Flory-Huggins spinodal

Figure 7 Precipitation temperatures (cloud points) vs. composition for polyisobutylene fractions with the indicated number average molecular weights in diisobutyl ketone. Dashed curves are Flory-Huggins binodals with parameters u and determined from a plot of the measured critical points according to equation (158). The dotted curves are the corresponding Flory-Huggins spinodals (reproduced by permission of Wiley, from J. Polym. ScL, Polym. Symp., 1976, 54, 62)...

AG/6V. =0 (spinodal) using the Flory-Huggins expression (14,7) for a 50 50 AB polymer blend. Here M is the number average molecular weight for the two component system, which allows for the possibility that the two monodisperse homopolymers in the blend may not be of identical molecular weight. 

Rigby et al. (1985) showed using a Flory-Huggins model that for symmetric blends, the spinodal and critical temperatures decrease linearly with increasing content of a symmetric diblock for blends with equal volume fractions of homopolymers (with the same molecular weight). The condition for a linear decrease of the binodal was less restrictive, not requiring equal concentrations of homopolymer in the blend. 

Figure 5. Examples of moment free energy (70) for Flory-Huggins theory of length-polydisperse polymers, with one moment density, p, retained. The parent is of the Schulz form (65), with pf = 0.03, Lu = 100 (hence p = p, /Lv = 3 x 10-4), and a = 2 (hence Lw = 150) the point pt = pj° is marked by the filled circles. In plot (a), the value of x = 0.55 is sufficiently small for the parent to be stable The moment free energy is convex. Plot (b) shows the cloud point, % 0.585, where the parent lies on one endpoint of a double tangent the other endpoint gives the polymer volume fraction p, in the shadow phase. Increasing x further, the parent eventually becomes spinodally unstable [x 0.62, plot (c)]. Note that for better visualization, linear terms have been added to all free energies to make the tangent at the parent coincide with the horizontal axis.

Figure 6. Conventional two-component phase behavior in poly disperse Flory-Huggins theory, shown in the (p, p0) plane for three values of % As in Fig. 5, the parent has Ln = 100 and Ly/ = 150 (hence a = 2). Along the y-axis, we plot L//p0 rather than p0 so that the dilution line p = LNp0, shown as the thick solid line in (a-c), is simply along the diagonal. With x considered as an additional variable, the dilution line constraint defines a plane (p, = L/vPq, x)- The last plot, (d), shows the cut by this plane through the phase behavior in (a-c) the solid line is the cloud point curve, and the dashed line is the spinodal stability condition.

To analyze the stability of the ordered microphases, the simplest incompressible random-phase approximation [132] can be employed. Using this approach, the critical value of the Flory-Huggins parameter, x > and the corresponding spinodal temperature, T = l/x > can be determined by the condition that the scattering intensity S(q) reaches its maximum value at a nonzero wave vector q. Within the RPA the scattering intensity is given by [132,142]... 

The thermodynamic definition of the spinodal, binodal and critical point were given earlier by Eqs. (9), (7) and (8) respectively. The variation of AG with temperature and composition and the resulting phase diagram for a UCST behaviour were illustrated in Fig. 1. It is well known that the classical Flory-Huggins theory is incapable of predicting an LCST phase boundary. If has, however, been used by several authors to deal with ternary phase diagrams Other workers have extensively used a modified version of the classical model to explain binary UCST or ternary phase boundaries The more advanced equation-of-state theories, such as the theory... 

In polymer-polymer mixtures, both Na and Nb in the Flory-Huggins equation (2-33) are large hence phase separation is predicted to occur even for very small values of x For Na = Ab ] > 1 and u = ua = i>b, the spinodal point where phase separation first... 

A. Sariban and K. Binder (1988) Phase-Separation of polymer mixtures in the presence of solvent. Macromolecules 21, pp. 711-726 ibid. (1991) Spinodal decomposition of polymer mixtures - a Monte-Carlo simulation. 24, pp. 578-592 ibid. (1987) Critical properties of the Flory-Huggins lattice model of polymer mixtures. J. Chem. Phys. 86, pp. 5859-5873 ibid. (1988) Interaction effects on linear dimensions of polymer-chains in polymer mixtures. Makromol. Chem. 189, pp. 2357-2365... 

Temperature dependence of y for mixtures of hydrogenated polybutadiene (88% vinyl) and deuterated polybutadiene (78% vinyl) and the calculated phase diagram from Flory-Huggins theory with Aa = Ab = 2000 and Vo = 100 A. The binodal is the solid curve and the spinodal is dashed. Adapted from N. P. Balsara, Physical Properties of Polymers Handbook (J. E. Mark, editor), AIP Press, 1996, Chapter 19. 

FIGURE 6.17 Solubility of a homopolymer according to the Flory-Huggins theory. Variables are the excluded volume parameter ft (or the polymer-solvent interaction parameter y), the net volume fraction of polymer q>, and the polymer-to-solvent molecular volume ratio q. Solid lines denote binodal, the broken line spinodal decomposition. Critical points for decomposition (phase separation) are denoted by . See text. 

The critical point Equation is derived from the spinodal Equation using the critical condition given earlier. The differential of the spinodal Equation with respect to 2 is zero at the critical point. This calculation is perfectly feasible but so far no one has made any attempt to use these Equations to predict the critical point. In the past it has usually been approximated using the simple Flory-Huggins expression for the critical point. 

The occurrence of a secondary phase separation inside dispersed phase particles, associated with the low conversion level of the p-phase when compared to the overall conversion, explains the experimental observation that phase separation is still going on in the system even after gelation or vitrification of the a-phase [26-31]. A similar thermodynamic analysis was performed by Clarke et al. [105], who analyzed the phase behaviour of a linear monodisperse polymer with a branched polydisperse polymer, within the framework of the Flory-Huggins lattice model. The polydispersity of the branched polymer was treated with a power law statistics, cut off at some upper degree of polymerization dependent on conversion and functionality of the starting monomer. Cloud-point and coexistence curves were calculated numerically for various conversions. Spinodal curves were calculated analytically up to the gel point. It was shown that secondary phase separation was not only possible but highly probable, as previously discussed. 

P.L. San Biagio and M.U. Palma, Spinodal Lines and Flory-Huggins Free-energies for Solutions of Human Hemoglobins HbS and HbA. Biophys. J., 60,508-512,1991. 

Figure 3.9. State diagram of the binary system P+LMWL from Flory-Huggins lattice model. Plot vs v-i (solid lined are binodals, dotted lines are spinodals, and the digits at the curves are the values of z) (Tompa, 1956) [H.Tompa Polymer Solutions. Copyright 1956 by Academic Press]...

Determine the critical indices (3 (for the binodal) and P,p (for the spinodal) according to the above approximations of Flory-Huggins theory. 

---