Coil dimensions of macromolecules in the presence of crowding colloids: Impact of crowder size

HYPOTHESIS

The size of a macromolecule in solution is strongly influenced by the size and concentration of added colloidal particles. Previous experimental and computer simulation studies have shown conflicting results regarding the influence of colloid size on coil compaction. We suggest the coil size depends on the Kuhn segment / nanoparticle size ratio and argue its subtle influence on the shrinking and expansion of a polymer chain.

METHODS

Based upon the work of van der Schoot (1998) [42] we propose theory that predicts how the colloid size mediates the compaction of macromolecules in crowded environments. The theoretical predictions are compared to self-consistent field (SCF) lattice computations and scattering experiments on polymer solutions exposed to crowders.

FINDINGS

The theoretical approach predicts that the shrinking of a polymer coil upon adding colloidal particles varies with the size of the colloids. We find coil shrinking is weakest when the colloidal particles are approximately the same size as the Kuhn segment length. The extent of coil shrinking passes a minimum at a specific colloid size relative to the Kuhn segment length, which is confirmed by SCF computations. Comparison with scattering experiments reveals that these experiments corroborate the extent of polymer shrinking at a given volume fraction of colloids. Our theoretical approach reproduces the functional dependence of the collapse on the crowder volume fraction.

Colloidal Monolayers with Short-Range Attractions and Dipolar Repulsions

This study investigates the structure and phase behavior of monolayers composed of model colloids that exhibit both short-range attractions and long-range repulsions. The model is motivated by recent experiments of hard-sphere-like microspheres that effectively interact via depletion attractions and dynamically tunable dipolar repulsions. Brownian dynamics simulations, grand canonical Monte Carlo methods, and approximate thermodynamic models are combined to investigate how the model's equilibrium and kinetically accessible solid and cluster fluid states depend on the relative strengths of the attractive and repulsive interactions. The results may help guide the understanding and design of quasi-two-dimensional colloidal assemblies with magnetic field tunable interactions.

Kagome Lattice Formation through Preliminary Structures in Colloidal Heteroepitaxy

Brownian dynamics simulations were conducted to investigate the formation of a Kagome lattice in the first epitaxial layer during colloidal heteroepitaxy. When the epitaxial particles were smaller than the substrate particles and the interactions were predominantly governed by the depletion force, a kagome lattice structure was successfully formed in the first epitaxial layer when the epitaxial particles were a specific size. The formation process of the Kagome lattice in the first layer was analyzed in detail. Initially, the epitaxial particles adhered to positions where they interacted with three substrate particles. When the density of epitaxial particles on the substrate increased by sedimentation, the epitaxial particles changed locations to attach to the sites where they interacted with one or two substrate particles, and a hexagonal structure formed. Subsequently, particles interacting with only one substrate particle detached from the substrate, resulting in the formation of a kagome lattice.

Assessing depletion attractions between colloidal nanocrystals

Adding nonadsorbing polymers to hard microsphere dispersions generates osmotic depletion attractions that can be quantitatively predicted and designed to manipulate colloidal phase behavior. Whether depletion described by classical theories is the mechanism for polymer-mediated nanosphere attractions is less evident. Colloidal hard nanospheres and nonadsorbing polymers are challenging to realize given the diverse interactions typically present in nanoparticle dispersions. Here, we use small-angle x-ray scattering to assess whether the depletion mechanism holds at the nanoscale, leveraging a recent finding that uncharged, oleate-capped indium oxide nanocrystals exhibit near-hard-sphere interactions in toluene. Classical modeling of polystyrene depletant as penetrable spheres predicts depletion-induced phase boundaries, nanocrystal second osmotic virial coefficients, and colloidal structuring in agreement with experiments for polymer radii of gyration up to 80% of the nanocrystal radius. Experimentally observed weakening of depletion interactions for larger polymer-to-nanocrystal size ratios qualitatively follows theoretical predictions that account for how polymer physics influences depletant interactions.

Kinetic phase diagram for two-step nucleation in colloid-polymer mixtures

Two-step crystallization via a metastable intermediate phase is often regarded as a non-classical process that lies beyond the framework of classical nucleation theory (CNT). In this work, we investigate two-step crystallization in colloid-polymer mixtures via an intermediate liquid phase. Using CNT-based seeding simulations, we construct a kinetic phase diagram that identifies regions of phase space where the critical nucleus is either liquid or crystalline. These predictions are validated using transition path sampling simulations at nine different relevant state points. When the critical nucleus is liquid, crystallization occurs stochastically during the growth phase, whereas for a crystalline critical nucleus, the crystallization process happens pre-critically at a fixed nucleus size. We conclude that CNT-based kinetic phase diagrams are a powerful tool for understanding and predicting "non-classical" crystal nucleation mechanisms.

Theoretical study of the impact of dilute nanoparticle additives on the shear elasticity of dense colloidal suspensions

Motivated by basic issues in soft matter physics and new experimental work on granule-nanoparticle mixtures, we systematically apply naive mode coupling theory with accurate microstructural input to investigate the elastic shear modulus of highly size asymmetric, dense, chemically complex, colloid-nanoparticle mixtures. Our analysis spans four equilibrium microstructural regimes: (i) entropic depletion induced colloid clustering, (ii) discrete adsorbed nanoparticle layers that induce colloid spatial dispersion, (iii) nanoparticle-mediated tight bridging network formation, and (iv) colloidal contact aggregation via direct attractions. Each regime typically displays a distinctive mechanical response to changing colloid-nanoparticle size ratio, packing fractions, and the strength and spatial range of interparticle attractive and repulsive interactions. Small concentrations of nanoparticles can induce orders of magnitude elastic reinforcements typically involving single or double exponential growth with increasing colloid and/or nanoparticle packing fraction. Depending on the system, the elementary stress scale can be controlled by the colloid volume, the nanoparticle volume, or a combination of both. Connections between local microstructural organization and the mixture elastic shear modulus are established. The collective structure factor of the relatively dilute nanoparticle subsystem exhibits strong spatial ordering and large osmotic concentration fluctuations imprinted by the highly correlated dense colloidal subsystem. The relevance of the theoretical results for experimental mixtures with large size asymmetry, particularly in the context of 3D ink printing and additive manufacturing, are discussed.

Mechanism of polymer molecular weight-dependent suppression and promotion of liquid-liquid phase separation of a protein solution by the addition of polymer

Polyethylene glycol (PEG) is a widely used precipitant to concentrate proteins. The effect of PEG is generally understood to be an entropic attraction between proteins due to the depletion effect of PEG around proteins. However, measurements by Bloustine et al. [Phys. Rev. Lett. 96, 087803 (2006)] of the liquid-liquid phase separation (LLPS) temperature have shown that a lysozyme solution is stabilized and destabilized by the addition of low and high molecular-weight PEG, respectively. They also presented a theoretical model of the LLPS temperature as a virial expansion of the free energy and concluded that, in addition to the depletion effect, the attractive interaction between protein and PEG is necessary to explain the experiments. In the present study, theoretical calculations based on liquid-state density functional theory utilizing coarse-grained models are conducted to demonstrate that the protein-PEG effective attraction is responsible for the suppression and promotion of LLPS upon the addition of low- and high-weight PEG, respectively. In contrast, if the interactions between the protein and the PEG are solely due to the excluded volume effect, PEG of any molecular weight destabilizes the solution. These results suggest the necessity to reconsider the conventional understanding of the effects of polymer addition, which have been historically attributed to solely the depletion force.

Accelerated amyloid fibril formation at the interface of liquid-liquid phase-separated droplets by depletion interactions

Amyloid fibril formation of α-synuclein (αSN) is a hallmark of synucleinopathies. Although the previous studies have provided numerous insights into the molecular basis of αSN amyloid formation, it remains unclear how αSN self-assembles into amyloid fibrils in vivo. Here, we show that αSN amyloid formation is accelerated in the presence of two macromolecular crowders, polyethylene glycol (PEG) (MW: ~10,000) and dextran (DEX) (MW: ~500,000), with a maximum at approximately 7% (w/v) PEG and 7% (w/v) DEX. Under these conditions, the two crowders induce a two-phase separation of upper PEG and lower DEX phases with a small number of liquid droplets of DEX and PEG in PEG and DEX phases, respectively. Fluorescence microscope images revealed that the interfaces of DEX droplets in the upper PEG phase are the major sites of amyloid formation. We consider that the depletion interactions working in micro phase-segregated state with DEX and PEG systems causes αSN condensation at the interface between solute PEG and DEX droplets, resulting in accelerated amyloid formation. Ultrasonication further accelerated the amyloid formation in both DEX and PEG phases, confirming the droplet-dependent amyloid formation. Similar PEG/DEX-dependent accelerated amyloid formation was observed for amyloid β peptide. In contrast, amyloid formation of β2-microglobulin or hen egg white lysozyme with a native fold was suppressed in the PEG/DEX mixtures, suggesting that the depletion interactions work adversely depending on whether the protein is unfolded or folded.

Structural analysis of physical gel networks using graph neural networks

We employ graph neural networks (GNN) to analyse and classify physical gel networks obtained from Brownian dynamics simulations of particles with competing attractive and repulsive interactions. Conventionally such gels are characterized by their position in a state diagram spanned by the packing fraction and the strength of the attraction. Gel networks at different regions of such a state diagram are qualitatively different although structural differences are subtile while dynamical properties are more pronounced. However, using graph classification the GNN is capable of positioning complete or partial snapshots of such gel networks at the correct position in the state diagram based on purely structural input. Furthermore, we demonstrate that not only supervised learning but also unsupervised learning can be used successfully. Therefore, the small structural differences are sufficient to classify the gel networks. Even the trend of data from experiments with different salt concentrations is classified correctly if the GNN was only trained with simulation data. Finally, GNNs are used to compute backbones of gel networks. As the node features used in the GNN are computed in linear time O ( N ) , the use of GNN significantly accelerates the computation of reduced networks on a particle level.

Simple method for the direct measurement of cohesive forces between microscopic particles

We present a simple and inexpensive method for measuring weak cohesive interactions. This technique is applied to the specific case of oil droplets with a depletion interaction, dispersed in an aqueous solution. The experimental setup involves creating a short string of droplets while immobilizing a single droplet. The droplets are held together via depletion interactions, and a single cohesive bond holds together nearest neighbours. Initially, the buoyant droplets are held in a flat horizontal chamber. The droplets float to the top of the chamber and are in contact with a flat glass interface. In the horizontal configuration, there is no component of the effective buoyant force acting in the plane of the chamber. The angle of the chamber is gradually increased, and the effective buoyant force acting on the string of droplets slowly increases. At a critical point, when the combination of gravity and buoyancy is equal to the cohesive force, the droplet string will detach from the immobile droplet. Our method allows for a simple direct measurement of cohesive forces on the tens of pico-Newton scale. To illustrate the validity of this technique, the droplet radii and concentration of depletant are varied, and their impact on the cohesive force is measured. This method offers a simple, accessible, and reproducible means of exploring cohesive interactions beyond the specific case of oil droplets and a depletion interaction.

Depletion potential, correlation functions and demixing transition in model colloid-polymer mixtures

We describe a theoretical framework to calculate depletion potential between colloid particles induced by non-adsorbing ideal polymer chains (s-species) and correlation functions in a coarse-grained one-component system of colloids (c-species). A Padé approximant is used to express the depletion potential as a pair potential with many-body contributions subsumed in it. The depletion potential and correlation functions of c-species are calculated using a self-consistent procedure. Results for several values of size ratio q=σsσc (σs and σc are, respectively diameters of the polymer chain and a colloid particle) and packing fractions of s- and c-species are reported. The spinodal curve and critical point of demixing transition are determined for several values of q. Calculated values are compared with values found from other theories and simulations.

Viscoelastic properties of colloidal systems with attractive solid particles at low concentration: A review, new results and interpretations

This paper concerns the viscoelastic properties and the resulting structure of colloidal systems with short-range attractions in the regime where the volume fraction f is small. Unlike the high ϕ regime, which is well understood in terms of mode-coupling theory (MCT), the low ϕ regime is still the subject of a debate based on different concepts such as percolation, diffusion-limited colloidal aggregation (DLCA), jamming, or cluster mode-coupling approach. Prior to the analysis of three examples of attractive systems at low ϕ values, a summary of concepts relevant to understanding the formation and properties of such attractive particles is discussed in the present study. Afterwards, we re-analyze the behaviour at a low ϕ of i) suspensions of carbon black (CB) particles, ii) suspensions of poly(methyl methacrylate) (PMMA) hard spheres with a depletion attraction induced by the addition of polystyrene (PS), and iii) suspensions of amino acid organogelator molecules which form rod-like objects. The rheological properties of these systems have been studied in detail and their response has been interpreted as being due either to a solid network discussed in relation to the jamming state diagram or to a suspension formed by jamming of clusters. Our analysis shows that these three systems are in fact cluster fluids and that their solid-like response corresponds to a change in their viscoelastic response, the elastic component G' becoming greater than the viscous component G" at low frequencies. Due to the presence of weak interparticle interactions in the tens range from 1 to 15 kBT, a liquid-like state is reversibly achieved at high frequencies, as indicated by the crossover of G' and G" as a function of frequency for a given concentration. Moreover, all these attractive particle systems at low ϕ show for both moduli a master curve which characterizes these cluster fluids and allows for the classification of these attractive particle systems.

Structures of the First Epitaxial Layer Created in Colloidal Heteroepitaxy

Brownian dynamics simulations have been performed to investigate the structural dependence of the first epitaxial layer in colloidal heteroepitaxy. When the epitaxial particles were larger than the substrate particles and the interactions were dominated by the depletion force, a hexagonal structure formed on a closely packed hexagonal substrate. The orientation of this hexagonal structure varied with the size ratio of the epitaxial to substrate particles to make the interaction between the substrate and epitaxial particles strong. When the sizes of the substrate and epitaxial particles were similar, long-period structures formed instead of hexagonal structures to strengthen the interaction between the substrate and epitaxial layer at the expense of the interaction between particles in the first epitaxial layer.

Potential effects on effective attraction between probes diffusing in colloidal crystal

Considering the significant influence of interparticle potentials on traditional depletion forces, we here employ computer simulations to investigate how varying potentials between particles affect the effective interaction of probes diffusing freely in a 2D colloidal crystal. Our results reveal that attractive potentials between the background particles and probes significantly modify the interprobe effective interactions, whereas long-range repulsive tails among the background particles have minimal impact. Furthermore, we observe contrary temperature dependencies of the effective force for soft and stiff repulsions between the background particles. These findings provide deeper insights into how direct interparticle potentials shape entropic-dominated effective forces mediated by colloidal crystals.

Entropy-Driven Crystallization of Hard Colloidal Mixtures of Polymers and Monomers

The most trivial example of self-assembly is the entropy-driven crystallization of hard spheres. Past works have established the similarities and differences in the phase behavior of monomers and chains made of hard spheres. Inspired by the difference in the melting points of the pure components, we study, through Monte Carlo simulations, the phase behavior of athermal mixtures composed of fully flexible polymers and individual monomers of uniform size. We analyze how the relative number fraction and the packing density affect crystallization and the established ordered morphologies. As a first result, a more precise determination of the melting point for freely jointed chains of tangent hard spheres is extracted. A synergetic effect is observed in the crystallization leading to synchronous crystallization of the two species. Structural analysis of the resulting ordered morphologies shows perfect mixing and thus no phase separation. Due to the constraints imposed by chain connectivity, the local environment of the individual spheres, as quantified by the Voronoi polyhedron, is systematically more spherical and more symmetric compared to that of spheres belonging to chains. In turn, the local environment of the ordered phase is more symmetric and more spherical compared to that of the initial random packing, demonstrating the entropic origins of the phase transition. In general, increasing the polymer content reduces the degree of crystallinity and increases the melting point to higher volume fractions. According to the present findings, relative concentration is another determining factor in controlling the phase behavior of hard colloidal mixtures based on polymers.

Freezing-mediated formation of supraproteins using depletion forces

Hypothesis Long-acting formulations such as microparticles, injectable depots and implantable devices can realize spatiotemporally controlled delivery of protein drugs to extend their therapeutic in vivo half-lives. To efficiently encapsulate the protein drugs into such drug delivery systems, (sub)micron-sized protein particles are needed. The formation of micronized supraproteins can be induced through the synergistic combination of attractive depletion forces and freezing. The size of the supraproteins can be fine-tuned from submicron to several microns by adjusting the ice crystallization rate through the freeze-quench depth, which is set by the target temperature. Methods Supraprotein micron structures were prepared from protein solutions under various conditions in the presence and absence of nonadsorbing polyethylene glycol. Scanning electron microscopy and dynamic light scattering were employed to determine the sizes of the supraproteins and real-time total internal reflection fluorescent microscopy was used to follow the supraprotein formation during freezing. The protein secondary structure was measured before and after micronization by circular dichroism. A phase diagram of a protein-polyethylene glycol mixture was theoretically predicted to investigate whether the depletion interaction can elucidate the phase behavior. Findings Micronized protein supraparticles could be prepared in a controlled manner by rapid freeze-drying of aqueous mixtures of bovine serum albumin, horseradish peroxidase and lysozyme mixed with polyethylene glycol. Upon freezing, the temperature quench initiates a phase separation process which is reminiscent of spinodal decomposition. This demixing is subsequently arrested during droplet phase separation to form protein-rich microstructures. The final size of the generated protein microparticles is determined by a competition between phase separation and cooling rate, which can be controlled by target temperature. The experimental phase diagram of the aqueous protein-polyethylene glycol dispersion aligns with predictions from depletion theory for charged colloids and nonadsorbing polymers.

Origin of Anomalously Large Depletion Zones in Like-Charged Colloid-Polyelectrolyte Mixtures

Depletion zones in polyelectrolyte solutions in contact with like-charged flat surfaces are investigated. Using a coupled self-consistent field and Debye-Hückel approach, an explicit expression for the thickness δ of the depletion layer is derived. It is found that δ∼δ_{n}+cκ^{-1}, where δ_{n} is the depletion thickness at a neutral surface, c is a function of the electrostatic characteristics of the system, and κ^{-1} is the Debye length. It is argued that the theory still holds beyond the mean-field approximation, which is confirmed by quantitative agreement between our theoretical results and experiments.

Nanoscopic Interfacial Hydrogel Viscoelasticity Revealed from Comparison of Macroscopic and Microscopic Rheology

Deviations between macrorheological and particle-based microrheological measurements are often considered to be a nuisance and neglected. We study aqueous poly(ethylene oxide) (PEO) hydrogels for varying PEO concentrations and chain lengths that contain microscopic tracer particles and show that these deviations reveal the nanoscopic viscoelastic properties of the particle-hydrogel interface. Based on the transient Stokes equation, we first demonstrate that the deviations are not due to finite particle radius, compressibility, or surface-slip effects. Small-angle neutron scattering rules out hydrogel heterogeneities. Instead, we show that a generalized Stokes-Einstein relation, accounting for an interfacial shell around tracers with viscoelastic properties that deviate from bulk, consistently explains our macrorheological and microrheological measurements. The extracted shell diameter is comparable to the PEO end-to-end distance, indicating the importance of dangling chain ends. Our methodology reveals the nanoscopic interfacial rheology of hydrogels and is applicable to different kinds of viscoelastic fluids and particles.

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.

Molecular Crowding: The History and Development of a Scientific Paradigm

It is now generally accepted that macromolecules do not act in isolation but "live" in a crowded environment, that is, an environment populated by numerous different molecules. The field of molecular crowding has its origins in the far 80s but became accepted only by the end of the 90s. In the present issue, we discuss various aspects that are influenced by crowding and need to consider its effects. This Review is meant as an introduction to the theme and an analysis of the evolution of the crowding concept through time from colloidal and polymer physics to a more biological perspective. We introduce themes that will be more thoroughly treated in other Reviews of the present issue. In our intentions, each Review may stand by itself, but the complete collection has the aspiration to provide different but complementary perspectives to propose a more holistic view of molecular crowding.

Nonequilibrium interactions between multi-scale colloids regulate the suspension microstructure and rheology

Understanding nonequilibrium interactions of multi-component colloidal suspensions is critical for many dynamical settings such as self-assembly and material processing. A key question is how the nonequilibrium distributions of individual components influence the effective interparticle interactions and flow behavior. In this work, we develop a first-principle framework to study a bidisperse suspension of colloids and depletants using a Smoluchowski equation and corroborated by Brownian dynamics (BD) simulations. Using nonlinear microrheology as a case study, we demonstrate that effective depletion interactions between driven colloids are sensitive to particle timescales out of equilibrium and cannot be predicted by equilibrium-based pair potentials like Asakura-Oosawa. Furthermore, we show that the interplay between Brownian relaxation timescales of different species plays a critical role in governing the viscosity of multi-component suspensions. Our model highlights the limitations of using equilibrium pair potentials to approximate interparticle interactions in nonequilibrium processes such as hydrodynamic flows and presents a useful framework for studying the transport of driven, interacting suspensions.

Phase behavior of mixtures of hard colloids and soft coarse-grained macromolecules

Effective "soft" interactions between macromolecules such as polymers, amphiphilic dendrimers, and suitably designed DNA based dendritic molecules have been shown to be purely repulsive and non-diverging. We report the structure and phase behavior of a mixture of hard colloids and soft coarse-grained macromolecules. Through the use of Reference Interaction Site Model theory and molecular dynamics simulations we find that hard colloids and soft macromolecules act as depletants toward each other, generating a medium-induced effective attraction. This effective attraction leads to the formation of non-dispersed phases at high densities. At low and high fractions of hard colloids the system macrophase separates into two disparate regions of hard colloids and soft macromolecules. However, this system microphase separates into a hard-rich and soft-rich self-assembled domains at intermediate compositions. The formation of microphase separated structure in this system of isotropic, disconnected, and purely repulsive colloids is surprising and quite novel. This behavior is likely due to a softening of the interface between hard-rich and soft-rich self-assembled domains. Molecular dynamics simulations have revealed that the microphase separated state has an overall disordered bicontinuous morphology. The hard-rich domain forms an ordered FCC structure and the soft-rich domain forms a disordered cluster-fluid, making the structure simultaneously ordered and disordered.

Phase behavior and dynamics in a colloid-polymer mixture under spherical confinement

From studies via molecular dynamics simulations, we report results on structure and dynamics in mixtures of active colloids and passive polymers that are confined inside a spherical container with a repulsive boundary. All interactions in the fully passive limit are chosen in such a way that in equilibrium coexistence between colloid-rich and polymer-rich phases occurs. For most part of the studies the chosen compositions give rise to Janus-like structure: nearly one side of the sphere is occupied by the colloids and the rest by the polymers. This partially wet situation mimics approximately a neutral wall in the fully passive scenario. Following the introduction of a velocity-aligning activity to the colloids, the shape of the polymer-rich domain changes to that of an ellipsoid, around the long axis of which the colloid-rich domain attains a macroscopic angular momentum. In the steady state, the orientation of this axis evolves via diffusion, magnitude of which depends upon the strength of activity, but only weakly.

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.

Blood Protein Exclusion from Polymer Brushes

We report interactions between adsorbed copolymers of poly(ethylene glycol) (PEG) in the presence of two abundant blood proteins, serum albumin and an immunoglobulin G, up to physiological blood concentrations. We directly and nonintrusively measure interactions between PEG triblock copolymers (PEG-PPO-PEG) adsorbed to hydrophobic colloids and surfaces using Total Internal Reflection Microscopy, which provides kT- and nanometer-scale resolution of interaction potentials (energy vs separation). In the absence of protein, adsorbed PEG copolymer repulsion is consistent with dimensions and architectures of PEG brushes on both colloids and surfaces. In the presence of proteins, we observe concentration dependent depletion attraction and no change to brush repulsion, indicating protein exclusion from PEG brushes. Because positive and negative protein adsorption are mutually exclusive, our observations of concentration dependent depletion attraction with no change to brush repulsion unambiguously indicate the absence of protein coronas at physiological protein concentrations. These findings demonstrate a direct sensitive approach to determine interactions between proteins and particle/surface coatings important to diverse biotechnology applications.

Depletion interaction mediated by semiflexible polymers

We present a simple mean-field theory to describe the polymer-mediated depletion attraction between colloidal particles that accounts for the polymer's chain stiffness. We find that for fixed polymer radius of gyration and volume fraction, the strength of this attraction increases with increasing chain stiffness in both dilute and semidilute concentration regimes. In contrast, the range of attraction monotonically decreases with chain stiffness in the dilute regime, while it attains a maximum in the semidilute regime. The obtained analytical expressions for the depletion interaction were compared with numerical self-consistent field lattice computations and shown to be in quantitative agreement. From the interaction potential between two spheres, we calculated the second osmotic virial coefficient B₂, which appears to be a convex function of chain stiffness. A minimum of B₂ as a function of chain stiffness was observed both in the numerical self-consistent field computations and the analytical theory. These findings help explain the general observation that semiflexible polymers are more effective depletants than flexible polymers and give insight into the phase behavior of mixtures containing spherical colloids and semiflexible polymers.

Heteroepitaxial Growth of Colloidal Crystals: Dependence of the Growth Mode on the Interparticle Interactions and Lattice Spacing

Epitaxial growth is one of the most important techniques for the control of crystal growth, especially for growing thin-film semiconductor crystals. Similarly, colloidal epitaxy, a template-assisted self-assembly method, is a powerful technique for controlling the structure of colloidal crystals. In this study, heteroepitaxial growth, which differs from homoepitaxial growth of conventional colloidal epitaxy, using foreign colloidal crystals as a substrate, was used to grow single-component colloidal crystal films. The Frank-van der Merwe (FM), Stranski-Krastanov (SK), and Volmer-Weber (VW) modes were observed, and the mode varied with the lattice-misfit ratio and interparticle interactions between the substrate and epitaxial phase. The transition of the growth mode (from SK to VW) and the coexistence of different growth modes (FM and VW) were observed, and their processes were revealed by in situ observation. Colloidal heteroepitaxy was confirmed to be useful for controlling structure, which will enable exploration of novel colloidal self-assembly structures.

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.

Machine-learning effective many-body potentials for anisotropic particles using orientation-dependent symmetry functions

Spherically-symmetric atom-centered descriptors of atomic environments have been widely used for constructing potential or free energy surfaces of atomistic and colloidal systems and to characterize local structures using machine learning techniques. However, when particle shapes are non-spherical, as in the case of rods and ellipsoids, standard spherically-symmetric structure functions alone produce imprecise descriptions of local environments. In order to account for the effects of orientation, we introduce two- and three-body orientation-dependent particle-centered descriptors for systems composed of rod-like particles. To demonstrate the suitability of the proposed functions, we use an efficient feature selection scheme and simple linear regression to construct coarse-grained many-body interaction potentials for computationally-efficient simulations of model systems consisting of colloidal particles with anisotropic shape: mixtures of colloidal rods and nonadsorbing polymer, hard rods enclosed by an elastic microgel shell, and ligand-stabilized nanorods. We validate the machine-learning (ML) effective many-body potentials based on orientation-dependent symmetry functions by using them in direct coexistence simulations to map out the phase behavior of colloidal rods and non-adsorbing polymer. We find good agreement with results obtained from simulations of the true binary mixture, demonstrating that the effective interactions are well-described by the orientation-dependent ML potentials.

Compact Peptoid Molecular Brushes for Nanoparticle Stabilization

Controlling the interfaces and interactions of colloidal nanoparticles (NPs) via tethered molecular moieties is crucial for NP applications in engineered nanomaterials, optics, catalysis, and nanomedicine. Despite a broad range of molecular types explored, there is a need for a flexible approach to rationally vary the chemistry and structure of these interfacial molecules for controlling NP stability in diverse environments, while maintaining a small size of the NP molecular shell. Here, we demonstrate that low-molecular-weight, bifunctional comb-shaped, and sequence-defined peptoids can effectively stabilize gold NPs (AuNPs). The generality of this robust functionalization strategy was also demonstrated by coating of silver, platinum, and iron oxide NPs with designed peptoids. Each peptoid (PE) is designed with varied arrangements of a multivalent AuNP-binding domain and a solvation domain consisting of oligo-ethylene glycol (EG) branches. Among designs, a peptoid (PE5) with a diblock structure is demonstrated to provide a superior nanocolloidal stability in diverse aqueous solutions while forming a compact shell (∼1.5 nm) on the AuNP surface. We demonstrate by experiments and molecular dynamics simulations that PE5-coated AuNPs (PE5/AuNPs) are stable in select organic solvents owing to the strong PE5 (amine)-Au binding and solubility of the oligo-EG motifs. At the vapor-aqueous interface, we show that PE5/AuNPs remain stable and can self-assemble into ordered 2D lattices. The NP films exhibit strong near-field plasmonic coupling when transferred to solid substrates.

The depletion thickness in solutions of semi-flexible polymers near colloidal surfaces: analytical approximations

We derive a simple, yet accurate approximate mean-field expression for the depletion thickness δsf of a solution of dilute semi-flexible polymers next to a hard surface. In the case of a hard wall this equation has the simple form δsf = δ0[1 - tanh(psf/δ0)], where psf accounts for the degree of flexibility and δ0 is the depletion thickness in the case of fully flexible polymers. For fixed polymer coil size, increasing the chain stiffness leads to a decrease in the depletion thickness. The approach is also extended to include higher polymer concentrations in the semidilute regime. The analytical expressions are in quantitative agreement with numerical self-consistent field computations. A remarkable finding is that there is a maximum in the depletion thickness as a function of the chain stiffness in the semidilute concentration regime. This also means that depletion attractions between colloidal particles reach a maximum for a certain chain stiffness, which may have important implications for the phase stability of colloid-polymer mixtures. The derived equations could be useful for the description of interactions in- and phase stability of mixtures of colloids and semi-flexible polymers.

How enzymatic activity is involved in chromatin organization

Spatial organization of chromatin plays a critical role in genome regulation. Previously, various types of affinity mediators and enzymes have been attributed to regulate spatial organization of chromatin from a thermodynamics perspective. However, at the mechanistic level, enzymes act in their unique ways and perturb the chromatin. Here, we construct a polymer physics model following the mechanistic scheme of Topoisomerase-II, an enzyme resolving topological constraints of chromatin, and investigate how it affects interphase chromatin organization. Our computer simulations demonstrate Topoisomerase-II's ability to phase separate chromatin into eu- and heterochromatic regions with a characteristic wall-like organization of the euchromatic regions. We realized that the ability of the euchromatic regions to cross each other due to enzymatic activity of Topoisomerase-II induces this phase separation. This realization is based on the physical fact that partial absence of self-avoiding interaction can induce phase separation of a system into its self-avoiding and non-self-avoiding parts, which we reveal using a mean-field argument. Furthermore, motivated from recent experimental observations, we extend our model to a bidisperse setting and show that the characteristic features of the enzymatic activity-driven phase separation survive there. The existence of these robust characteristic features, even under the non-localized action of the enzyme, highlights the critical role of enzymatic activity in chromatin organization.

Depletion Induced Demixing and Crystallization in Binary Colloids Subjected to an External Potential Barrier

Depletion interaction plays an important role in determining the structural and dynamical properties of binary colloidal mixtures. We have investigated the effect of the attractive depletion interaction between an external potential barrier and larger species in the binary mixture on the phase behavior of a binary colloidal mixture using canonical-isokinetic ensemble molecular dynamics simulations. The demixing of the binary mixture due to this depletion interaction increases as the volume fraction increases, and a pure phase of larger particles forms in the region of the potential barrier. The local density of this pure phase is high enough that a face centered cubic crystalline domain is formed at this region. This crystalline phase diffuses perpendicular to the external potential barrier, indicating that moving crystals can be obtained in an equilibrium system. The temperature dependence of diffusivity of larger particles is non-Arrhenius and changes from sub-Arrhenius to super-Arrhenius as the volume fraction increases. This crossover from sub-Arrhenius to super-Arrhenius diffusion coincides with the crystalline formation near the potential barrier.

Second-virial theory for shape-persistent living polymers templated by disks

Living polymers composed of noncovalently bonded building blocks with weak backbone flexibility may self-assemble into thermoresponsive lyotropic liquid crystals. We demonstrate that the reversible polymer assembly and phase behavior can be controlled by the addition of (nonadsorbing) rigid colloidal disks which act as an entropic reorienting "template" onto the supramolecular polymers. Using a particle-based second-virial theory that correlates the various entropies associated with the polymers and disks, we demonstrate that small fractions of discotic additives promote the formation of a polymer nematic phase. At larger disk concentrations, however, the phase is disrupted by collective disk alignment in favor of a discotic nematic fluid in which the polymers are dispersed antinematically. We show that the antinematic arrangement of the polymers generates a nonexponential molecular-weight distribution and stimulates the formation of oligomeric species. At sufficient concentrations the disks facilitate a liquid-liquid phase separation which can be brought into simultaneously coexistence with the two fractionated nematic phases, providing evidence for a four-fluid coexistence in reversible shape-dissimilar hard-core mixtures without cohesive interparticle forces. We stipulate the conditions under which such a phenomenon could be found in experiment.

Machine learning many-body potentials for colloidal systems

Simulations of colloidal suspensions consisting of mesoscopic particles and smaller species such as ions or depletants are computationally challenging as different length and time scales are involved. Here, we introduce a machine learning (ML) approach in which the degrees of freedom of the microscopic species are integrated out and the mesoscopic particles interact with effective many-body potentials, which we fit as a function of all colloid coordinates with a set of symmetry functions. We apply this approach to a colloid-polymer mixture. Remarkably, the ML potentials can be assumed to be effectively state-independent and can be used in direct-coexistence simulations. We show that our ML method reduces the computational cost by several orders of magnitude compared to a numerical evaluation and accurately describes the phase behavior and structure, even for state points where the effective potential is largely determined by many-body contributions.

Self-consistent field theory study of polymer-mediated colloidal interactions in solution: Depletion effects and induced forces

Polymer-mediated colloidal interactions control the stability and phase properties of colloid-polymer mixtures that are critical for a wide range of important applications. In this work, we develop a versatile self-consistent field theory (SCFT) approach to study this type of interaction based on a continuum confined polymer solution model with explicit solvent and confining walls. The model is formulated in the grand canonical ensemble, and the potential of mean force for the polymer-mediated interaction is computed from grand potentials. We focus on the case of non-adsorbing linear polymers and present a systematic investigation on depletion effects using SCFT. The properties of confined polymer solutions are probed, and mean-field profiles of induced interactions are shown across different physical regimes. We expose a detailed parametric dependence of the interaction, concerning both attractive and repulsive parts, on polymer concentration, chain length, and solvent quality and explore the effect of wall surface roughness, demonstrating the versatility of the proposed approach. Our findings show good agreement with previous numerical studies and experiments, yet extend prior work to new regimes. Moreover, the mechanisms of depletion attraction and repulsion, along with the influence of individual control factors, are further discussed. We anticipate that this study will provide useful insights into depletion forces and can be readily extended to examine more complex colloid-polymer mixtures.

Phase stability of colloidal spheres mixed with semiflexible supramolecular polymers

HYPOTHESIS

Mixtures of colloids and supramolecular polymers may exhibit stimuli-responsive phase behaviour. However, in theoretical descriptions of such systems, the polymers are commonly described either as flexible chains or as rigid rods, while in experimental systems supramolecular polymers usually fall in between these two limits. We expect the flexibility of the polymers to have a profound effect on the stimuli-responsive phase behaviour.

THEORY

We propose a general approach to predict the phase behaviour of colloidal hard spheres mixed with covalent or supramolecular polymers of arbitrary persistence length using free volume theory and an interpolation between flexible and rigid chains.

FINDINGS

The binodals are predicted to shift to lower monomer concentrations as the persistence length is increased, making the polymers more efficient depletants. The persistence length is therefore an additional degree of freedom for manipulating the phase behaviour of colloid-polymer mixtures. We show that by manipulating the persistence length of temperature responsive supramolecular polymers, a wide range of phase diagrams with various topologies can be obtained. For example, we find phase diagrams with a critical point but no triple point or displaying two triple points for temperature-sensitive supramolecular polymers mixed with hard spheres.

Influence of solvent quality on depletion potentials in colloid-polymer mixtures

As first explained by the classic Asakura-Oosawa (AO) model, effective attractive forces between colloidal particles induced by depletion of nonadsorbing polymers can drive demixing of colloid-polymer mixtures into colloid-rich and colloid-poor phases, with practical relevance for purification of water, stability of foods and pharmaceuticals, and macromolecular crowding in biological cells. By idealizing polymer coils as effective penetrable spheres, the AO model qualitatively captures the influence of polymer depletion on thermodynamic phase behavior of colloidal suspensions. In previous work, we extended the AO model to incorporate aspherical polymer conformations and showed that fluctuating shapes of random-walk coils can significantly modify depletion potentials [W. K. Lim and A. R. Denton, Soft Matter 12, 2247 (2016); J. Chem. Phys. 144, 024904 (2016)]. We further demonstrated that the shapes of polymers in crowded environments sensitively depend on solvent quality [W. J. Davis and A. R. Denton, J. Chem. Phys. 149, 124901 (2018)]. Here, we apply Monte Carlo simulation to analyze the influence of solvent quality on depletion potentials in mixtures of hard-sphere colloids and nonadsorbing polymer coils, modeled as ellipsoids whose principal radii fluctuate according to random-walk statistics. We consider both self-avoiding and non-self-avoiding random walks, corresponding to polymers in good and theta solvents, respectively. Our simulation results demonstrate that depletion of polymers of equal molecular weight induces much stronger attraction between colloids in good solvents than in theta solvents and confirm that depletion interactions are significantly influenced by aspherical polymer conformations.

Structural and short-time vibrational properties of colloidal glasses and supercooled liquids in the vicinity of the re-entrant glass transition

We investigate the short-time vibrational properties and structure of two-dimensional, bidisperse, colloidal glasses and supercooled liquids in the vicinity of the re-entrant glass transition, as a function of interparticle depletion attraction strength. The long-time spatiotemporal dynamics of the samples are measured to be non-monotonic, confirming that the suspensions evolve from repulsive glass to supercooled liquid to attractive glass with increasing depletion attraction. Here, we search for vibrational signatures of the re-entrant behavior in the short-time spatiotemporal dynamics, i.e., dynamics associated with particle motion inside its nearest-neighbor cage. Interestingly, we observe that the anharmonicity of these in-cage vibrations varies non-monotonically with increasing attraction strength, consistent with the non-monotonic long-time structural relaxation dynamics of the re-entrant glass. We also extract effective spring constants between neighboring particles; we find that spring stiffness involving small particles also varies non-monotonically with increasing attraction strength, while stiffness between large particles increases monotonically. Last, from study of depletion-dependent local structure and vibration participation fractions, we gain microscopic insight into the particle-size-dependent contributions to short-time vibrational modes in the glass and supercooled liquid states.

Diffusion, phase behavior, and gelation in a two-dimensional layer of colloids in osmotic equilibrium with a polymer reservoir

The addition of enough non-adsorbing polymers to an otherwise stable colloidal suspension gives rise to a variety of phase behaviors and kinetic arrest due to the depletion attraction induced between the colloids by the polymers. We report a study of these phenomena in a two-dimensional layer of colloids. The three-dimensional phenomenology of crystal-fluid coexistence is reproduced, but gelation takes a novel form, in which the strands in the gel structure are locally crystalline. We compare our findings with a previous simulation and theory and find substantial agreement.

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.