Depletion interaction between two spheres

1 Interaction Potential Between Two Spheres Using the Force Method 

When the depletion zones with thickness r/2 around spherical colloidal particles with radius R start to overlap, i.e., when the distance r between the centers of the colloidal particles is smaller than 2R+ a = IRj, a net force arises between the colloidal particles. For a convenient notation we defined an effective depletion radius Rd [9] 

This (attractive) force originates from an uncompensated (osmotic) pressure due to the depletion of penetrable hard spheres from the gap between the colloidal particles. This is depicted in Fig. 2.4 from which we immediately deduce that the uncompensated pressure acts on the surface between 9 = 0 and Oq = arc cos r/2Rd). 

For obvious symmetry reasons only the component along the line connecting the centers of the colloidal spheres contributes to the total force. For the angle 0 this component is P cos 6 where the pressure is P = ribkT. The surface on which this force acts between 6 and 6 + dO equals 2nR sin 0 d0. The total force between the colloidal spheres is obtained by integration over 0 from 0 to 0o 

This result was also obtained by Asakura and Oosawa [1]. The minus sign in the right-hand side of (2.17) implies that the force is attractive. 

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Fig. 1.8 Sketch of the depletion interaction between two 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. 

The depletion interaction between two large spheres induced by oblate spheroids was first investigated in the low density limit by Kech and Walz. " Applying the Derjaguin approximation, they found for the limiting case of infinitely thin circular discs... 

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. 

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 effective pair interactions measured with these techniques are the direct pair interactions between two colloidal particles plus the interactions mediated by the depletants. In practice depletants are poly disperse, for which there are sometimes theoretical results available. For the interaction potential between hard spheres we quote references for the depletion interaction in the presence of polydisperse penetrable hard spheres [74], poly disperse ideal chains [75], poly-disperse hard spheres [76] and polydisperse thin rods [77]. 

Walz and Sharma [1439] calculated the depletion force between two charged spheres in a solution of charged spherical macromolecules. Compared to the case of hard-sphere interactions only, the presence of a long-range electrostatic repulsion increases greatly both the magnitude and the range of the depletion effect. Simulations and density functional calculations for polyelectrolytes between two planar surfaces extend these results [1440]. 

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... 

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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