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, ${\rho }_b\to 0,$ 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 $s=0.0755\mathrm{}$ 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 ${\eta }_s.$