Direct Simulation of the Phase Behavior of Binary Hard-Sphere Mixtures: Test of the Depletion Potential Description (original) (raw)
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The Journal of Chemical Physics, 2021
Comprehensive calculations were performed to predict the phase behaviour of large spherical colloids mixed with small spherical colloids that act as depletant. To this end, the free volume theory (FVT) of Lekkerkerker et al. [Europhys. Lett. 20 (1992) 559] is used as a basis and is extended to explicitly include the hard-sphere character of colloidal depletants into the expression for the free volume fraction. Taking the excluded volume of the depletants into account in both the system and the reservoir provides a relation between the depletant concentration in the reservoir and in the system that accurately matches with computer simulation results of Dijkstra et al. [Phys. Rev. E 59 (1999) 5744]. Moreover, the phase diagrams for highly asymmetric mixtures with size ratios q 0.2 obtained by using this new approach corroborates simulation results significantly better than earlier FVT applications to binary hard-sphere mixtures. The phase diagram of a binary hard-sphere mixture with a size ratio of q = 0.4, where a binary interstitial solid solution is formed at high densities, is investigated using a numerical free volume approach. At this size ratio, the obtained phase diagram is qualitatively different from previous FVT approaches for hard-sphere and penetrable depletants, but again compares well with simulation predictions.
Asymmetric binary mixtures of hard-spheres exhibit several interesting thermodynamic phenom- ena, such as multiple kinds of glassy states. When the degrees of freedom of the small spheres are integrated out from the description, their effects are incorporated into an effective pair inter- action between large spheres known as the depletion potential. The latter has been widely used to study both the phase behavior and dynamic arrest of the big particles. Depletion forces can be ac- counted for by a contraction of the description in the multicomponent Ornstein-Zernike equation [R. Castañeda-Priego, A. Rodríguez-López, and J. M. Méndez-Alcaraz, Phys. Rev. E 73, 051404 (2006)]. Within this theoretical scheme, an approximation for the difference between the effective and bare bridge functions is needed. In the limit of infinite dilution, this difference is irrelevant and the typical Asakura-Osawa depletion potential is recovered. At higher particle concentrations, how- ever, this difference becomes important, especially where the shell of first neighbors is formed, and, as shown here, cannot be simply neglected. In this work, we use a variant of the Verlet expression for the bridge functions to highlight their importance in the calculation of the depletion potential at high densities and close to the spinodal decomposition. We demonstrate that the modified Verlet closure predicts demixing in binary mixtures of hard spheres for different size ratios and compare its predic- tions with both liquid state and density functional theories, computer simulations, and experiments. We also show that it provides accurate correlation functions even near the thermodynamic instability; this is explicitly corroborated with results of molecular dynamics simulations of the whole mixture. Particularly, our findings point toward a possible universal behavior of the depletion potential around the spinodal line.
Phase diagram of highly asymmetric binary hard-sphere mixtures
Physical Review E, 1999
We study the phase behavior and structure of highly asymmetric binary hard-sphere mixtures. By first integrating out the degrees of freedom of the small spheres in the partition function we derive a formal expression for the effective Hamiltonian of the large spheres. Then using an explicit pairwise ͑depletion͒ potential approximation to this effective Hamiltonian in computer simulations, we determine fluid-solid coexistence for size ratios qϭ0.033, 0.05, 0.1, 0.2, and 1.0. The resulting two-phase region becomes very broad in packing fractions of the large spheres as q becomes very small. We find a stable, isostructural solid-solid transition for qр0.05 and a fluid-fluid transition for qр0.10. However, the latter remains metastable with respect to the fluid-solid transition for all size ratios we investigate. In the limit q→0 the phase diagram mimics that of the sticky-sphere system. As expected, the radial distribution function g(r) and the structure factor S(k) of the effective one-component system show no sharp signature of the onset of the freezing transition and we find that at most points on the fluid-solid boundary the value of S(k) at its first peak is much lower than the value given by the Hansen-Verlet freezing criterion. Direct simulations of the true binary mixture of hard spheres were performed for qу0.05 in order to test the predictions from the effective Hamiltonian. For those packing fractions of the small spheres where direct simulations are possible, we find remarkably good agreement between the phase boundaries calculated from the two approaches-even up to the symmetric limit qϭ1 and for very high packings of the large spheres, where the solid-solid transition occurs. In both limits one might expect that an approximation which neglects higher-body terms should fail, but our results support the notion that the main features of the phase equilibria of asymmetric binary hard-sphere mixtures are accounted for by the effective pairwise depletion potential description. We also compare our results with those of other theoretical treatments and experiments on colloidal hard-sphere mixtures. ͓S1063-651X͑99͒07805-8͔
Phase Behavior and Structure of Binary Hard-Sphere Mixtures
Physical Review Letters, 1998
By integrating out the degrees of freedom of the small spheres in a binary mixture of large and small hard spheres, we derive an explicit effective Hamiltonian for the large spheres. Using the two-body (depletion potential) contribution to this effective Hamiltonian in simulations, we find stable fluid-solid and both metastable fluid-fluid and solid-solid coexistence in a mixture with size ratio q 0.1. For q 0.05 the solid-solid coexistence becomes stable. [S0031-9007(98)07074-4]
Phase behavior of nonadditive hard-sphere mixtures
Physical Review E, 1998
We show the existence of a fluid-fluid demixing transition in binary mixtures of nonadditive asymmetric hard-sphere mixtures by performing Gibbs ensemble Monte Carlo simulations for a size ratio of 0.1 and varying degrees of nonadditivity. We compare our results with the theoretical binodals obtained from the equation of state proposed by Barboy and Gelbart ͓J. Chem. Phys. 71, 3053 ͑1979͔͒ and we find reasonable agreement for sufficiently large values of the nonadditivity parameter. Upon decreasing the nonadditivity parameter, we find that the fluid-fluid demixing region shifts to higher pressures and becomes narrower. For sufficiently small nonadditivities, we do not find a fluid-fluid demixing transition for total packing fractions Ͻ0.5. ͓S1063-651X͑98͒02412-X͔
Large attractive depletion interactions in soft repulsive–sphere binary mixtures
The Journal of Chemical Physics, 2007
We consider binary mixtures of soft repulsive spherical particles and calculate the depletion interaction between two big spheres mediated by the fluid of small spheres, using different theoretical and simulation methods. The validity of the theoretical approach, a virial expansion in terms of the density of the small spheres, is checked against simulation results. Attention is given to the approach toward the hard-sphere limit, and to the effect of density and temperature on the strength of the depletion potential. Our results indicate, surprisingly, that even a modest degree of softness in the pair potential governing the direct interactions between the particles may lead to a significantly more attractive total effective potential for the big spheres than in the hard-sphere case. This might lead to significant differences in phase behavior, structure and dynamics of a binary mixture of soft repulsive spheres. In particular, a perturbative scheme is applied to predict the phase diagram of an effective system of big spheres interacting via depletion forces for a size ratio of small and big spheres of 0.2; this diagram includes the usual fluid-solid transition but, in the soft-sphere case, the metastable fluid-fluid transition, which is probably absent in hard-sphere mixtures, is close to being stable with respect to direct fluid-solid coexistence. From these results the interesting possibility arises that, for sufficiently soft repulsive particles, this phase transition could become stable. Possible implications for the phase behavior of real colloidal dispersions are discussed.
Physical Review E, 2011
We report a detailed study, using state-of-the-art simulation and theoretical methods, of the effective (depletion) potential between a pair of big hard spheres immersed in a reservoir of much smaller hard spheres, the size disparity being measured by the ratio of diameters q ≡ σ s /σ b . Small particles are treated grand canonically, their influence being parameterized in terms of their packing fraction in the reservoir η s r . Two Monte Carlo simulation schemes-the geometrical cluster algorithm, and staged particle insertion-are deployed to obtain accurate depletion potentials for a number of combinations of q � 0.1 and η s r . After applying corrections for simulation finite-size effects, the depletion potentials are compared with the prediction of new density functional theory (DFT) calculations based on the insertion trick using the Rosenfeld functional and several subsequent modifications. While agreement between the DFT and simulation is generally good, significant discrepancies are evident at the largest reservoir packing fraction accessible to our simulation methods, namely, η s r = 0.35. These discrepancies are, however, small compared to those between simulation and the much poorer predictions of the Derjaguin approximation at this η s r . The recently proposed morphometric approximation performs better than Derjaguin but is somewhat poorer than DFT for the size ratios and small-sphere packing fractions that we consider. The effective potentials from simulation, DFT, and the morphometric approximation were used to compute the second virial coefficient B 2 as a function of η s r . Comparison of the results enables an assessment of the extent to which DFT can be expected to correctly predict the propensity toward fluid-fluid phase separation in additive binary hard-sphere mixtures with q � 0.1. In all, the new simulation results provide a fully quantitative benchmark for assessing the relative accuracy of theoretical approaches for calculating depletion potentials in highly size-asymmetric mixtures.
Phase-Stability of Binary Nonadditive Hard-Sphere Mixtures-A Self-Consistent Integral-Equation Study
Journal of Chemical Physics, 1996
We have tested the capabilities of a new self-consistent integral equation, closely connected with Verlet's modified closure, for the study of fluid-fluid phase separation in symmetric non-additive hard-sphere mixtures. New expressions to evaluate the chemical potential of mixtures are presented and play a key role in the construction of the phase diagram. The new integral equation, which implements consistency between virial and fluctuation theorem routes to the isothermal compressibility, together with chemical potential and virial pressure consistency via the Gibbs-Duhem relation, yields a phase diagram which especially at high densities agrees remarkably well with the new semi-Grand Ensemble Monte Carlo simulation data also presented in this work. Deviations close to the critical point can be understood as a consequence of the inability to enforce virial-fluctuation consistency in the neighborhood of the spinodal decomposition curve.
Fractionation effects in phase equilibria of polydisperse hard-sphere colloids
Physical Review E, 2004
The equilibrium phase behaviour of hard spheres with size polydispersity is studied theoretically. We solve numerically the exact phase equilibrium equations that result from accurate free energy expressions for the fluid and solid phases, while accounting fully for size fractionation between coexisting phases. Fluids up to the largest polydispersities that we can study (around 14%) can phase separate by splitting off a solid with a much narrower size distribution. This shows that experimentally observed terminal polydispersities above which phase separation no longer occurs must be due to non-equilibrium effects. We find no evidence of re-entrant melting; instead, sufficiently compressed solids phase separate into two or more solid phases. Under appropriate conditions, coexistence of multiple solids with a fluid phase is also predicted. The solids have smaller polydispersities than the parent phase as expected, while the reverse is true for the fluid phase, which contains predominantly smaller particles but also residual amounts of the larger ones. The properties of the coexisting phases are studied in detail; mean diameter, polydispersity and volume fraction of the phases all reveal marked fractionation. We also propose a method for constructing quantities that optimally distinguish between the coexisting phases, using Principal Component Analysis in the space of density distributions. We conclude by comparing our predictions to perturbative theories for near-monodisperse systems and to Monte Carlo simulations at imposed chemical potential distribution, and find excellent agreement.
Phase equilibria of asymmetric hard sphere mixtures
Physical Review E, 1999
The phase diagram of mixtures of hard spheres with additive diameters is studied. The case of very different sizes is treated by means of mapping the two component system on a one component problem. In this monocomponent system large particles are explicitly considered, whereas the effects of the small component are included through an additional effective interaction potential between large particles. The effective potential is used to analyze the phase diagram of the mixture by means of computer simulation techniques. Results for the behavior at low density of small spheres seem to indicate that no fluid-fluid equlibria occur. On the other hand, the results show how this kind of mixture can exhibit equilibria between isostructural crystalline phases. ͓S1063-651X͑99͒08204-5͔