chapter 2 section 2.4

DISCUSSION

When infinitely thin needles are added to a hard sphere solution a phase separation will occur in a dense, hard-sphere-rich, liquid-like phase and a dilute vapor-like phase that contains few hard spheres. This is a pure entropic effect as only hard core interactions are involved. The phase separation is caused by an increase in the available volume for the needles as the spheres cluster. This increase in entropy of the rods will outweigh the loss in entropy of the spheres at high rod fugacity. The fluid-solid transition will preempt this phase separation for a system with a rod length 0.3 and a liquid phase will not be stable for these rod lengths. We expect that for very short rods the critical point will reappear in the solid phase. In the next chapter we show that in systems with very short ranged attraction a phase transition can occur between a dense packed crystal and a less dense crystal (see also ref. [41]). This phase transition is first order, ends in a critical point and is similar to the liquid-vapor transition.
The features of the phase diagrams of the rodlike polymer are completely comparable with that of the fully flexible chains [10], although in the latter case the disappearing of the liquid region is predicted at polymer radii of gyration 20.45. However, the latter estimate is quite crude. It is therefore not obvious that the difference with the value obtained here is significant.
The liquid-gas separation in the rod like polymer colloid mixture can be described by first order perturbation theory. This theory predicts the location of the critical point well at rod lengths 4, although this accuracy decreases for higher values of . The perturbation theory is accurate because the rods do not significantly affect the structure of the hard sphere dispersion. Hence, the free volume for a rod in a dispersion of spheres is effectively independent of the concentration of rods. Of course this is only valid in the case of ideal polymers. If the polymers would interact with each other the free volume would decrease considerably with the fugacity.
In the limit , the critical point shifts to zero density and zero polymer fugacity. The binodal curve is in this limit equivalent to the equation of state for hard spheres. The fugacity of the rods is equal to the pressure of the coexisting hard sphere fluid phase because in the vapor phase no spheres are present and the liquid phase is free of needles. This fact may provide an alternative route to obtain the equation of state of hard spheres experimentally by measuring the concentration of rods in the vapor phase of a phase separated mixture of colloids and stiff polymers as a function of the volume fraction of spheres in the liquid phase.
The simulations discussed in this chapter show that for mixtures of hard spherical particles and thin needles liquid-vapor phase separation will occur. However, one could object that the limiting case of infinitely thin needles is unrealistic. Hence, it is important to know whether the demixing transition will still occur if the needles have a finite diameter D. Recently, Mao et al. [42] have investigated the effect of the finite thickness of the rods on the potential of mean force between two spheres. They find that, for rods of finite thickness, the effective pair potential develops a repulsive barrier at finite separation, in addition to the attractive well at contact. In particular for small values, this barrier can be high enough to lead to kinetic stabilization of the suspension and prevent any phase separation.

Moreover, one should also consider the possibility that the rodlike particles will undergo a transition from the isotropic to the nematic phase. This transition will take place when , where is the second virial coefficient of the rods and their number density. To see at what value of the rod osmotic pressure the isotropic-nematic transition occurs, we consider the Onsager limit of rods with high ratio [25]. In this limit the osmotic pressure of the rods can be written as

where we used the diameter of the spheres as unit of length. The isotropic phase becomes unstable at a density of 3.3 [25], which corresponds to an osmotic pressure . To observe the liquid-vapor phase separation, this pressure has to be larger than the critical pressure for the phase separation.
In the limit that while L remains finite, the I-N transition occurs at infinite pressure. For rods with a finite diameter, demixing can compete with the I-N transition. To be specific, on basis of the perturbation theory and eqn. 2.19 we assume that for , . If we equate to , we find that demixing is preempted by an I-N transition if . This implies that for small the liquid-vapor phase separation occurs well before the I-N transition.
Of course, the finite thickness of the rods will alter the simulated phase diagrams, especially for small ratios. This will probably shift the phase separation region as well as the isotropic-nematic transition to higher fugacities. So we expect that the liquid-vapor phase separation still occurs at lower fugacities than the isotropic-nematic transition for reasonable ratios (5), provided that the transition is not prevented by any kinetic stabilization [42] .

chapter 2 section 2.4