Density functional theory of nonuniform colloidal suspensions: 3D density distributions and depletion forces

Long-range weight functions in fundamental measure theory of the non-uniform hard-sphere fluid

We introduce long-range weight functions to the framework of fundamental measure theory (FMT) of the non-uniform, single-component hard-sphere fluid. While the range of the usual weight functions is equal to the hard-sphere radius R, the modified weight functions have range 3R. Based on the augmented FMT, we calculate the radial distribution function g(r) up to second order in the density within Percus' test particle theory. Consistency of the compressibility and virial routes on this level allows us to determine the free parameter γ of the theory. As a side result, we obtain a value for the fourth virial coefficient B 4 which deviates by only 0.01% from the exact result. The augmented FMT is tested for the dense fluid by comparing results for g(r) calculated via the test particle route to existing results from molecular dynamics simulations. The agreement at large distances (r  >  6R) is significantly improved when the FMT with long-range weight functions is used. In order to improve agreement close to contact (r  =  2R) we construct a free energy which is based on the accurate Carnahan-Starling equation of state, rather than the Percus-Yevick compressibility equation underlying standard FMT.

Extension of scaled particle theory to inhomogeneous hard particle fluids. III. Entropic force exerted on a cavity that intersects a hard wall

We present a further development of an inhomogeneous scaled particle theory (I-SPT) for hard particle fluids confined by hard walls, such that the reversible work of cavity insertion can now be determined for all cavities that intersect one of the walls. Building upon a previous version of I-SPT [D. W. Siderius and D. S. Corti, Phys. Rev. E, 71, 036141 (2005)], a new function, F[over ] , is introduced, which is proportional to the net force on the surface of the cavity in the direction normal to the wall. The reversible work of cavity insertion is then determined by an integral over the force required to "push" the cavity of fixed size into the fluid starting from a position behind the wall. An exact relation for F[over ] at certain cavity locations and radii is derived and an accurate interpolation scheme is proposed for the computation of F[over ] beyond these exact limits. The chosen interpolation incorporates a large number of exact and nearly exact conditions, several of which follow from the surface thermodynamics of macroscopic cavities. Work predictions using F[over ] are highly accurate as compared to simulation results at low to moderate fluid densities. Good agreement still persists at densities near the hard-sphere freezing transition. The interpolation of F[over ] is also used to estimate the depletion force between a hard sphere solute and the wall. The I-SPT entropic force predictions are in good agreement with simulation results presented in the literature. Due to its reliance upon physical and geometric arguments, I-SPT provides important insights into the origin of various depletion effects such as how the interplay between geometry and the varying local density at the cavity surface gives rise to the appearance of multiple attractive regions at intermediate solute sizes and a universal repulsive region, both within solute to wall separations that are less than the diameter of a solvent particle. Finally, all of the scaled particle theory-based methods presented here can, in principle, be extended to describe hard particle fluids confined by nonplanar surfaces, thereby providing estimates of the depletion force between a solute and a variety of surfaces of interest.

Geometrical confinements and depletion interactions

In the system with two large spheres confined between two parallel plates, there are depletion interactions between the two large spheres and between one large sphere and the closely placed plate. Obviously, the depletion interactions exerted on one large sphere will be strongly affected by the presence of the closely placed plate or the other large sphere. This prediction is confirmed by the numerical results obtained through the acceptance ratio method (ARM) or density integration method (DIM), i.e., they are strengthened when two large spheres are contacted. Furthermore, it is found that the influences on the depletion forces are also sensitive to the angle of the centers' connection line between the two large spheres and the confining walls. In addition, the numerical results show that the total depletion force exerted on one large sphere from both the other large sphere and the closely placed plate can be determined through ARM or DIM from the interactions between the two large spheres or between one large sphere and the corresponding closely placed plate.

Entropic forces in dilute colloidal systems

Depletion forces can be accounted for by a contraction of the description in the framework of the integral equations theory of simple liquids. This approach includes, in a natural way, the effects of the concentration on the depletion forces, as well as energetic contributions. In this paper we systematically study this approach in a large variety of dilute colloidal systems composed of spherical and nonspherical hard particles, in two and in three dimensions, in the bulk and in front of a hard wall with a relief pattern. We show by this way the form in which concentration and geometry determine the entropic interaction between colloidal particles. The accuracy of our results is corroborated by comparison with computer simulations.

Potential of mean force in confined colloids: Integral equations with fundamental measure bridge functions

The potential of mean force for uncharged macroparticles suspended in a fluid confined by a wall or a narrow pore is computed for solvent-wall and solvent-macroparticle interactions with attractive forces. Bridge functions taken from Rosenfeld's density-functional theory are used in the reference hypernetted chain closure of the Ornstein-Zernike integral equations. The quality of this closure is assessed by comparison with simulation. As an illustration, the role of solvation forces is investigated. When the "residual" attractive tails are given a range appropriate to "hard sphere-like" colloids, the unexpected role of solvation forces previously observed in bulk colloids is confirmed in the confinement situation.

Extension of scaled particle theory to inhomogeneous hard particle fluids. I. Cavity growth at a hard wall

The methods of traditional scaled particle theory (SPT) are used to develop an extended scaled particle theory that is now applicable to hard particle fluids confined by hard walls. The new theory, labeled inhomogeneous SPT (I-SPT), introduces the function G that describes the average value of the anisotropic density of hard particle centers contacting a cavity placed at or behind a hard wall. We present an exact relation describing G for certain cavity sizes and, similar to the original SPT, an accurate interpolation scheme for larger cavity radii. Given G , the reversible work of inserting a cavity centered at or behind the hard wall can be estimated. The work predictions at low to moderate packing fractions are extremely accurate, though small deviations from simulation results become apparent at packing fractions close to the bulk hard sphere freezing transition. I-SPT also reveals the importance of the line tension in determining the free energy of cavity formation for cavities intersecting a hard wall, a term which has been previously neglected. Furthermore, this paper provides the initial groundwork needed to develop a more complete SPT-based theory that can accurately predict the depletion force between a hard particle and a hard structureless wall.

Solvent mediated interactions close to fluid-fluid phase separation: Microscopic treatment of bridging in a soft-core fluid

Using density functional theory we calculate the density profiles of a binary solvent adsorbed around a pair of big solute particles. All species interact via repulsive Gaussian potentials. The solvent exhibits fluid-fluid phase separation, and for thermodynamic states near to coexistence the big particles can be surrounded by a thick adsorbed "wetting" film of the coexisting solvent phase. On reducing the separation between the two big particles we find there can be a "bridging" transition as the wetting films join to form a fluid bridge. The effective (solvent mediated) potential between the two big particles becomes long ranged and strongly attractive in the bridged configuration. Within our mean-field treatment the bridging transition results in a discontinuity in the solvent mediated force. We demonstrate that accounting for the phenomenon of bridging requires the presence of a nonzero bridge function in the correlations between the solute particles when our model fluid is described within a full mixture theory based upon the Ornstein-Zernike equations.

Interactions between colloidal particles in polymer solutions: A density functional theory study

We present a density functional theory study of colloidal interactions in a concentrated polymer solution. The colloids are modeled as hard spheres and polymers are modeled as freely jointed tangent hard sphere chains. Our theoretical results for the polymer-mediated mean force between two dilute colloids are compared with recent simulation data for this model. Theory is shown to be in good agreement with simulation. We compute the colloid-colloid potential of mean force and the second virial coefficient, and analyze the behavior of these quantities as a function of the polymer solution density, the polymer chain length, and the colloid/polymer bead size ratio.

Effect of repulsive and attractive interactions on depletion forces in colloidal suspensions: a density functional theory treatment

The author employs density functional theory to study colloidal interactions in solution. Hardcore Yukawa potentials with soft tails, either repulsive or attractive, are used to model colloid-solvent and solvent-solvent interactions. We analyze the effect of these interactions on the solvent-mediated potential of mean force between two colloids in solution. Overall, theory is shown to be in good agreement with recent simulation data. We use the theory to study the density dependence of the colloid-colloid second virial coefficient.

Attractive forces in sterically stabilized colloidal suspensions: from the effective potential to the phase diagram

The potential of mean force for macroparticles at infinite dilution is computed for several models of solvent-solvent and solvent-macroparticle interactions by using the reference hypernetted chain (RHNC) integral equations with Rosenfeld's density functional theory bridge functions. The phase diagram of the associated effective fluid is obtained from the RHNC free energy for the fluid branch and the perturbation theory for the solid one. The computation of the effective potential and of the fluid branch is tested by comparison with Monte Carlo simulation. The important modifications with respect to the pure hard spheres that were previously reported are confirmed. The possibility of inverting the relative stability of the fluid-fluid and the fluid-solid transitions by appropriate combination of the interaction parameters is shown. The importance of a fine description of the interactions is illustrated in the example of the role of the range of the solvent-solvent interaction potential.

Effective forces in colloidal mixtures: from depletion attraction to accumulation repulsion

Computer simulations and theory are used to systematically investigate how the effective force between two big colloidal spheres in a sea of small spheres depends on the basic (big-small and small-small) interactions. The latter are modeled as hardcore pair potentials with a Yukawa tail which can be either repulsive or attractive. For a repulsive small-small interaction, the effective force follows the trends as predicted by a mapping onto an effective nonadditive hardcore mixture: both a depletion attraction and an accumulation repulsion caused by small spheres adsorbing onto the big ones can be obtained depending on the sign of the big-small interaction. For repulsive big-small interactions, the effect of adding a small-small attraction also follows the trends predicted by the mapping. But a more subtle "repulsion through attraction" effect arises when both big-small and small-small attractions occur: upon increasing the strength of the small-small interaction, the effective potential becomes more repulsive. We have further tested several theoretical methods against our computer simulations: The superposition approximation works best for an added big-small repulsion, and breaks down for a strong big-small attraction, while density functional theory is very accurate for any big-small interaction when the small particles are pure hard spheres. The theoretical methods perform most poorly for small-small attractions.

Energetic contributions to wall-particle depletion forces

Recently, depletion forces were accounted for by a contraction of the description based on the integral equations theory of simple liquids [Phys. Rev. E 61, 4095 (2000)]. The extension of those results to the case of inhomogeneous systems is reported here. Besides, the energetic contributions to the wall-particle depletion forces are studied, as they arise as soon as charge is put on some of the components of a binary mixture of hard spheres on the front of a hard wall. By charging the small particles the amplitude of the depletion attraction between wall and large particles is reduced, and can even become a repulsion. A similar effect is observed if an attractive interaction between wall and small particles is present.

Depletion potential in hard-sphere mixtures: theory and applications

We present a versatile density functional approach (DFT) for calculating the depletion potential in general fluid mixtures. For the standard situation of a single big particle immersed in a sea of small particles near a fixed object, the system is regarded as an inhomogeneous binary mixture of big and small particles in the external field of the fixed object, and the limit of vanishing density of the big species, rho(b)-->0, is taken explicitly. In this limit our approach requires only the equilibrium density profile of a one-component fluid of small particles in the field of the fixed object, and a knowledge of the density independent weight functions which characterize the mixture functional. Thus, for a big particle near a planar wall or a cylinder or another fixed big particle, the relevant density profiles are functions of a single variable, which avoids the numerical complications inherent in brute force DFT. We implement our approach for additive hard-sphere mixtures, comparing our results with computer simulations for the depletion potential of a big sphere of radius R(b) in a sea of small spheres of radius R(s) near (i) a planar hard wall, and (ii) another big sphere. In both cases our results are accurate for size ratios s=R(s)/R(b) as small as 0.1, and for packing fractions of the small spheres eta(s) as large as 0.3; these are the most extreme situations for which reliable simulation data are currently available. Our approach satisfies several consistency requirements, and the resulting depletion potentials incorporate the correct damped oscillatory decay at large separations of the big particles or of the big particle and the wall. By investigating the depletion potential for high size asymmetries we assess the regime of validity of the well-known Derjaguin approximation for hard-sphere mixtures and argue that this fails, even for very small size ratios s, for all but the smallest values of eta(s) where the depletion potential reduces to the Asakura-Oosawa potential. We provide an accurate parametrization of the depletion potential in hard-sphere fluids, which should be useful for effective Hamiltonian studies of phase behavior and colloid structure. Our results for the depletion potential in a hard-sphere system, with a size ratio s=0.0755 chosen to mimic a recent experiment on a colloid-colloid mixture, are compared with the experimental data. Although there is good overall agreement, in particular for the dependence of the oscillations on eta(s), there are some significant differences at high values of eta(s).

Depletion forces in colloidal mixtures

Depletion forces are accounted for by a contraction of the description of colloidal mixtures based on the integral equations theory of simple liquids. The applicability of this treatment is illustrated for binary mixtures of hard spheres, in the bulk and near a hard wall. The Asakura and Oosawa potential is obtained as the dilute limit of our equations. At higher concentrations the depletion potential has an oscillatory behavior and becomes more long ranged. If charge is put on the small particles there are energy-driven depletion forces in addition to those of entropic origin, which result in repulsive interaction at contact.