Depletion Interaction Due to Colloid 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]) ... 

The polymer density profile of ideal chains next to a hard sphere for arbitrary size ratio q was first ealeulated by Taniguchi et al. [125] and later independently by Eisenriegler et al. [126]. Eisenriegler also considered the pair interaction between two colloids for Rg< R [127] and for Rg R [128], as well as the interaction between a sphere and a flat wall due to ideal chains [129]. Depletion of excluded volume polymer chains at a wall and near a sphere was considered by Hanke et al. [130]. One of their results is that the ratio /Rg at a flat plate, which is 1.13 for ideal chains [118, 119], is slightly smaller (1.07) for excluded-volume chains. 

The large reduction in percolation threshold can be theoretically predicted on the basis of different depletion-induced interaction forces at play in the two systems. For this model, it is assumed here that the percolation network structure is largely determined by the initial colloidal system. The latex spheres are colloidal structures that can induce attraction between the dispersed CNTs due to depletion [osmotic pressure due to presence of hard-spheres). Consequently, these attractive forces can lead to changes in the... 

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