Experimental phase diagram of a model colloid-polymer mixture in the protein limit

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.

Phase diagram and aggregation dynamics of a monolayer of paramagnetic colloids

We have developed a tunable colloidal system and a corresponding theoretical model for studying the phase behavior of particles assembling under the influence of long-range magnetic interactions. A monolayer of paramagnetic particles is subjected to a spatially uniform magnetic field with a static perpendicular component and a rapidly rotating in-plane component. The sign and strength of the interactions vary with the tilt angle θ of the rotating magnetic field. For a purely in-plane field, θ=90^{∘}, interactions are attractive and the experimental results agree well with both equilibrium and out-of-equilibrium predictions based on a two-body interaction model. For tilt angles 50^{∘}≲θ≲55^{∘}, the two-body interaction gives a short-range attractive and long-range repulsive interaction, which predicts the formation of equilibrium microphases. In experiments, however, a different type of assembly is observed. Inclusion of three-body (and higher-order) terms in the model does not resolve the discrepancy. We further characterize the anomalous regime by measuring the time-dependent cluster size distribution.

Phase behaviour of colloids plus weakly adhesive polymers

The phase behaviour of a colloidal dispersion mediated by weakly adhesive polymers is considered. The polymers are depleted but are weakly adhesive and hence comprise a non-zero polymer concentration at the colloid's surface, in contrast to the classical assumption in depletion theories involving a zero polymer concentration at the surface. The theory is composed of a generalized free-volume theory for colloid-polymer mixtures and a self-consistent mean-field theory for polymers at surfaces. It is found that the weak adhesion of the polymers shifts the phase stability of the colloid-polymer mixtures to higher polymer concentrations as compared to assuming a full depletion effect. The predicted phase diagrams employing the new theory are consistent with experiments on mixtures of silica spheres coated with stearyl alcohol and polydimethylsiloxane in cyclohexane and with Monte Carlo simulation results.

Pressure and density scaling for colloid-polymer systems in the protein limit

Grand canonical Monte Carlo and histogram reweighting techniques are used to study the fluid-phase behavior of an athermal system of colloids and nonadsorbing polymers on a fine lattice in the "protein limit," where polymer dimensions exceed those of the colloids. The main parameters are the chains' radius of gyration, R_{g}, the diameter of the colloids, σ_{c}, and the monomer diameter, σ_{s}. The phase behavior is controlled by the macroscopic size ratio, q_{r}=2R_{g}/σ_{c}, and the microscopic size ratio, d=σ_{s}/σ_{c}. The latter ratio is found to play a significant role in determining the critical monomer concentration for q_{r}≲4 and the critical colloid density for all chain lengths. However, the critical (osmotic) pressure is independent of the microscopic size ratio at all macroscopic size ratios studied. Quantitative agreement is observed between our simulation results and experimental data. We scale our results based on the polymer correlation length, which has previously been suggested to universally collapse these binodals [Bolhuis et al., Phys. Rev. Lett. 90, 068304 (2003); Fleer and Tuinier, Phys. Rev. E 76, 041802 (2007)]. While the density binodals exhibit universal characteristics along the low-colloid-density branch, such features are not present in the corresponding high-density phase. However, pressure binodals do collapse nicely under such a scaling, even far from the critical point, which allows us to produce a binodal curve whose shape is independent of either size ratio.

Poly(ethylene oxide) (PEO) and poly(vinyl pyrolidone) (PVP) induce different changes in the colloid stability of nanoparticles

The phase behavior of model polymer-colloid mixtures is measured for solutions approaching the "protein limit", that is, when the radius of gyration of the polymer (R(g)) is greater than or approximately equal to the radius of the colloid (R). Cationic nanoparticles are mixed with poly(ethylene oxide) (PEO) or poly(vinyl pyrolidone) (PVP) at size ratios of R(g)/R = 0.7 and 1.8. The addition of PEO to stable nanoparticle dispersions leads to depletion flocculation in both deionized water and buffer solutions. The instability mechanism for the PVP-nanoparticle system depends on the suspension medium. In water, bridging occurs below the saturation adsorption of PVP, whereas depletion phase separation is evident at concentrations exceeding those necessary to saturate the particle surface. In acidic buffer, PVP addition results in depletion phase separation. The difference between bridging and depletion is distinguished by both visual appearances and rheological measurements. There is no trend (within error bars) in the polymer concentration required to induce instability with increasing R(g)/R in contrast with theoretical predictions. This is most likely due to adsorption of polymer onto the particle surface.

Scaling the structure factors of protein limit colloid-polymer mixtures

The scaling of the phase boundaries and structure factors of protein limit colloid-polymer mixtures has been investigated through the addition of large nonadsorbing polymer chains to a solution of small microemulsion droplets. The colloid-polymer size ratio has been varied between 10 and 16 by changing the microemulsion droplet size; the phase boundaries were shown previously to observe theoretical scaling relations very well [Langmuir 2009, 25 (7), 3944-3952]. These thermodynamic scaling relations are now shown to also hold extremely well for the individual and cross-term partial structure factors. The structure factors for systems with different size ratios occupying the same point in scaled phase space show extremely good agreement. The properties and stability of these mixtures are governed by the polymer mesh.

Low energy methods of phase separation in colloidal dispersions and microemulsions

The majority of work on phase separation of colloidal systems has been concerned with the energy intensive approaches such as ultracentrifugation, solvent evaporation, changes of temperature and pressure etc. However, in modern nanotechnology it is desirable to minimize environmental impact in order to achieve separation and recovery of colloidal products. In this review recent research on phase separation methods, requiring relatively lower energy consumption are summarized. These include polymer-, solvent- and photo-induced approaches to phase separation.

Testing the scaling behavior of microemulsion-polymer mixtures

The phase behavior and structural properties of "protein limit" mixtures of small (radius 20-30 A) water-in-oil microemulsion droplets (colloids) and large (radius 130-580 A) nonadsorbing polymer chains have been investigated. Accepted theoretical scaling relations for describing correlations have been applied and do not account fully for the observations; solvency effects may account for the deviations. The polymer/colloid size ratio has been varied from around 4 to 19 by using three different molecular weights of polyisoprene. Small-angle neutron scattering (SANS) has been used to determine partial structure factors (PSF) through contrast variation. The structure factors describing colloid-colloid interactions for the three polymers at fixed polymer concentration are shown to exhibit the same scaling behavior as the phase boundaries, provided that samples are sufficiently far from the demixing phase transition. The structure factors show a dramatic increase at low wavevectors on approaching the phase boundary, and behavior in this region does not obey expected scaling relations. By calculating effective polymer Flory-Huggins parameters, the effect of apparent solvent properties on adding microemulsion are shown to be less dramatic for the higher molecular weight polymers. This study extends previous work carried out on microemulsion-polymer mixtures.

Analytical phase diagrams for colloids and non-adsorbing polymer

We review the free-volume theory (FVT) of Lekkerkerker et al. [Europhys. Lett. 20 (1992) 559] for the phase behavior of colloids in the presence of non-adsorbing polymer and we extend this theory in several aspects: (i) We take the solvent into account as a separate component and show that the natural thermodynamic parameter for the polymer properties is the insertion work Pi(v), where Pi is the osmotic pressure of the (external) polymer solution and v the volume of a colloid particle. (ii) Curvature effects are included along the lines of Aarts et al. [J. Phys.: Condens. Matt. 14 (2002) 7551] but we find accurate simple power laws which simplify the mathematical procedure considerably. (iii) We find analytical forms for the first, second, and third derivatives of the grand potential, needed for the calculation of the colloid chemical potential, the pressure, gas-liquid critical points and the critical endpoint (cep), where the (stable) critical line ends and then coincides with the triple point. This cep determines the boundary condition for a stable liquid. We first apply these modifications to the so-called colloid limit, where the size ratio q(R)=R/a between the radius of gyration R of the polymer and the particle radius a is small. In this limit the binodal polymer concentrations are below overlap: the depletion thickness delta is nearly equal to R, and Pi can be approximated by the ideal (van't Hoff) law Pi=Pi(0)=phi/N, where phi is the polymer volume fraction and N the number of segments per chain. The results are close to those of the original Lekkerkerker theory. However, our analysis enables very simple analytical expressions for the polymer and colloid concentrations in the critical and triple points and along the binodals as a function of q(R). Also the position of the cep is found analytically. In order to make the model applicable to higher size ratio's q(R) (including the so-called protein limit where q(R)>1) further extensions are needed. We introduce the size ratio q=delta/a, where the depletion thickness delta is no longer of order R. In the protein limit the binodal concentrations are above overlap. In such semidilute solutions delta approximately xi, where the De Gennes blob size (correlation length) xi scales as xi approximately phi(-gamma), with gamma=0.77 for good solvents and gamma=1 for a theta solvent. In this limit Pi=Pi(sd) approximately phi(3gamma). We now apply the following additional modifications: With these latter two modifications we obtain again a fully analytical model with simple equations for critical and triple points as a function of q(R). In the protein limit the binodal polymer concentrations scale as q(R)(1/gamma), and phase diagrams phiq(R)(-1/gamma) versus the colloid concentration eta become universal (i.e., independent of the size ratio q(R)). The predictions of this generalized free-volume theory (GFVT) are in excellent agreement with experiment and with computer simulations, not only for the colloid limit but also for the protein limit (and the crossover between these limits). The q(R)(1/gamma) scaling is accurately reproduced by both simulations and other theoretical models. The liquid window is the region between phi(c) (critical point) and phi(t) (triple point). In terms of the ratio phi(t)/phi(c) the liquid window extends from 1 in the cep (here phi(t)-phi(c)=0) to 2.2 in the protein limit. Hence, the liquid window is narrow: it covers at most a factor 2.2 in (external) polymer concentration.

Colloid-polymer mixtures in the protein limit

This review discusses the structure and phase behaviour of mixtures of colloidal particles and non-adsorbing polymers in the protein limit of large polymers and small colloids. The vast majority of work on colloid-polymer mixtures has been concerned with the colloid limit of large colloidal particles and small polymer chains. In this regime, the diameter of the colloidal particles, , is larger than the characteristic size of the polymer-taken as twice their radius of gyration, . The opposite limit, of size ratios , is called the protein limit due to the common practice of adding polymer to protein solutions in order to aid protein crystallisation. Theoretical predictions for systems in the protein limit are considered briefly and then the main focus is on recent experimental studies of mixtures in the protein limit.

Spinodal decomposition in a food colloid-biopolymer mixture: evidence for a linear regime

We investigate phase separation and structural evolution in a complex food colloid (casein micelles) and biopolymer (xanthan) mixture using small-angle light scattering. We demonstrate that phase separation is induced by a depletion mechanism, and that the resulting coexistence curve can be described by osmotic equilibrium theory for mixtures of colloids and polymer chains in a background solvent, taking into account interactions between the polymer chains in the excluded volume limit. We show that the light scattering pattern of an unstable mixture exhibits the typical behaviour of spinodal decomposition, and we are able to confirm the validity of dynamic similarity scaling. We find three distinct regimes (initial or linear, intermediate and transition stage) for the decomposition kinetics that differ in the time dependence of the peak position of the structure factor. In particular we find clear evidence for the existence of an initial linear regime, where the peak position remains constant and the amplitude grows. The existence of spinodal-like decomposition and the validity of universal scaling in the intermediate and transition stages have been found in previous studies of phase separation in attractive colloidal suspensions. However, to our knowledge the initial linear regime has never been observed in colloidal suspensions, and we attribute this at least partly to the effect of hydrodynamic interactions which are efficiently screened in our system due to the fact that the measurements were performed at high polymer concentrations, i.e. in the semi-dilute regime.