Free Volume Theory for Sphere-Rod Mixtures

In the introduction of this chapter we mentioned that rod-like colloids influence the phase behaviour of suspensions of spherical colloids significantly at very low concentration. For a review see [32]. This is not surprising as rod-like colloids give rise to a strong depletion interaction at low concentration, see (2.107). Here we will see that FVT (correctly) captures the above mentioned pronounced depletion effect caused by rod-like particles. 

Again we start from an osmotic equilibrium where now the reservoir contains colloidal rods and the system contains colloidal spheres and rods. The osmotic equilibrium system considered is depicted in a schematic way in Fig. 5.9. Following the same steps as Sect. 3.3 we obtain for the semi-grand potential for 

In this equation Ni is the number of large hard spheres, 2 chemical potential of the hard rods imposed by the (hypothetical) reservoir, Fq is the free energy of the hard sphere system without added rods, P is the pressure of the hard rods in the reservoir and (Vfree)o is the free volume of an added rod in the system of N hard spheres in a volume V. 

Since we are now dealing with hard rods as the depletion agent both the pressure in the reservoir and the free volume differ from the case of spheres as depletion agent. Both quantities can again be calculated conveniently using SPT [33]. 

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