental binary mixture systems and help tailor the properties of novel nanomaterials.

Local order and cluster formation in model fluids with competing interactions: a simulation and theoretical study

In SALR fluids, theory and simulation predict tiny morphological changes in the density profile occurring at the onset of clustering.

Evidence of Structural Inhomogeneities in Hard-Soft Dimeric Particles without Attractive Interactions

We perform Monte Carlo simulations of a simple hard-soft dimeric model constituted by two tangent spheres experiencing different interactions. Specifically, two hard spheres belonging to different dimers interact via a bare hard-core repulsion, whereas two soft spheres experience a softly repulsive Hertzian interaction. The cross correlations are soft as well. By exploring a wide range of temperatures and densities we investigate the capability of this model to document the existence of structural inhomogeneities indicating the possible onset of aggregates, even if no attraction is set. The fluid phase behavior is studied by analyzing structural and thermodynamical properties of the observed structures, in particular by computing radial distribution functions, structure factors and cluster size distributions. The numerical results are supported by integral equation theories of molecular liquids which allow for a finer and faster spanning of the temperature-density diagram. Our results may serve as a framework for a more systematic investigation of self-assembled structures of functionalized hard-soft dimers able to aggregate in a variety of structures widely oberved in colloidal dispersion.

Determination of Protein-Protein Interactions in a Mixture of Two Monoclonal Antibodies

The coformulation of monoclonal antibody (mAb) mixtures provides an attractive route to achieving therapeutic efficacy where the targeting of multiple epitopes is necessary. Controlling and predicting the behavior of such mixtures requires elucidating the molecular basis for the self- and cross-protein-protein interactions and how they depend on solution variables. While self-interactions are now beginning to be well understood, systematic studies of cross-interactions between mAbs in solution do not exist. Here, we have used static light scattering to measure the set of self- and cross-osmotic second virial coefficients in a solution containing a mixture of two mAbs, mAbA and mAbB, as a function of ionic strength and pH. mAbB exhibits strong association at a low ionic strength, which is attributed to an electrostatic attraction that is enhanced by the presence of a strong short-ranged attraction of nonelectrostatic origin. Under all solution conditions, the measured cross-interactions are intermediate self-interactions and follow similar patterns of behavior. There is a strong electrostatic attraction at higher pH values, reflecting the behavior of mAbB. Protein-protein interactions become more attractive with an increasing pH due to reducing the overall protein net charges, an effect that is attenuated with an increasing ionic strength due to the screening of electrostatic interactions. Under moderate ionic strength conditions, the reduced cross-virial coefficient, which reflects only the energetic contribution to protein-protein interactions, is given by a geometric average of the corresponding self-coefficients. We show the relationship can be rationalized using a patchy sphere model, where the interaction energy between sites i and j is given by the arithmetic mean of the i-i and j-j interactions. The geometric mean does not necessarily apply to all mAb mixtures and is expected to break down at a lower ionic strength due to the nonadditivity of electrostatic interactions.

Self-assembly in amphiphilic macromolecules with solvent exposed hydrophobic moieties

Self-assembly by amphiphilic molecules with solvent exposed hydrophobic groups are relevant in biomolecular systems as well as in technological applications. Here we study such self-assembly in these systems using a model system of spherical particles having charge at core but solvent repelling surface, using Monte-Carlo simulations and mean field treatment. We find that solvophobicity mediated attraction leads aggregation, while electrostatic repulsions control stability of finite clusters. The aggregation threshold relates the parameters of two interactions through an algebraic dependence. The study also qualitatively explains experimental observations on aggregation of misfolded proteins and can be useful guide to tune stability of nm sized self-assembly in systems with exposed hydrophobic groups.

Predicting High-Concentration Interactions of Monoclonal Antibody Solutions: Comparison of Theoretical Approaches for Strongly Attractive Versus Repulsive Conditions

Nonspecific protein-protein interactions of a monoclonal antibody were quantified experimentally using light scattering from low to high protein concentrations (c₂) and compared with prior work for a different antibody that yielded qualitatively different behavior. The c₂ dependence of the excess Rayleigh ratio (R^ex) provided the osmotic second virial coefficient (B₂₂) at low c₂ and the static structure factor (S_q=0) at high c₂, as a function of solution pH, total ionic strength (TIS), and sucrose concentration. Net repulsive interactions were observed at pH 5, with weaker repulsions at higher TIS. Conversely, attractive electrostatic interactions were observed at pH 6.5, with weaker attractions at higher TIS. Refined coarse-grained models were used to fit model parameters using experimental B₂₂ versus TIS data. The parameters were used to predict high-c₂ R^ex values via Monte Carlo simulations and separately with Mayer-sampling calculations of higher-order virial coefficients. For both methods, predictions for repulsive to mildly attractive conditions were quantitatively accurate. However, only qualitatively accurate predictions were practical for strongly attractive conditions. An alternative, higher resolution model was used to show semiquantitatively and quantitatively accurate predictions of strong electrostatic attractions at low c₂ and low ionic strength.

Effect of aggregation on adsorption phenomena

Adsorption at an attractive surface in a system with particles self-assembling into small clusters is studied by molecular dynamics simulation. We assume Lennard-Jones plus repulsive Yukawa tail interactions and focus on small densities. The relative increase in the temperature at the critical cluster concentration near the attractive surface (CCCS) shows a power-law dependence on the strength of the wall-particle attraction. At temperatures below the CCCS, the adsorbed layer consists of undeformed clusters if the wall-particle attraction is not too strong. Above the CCCS or for strong attraction leading to flattening of the adsorbed aggregates, we obtain a monolayer that for strong or very strong attraction consists of flattened clusters or stripes, respectively. The accumulated repulsion from the particles adsorbed at the wall leads to a repulsive barrier that slows down the adsorption process, and the accession time grows rapidly with the strength of the wall-particle attraction. Beyond the adsorbed layer of particles, a depletion region of a thickness comparable with the range of the repulsive tail of interactions occurs, and the density in this region decreases with increasing strength of the wall-particle attraction. At larger separations, the exponentially damped oscillations of density agree with theoretical predictions for self-assembling systems. Structural and thermal properties of the bulk are also determined. In particular, a new structural crossover associated with the maximum of the specific heat and a double-peaked histogram of the cluster size distribution are observed.

Using symmetry to elucidate the importance of stoichiometry in colloidal crystal assembly

We demonstrate a method based on symmetry to predict the structure of self-assembling, multicomponent colloidal mixtures. This method allows us to feasibly enumerate candidate structures from all symmetry groups and is many orders of magnitude more computationally efficient than combinatorial enumeration of these candidates. In turn, this permits us to compute ground-state phase diagrams for multicomponent systems. While tuning the interparticle potentials to produce potentially complex interactions represents the conventional route to designing exotic lattices, we use this scheme to demonstrate that simple potentials can also give rise to such structures which are thermodynamically stable at moderate to low temperatures. Furthermore, for a model two-dimensional colloidal system, we illustrate that lattices forming a complete set of 2-, 3-, 4-, and 6-fold rotational symmetries can be rationally designed from certain systems by tuning the mixture composition alone, demonstrating that stoichiometric control can be a tool as powerful as directly tuning the interparticle potentials themselves.

Experimental Evidence for a Cluster Glass Transition in Concentrated Lysozyme Solutions

Lysozyme is known to form equilibrium clusters at pH ≈ 7.8 and at low ionic strength as a result of a mixed potential. While this cluster formation and the related dynamic and static structure factors have been extensively investigated, its consequences on the macroscopic dynamic behavior expressed by the zero shear viscosity η₀ remain controversial. Here we present results from a systematic investigation of η₀ using two complementary passive microrheology techniques, dynamic light scattering based tracer microrheology, and multiple particle tracking using confocal microscopy. The combination of these techniques with a simple but effective evaporation approach allows for reaching concentrations close to and above the arrest transition in a controlled and gentle way. We find a strong increase of η₀ with increasing volume fraction ϕ with an apparent divergence at ϕ ≈ 0.35, and unambiguously demonstrate that this is due to the existence of an arrest transition where a cluster glass forms. These findings demonstrate the power of tracer microrheology to investigate complex fluids, where weak temporary bonds and limited sample volumes make measurements with classical rheology challenging.

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.

Dynamic properties of different liquid states in systems with competing interactions studied with lysozyme solutions

Recent studies of colloidal systems with a short-range attraction and long-range repulsion (SALR) have been demonstrated to have a generalized phase diagram with multiple liquid states defined by their structures. In this paper, we identify the different liquid states of previous experimentally studied lysozyme samples within this proposed generalized state diagram and explore the dynamic properties of each liquid state. We show that most lysozyme samples studied here and previously at low and intermediate concentrations are dispersed fluids while a few high concentration samples are randomly percolated liquids. In the dispersed fluid region, the short-time diffusion coefficient measured by neutron spin echo agrees well with the long time diffusion coefficient estimated with the solution viscosity. This dynamic feature is maintained even for some samples in the random percolated region. However, the short-time and long-time diffusion coefficients of random percolated fluids deviate at larger concentration and attraction strength. At high enough concentrations, the mean square displacement can be as slow as those of many glassy colloidal systems at time scales near the characteristic diffusion time even though these lysozyme samples remain in liquid states at the long-time limit. We thus identify the region in the generalized phase diagram where these equilibrium states with extremely slow local dynamics exist relative to bulk percolation and kinetic arrest (gel and glassy) transitions.