Statistical geometric models of hard-sphere colloidal dispersions: Application to interfacial thermodynamics and the calculation of depletion forces

Despite the lack of attractive interaction potentials in hard-sphere (HS) colloidal dispersions, a colloid immersed in a solvent of smaller hard-spheres may still experience a net force toward a surface due to imbalanced collisions. Consequently, depletion forces may be utilized to control self-assembly of colloidal structures on various surfaces or control aggregation of colloidal dispersions. To begin to better understand the ability of depletion forces to generate colloidal structures, guide colloidal motion, and control aggregation, accurate theoretical descriptions of depletion forces are necessary. Here, we discuss models of HS colloidal dispersions based on the ideas of Scaled Particle Theory (SPT) to provide methods of computing both HS thermophysical properties and depletion forces in HS fluids. We begin by introducing a new SPT interpolation that accurately provides many HS fluid properties, including the surface tension. We then further develop the inhomogeneous SPT (I-SPT) that describes cavities grown near a planar surface that confines a HS fluid, thereby providing a complete description of cavities near a planar surface, i.e., cavities that are centered at any position relative to the wall. The surface thermodynamics of HS cavities are then reexamined using a Gibbs dividing surface analysis, which produces thermodynamic expressions related to, among other things, the line tension of a HS cavity. Subsequently, we utilize I-SPT to compute the HS line tension and explore its behavior for different cavity locations. Using the accurate description of HS surface thermodynamics from SPT and I-SPT, we then construct a geometric model of depletion forces that is generalizable to many different surface structures and is based on different thermodynamic approximations. Versions of the geometric model based on HS surface thermodynamics (including the line tension) are demonstrated to be highly accurate, though simpler versions based on ideal gas arguments are often sufficient and more easily implemented for complex surfaces. Finally, using the geometric model, we perform stochastic simulations of HS colloids to investigate the dynamics of depletion interactions. The combination of the geometric model and stochastic simulations serves as an engineering tool, allowing one to design surfaces and examine their suitability for controlling colloidal dispersions