Phase separation binodal

The statistical thermodynamic theory makes a mean-field treatment to the crystallization energy Ep. However, in practice, thermal fluctuations for parallel ordering of local chains generally exist in the melt states of blends. Such an additional anisotropic contribution of thermal fluctuations will make the practical parallel-packing interactions larger than the mean-field estimation on the basis of the isotropic liquid, which leads to a deviation from the theoretical predictions of the phase separation binodals. 

The investigation of the Han plots, which is the log-log plot of storage modulus versus loss modulus, is another effective method to determine the onset of phase separation. This method is more sensitive to concentration fluctuations than data obtained from time-temperature superposition. The Han plot of homogeneous phases shows two main features temperature independence and terminal slope of two (Han et al. 1990, 1995). Deviations from these two criteria were reported only for Han plots above the LCST and below the UCST (Kim et al. 1998 Sharma and Clarke 2004). Therefore, it has been suggested to use this method to infer the phase-separation (binodal) temperature rheologicaUy. 

The interception of these two curves is known as Berghmans point (BP) and defined as the point where the liquid-liquid phase separation binodal line is intercepted by the vitrification curve (9,10) see Figure 4.4. When the vitrification curve intercepts the binodal line, the development of ordinary tie lines is inhibited because molecular diffusion in the glassy state is far slower than in the liquid state see Figure 4.4. 

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). 

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). 

At concentrations above ( ), a phase separation would lead to the formation of particles dispersed in a liquid matrix. The composition of such particles should be given by the binodal line. Thus such particles will still contain enough solvent to undergo a phase separation. Indeed such an internal phase separation can be used to prepare porous polymeric particles with potential for application as chromatography beads [48]. 

If hexane is used as the low molecular weight liquid, the desired phase separation is observed when precursor mixtures containing 6-15 wt % hexane are cured isothermally at 40 °C. Further discussion of the phase separation behavior requires more detailed consideration of the schematic phase diagram, as presented in Fig. 17, which resembles the real phase diagram shown in Fig. 13. Experimentally it is found, that no phase separation occurs with hexane concentrations equal to or lower than 5 wt %. Hence the critical amount for phase separation, (j)p, is given by the intercept of the binodal line and the imaginary value of Hence no phase separation occurs if is reached before the metastable region is entered. 

The growth rate, characterized by the change of the radius with time, is proportional to the driving force for the phase separation, given by the differences between 2 > the chemical composition of the second phase in the continuous phase at any time, and, its equihbrium composition given by the binodal line. The proportionahty factor, given by the quotient of the diffusion constant, D, and the radius, r, is called mass transfer coefficient. Furthermore the difference between the initial amount of solvent, (])o, and c]) must be considered. The growth rate is mathematically expressed by [101]... 

If k - oo then equilibrium is instantaneously reached and the system evolves along the binodal curve (trajectory a and a ). On the other hand, if k —> 0, then no phase separation will... 

Figure 3.3 Illustration of the calculation of the phase diagram of a mixed biopolymer solution from the experimentally determined osmotic second virial coefficients. The phase diagram of the ternary system glycinin + pectinate + water (pH = 8.0, 0.3 mol/dm3 NaCl, 0.01 mol/dm3 mercaptoethanol, 25 °C) —, experimental binodal curve —, calculated spinodal curve O, experimental critical point A, calculated critical point O—O, binodal tielines AD, rectilinear diameter,, the threshold of phase separation (defined as the point on the binodal curve corresponding to minimal total concentration of biopolymer components). Reproduced from Semenova et al. (1990) with permission.

Nowadays it is established that confocal microscopy observation can be a more sensitive method to assess die phase state of mixed biopolymer systems than the traditional centrifugation or viscometric methods (Alves et al., 1999, 2001 Vega et al., 2005). Indeed, microscopy can demonstrate that a system may be already phase-separated at compositions well below the apparent binodal line (as determined by these other methods). The report of Alves et al. (2001) demonstrates the relationship between specific compositional points in the phase diagram (Figure 7.1) and the observed microstructure (Figures 7.2 and 7.3) for water + gelatin + locust bean gum (LBG). The white areas in Figures 7.2 and 7.3 corre-... 

Figure 7.10 Effect of the thermodynamic incompatibility of otsi/p-casein + high-methoxy pectin (pH = 7.0, / = 0.01 M) on phase diagram of the mixed solutions and elastic modulus of corresponding casein-stabilized emulsions (40 vol% oil, 2 wt% protein), (a) (O) Binodal line for p-casein + pectin solution with critical point ( ) ( ) binodal line for asi-casein + pectin solution with critical point ( ). (b) Complex shear modulus G (1 Hz) is plotted against the pectin concentration (O) p-casein ( ) o i -casein. Dotted lines indicate the range of pectin concentration for phase separation in the mixed solutions. The pectin was added to the protein solution before emulsion preparation. Data are taken front Semenova et al. (1999a).

The solutions in the region inside the spinodal domain are unstable, whereas the solid solutions in the region between the binodal and spinodal lines are metastable. The presence of a miscibility gap limits the potential usefulness of these materials in device applications. Solutions with compositions lying inside the spinodal domain cannot be grown by LPE, whereas metastable solid solutions have a tendency toward phase separation and, eventually, device degradation. 

An example of an application to polysiloxane elastomers is the characterization of binodal and spinodal phase-separated structures occurring in model PDMS networks.310-312... 

In contrast to the critical temperature Tc, the spinodal temperature Tsp is well below the binodal temperature for off-critical mixtures and can hardly be reached due to prior phase separation. The diffusion coefficients in the upper left part of Fig. 8 have been fitted by (23) with a fixed activation temperature determined from Dj. The binodal points in Fig. 8 mark the boundary of the homogeneous phase at the binodal. The spinodal temperatures Tsp are obtained as a fit parameter for every concentration and together define the (pseudo)spinodal line plotted in the phase diagram in Fig. 7. The Soret coefficient is obtained from (11) and (23) as... 

The coupling of the order parameter to the temperature gradient also leads to unexpected excursions along the concentration axis in the case of off-critical mixtures. As a consequence, equilibrium phase diagrams lose their usual meaning in thermal nonequilibrium situations, and even an off-critical blend with a temperature above the binodal can be quenched into phase separation by local heating with a laser beam. 

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