Colloid-polymer Mixtures Research Articles

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.

Scaling variation in the pinch-off of colloid-polymer mixtures

Hypothesis: The scaling laws of drop pinch-off are known to be affected by drop compositions including dissolved polymers and non-Brownian particles. When the size of the particles is comparable to the characteristic length scale of the polymer network, these particles may interact strongly with the polymer environment, leading to new types of scaling behaviors not reported before.Experiments: Using high-speed imaging, we experimentally studied the time evolution of the neck diameter hmin of drops composed of silica nanoparticles dispersed in PEO solution when extruded from a nozzle.Findings: After initial Newtonian necking with hmin ∼ t2/3, the subsequent stage may exhibit scaling variation, characterized by either exponential or power-law decay, depending on the nanoparticle volume fraction ϕ. The exponential decay hmin ∼ e-t/τ signifies the coil-stretch transition in typical viscoelastic suspensions. We conducted an analysis of the power-law scenario hmin ∼ tα at high ϕ, categorizing the entire process into three distinct regimes based on the exponents α. The dependences of critical thicknesses at transition points and exponents on polymer concentration offer initial insights into the potential transition from heterogeneous to homogeneous thinning in the mixture. This novel scaling variation bears implications for accurately predicting and controlling droplet fragmentation in industrial applications.

Percolation of nonequilibrium assemblies of colloidal particles in active chiral liquids.

The growing interest in the non-equilibrium assembly of colloidal particles in active liquids is driven by the motivation to create novel structures endowed with tunable properties unattainable within the confines of equilibrium systems. Here, we present an experimental investigation of the structural features of colloidal assemblies in active liquids of chiral E. coli. The colloidal particles form dynamic clusters due to the effective interaction mediated by active media. The activity and chirality of the swimmers strongly influence the dynamics and local ordering of colloidal particles, resulting in clusters with persistent rotation, whose structure differs significantly from those in equilibrium systems with attractive interactions, such as colloid-polymer mixtures. Our colloid-bacteria mixture displays several hallmark features of a percolation transition at a critical density, where the clusters span the system size. A closer examination of the critical exponents associated with cluster size distribution, the average cluster size, and the correlation length in the vicinity of the critical density shows deviations from the prediction of the standard continuum percolation model. Therefore, our experiments reveal a richer phase behavior of colloidal assemblies in active liquids.

Pinching dynamics and extensional rheology of dense colloidal suspensions with depletion attractions

We study the extensional flow properties by characterizing the capillarity-driven pinching dynamics of dense colloidal suspensions at a constant volume fraction ϕ=0.40 with polymer-induced depletion interactions using a dripping-onto-substrate (DoS) protocol. Methacrylate copolymer particles with dimethylacrylamide copolymer brushes are suspended in a refractive-index- and density-matched mixture of 80 (w/w)% glycerol in water with NaCl added to screen the electrostatic repulsions. Depletion attractions between the colloids are introduced by adding polyacrylamide polymers of weight and dispersity. The addition of polymer delays and modifies the pinch-off dynamics of the dense suspensions, depending on the size and dispersity of the polymer. The extensional relaxation time λE of suspensions collapses as a function of the normalized free volume polymer concentration c/c∗ with the corresponding polymer solutions, indicating that the elastic properties of the polymer solutions control the extensional time scale. Following the results of our previous study [Soetrisno et al., Macromolecules 56, 4919–4928 (2023)], the polymer size determines the scaling exponent of λE for colloid-polymer mixtures in the dilute regime and high dispersity shifts the concentration where the scaling of λE transitions from power-law to linear. The filament lifespans tf of colloid-polymer mixtures and of polymer solutions collapse onto a master curve as a function of c/c∗ when normalized by the filament lifespan of the corresponding fluid without polymer tf,0. These results provide insight into the role of the polymer size in dictating the pinching dynamics and extensional rheology of colloid-polymer mixtures and further suggest that the shear and extensional responses of these mixtures can be separately tuned through the concentrations of the two constituents.

Generalized geometric criteria for the absence of effective many-body interactions in the Asakura–Oosawa model

We investigate variants of the Asakura–Oosawa (AO) model for colloid-polymer mixtures, represented by hard classical particles interacting via their excluded volume. The interaction between the polymers is neglected but the colloid-polymer and colloid-colloid interactions are present and can be condensed into an effective depletion interaction among the colloids alone. The original AO model involves hard spherical particles in three spatial dimensions with colloidal radii R and the so-called depletion radius δ of the polymers, such that the minimum possible center-to-center distance between polymers and colloids allowed by the excluded-volume constraints is R + δ. It is common knowledge among physicists that there are only pairwise effective depletion interactions between the colloids if the geometric condition δ/R<2/3−1 is fulfilled. In this case, triplet and higher-order many body interactions are vanishing and the equilibrium statistics of the binary mixture can exactly be mapped onto that of an effective one-component system with the effective depletion pair-potential. Here we rigorously prove that the criterion δ/R<2/3−1 is both sufficient and necessary to guarantee the absence of triplet and higher-order many body interactions among the colloids. For an external hard wall confining the system, we also include a criterion which guarantees that the system can be exactly mapped onto one with effective external one-body interactions. Our general formulation also accounts for polydisperse mixtures and anisotropic shapes of colloids in any spatial dimension. In those cases where the resulting condition is only sufficient, we further demonstrate how to specify improved bounds.

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.

High salinity effects on the depletion attraction in colloid-polymer mixtures

HypothesisDepletion attraction induced by polymers can be employed across multiple disciplines. Previous studies implied that besides screening the electrostatic repulsion between colloids, adding salt may also affect the polymers even in a nonpolar solvent. Here, we study the depletion-induced gelation of a colloid-polymer model system, focusing on the salt effects on the depletion attraction. ExperimentsConfocal microscopy was used to quantitatively characterize the colloidal gels formed by the polymethylmethacrylate (PMMA)/polystyrene (PS) model system. The attraction experienced by colloids was estimated by correlating the colloidal dynamics with the local structure. Correspondingly, static light scattering was employed to systematically investigate the polymers. The resulting radius of gyration Rg and osmotic pressure were used to evaluate the depletion attraction offered by polymers. FindingsSalt was discovered to lower the strength of inter-particle attraction, which can be attributed to the salt-induced decrease in Rg. The depletion attraction grew sublinearly with c, owing to the considerable decrease in Rg in the good solvent as c increased. We demonstrated how the close form equations in the framework of renormalization group theory can be employed to predict the depletion interaction using the properly determined zero-concentration radius of gyration.

Shear-induced vorticity aligned flocs in a temperature responsive colloid-polymer mixture

Shear driven patterning is seen in many soft matter systems. We use rheology and optical microscopy to probe the structures formed when we shear a colloid-polymer mixture containing temperature-sensitive microgel particles. By increasing the temperature, we can increase the particle attraction and transition from liquid-like to gel-like behavior. And by applying shear flow to the sample as the temperature and, hence, state of the system changes, we can affect the morphology of mesoscopic colloidal clusters. We can produce gels comprised of fibrous, elongated colloid-dense clusters, or we can form more isotropic clusters. The rheology is measured and shear-induced flocculation observed for colloid-polymer systems with different cluster morphologies. At shear rates high enough to produce elongated clusters but low enough to not break clusters apart, we observe log-like flocs that are aligned with the vorticity direction and roll between the parallel plates of our rheometer.

Response to "Comment on 'Communication: Is directed percolation in colloid-polymer mixtures linked to dynamic arrest?'" [J. Chem. Phys. 157, 027101 (2022)

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Comment on "Communication: Is directed percolation in colloid-polymer mixtures linked to dynamic arrest?" [J. Chem. Phys. 148, 241101 (2018)

In their Communication [J. Chem. Phys. 148, 241101 (2018)], Richard et al. state that in the work of Kohl et al. [Nat. Commun. 7, 11817 (2016)], a mechanism for dynamical arrest in temporal networks has been proposed that actually has never been proposed (and would be obviously wrong) in this context. The actual findings of Kohl et al. are not tested nor affected by the communication. The work of Richard et al. rests on simulations in a regime of the phase diagram that significantly differs from the one that Kohl et al. consider. In this Comment, it is shown that both the effective density and the rescaled second virial coefficient indicate that the comparison presented by Richard et al. is invalid. Therefore, the implications that are based on this comparison are incorrect. There is no indication for a disagreement between the simulations of Richard et al. and those of Kohl et al., and I am confident that upon consistent comparison and interpretation of the results, both works can contribute to a more comprehensive picture of gel-forming systems.

Wetting Behavior of a Three-Phase System in Contact with a Surface.

We extend the Cahn-Landau-de Gennes mean field theory of wetting in binary mixtures to understand the wetting thermodynamics of a three phase system (e.g., polymer dispersed liquid crystals or polymer-colloid mixtures) that is in contact with an external surface, which prefers one of the phases. Using a model free-energy, which has three minima in its landscape, we show that as the central minimum becomes more stable compared to the remaining ones, the bulk phase diagram encounters a triple point and then bifurcates and we observe a novel non-monotonic dependence of the surface tension as a function of the stability of the central minimum. We show that this non-monotonicity in surface tension is associated with a complete to partial wetting transition. We obtain the complete wetting phase behavior as a function of phase stability and the surface interaction parameters when the system is close to the bulk triple point. The model free-energy that we use is qualitatively similar to that of a renormalized free energy, which arises in the context of polymer-liquid crystal mixtures. Finally, we study the thermodynamics of wetting for an explicit polymer-liquid crystal mixture and show that its thermodynamics is similar to that of our model free-energy.

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.

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

HypothesisMixtures 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. TheoryWe 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. FindingsThe 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.

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.

Structural and dynamical properties of dilute gel networks in colloid-polymer mixtures.

The competition of short-ranged depletion attraction and long-ranged repulsion between colloidal particles in colloid-polymer mixtures leads to the formation of heterogeneous gel-like structures. Our special focus will be on the states where the colloids arrange in thin strands that span the whole system and that we will refer to as dilute gel networks. These states occur at low packing fractions for attractions that are stronger than those at both the binodal line of the equilibrium gas-liquid phase separation and the directed percolation transition line. By using Brownian dynamics simulations, we explore the formation, structure, and aging dynamics of dilute gel networks. The essential connections in a dilute gel network are determined by constructing reduced networks. We compare the observed properties to those of clumpy gels or cluster fluids. Our results demonstrate that both the structure and the (often slow) dynamics of the stable or meta-stable heterogeneous states in colloid-polymer mixtures possess distinct features on various length and time scales and thus are richly diverse.

Liquid-liquid phase separation in an inhomogeneous ternary colloid-polymer mixture.

Suspended colloids are often considered as models for molecules, which are sufficiently big so that they can be observed directly in (light) microscopes and for which the effective interaction among each other can be tailored. The Asakura-Oosawa model of ideal colloid-polymer mixtures captures the idea of tuning the interaction between the colloids via a potential, which possesses a range set by the size of the polymers and an attractive strength characterized by the (reservoir) number density of the polymers, which plays the role of an inverse temperature. The celebrated Asakura-Oosawa depletion potential allows one to recreate the bulk phase diagram of a simple fluid by employing a colloid-polymer mixture. This has been verified in theory, by computer simulations, and via experiments. Here, we study the phase behavior of a confined colloid-polymer mixture with two polymer species. The sizes and densities are chosen such that the resulting bulk phase diagram exhibits a second stable critical point within the framework of the classical density functional theory. Our results suggest that a suitably tuned colloid-polymer mixture can be an interesting model system to study fluids with two critical points.

Phase transitions and phase coexistence: equilibrium systems versus externally driven or active systems - Some perspectives

ABSTRACT A tutorial introduction to the statistical mechanics of phase transitions and phase coexistence is presented, starting out from equilibrium systems and nonequilibrium steady-state situations in externally driven systems, such as unmixing of sheared binary fluid mixtures, the driven lattice gas model, and the onset of Rayleigh-Bénard convection. Then, some models for phase separation in models for active systems, where particles possess internal motility, are discussed, emphasizing what one can learn by extending analysis methods to study phase transitions in equilibrium systems by computer simulations to active systems. Specific examples will include colloid-polymer mixtures where the colloids are assumed to be active particles, and active Brownian particles. The extent to which concepts familiar from the study of equilibrium systems are still useful will be critically discussed.