Depletion Interaction Due to Penetrable Hard Spheres

In this chapter we consider the depletion interaction between two flat plates and between two spherical colloidal particles for different depletants (polymers, small colloidal spheres, rods and plates). First of all we focus on the depletion interaction due to a somewhat hypothetical model depletant, the penetrable hard sphere (phs), to mimic a (ideal) polymer molecule. This model, implicitly introduced by Asakura and Oosawa [1] and considered in detail by Vrij [2], is characterized by the fact that the spheres freely overlap each other but act as hard spheres with diameter a when interacting with a wall or a colloidal particle. The thermodynamic properties of a system of hard spheres plus added penetrable hard spheres have been considered by Widom and Rowlinson [3] and provided much of the inspiration for the theory of phase behavior developed in Chap. 3. 

We now consider the depletion interaction due to (small) colloidal hard spheres with diameter interaction between the spheres so the system can considered to be thermodynamically ideal, the results for the depletion interaction are identical to those for penetrable hard spheres. At higher concentrations, say at volmne fractions larger than a few percent, the interactions between the spheres cannot be neglected. This has two important consequences for the depletion interaction. First of all the pressure and chemical potential are no longer given by the ideal expressions. The corrections to ideal behaviour can be written in terms of the virial series (see textbooks on statistical thermodynamics, e.g.. Hill [42] or Widom [43]) ... 

For the calculation of the depletion interaction due to hard spheres we need the concentration profile between two confining walls. This problem was treated analytically by Glandt [45] and by Antonchenko et al. [46] using Monte Carlo computer simulations. Like for a single waU we present the calculation of the concentration profile between two confining walls to order n. For hdepletion zone of a sphere overlaps with the depletion zones of both walls (see Fig. 2.22) and we can write... 

Theoretical work on depletion interactions and their effects on macroscopic properties such as phase stability commenced along various routes. First, Vrij [40] considered the depletion interaction between hard spheres due to dilute non-ad-sorbing polymers such as penetrable hard spheres (see Sect. 1.2.5 and Sect. 2.1). Vrij [40] referred to the work of Vester [82], Li-In-On et al. [55] and preliminary experiments at the Van t Hoff Laboratory on micro-emulsion droplets mixed with free polymer [40] for experimental evidence of depletion effects. 

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