Phase Separation in Binary Mixtures

The previous section reviewed the elements of statistical mechanics that are important in thinking about the structures, fluctuations, and phase behavior of surfaces, interfaces, and membranes. In this section, we consider an important application of these ideas to the problem of phase separation in binary mixtures. This problem is analogous to other types of phase transitions, such as those found in Ising magnets. It is important to understand the specific problem of phase separation because it is this phenomenon that results in the equilibrium between two coexisting states, which naturally gives rise to the existence of interfaces. 

The negative sign indicates that these interactions are attractive the Jij are positive quantities that are a measure of the magnitude of the attractive interactions. The first term is nonzero only when j,- and sj are both zero, indicating the presence of two A molecules on sites separated by a distance 1, - similar considerations apply to the second and third terms. 

One can multiply all the occupation variables 5, and collect terms in powers of s,. The result yields a constant term that just redefines the zero of energy for the system, a linear term that multiplies the average volume fraction of solute molecules (which is either fixed or determined by a chemical potential), and a quadratic term. Adding and subtracting terms linear in, the net interaction in the system can be written 

if the magnitude of the attractive interactions between AA and BB pairs are larger than those between the AB pairs, the system will tend to phase separate and maximize the number of AA pairs and BB pairs in the two coexisting phases. One therefore defines 

To calculate the partition function, one includes a chemical potential to conserve the number of B particles (all of the other terms linear in the i, are included here). The constrained partition function, Z, is 

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Figure 8.2 Phase separation in binary mixtures of model spherical particles at a planar interface generated by Brownian dynamics simulation. The three 2-D images refer to systems in which (A) light particles form irreversible bonds, (B) light particles form reversible bonds, and (C) neither dark nor light particles form bonds, but they repel each other. Picture D shows a 3-D representation. Reproduced from Pugnaloni et al. (2003b) with permission.

Heterogeneity, as in polyblends, has also been observed in random copolymers. F. Kollinsky and G. Markert found phase separation in binary mixtures of copolymers of methyl methacrylate and butyl acrylate. C. Kraus and K. W. Rollmann discovered heterogeneity in blends of random copolymers of butadiene and styrene if they differ by more than 20% in composition. 

Now, we turn our attention to late stages of spinodal decomposition. Since the phase separation in binary mixture is intrinsically a nonlinear phenomenon, a number of nonlinear theories have been put forward on the basis of statistical consideration, notably the LBM (Danger, Baron k Miller) (H) and BS (Binder k Stauffer) (12) theories. Both theories predicted the power law scheme rather than the exponential growth of the structure... 

C. Behavior of EELS near Point of Phase Separation in Binary Mixtures References... 

However if, over some range of compositions, the mixture splits into two phases, then the single-phase equilibrium curve for g "(xi) will not be convex over all Xj. Similarly, the monotonicity of f x ) will be disrupted either by oscillations or by branching. These possibilities appear in Figure 8.13 oscillations occur in the f xi) curves for 30 and 60, while at 10 bar, the /i(xi) curve has divided into two distinct branches. These phenomena are caused by bifurcations in either the equation of state or the fugacity equation or both. Here we use those possibilities to identify four classes of instabilities that can lead to vapor-liquid phase separations in binary mixtures. 

Dahmen, N. and Schneider, G.M. (1993) Phase separation in binary mixtures of trifluo-romethane with propane, butane and xenon at low temperatures between 200 and 280 K, and at pressures up to 200 MPa, Fluid Phase Equilibria 87, 295-308. 

Let us now look at the phase separation in binary mixtures from the viewpoint of stability. The separation of phases occurs when the system becomes unstable with respect to diffusion of the two components, i.e. if the separation of the two components produces an increase in entropy then the fluctuations in the mole number due to diffusion in a given volume grow, resulting in the separation of the two components. As we have seen in section 12.4, the condition for stability against diffusion of the components is... 

Phase separations in binary mixtures of a flexibie polymer and a liquid crystal... 

Ginzburg, V.V. et al. (1999) Kinetic model of phase separation in binary mixtures with hard mobile impurities. Physical Review E, 60 (4), 4352 359. 

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