Depletion-induced phase separation in colloid-polymer mixtures

Phase separation can be induced in a colloidal dispersion by adding non-adsorbing polymers. Depletion of polymer around the colloidal particles induces an effective attraction, leading to demixing at sufficient polymer concentration. This communication reviews theoretical and experimental work carried out on the polymer-mediated attraction between spherical colloids and the resulting phase separation of the polymer-colloid mixture. Theoretical studies have mainly focused on the limits where polymers are small or large as compared to the colloidal size. Recently, however, theories are being developed that cover a wider colloid-polymer size ratio range. In practical systems, size polydispersity and polyelectrolytes (instead of neutral polymers) and/or charges on the colloidal surfaces play a role in polymer-colloid mixtures. The limited amount of theoretical work performed on this is also discussed. Finally, an overview is given on experimental investigations with respect to phase behavior and results obtained with techniques enabling measurement of the depletion-induced interaction potential, the structure factor, the depletion layer thickness and the interfacial tension between the demixed phases of a colloid-polymer mixture.     

Phase behavior and particle size cutoff effects in polydisperse fluids

We report a joint simulation and theoretical study of the liquid-vapor phase behavior of a fluid in which polydispersity in the particle size couples to the strength of the interparticle interactions. Attention is focused on the case in which the particle diameters are distributed according to a fixed Schulz form with degree of polydispersity delta = 14%. The coexistence properties of this model are studied using grand canonical ensemble Monte Carlo simulations and moment free energy calculations. We obtain the cloud and shadow curves as well as the daughter phase density distributions and fractional volumes along selected isothermal dilution lines. In contrast to the case of size-independent interaction [N. B. Wilding et al., J. Chem. Phys. 121, 6887 (2004)], the cloud and shadow curves are found to be well separated, with the critical point lying significantly below the cloud curve maximum. For densities below the critical value, we observe that the phase behavior is highly sensitive to the choice of upper cutoff on the particle size distribution. We elucidate the origins of this effect in terms of extremely pronounced fractionation effects and discuss the likely appearance of new phases in the limit of very large values of the cutoff.     

Isotropic–nematic phase separation and demixing in mixtures of spherical nanoparticles with length‐polydisperse nanorods

An extension of Onsager theory is developed to simulate isotropic–nematic phase separation in a mixture of spheres with length‐polydisperse system of rods. This work is motivated by recent experimental data on nanorod liquid crystals. Prior theoretical investigations indicate that both polydispersity and the presence of spheres should increase the biphasic–nematic phase transition, that is, the nematic cloud point. Results indicate that the phase diagrams undergo drastic changes depending upon both particle geometry and rod length polydispersity. The key geometric factor is the ratio between the sphere diameter and the rod diameter. In general, length fractionation is enhanced by the addition of spheres, which may be experimentally advantageous for separating short nanorods from a polydisperse population. Simulation results also indicate that the nematic cloud and shadow curves may cross one another because of the scarcity of spheres in the shadow phase. In general, these results do indicate that the nematic cloud point increases as a function of sphere loading; however, in certain areas of phase space, this relationship is nonmonotonic such that the nematic cloud point may actually decrease with the addition of spheres. This work has application to a wide range of nanoparticle systems, including mixtures of spherical nanoparticles with nanorods or nanotubes. Additionally, a number of nonspherical particles and structures may behave as spheres, including crumpled graphene and tightly coiled polymers. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Liquid-gas coexistence and critical point shifts in size-disperse fluids

Specialized Monte Carlo simulations and the moment free energy (MFE) method are employed to study liquid-gas phase equilibria in size-disperse fluids. The investigation is made subject to the constraint of fixed polydispersity, i.e., the form of the "parent" density distribution rho(0)(sigma) of the particle diameters sigma, is prescribed. This is the experimentally realistic scenario for, e.g., colloidal dispersions. The simulations are used to obtain the cloud and shadow curve properties of a Lennard-Jones fluid having diameters distributed according to a Schulz form with a large (delta approximately 40%) degree of polydispersity. Good qualitative accord is found with the results from a MFE method study of a corresponding van der Waals model that incorporates size dispersity both in the hard core reference and the attractive parts of the free energy. The results show that polydispersity engenders considerable broadening of the coexistence region between the cloud curves. The principal effect of fractionation in this region is a common overall scaling of the particle sizes and typical interparticle distances, and we discuss why this effect is rather specific to systems with Schulz diameter distributions. Next, by studying a family of such systems with distributions of various widths, we estimate the dependence of the critical point parameters on delta. In contrast to a previous theoretical prediction, size dispersity is found to raise the critical temperature above its monodisperse value. Unusually for a polydisperse system, the critical point is found to lie at or very close to the extremum of the coexistence region in all cases. We outline an argument showing that such behavior will occur whenever polydispersity affects only the range, rather than the strength of the interparticle interactions.     

Phase behavior of charge stablized colloid dispersion with added water soluble polymers

Demixing and colloidal crystallization in the mixture of charge stabilized colloidal poly(methyl methacrylate) particles and soluble poly(ethylene oxide) were investigated by means of synchrotron small-angle X-ray scattering (SAXS) technique. Phase diagram of the mixture was obtained based on visual inspection and SAXS results. The phase behavior is determined as a function of the concentration of the polymer as well as the volume fraction of the colloidal particles. The system shows a one phase region when the concentration of the polymer is low, whereas a two-phase region is present when the concentration of the polymer is larger than a critical concentration at certain volume fraction of the colloids. Interestingly, a face centered cubic colloidal crystalline structure was formed under certain conditions, which has been rarely observed in experiments of colloid-polymer mixtures with competing interactions.     

Preparation and characterization of particles with small differences in polydispersity

Colloidal particles are widely used both in fundamental research and in materials science. One important parameter influencing the physical properties of colloidal materials is the particle size distribution (polydispersity) of the colloidal particles. Recent work on colloidal crystallization has demonstrated that even subtle changes in polydispersity can have significant effects. In this study we present centrifugation techniques for subtly manipulating the width and the shape of the particle size distribution, for polydispersities less than 10%. We use scanning electron microscopy as well as dynamic and static light scattering to characterize the particle size distributions. We compare the results and highlight the difficulties associated with the determination of accurate particle size distributions.     

Stimuli‐Responsive Shape Switching of Polymer Colloids by Temperature‐Sensitive Absorption of Solvent

The dynamic manipulation of colloidal particle shape offers a novel design mechanism for the creation of advanced responsive materials. To this end, we introduce a versatile new strategy for shape control of anisotropic polymeric colloidal particles. The concept utilizes temperature‐sensitive absorption of a suitable solvent from a binary mixture. Specifically, increasing the temperature in the vicinity of the demixing transition of a binary mixture causes more solvent to be absorbed into the polymeric colloidal particle, which, in turn, lowers the glass transition temperature of the polymer inside the particle, with a concomitant decrease in viscosity. The balance between the internal viscosity and surface tension of the particle is thus disrupted, and the anisotropic shape of the particle shifts to become more spherical. Subsequent rapid temperature quenching can halt the process, leaving the particle with an intermediate anisotropy. The resultant shape anisotropy control provides new routes for studies of the phase transitions of anisotropic colloids and enables the fabrication of unique particles for materials applications.     

A coarse-grained explicit solvent simulation of rheology of colloidal suspensions

We use a simple extension of the dissipative particle dynamics (DPD) model to address the dynamical properties of macrosolutes immersed in complex fluid solvents. In this approach, the solvent particles are still represented as DPD particles, thereby retaining the time and length scale advantages offered by the DPD approach. In contrast, the solute particles are represented as hard particles of the appropriate size. We examine the applicability of this simulation approach to reproduce the correct hydrodynamical characteristics of the mixture. Our results focus on the equilibrium dynamics and the steady-state shear rheological behaviors for a range of volume fractions of the suspension, and demonstrate excellent agreement with many published experimental and theoretical results. Moreover, we are also able to track the glass transition of our suspension and the associated dynamical signatures in both the diffusivities and the rheological properties of our suspension. Our results suggest that the simulation approach can be used as a one-parameter model to examine quantitatively the rheological properties of colloidal suspensions in complex fluid solvents such as polymeric melts and solutions, as well as allied dynamical phenomena such as phase ordering in mixtures of block copolymers and particles.     

Phase behavior of weakly polydisperse sticky hard spheres: perturbation theory for the Percus-Yevick solution

We study the effects of size polydispersity on the gas-liquid phase behavior of mixtures of sticky hard spheres. To achieve this, the system of coupled quadratic equations for the contact values of the partial cavity functions of the Percus-Yevick solution [R. J. Baxter, J. Chem. Phys. 49, 2770 (1968)] is solved within a perturbation expansion in the polydispersity, i.e., the normalized width of the size distribution. This allows us to make predictions for various thermodynamic quantities which can be tested against numerical simulations and experiments. In particular, we determine the leading order effects of size polydispersity on the cloud curve delimiting the region of two-phase coexistence and on the associated shadow curve; we also study the extent of size fractionation between the coexisting phases. Different choices for the size dependence of the adhesion strengths are examined carefully; the Asakura-Oosawa model [J. Chem. Phys. 22, 1255 (1954)] of a mixture of polydisperse colloids and small polymers is studied as a specific example.     

Reversible addition–fragmentation chain transfer polymerization of methyl methacrylate in emulsion

Theoretical simulations showed that for controlled/living radical polymerization in an emulsion system, some of the earliest born particles could be superswollen to a size close to 1 μm. We hypothesized that the superswelling of these particles would lead to colloidal instability. Under the guidance of the simulation results, reversible addition–fragmentation chain transfer (RAFT) emulsion polymerization of methyl methacrylate (MMA) was carried out. Experimental results showed that increasing the initiation rate, surfactant level, and targeted molecular weight could improve the colloidal stability of the RAFT polymerization of MMA in an emulsion. The experimental results were in full accord with the theoretical predictions. The poor control of the molecular weight and polydispersity index was found to have a close relationship with the colloidal instability. For the first time, we demonstrated that RAFT polymerization could successfully be implemented with little coagulum, good control of the molecular weight, and a low polydispersity index with the same process used for traditional emulsion polymerization but with higher surfactant levels and initiation rates. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44:2837–2847, 2006     

Phase coexistence in polydisperse multi-Yukawa hard-sphere fluid: high temperature approximation

High temperature approximation (HTA) is used to describe the phase behavior of polydisperse multi-Yukawa hard-sphere fluid mixtures. It is demonstrated that in the frames of the HTA the model belongs to the class of "truncatable free energy models," i.e., the models with thermodynamical properties (Helmholtz free energy, chemical potential, and pressure) defined by the finite number of generalized moments. Using this property we were able to calculate the complete phase diagram (i.e., cloud and shadow curves as well as binodals) and size distribution functions of the coexisting phases of several different models of polydisperse fluids. In particular, we consider polydisperse one-Yukawa hard-sphere mixture with factorizable Yukawa coefficients and polydisperse Lennard-Jones (LJ) mixture with interaction energy parameter and/or size polydispersity. To validate the accuracy of the HTA we compare theoretical results with previously published results of more advanced mean spherical approximation (MSA) for the one-Yukawa model and with the Monte Carlo (MC) computer simulation results of [Wilding et al. J. Chem. Phys. 121, 6887 (2004); Phys. Rev. Lett. 95, 155701 (2005)] for the LJ model. We find that overall predictions of the HTA are in reasonable agreement with predictions of the MSA and MC, with the accuracy range from semiquantitative (for the phase diagram) to quantitative (for the size distribution functions).     

Examining the effect of chain length polydispersity on the phase behavior of polymer solutions with the statistical associating fluid theory (Wertheim TPT1) using discrete and continuous distributions

Polymers are naturally polydisperse. Polydispersity may have a large effect on the phase behavior of polymer solutions, in particular, on the liquid-liquid phase equilibria. In this paper, we determine the cloud and shadow curves bounded by lower critical solution temperatures for a number of polymer+solvent systems where the polymer is polydisperse in terms of molecular weight (chain length). The moment method [P. Sollich, P. B. Warren, and M. E. Cates, Adv. Chem. Phys. 116, 265 (2001)] is applied with the SAFT approach to determine cloud and shadow curves with continuous Schulz-Flory distributions. It is seen that chain length polydispersity always enhances the extent of liquid-liquid phase equilibria. The predicted cloud curves obtained for continuous distributions are very similar to those obtained for simple ternary mixtures with the same polydispersity index, while the corresponding shadow curves can be very different depending on the composition of the parent distribution. The ternary phase behavior can be used to provide an understanding of the shape of the cloud and shadow curves. Regions of phase equilibria between three liquid phases are found for ternary systems when the chain length distribution is very asymmetrical; such regions are not observed for Schulz-Flory distributions even in the case of a large degree of polydispersity.     

Colloidal glasses

Colloidal suspensions are important model systems for the study of phase transitions. The glass transition, especially, can be followed more directly with colloidal systems and compared to theoretical predictions. At high volume fractions of the colloidal particles the density fluctuations are partially frozen in. This can be identified by the typical plateau values in the time correlation function. If one compares the experimental results with the mode coupling theory, a very good agreement can be obtained. Currently, some new experimental results concerning the dynamical heterogeneity in colloidal systems are under discussion and will certainly initiate new theoretical developments.     

Effect of quenched size polydispersity on the fluid-solid transition in charged colloidal suspensions

We study the effect of quenched size polydispersity on the phase behavior of charged colloidal suspensions using free-energy calculations in Monte Carlo simulations. The colloids are assumed to interact with a hard-core repulsive Yukawa (screened-Coulomb) interaction with constant surface potential, so that the particles are polydisperse both in size and charge. In addition, we take the size distribution to be fixed in both the fluid and crystal phase (no size fractionation is allowed). We study the fluid-solid transition for various screening lengths and surface potentials, finding that upon increasing the size polydispersity the freezing transition shifts toward higher packing fractions and the density discontinuity between the two coexisting phases diminishes. Our results provide support for a terminal polydispersity above which the freezing transition disappears.     

Kinetic Study of the Aggregation of Colloids Immersed in a Binary Mixture of Polymers

Summary: We consider an assembly of colloidal particles, which are immersed in a binary mixture of polymers. At temperatures close to the critical point of the free mixture, the fluctuations of the composition induce a critical Casimir force between colloids. This force naturally depends on their distance and it is responsible for the colloidal aggregation. In this paper, we propose to study the kinetics of such a transition leading colloids from a dispersed phase (gas) to a dense one (liquid). This kinetics is studied through the relaxation rate, which is a function of the transfer wave-vector.     

Refined Continuous Thermodynamic Treatment for the Liquid-Liquid Equilibrium of Copolymer Solutions

Based on an improved calculation of activity coefficients, continuous thermodynamics using a generalized divariate Stockmayer distribution is applied to the liquid-liquid equilibrium of random copolymer solutions. The effects of the chemical polydispersity on the cloud-point curve, the shadow curve, the spinodal, and the critical point are discussed. The theory can account for the occurrence of three-phase equilibria as well as for the phase separation in pure copolymers.     

Role of polydispersity in anomalous interactions in electrostatically levitated colloidal systems

In this paper, we investigate the effects of using inverse analyses developed for monodisperse particles to extract particle-particle and particle-surface potentials from simulated interfacial colloidal configurations having finite-size polydispersity. Forward Monte Carlo simulations are used to generate three-dimensional equilibrium configurations of log normal-distributed polydisperse particles confined by gravity near an underlying surface. Particles remain levitated above the substrate and stabilized against aggregation by repulsive electrostatic Derjaguin-Landau-Verwey-Overbeek pair potentials. An inverse Ornstein-Zernike analysis and an inverse Monte Carlo simulation method are used to obtain interactions from simulated distribution functions as a function of polydispersity (sigma), relative range of repulsion (kappa a), and projected interfacial concentration (rho). Both inverse analyses successfully recover input potentials for all monodisperse cases, but fail for polydispersities often encountered in experiments. For different conditions (sigma, kappa a, and rho), our results indicate softened short-range repulsion, anomalous long-range attraction, and apparent particle overlaps, which are similar to commonly reported observations in optical microscopy measurements of quasi-two-dimensional interfacial colloidal ensembles. By demonstrating signatures of, and limitations due to, polydispersity when extracting pair potentials from measured distribution functions, our specific goal is to provide a basis to objectively interpret and resolve the effects of polydispersity in optical microscopy experiments.     

Sol–gel derived mesoporous and microporous alumina membranes

Stable polymeric and colloidal boehmite sols were prepared by sol–gel process through controlled hydrolysis/condensation reactions. The particle sizes of the colloidal sols were in the 12–25 nm range depending on the process parameters and about 2 nm for polymeric sols. The presence of a significant increase in the microporosity content of the heat treated polymeric membranes relative to the mesoporous colloidal membranes might make the design of thermally stable microporous alumina membranes with controlled pore structures possible. The phase structure evolution in the 600–800 °C range had shown that the crystallization of the gamma alumina in the amorphous matrix starts at about 800 °C. This indicated that the pore structure stability may be enhanced through processing up to this relatively high temperature in polymeric alumina derived unsupported membranes. The permeance values of the two and three layered colloidal alumina membranes were observed to be independent of pressure which implies that the dominant gas transport mechanism is Knudsen diffusion in these structures. This was also supported by the 2.8 nm BJH pore sizes of the colloidal membranes. The Knudsen diffusion equation derived permeances of the polymeric alumina membranes with thicknesses of about 300 nm were determined to be very close to the experimentally determined permeance values.     

Anomalous potentials from inverse analyses of interfacial polydisperse attractive colloidal fluids

This paper investigates effects of using monodisperse inverse analyses to extract particle-particle and particle-surface potentials from simulated interfacial colloidal fluids of polydisperse attractive particles. Effects of polydispersity are investigated as functions of particle concentration and attractive well depth and range for van der Waals and depletion potentials. Forward Monte Carlo simulations are used to generate particle distribution functions for polydisperse interfacial colloidal fluids from which inverted potentials are obtained using an inverse Ornstein-Zernike analysis and an inverse Monte Carlo simulation method. Attractive potentials are successfully recovered for monodisperse colloidal fluids, but polydispersity that is unaccounted for in inverse analyses produces (1) apparent softening of strong forces, (2) anomalous repulsive and attractive interactions, and (3) aphysical particle overlaps. This investigation provides insights into the role of polydispersity in altering the equilibrium structure and corresponding inverted potentials of attractive colloidal fluids near surfaces. These findings should assist the design and interpretation of optical microscopy experiments involving interfacial colloidal fluids similar to the simulated experiments reported here.     

Colloidal Rings by Site‐Selective Growth on Patchy Colloidal Disc Templates

Anisotropic colloidal building blocks are quite attractive as they enable the self‐assembly towards new materials with designated hierarchical structures. Although many advances have been achieved in colloidal synthetic methodology, synthesis of colloidal rings with low polydispersity and on a large scale remains a challenge. To address this issue we introduce a new site‐selective growth strategy, which relies on using patchy particles. For example, by using patchy discs as templates, silica can selectively be grown on only side surfaces, resulting in formation of silica rings. We demonstrate that shape parameters are tunable and find that these silica rings can be used as secondary template to synthesize other types of rings. This method for synthesizing ring‐like colloids provides possibilities for studying their self‐assembly and associated phase transitions, and this patchy particles template strategy paves a new route for fabricating other new colloidal particles.