Effects of colloid polydispersity on the phase behavior of colloid-polymer mixtures

We study theoretically the equilibrium phase behavior of a mixture of polydisperse hard-sphere colloids and monodisperse polymers, modeled using the Asakura-Oosawa model [S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 (1954)] within the free volume approximation of H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A. Stroobants, and P. B. Warren [Europhys. Lett. 20, 559 (1992)]. We compute full phase diagrams in the plane of colloid and polymer volume fractions, using the moment free energy method. The intricate features of phase separation in pure polydisperse colloids combine with the appearance of polymer-induced gas-liquid coexistence to give a rich variety of phase diagram topologies as the polymer-colloid size ratio xi and the colloid polydispersity delta are varied. Quantitatively, we find that polydispersity disfavors fluid-solid against gas-liquid separation, causing a substantial lowering of the threshold value xi(c) above which stable two-phase gas-liquid coexistence appears. Phase splits involving two or more solids can occur already at low colloid concentration, where they may be kinetically accessible. We also analyze the strength of colloidal size fractionation. When a solid phase separates from a fluid, its polydispersity is reduced most strongly if the phase separation takes place at low colloid concentration and high polymer concentration, in agreement with experimental observations. For fractionation in gas-liquid coexistence we likewise find good agreement with experiment, as well as with perturbative theories for near-monodisperse systems.     

Theoretical and numerical study of the phase diagram of patchy colloids: ordered and disordered patch arrangements

We report theoretical and numerical evaluations of the phase diagram for a model of patchy particles. Specifically, we study hard spheres whose surface is decorated by a small number f of identical sites ("sticky spots") interacting via a short-ranged square-well attraction. We theoretically evaluate, solving the Wertheim theory, the location of the critical point and the gas-liquid coexistence line for several values of f and compare them to the results of Gibbs and grand canonical Monte Carlo simulations. We study both ordered and disordered arrangements of the sites on the hard-sphere surface and confirm that patchiness has a strong effect on the phase diagram: the gas-liquid coexistence region in the temperature-density plane is significantly reduced as f decreases. We also theoretically evaluate the locus of specific heat maxima and the percolation line.     

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.     

Connectivity percolation of polydisperse anisotropic nanofillers

We present a generalized connectedness percolation theory reduced to a compact form for a large class of anisotropic particle mixtures with variable degrees of connectivity. Even though allowing for an infinite number of components, we derive a compact yet exact expression for the mean cluster size of connected particles. We apply our theory to rodlike particles taken as a model for carbon nanotubes and find that the percolation threshold is sensitive to polydispersity in length, diameter, and the level of connectivity, which may explain large variations in the experimental values for the electrical percolation threshold in carbon-nanotube composites. The calculated connectedness percolation threshold depends only on a few moments of the full distribution function. If the distribution function factorizes, then the percolation threshold is raised by the presence of thicker rods, whereas it is lowered by any length polydispersity relative to the one with the same average length and diameter. We show that for a given average length, a length distribution that is strongly skewed to shorter lengths produces the lowest threshold relative to the equivalent monodisperse one. However, if the lengths and diameters of the particles are linearly correlated, polydispersity raises the percolation threshold and more so for a more skewed distribution toward smaller lengths. The effect of connectivity polydispersity is studied by considering nonadditive mixtures of conductive and insulating particles, and we present tentative predictions for the percolation threshold of graphene sheets modeled as perfectly rigid, disklike particles.     

Particle connectedness and cluster formation in sequential depositions of particles: integral-equation theory

We applied the integral-equation theory to the connectedness problem. The method originally applied to the study of continuum percolation in various equilibrium systems was modified for our sequential quenching model, a particular limit of an irreversible adsorption. The development of the theory based on the (quenched-annealed) binary-mixture approximation includes the Ornstein-Zernike equation, the Percus-Yevick closure, and an additional term involving the three-body connectedness function. This function is simplified by introducing a Kirkwood-like superposition approximation. We studied the three-dimensional (3D) system of randomly placed spheres and 2D systems of square-well particles, both with a narrow and with a wide well. The results from our integral-equation theory are in good accordance with simulation results within a certain range of densities.     

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 coexistence in a polydisperse charged hard-sphere fluid: polymer mean spherical approximation

We have reconsidered the phase behavior of a polydisperse mixture of charged hard spheres (CHSs) introducing the concept of minimal size neutral clusters. We thus take into account ionic association effects observed in charged systems close to the phase boundary where the properties of the system are dominated by the presence of neutral clusters while the amount of free ions or charged clusters is negligible. With this concept we clearly pass beyond the simple level of the mean spherical approximation (MSA) that we have presented in our recent study of a polydisperse mixture of CHS [Yu. V. Kalyuzhnyi, G. Kahl, and P. T. Cummings, J. Chem. Phys. 120, 10133 (2004)]. Restricting ourselves to a 1:1 and possibly size-asymmetric model we treat the resulting polydisperse mixture of neutral, polar dimers within the framework of the polymer MSA, i.e., a concept that--similar as the MSA--readily can be generalized from the case of a mixture with a finite number of components to the polydisperse case: again, the model belongs to the class of truncatable free-energy models so that we can map the formally infinitely many coexistence equations onto a finite set of coupled, nonlinear equations in the generalized moments of the distribution function that characterizes the system. This allows us to determine the full phase diagram (in terms of binodals as well as cloud and shadow curves), we can study fractionation effects on the level of the distribution functions of the coexisting daughter phases, and we propose estimates on how the location of the critical point might vary in a polydisperse mixture with an increasing size asymmetry and polydispersity.     

Investigation of structure factors in dense colloidal suspensions using frequency domain photon migration: polydisperse systems

Frequency domain photon migration (FDPM) technique was employed to investigate the structure factors of dense, polydisperse colloidal suspensions. The angle-integrated structure factors, [S(q)], extracted from FDPM measurements of scattering properties at volume fractions ranging from 0.05 to 0.4, were compared with the values predicted from the polydisperse hard sphere Percus-Yevick (HSPY) model, as well as decoupling approximation (DA) and local monodisperse approximation (LMA) models that incorporated independently measured particle size information. Results show that the polydisperse HSPY model is the most suitable for accounting for particle interactions which predominantly arise from volume exclusion effects. Furthermore, the influence of size polydispersity upon [S(q)] is most significant at high volume fractions. The static structure factors at small wave vector q, S(0), were also assessed from dual wavelength FDPM measurements by using the small wave number approximation as well as the local monodisperse approximation. The measured S(0) agrees well with the values predicted by the polydisperse HSPY model.     

From gas-liquid to liquid crystalline phase behavior via anisotropic attraction: a computer simulation study

The partial phase behavior of a continuum molecular model for self-assembling semiflexible equilibrium polymers is studied via Monte Carlo and molecular dynamics simulation. We investigate the transfer from ordinary gas-liquid coexistence to the appearance of liquid crystallinity driven by excluded volume interaction between rodlike aggregates. The transfer between the two types of phase behavior is governed by a tunable anisotropic attractive interaction between monomer particles. The relation to dipolar fluid models, which are also known to form reversible chains, is discussed.     

Vapor-liquid coexistence of patchy models: relevance to protein phase behavior

The vapor-liquid coexistence boundaries of fluids composed of particles interacting with highly directional patchy interactions, in addition to an isotropic square well potential, are evaluated using grand canonical Monte Carlo simulations combined with the histogram reweighting and finite size scaling methods. We are motivated to study this more complicated model for two reasons. First, it is established that the reduced widths of the metastable vapor-liquid coexistence curve predicted by a model with only isotropic interparticle interactions are much too narrow when compared to the experimental phase behavior of protein solutions. Second, interprotein interactions are well known to be "patchy." Our results show that at a constant total areal density of patches, the critical temperature and the critical density increase monotonically with an increasing number of uniformly spaced patches. The vapor-liquid coexistence curves plotted in reduced coordinates (i.e., the temperature and the density scaled by their respective critical values) are found to be effectively independent of the number of patches, but are much broader than those found for the isotropic models. Our findings for the reduced width of the coexistence curve are almost in quantitative agreement with the available experimental data for protein solutions, stressing the importance of patchiness in this context.     

Nanoparticles in solutions of adsorbing polymers: pair interactions, percolation, and phase behavior

We study the polymer adsorption characteristics, pair-interaction potentials, and phase and percolation behavior in nanoparticle-polymer mixtures. We propose a "saturable" adsorption model to capture the effect of the finite surface saturation capacity for adsorption, and use polymer self-consistent field theory in combination with a McMillan-Mayer framework [McMillan, W. G., Jr.; Mayer, J. E. J. Chem. Phys. 1945, 13, 276] to compute the pair-interaction potentials. Our results demonstrate novel size effects that distinguish the adsorption characteristics of nanoparticles from that of larger particles. Specifically, we predict that the nanoparticle regime is characterized by a significant adsorbance of polymers, albeit distributed predominantly in the form of tails. We also demonstrate that an interplay between the surface saturation, polymer-to-particle size ratios, and the polymer concentrations governs the overall effective interactions between nanoparticles in the presence of an adsorbing polymer. We use simple, mean-field models to relate these characteristics to the phase and percolation behavior in such systems. Our results show that the percolation thresholds for smaller particles are significantly smaller (and, overall, correspond only to a few volume percent) compared to that of the larger particles. Further, with a decrease in the size of the particles, we also predict a considerable increase in the miscibility of the polymer-particle mixtures. Our results are qualitatively in accord with many experimental observations in the nanoparticle regime.     

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).     

De-mixing of polydisperse fluids: experimental test of a universal relation

Colloids, emulsions, polymer blends, and other important complex fluids, are polydisperse, i.e. there are variations among their constituent particles. Polydispersity is usually regarded as an ubiquitous, uncontrollable nuisance causing experimental inconsistencies. We have varied the polydispersity of a complex fluid, whilst keeping all other parameters constant, and report the first measurements of some universal physics. At coexistence (e.g. between liquid and vapour), fractionation occurs—each phase receives a different mix of the various ingredients, e.g. with the liquid disproportionately abundant with larger particles. Theory predicts, at low polydispersity, that this de-mixing becomes universal, irrespective of the material, with chemical differences between the phases proportional to polydispersity to the power two. We have studied colloid–polymer suspensions at two-phase coexistence and, using light scattering, measured the exponent as 2.16±0.44.     

Space-time thermodynamics and subsystem observables in a kinetically constrained model of glassy materials

In a recent article [M. Merolle et al., Proc. Natl. Acad. Sci. U.S.A. 102, 10837 (2005)], it was argued that dynamic heterogeneity in d-dimensional glass formers is a manifestation of an order-disorder phenomenon in the d+1 dimensions of space time. By considering a dynamical analog of the free energy, evidence was found for phase coexistence between active and inactive regions of space time, and it was suggested that this phenomenon underlies the glass transition. Here we develop these ideas further by investigating in detail the one-dimensional Fredrickson-Andersen (FA) model, in which the active and inactive phases originate in the reducibility of the dynamics. We illustrate the phase coexistence by considering the distributions of mesoscopic space-time observables. We show how the analogy with phase coexistence can be strengthened by breaking microscopic reversibility in the FA model, leading to a nonequilibrium theory in the directed percolation universality class.     

Gas-liquid phase coexistence in quasi-two-dimensional Stockmayer fluids: A molecular dynamics study

The Maxwell construction together with molecular dynamics simulation is used to study the gas-liquid phase coexistence of quasi-two-dimensional Stockmayer fluids. The phase coexistence curves and corresponding critical points under different dipole strength are obtained, and the critical properties are calculated. We investigate the dependence of the critical point and critical properties on the dipole strength. When the dipole strength is increased, the abrupt disappearance of the gas-liquid phase coexistence in quasi-two-dimensional Stockmayer fluids is not found. However, if the dipole strength is large enough, it does lead to the formation of very long reversible chains which makes the relaxation of the system very slow and the observation of phase coexistence rather difficult or even impossible.     

Dynamical arrest, percolation, gelation, and glass formation in model nanoparticle dispersions with thermoreversible adhesive interactions

We report an experimental study of the dynamical arrest transition for a model system consisting of octadecyl coated silica suspended in n-tetradecane from dilute to concentrated conditions spanning the state diagram. The dispersion's interparticle potential is tuned by temperature affecting the brush conformation leading to a thermoreversible model system. The critical temperature for dynamical arrest, T*, is determined as a function of dispersion volume fraction by small-amplitude dynamic oscillatory shear rheology. We corroborate this transition temperature by measuring a power-law decay of the autocorrelation function and a loss of ergodicity via fiber-optic quasi-elastic light scattering. The structure at T* is measured using small-angle neutron scattering. The scattering intensity is fit to extract the interparticle pair-potential using the Ornstein-Zernike equation with the Percus-Yevick closure approximation, assuming a square-well interaction potential with a short-range interaction (1% of particle diameter). (1) The strength of attraction is characterized using the Baxter temperature (2) and mapped onto the adhesive hard sphere state diagram. The experiments show a continuous dynamical arrest transition line that follows the predicted dynamical percolation line until ? ≈ 0.41 where it subtends the predictions toward the mode coupling theory attractive-driven glass line. An alternative analysis of the phase transition through the reduced second virial coefficient B(2)* shows a change in the functional dependence of B(2)* on particle concentration around ? ≈ 0.36. We propose this signifies the location of a gel-to-glass transition. The results presented herein differ from those observed for depletion flocculated dispersion of micrometer-sized particles in polymer solutions, where dynamical arrest is a consequence of multicomponent phase separation, suggesting dynamical arrest is sensitive to the physical mechanism of attraction.     

A THEORY OF THE FORMATION OF MICROEMULSIONS

A mathematical model was proposed (1) in a recently published paper in an attempt to represent the formation of microemulsion systems. However, the model fits microemulsion systems only up to certain critical dispersed phase volume, known as the percolation threshold. A single total free energy minimum corresponding to a stable system was found by the model for phase volumes below the percolation threshold. If the dispersed phase volume is further increased, the model predicts instability with no free energy minimum. However, experimental results (1,2) indicate percolation of microdroplets above the percolation threshold before the system eventually breaks down upon further increase of the phase volume. An extra free energy term from the interaction between microdroplets is introduced and incorporated into the model. This new model now predicts two free energy minima above the percolation threshold of the dispersed phase corresponding to polydispersity of the system. Further increase in dispersed phase volume will then cause the system to become unstable.     

Performance evaluation of third-order thermodynamic perturbation theory and comparison with existing liquid state theories

To evaluate the performance of a recently proposed third-order thermodynamic perturbation theory (TPT), we employ the third TPT for calculation of thermodynamic properties such as compressibility factor, internal energy, excess chemical potential, gas-liquid coexistence curve, and critical properties of several fluids. By comparing the third-order TPT results with corresponding simulation data available in literature and supplied in the present report and theoretical results from several other theoretical approaches, one concludes that the third-order TPT is, in general, more accurate than other approaches such as Barker-Henderson second-order TPT using a macroscopic compressibility approximation (MCA-TPT), self-consistent Ornstein-Zernike approach, Monte Carlo perturbation theory, and a specially devised equation of state. Specifically, the third-order TPT can predict quantitatively a double critical phenomena of gas-liquid transition and a low-density liquid (LDL)-high-density liquid (HDL) transition associated with a soft core (SC) potential fluid very satisfactorily, but the predictions for the LDL-HDL transition based on the second-order MCA-TPT are quantitatively very bad or qualitatively incorrect. The failure of the second-order MCA-TPT for the SC fluid can be ascribed to the facts that for the SC potential the second-order and third-order terms of the perturbation expansion are not small quantities and that the second-order term is underestimated by the MCA. It is concluded that the present third-order version of the TPT is reliable for varying model fluids.     

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.