Multiphase equilibria in solutions of polydisperse homopolymers. 4. Three-phase and four-phase separations in quaternary systems

Abstract

Criteria and the mechanism for the four-phase separation in quaternary systems solventpolydisperse polymer are established. Multiple critical points act as roots of multiphase regions. Thus, a critical point diagram displaying multiple critical points in the global space of all quaternary systems can serve as a basis for classification of their phase behavior. For instance, four-phase equilibria are potentially possible only in systems mapping beyond the quadruple critical line. A four-phase region is invariably tied to the existence of multiple three-phase regions. For instance, in systems with dx/dT < 0, two seemingly independent three-phase regions, TI and TB, grow closer with decreasing temperature, form a linear contact, and T3 penetrates into T1, thus creating the four-phase tetrahedron with four surrounding three-phase regions. Eventually T, is absorbed by T1; at this point the complex phase diagram degenerates into a single three-phase region that evolves then normally.

Molecular theory of solutions and blends of heteropolymers. I. Thermodynamics of amorphous multicomponent polymer systems

“…Rigorous thermodynamic consideration count of the repulsion at short distances, i.e., of the account of excluded volume. Written down in of multiphase equilibrium in such systems [58][59][60][61] allowed him to make a number of very interesting a compact matrix form formulas for both theories of polymer solutions based on the equation of state conclusions concerning possible mechanisms of phase separation under the variation of external are presented in ref. 23.…”

Phase equilibria of modified ethylene glycol–maleic anhydride–phtalic anhydride polyester resins

“…The normalization condition is accepted7-10: Equation ( 7 ) becomes Solc et al, [11][12][13][14] and Koningsveld and Staverman, l5 assumed dependence of x only on 42 and T. Matsuda and Kamide also considered dependence of X on the polymerization degree and take that [16][17][18][19][20] where xo0 = a + b / T; pj and k' are molecular weight and the concentration independent parameters, respectively; and p,, is the number-average polymerization degree.…”

“…Applying continuous thermodynamics, when polydisperse polymer is described by continuous molar mass distribution density function W ( M ) and when it is accepted that W ( M ) d M is a relative segment or mass fraction of all polymer species with molar masses between M and M + d, we acquire a real dependence description of the interaction parameter XA of polydisperse polyester on molecular weight distribution. The normalization condition is accepted7-10: Equation ( 7 ) becomes Solc et al, [11][12][13][14] and Koningsveld and Staverman, l5 assumed dependence of x only on 42 and T. Matsuda and Kamide also considered dependence of X on the polymerization degree and take that [16][17][18][19][20] where xo0 = a + b / T; pj and k' are molecular weight and the concentration independent parameters, respectively; and p,, is the number-average polymerization degree.…”

Statistical Thermodynamics of Copolymers and their Blends

“…The question of the number of such points and their stability has been thoroughly examined only for solutions and blends of homopolymers. [2][3][4][5][6][7][8][9] Phase diagrams feature a number of additional characteristics when a polymer contains macromolecules including rather long blocks of units of the same type. Then, provided specific conditions are met, there is a feasibility of the occurrence of such branches of the cloud-point curve (in the general case of a hypersurface of certain dimension), where inside the homogeneous principal phase the droplets of incipient phase (mesophase) possessing space periodicity will appear.…”

“…The tensor y (p) , to which here a gray p-gon corresponds, can be found by summing the contributions of all cactuses with p external colored apices in accordance with the rules formulated for the diagram technique of equation (81). Thus, for instance, the diagram expression for y (5) is presented in Figure 4(b), where 2 represents summation over all the different ways of coloring the cactus. When calculating the coefficients L p in terms of the RF formalism, the diagram technique (81) remains the same as that discussed above with the only difference being that other quantities will now correspond to the diagram elements m a s (86) v a G£.…”