Effect of the range of attractive interactions on crystallization, metastable phase transition, and percolation in colloidal dispersions

Role of interaction range on the microstructure and dynamics of attractive colloidal systems

Colloidal gelation phase diagram has been traditionally characterized using three key factors: particle volume fraction, strength of attraction, and range of attraction. While there's a rich body of literature on the role of attraction strength and particle volume fraction, majority of studies have been limited to short range interactions. Using Brownian dynamics simulations, we explored the effect that the range of attractions has on the microstructure and dynamics of both weakly and strongly attractive colloidal systems. Although gelation occurs significantly faster at high attraction strength, by an order of magnitude compared to low strength, we did not observe any clear trend in gelation-rate with respect to a change in the range of interaction. However, as the attraction range increases in both systems, the final structure undergoes a transition from a single connected network to a fluid of dense clusters. This results in a new gelation phase boundary for long range attractive colloids.

Extended law of corresponding states: square-well oblates

The vapour-liquid coexistence collapse in the reduced temperature,T_r=T/T_c, reduced density,ρ_r=ρ/ρ_c, plane is known as a principle of corresponding states, and Noro and Frenkel have extended it for pair potentials of variable range. Here, we provide a theoretical basis supporting this extension, and show that it can also be applied to short-range pair potentials where both repulsive and attractive parts can be anisotropic. We observe that the binodals of oblate hard ellipsoids for a given aspect ratio (κ= 1/3) with varying short-range square-well interactions collapse into a single master curve in theΔB2*-ρ_rplane, whereΔB2*=(B2(T)-B2(Tc))/v0,B₂is the second virial coefficient, andv₀is the volume of the hard body. This finding is confirmed by both REMC simulation and second virial perturbation theory for varying square-well shells, mimicking uniform, equator, and pole attractions. Our simulation results reveal that the extended law of corresponding states is not related to the local structure of the fluid.

Microdynamics and arrest of coarsening during spinodal decomposition in thermoreversible colloidal gels

Coarsening and kinetic arrest of colloidal systems undergoing spinodal decomposition (SD) is a conserved motif for forming hierarchical, bicontinuous structures. Although the thermodynamic origins of SD in colloids are widely known, the microstructural processes responsible for its coarsening and associated dynamics en route to arrest remain elusive. To better elucidate the underlying large-scale microdynamical processes, we study a colloidal system with moderate-range attractions which displays characteristic features of arrested SD, and study its dynamics during coarsening through a combination of differential dynamic microscopy and real-space tracking. Using these recently developed imaging techniques, we reveal directly that the coarsening arises from collective dynamics of dense domains, which undergo slow, intermittent, and ballistic motion. These collective motions indicate interfacial effects to be the driving force of coarsening. The nature of the gelation enables control of the arrested length scale of coarsening by the depths of quenching into the spinodal regime, which we demonstrate to provide an effective means to control the elasticity of colloidal gels.

Common behavior of the critical properties of the 2D and 3D square-well fluids

We have analyzed the behavior of the critical properties and second virial coefficient of the square well fluids in two (2D) and three dimensions (3D) as a function of the interaction range. In both systems, the critical density shows an oscillating-like behavior as the interaction range increases. The second virial coefficient evaluated at the critical temperature as a function of the interaction range shows a general behavior for both cases, and quite surprisingly, there is a minimum of this parameter, for the 2D and 3D fluids, located approximately at the same interaction range. These findings are discussed in terms of the structure of the fluids, via the analysis of the radial distribution function evaluated at the critical point.

Liquid-vapor phase diagram and surface properties in oppositely charged colloids represented by a mixture of attractive and repulsive Yukawa potentials

The liquid-vapor phase diagrams of equal size diameter σ binary mixtures of screened potentials have been reported for several ranges of interaction using Monte Carlo simulation methods [J. B. Caballero, A. M. Puertas, A. Ferńandez-Barbero, F. J. de las Nieves, J. M. Romero-Enrique, and L. F. Rull, J. Chem. Phys. 124, 054909 (2006); A. Fortini, A.-P. Hynninen, and M. Dijkstra, J. Chem. Phys. 125, 094502 (2006)]. Both works report controversial results about the stability of the phase diagram with the inverse Debye screening length κ. Caballero found stability for values of κσ up to 20 while Fortini reported stability for κσ up to 20 while Fortini reported stability for κσ ≤ 4. In this work a spinodal decomposition process where the liquid and vapor phases coexist through an interface in a slab geometry is used to obtain the phase equilibrium and surface properties using a discontinuous molecular dynamics simulations for mixtures of equal size particles carrying opposite charge and interacting with a mixture of attractive and repulsive Yukawa potentials at different values of κσ. An crude estimation of the triple point temperatures is also reported. The isothermal-isobaric method was also used to determine the phase stability using one phase simulations. We found that liquid-vapor coexistence is stable for values of κσ > 20 and that the critical temperatures have a maximum value at around κσ = 10, in agreement with Caballero et al. calculations. There also exists a controversy about the liquid-vapor envelope stability of the pure component attractive Yukawa model which is also discussed in the text. In addition, details about the equivalence between continuous and discontinuous molecular dynamics simulations are given, in the Appendix, for Yukawa and Lennard-Jones potentials.

Phase diagrams and kinetics of phase transitions in protein solutions

The phase behavior of proteins is of interest for fundamental and practical reasons. The nucleation of new phases is one of the last major unresolved problems of nature. The formation of protein condensed phases (crystals, polymers, and other solid aggregates, as well as dense liquids and gels) underlies pathological conditions, plays a crucial role in the biological function of the respective protein, or is an essential part of laboratory and industrial processes. In this review, we focus on phase transitions of proteins in their properly folded state. We first summarize the recently acquired understanding of physical processes underlying the phase diagrams of the protein solutions and the thermodynamics of protein phase transitions. Then we review recent findings on the kinetics of nucleation of dense liquid droplets and crystals. We explore the transition from nucleation to spinodal decomposition for liquid-liquid separation and introduce the new concept of solution-to-crystal spinodal. We review the two-step mechanism of protein crystal nucleation, in which mesoscopic metastable protein clusters serve as precursors to the ordered crystal nuclei. The concepts and mechanisms reviewed here provide powerful tools for control of the nucleation process by varying the solution thermodynamic parameters.

Effect of glycerol and dimethyl sulfoxide on the phase behavior of lysozyme: theory and experiments

Salt, glycerol, and dimethyl sulfoxide (DMSO) are used to modify the properties of protein solutions. We experimentally determined the effect of these additives on the phase behavior of lysozyme solutions. Upon the addition of glycerol and DMSO, the fluid-solid transition and the gas-liquid coexistence curve (binodal) shift to lower temperatures and the gap between them increases. The experimentally observed trends are consistent with our theoretical predictions based on the thermodynamic perturbation theory and the Derjaguin-Landau-Verwey-Overbeek model for the lysozyme-lysozyme pair interactions. The values of the parameters describing the interactions, namely the refractive indices, dielectric constants, Hamaker constant and cut-off length, are extracted from literature or are experimentally determined by independent experiments, including static light scattering, to determine the second virial coefficient. We observe that both, glycerol and DMSO, render the potential more repulsive, while sodium chloride reduces the repulsion.

Electrostatics and aggregation: how charge can turn a crystal into a gel

The crystallization of proteins or colloids is often hindered by the appearance of aggregates of low fractal dimension called gels. Here we study the effect of electrostatics upon crystal and gel formation using an analytic model of hard spheres bearing point charges and short range attractive interactions. We find that the chief electrostatic free energy cost of forming assemblies comes from the entropic loss of counterions that render assemblies charge-neutral. Because there exists more accessible volume for these counterions around an open gel than a dense crystal, there exists an electrostatic entropic driving force favoring the gel over the crystal. This driving force increases with increasing sphere charge, but can be counteracted by increasing counterion concentration. We show that these effects cannot be fully captured by pairwise-additive macroion interactions of the kind often used in simulations, and we show where on the phase diagram to go in order to suppress gel formation.

Structure and thermodynamics of hard-core Yukawa fluids: thermodynamic perturbation approaches

The thermodynamic perturbation theories, which are based on the power series of a coupling constant (λ-expansion), have been proposed for studying the structural and thermodynamic properties of a hard-core Yukawa (HCY) fluid: one (A1-approximation) is the perturbation theory based on the hard-sphere repulsion as a reference system. The other (A2-approximation) is the perturbation theory based on the reference system which incorporates both the repulsive and short-range attractive interactions. The first-order mean-spherical approximation (FMSA) provided by Tang and Lu [J. Chem. Phys. 99, 9828 (1993)] has been employed for investigating the thermodynamic properties of a HCY fluid using the alternative method via the direct correlation function. The calculated results show that (i) the A1 and A2 approximations are in excellent agreements with previous computer simulation results in the literature and compare with the semi-empirical works of Shukla including the higher-order free energy terms, (ii) the A1 and A2 approximations are better than the FMSA and the mean-spherical approximation, (iii) the A2-approximation compares with the A1-approximation, even though the perturbation effect of an A2-approximation is much smaller than that of an A1-approximation, and that (iv) the FMSA study is particularly of advantage in providing the structure and thermodynamics in a simple and analytic manner.

Cluster formation and bulk phase behavior of colloidal dispersions

We investigated cluster formation in model colloids where the interparticle potential beyond the collision diameter includes a short-ranged attraction and a longer-ranged repulsion. The structure and thermodynamic properties of the model system can be accurately described by the first-order mean-spherical approximation (FMSA) and a perturbation theory for bulk phases and by a nonlocal density-functional theory (DFT) for stable/metastable clusters. In corroboration with recent experiments and molecular simulations, we predicted that the bulk to cluster transition may occur at both one-phase and two-phase regions of the colloidal phase diagram. While formation of colloidal clusters appears uncorrelated with the bulk phase transitions, the local phenomena may nevertheless play an important role in the dynamics of stable colloids and the kinetics of colloidal phase transitions.

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.

Study on Vapor−Liquid Equilibria and Surface Tensions for Nonpolar Fluids by Renormalization Group Theory and Density Gradient Theory

An equation of state (EOS) applicable for both the uniform and nonuniform fluids is established by using the density-gradient theory (DGT). In the bulk phases, the EOS reduces to statistical associating fluid theory (SAFT). By combining the EOS with the renormalization group theory (RGT), the vapor-liquid-phase equilibria and surface tensions for 10 nonpolar chainlike fluids are investigated from low temperature up to the critical point. The obtained results agree well with the experimental data.

Spinodal for the solution-to-crystal phase transformation

The formation of crystalline nuclei from solution has been shown for many systems to occur in two steps: the formation of quasidroplets of a disordered intermediate, followed by the nucleation of ordered crystalline embryos within these droplets. The rate of each step depends on a respective free-energy barrier and on the growth rate of its near-critical clusters. We address experimentally the relative significance of the free-energy barriers and the kinetic factors for the nucleation of crystals from solution using a model protein system. We show that crystal nucleation is 8-10 orders of magnitude slower than the nucleation of dense liquid droplets, i.e., the second step is rate determining. We show that at supersaturations of three or four k(B)T units, crystal nuclei of five, four, or three molecules transform into single-molecule nuclei, i.e., the significant nucleation barrier vanishes below the thermal energy of the molecules. We show that the main factor, which determines the rate of crystal nucleation, is the slow growth of the near-critical ordered clusters within the quasidroplets of the disordered intermediate. Analogous to the spinodal in supersaturated fluids, we define a solution-to-crystal spinodal from the transition to single-molecule crystalline nuclei. We show that heterogeneous nucleation centers accelerate nucleation not only because of the wettinglike effects that lower the nucleation barrier, as envisioned by classical theory, but by helping the kinetics of growth of the ordered crystalline embryos.

Kinetics of two-step nucleation of crystals

When the nucleation of a stable crystalline phase directly in a supersaturated old phase is greatly retarded, the crystal nuclei might nucleate within faster-forming particles of an intermediate phase. Here we present a theoretical investigation of the kinetics of this two-step nucleation of crystals and derive general expressions for the time dependence of the number of crystals nucleated within the particles of the intermediate phase. The results reveal that crystal nucleation can be strongly delayed by the slow growth of the particles and/or by the slow nucleation of the crystals in them. Furthermore, the linear part of the time dependence of the number of nucleated crystals is determined by the formation rate of the intermediate particles. This is in contrast with the one-step nucleation of crystals when this linear part is determined by the rate of crystal nucleation directly in the old phase. Criteria are proposed for distinction between the one- and two-step nucleation mechanisms, based on the supersaturation dependence of the delay time for nucleation. The application of the theoretical approach to the analysis of experimental data on the nucleation of crystals and other ordered aggregates of protein and other soluble materials is discussed.

A Metastable Prerequisite for the Growth of Lumazine Synthase Crystals

Dense liquid phases, metastable with respect to a solid phase, form in solutions of proteins and small-molecule materials. They have been shown to serve as a prerequisite for the nucleation of crystals and other ordered solid phases. Here, using crystals of the protein lumazine synthase from Bacillus subtilis, which grow by the generation and spreading of layers, we demonstrate that within a range of supersaturations the only mechanism of generation of growth layers involves the association of submicrometer-size droplets of the dense liquid to the crystal surface. The dense liquid is metastable not only with respect to the crystals, but also with respect to the low-concentration solution: dynamic light scattering reveals that the droplets' lifetime is limited to several seconds, after which they decay into the low-concentration solution. The short lifetime does not allow growth to detectable dimensions so that liquid-liquid phase separation is not observed within a range of conditions broader than the one used for crystallization. If during their lifetime the droplets encounter a crystal surface, they lower their free energy not by decay, but by transformation into crystalline matter, ensuring perfect registry with the substrate. These observations illustrate two novel features of phase transformations in solutions: the existence of doubly metastable, short-lifetime dense phases and their crucial role for the growth of an ordered solid phase.

Thermodynamic and structural properties of repulsive hard-core Yukawa fluid: Integral equation theory, perturbation theory and Monte Carlo simulations

The thermodynamic and structural properties of purely repulsive hard-core Yukawa particles in the fluid state are determined through Monte Carlo simulation and modeled using perturbation theory and integral equation theory in the mean spherical approximation (MSA). Systems of particles with Yukawa screening lengths of 1.8, 3.0, and 5.0 are examined with results compared to variations of MSA and perturbation theory. Thermodynamic properties were predicted well by both theories in the fluid region up to the fluid-solid phase boundary. Further, we found that a simplified exponential version of the MSA is the most accurate at predicting radial distribution function at contact. Radial distribution function of repulsive hard-core Yukawa particles are also reported. The results show that methods based on MSA and perturbation theory that are typically applied to the attractive hard-core Yukawa potential can also be extended to the purely repulsive hard-core Yukawa potential.