Free Volume Theory of Hard Spheres and Depletants

3 Free Volume Theory of Hard Spheres and Depletants 3.3.1 System 

In the early nineties of the last century a theory that accounts for depletant partitioning over the coexisting phases was developed [27], which nowadays is commonly referred to as free volume theory (FVT) [28]. This theory is based on the osmotic equilibrium between a (hypothetical) depletant and the colloid + depletant system. The depletants were simplified as penetrable hard spheres. A pictorial representation is given in Fig. 3.6. 

In FVT multiple overlap of depletion zones with thickness 5, see Fig. 3.8, is taken into account. Multiple overlap occurs for 

3 Phase Transitions of Hard Spheres Plus Depletants Basics 

Exercise Show that multiple overlap only occurs for — -v3 — 1. 

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In this chapter we have presented the free volume theory for hard spheres plus depletants and focused on the simplest possible case of hard spheres + penetrable hard spheres. In the next chapters we will extend the free volume theory to more realistic situations (Chap. 4 hard spheres + polymers. Chap. 5 hard spheres -I- small colloidal particles. Chap. 6 hard rods -I- polymers) and compare the results with experiments and simulations. 

In this chapter we discuss the basics of the phase behaviour of hard spheres plus depletants. Phase transitions are the result of physical properties of a collection of particles depending on many-body interactions. In Chap. 2 we focused on two-body interactions. As we shall see, depletion elfects are commonly not pair-wise additive. Therefore, the prediction of phase transitions of particles with depletion interaction is not straightforward. As a starting point a description is required for the thermodynamic properties of the pure colloidal dispersion. Here the colloid-atom analogy, recognized by Einstein and exploited by Perrin in his classical experiments, is very useful. Subsequently, we explain the basics of the free volume theory for the phase behaviour of colloids -I- depletants. In this chapter we treat only simplest type of depletant, the penetrable hard sphere. 

An expression for the work of insertion W can be obtained from scaled particle theory (SPT) [31]. SPT was developed to derive expressions for the chemical potential and pressure of hard sphere fluids by relating them to the reversible work needed to insert an additional particle in the system. This work W is calculated is by expanding (scaling) the size of the sphere to be inserted from zero to its final size the size of the scaled particle is Act, with X running from 0 to 1. In the limit 2 0, the inserted sphere approaches a point particle. In this limiting case it is very unlikely that the depletion layers overlap. The free volume fraction in this limit can therefore be written as... 

We now incorporate the correct depletion thickness into free volume theory presented in Sect. 3.3. We consider the osmotic equilibrium between a polymer solution (reservoir) and the colloid-polymer mixture (system) of interest, see Fig. 4.6. The general expression for the semi-grand potential for Nc hard spheres plus interacting polymers as depletants, see (3.18), is... 

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