Binodal

Figure C2.1.10. (a) Gibbs energy of mixing as a function of the volume fraction of polymer A for a symmetric binary polymer mixture = Ag = N. The curves are obtained from equation (C2.1.9 ). (b) Phase diagram of a symmetric polymer mixture = Ag = A. The full curve is the binodal and delimits the homogeneous region from that of the two-phase stmcture. The broken curve is the spinodal.

Fig. 2. The phase diagrams and terminology of a microemulsion system close to its two critical end points, where the middle phase and one of the binodals...

The thermodynamics and physical properties of the mixture to be separated are examined. VLE nodes and saddles, LLE binodal curves, etc, are labeled. Critical features and compositions of interest are identified. A stream is selected from the source Hst. This stream is either identified as meeting all the composition objectives of a destination, or else as in need of further processing. Once an opportunistic or strategic operation is selected and incorporated into the flow sheet, any new sources or destinations are added to the respective Hsts. If a strategic separation for dealing with a particular critical feature has been implemented, then that critical feature is no longer of concern. Alternatively, additional critical features may arise through the addition of new components such as a MSA. The process is repeated until the source Hst is empty and all destination specifications have been satisfied. 

A convenient way of representing the T—x y phase diagram (Eig. 14b) is by projection onto the composition triangle at the base of the figure. It is understood that the temperature varies from point to point on the projected vapor line and on the projected boiling envelope. The latter looks like an isothermal Hquid—Hquid binodal envelope, but is not. Each tie line across the boiling envelope is associated with a different boiling temperature (Eig. 15). 

Residue cui ve maps and distillation region diagrams are very power-Ril tools for understanding all types of catch and continuous distillation operations, particularly when combined with other information such as hquid-liqiiid binod cui ves. Applications include ... 

Another kinetic jjhenomenon where Calm s critical waves can possibly be visualized and studied is the replication of interphase boundaries (IPB) illustrated in Figs. 8-10. Similarly to the replication of APBs. it can arise after a two-step quench of an initially uniform disordered alloy. First the alloy is quenched and annealed at temperature T in some two-phase state that can be either metastable or spinodally unstable with respect to phase separation. Varying the annealing time one can grow here precipitates ("droplets ) of a suitable size /. For sufficiently large /, the concentration c(r) within A-riched droplets is close to the equilibrium binodal value C(,(T ) (thin curve in Fig. 9). 

Figure 3 illustrates the thermodynamic interplay of polymer crystallization and liquid-liquid demixing in polymer solutions. The liquid-liquid binodal curve is primarily determined by the B value. With the increase of Ep values, the liquid-liquid binodal curves shift slightly upward. On the other hand, the... 

On the basis of the concept described above, we propose a model for the homogeneous crystallization mechanism of one component polymers, which is schematically shown in Fig. 31. When the crystallization temperature is in the coexistence region above the binodal temperature Tb, crystal nucleation occurs directly from the melt, which is the well-known mechanism of polymer crystal nucleation. However, the rate of crystallization from the coexistence region is considered to be extremely slow, resulting in single crystals in the melt matrix. Crystallization at a greater rate always involves phase separation the quench below Tb causes phase separations. The most popular case... 

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

Both the binodal line, defining the immiscibility gap, and the spinodal line are for a regular solution symmetrical about xA = xB = 0.5. This is shown in Figure 5.7(a), where theoretical predictions of the miscibility gaps in selected semiconductor systems are given [15],... 

Figure 5.7 (a) Theoretical predictions of the unstable regions (miscibility gap) of the solid solutions in the systems AlN-GaN, InN-GaN and AlN-InN [15]. For the system InN-GaN both the phase boundary (binodal) and spinodal lines are shown, (b) Gibbs energy of mixing for the solid solution InN-GaN at 1400 K. 

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