Distribution function divariate

After presenting the fundamental equations in the preceding chapter we now consider their applications. Continuous thermodynamics is always important if thermodynamic properties or processes are influenced by distribution functions. From the practical point of view the liquid-liquid equilibrium is the most important aspect [29]. The following considerations are first restricted to systems containing random copolymers, especially one copolymer ensemble (which is characterized by a divariate distribution function), and one solvent A. Blends and block copolymer solutions are considered in Sects. 4 and 5. Generalizations to systems with more than one copolymer ensemble and/or more than one solvent are given in Sect. 3.2 (for solutions of random copolymers) and in Sect. 4.2 (for blends of random copolymers). 

The second important point in calculating phase equilibria in polydisperse copolymer systems is the choice of the divariate distribution function. Experimental distribution functions may be employed, but at present they are difficult to be measured with sufficient accuracy. Furthermore, when they are used, calculations have to be performed numerically. 

The simplest analytical divariate distribution function for random copolymers was the one proposed by Stockmayer [44]. In a generalized form, this distribution function may be written as... 

For calculating the spinodal curve and the critical point, there are two possible ways in the framework of continuous thermodynamics. The most general one is the application of the stability theory of continuous thermodynamics [45-47]. The other way is based on a power series expansion of the phase equilibrium conditions at the critical point. Following the second procedure. Sole et al. [48] studied multiple critical points in homopolymer solutions. However, in the case of divariate distribution functions the method by Sole has to be modified as outlined in the text below. 

Some results obtained by the different approaches are compared below. The systems considered contain a sample of copolyfethene vinyl acetate) which is characterized by = 15500 g mol" and = 40800gmol and B = 0.677, 6b = 0.232. Applicability of the generalized Stockmayer distribution function, Eq. (84), is assumed for the divariate case and, hence, analytical integrability of the double integrals in the scalar equations for ipJJ and is obtained. The... 

The application of continuous thermodynamics to block copolymer systems appears to be a difficult task. This is mainly due to the sophisticated thermodynamic models which have to be used, i.e. to the calculation of f and fg. Since there have been only first attempts [94], we will restrict ourselves to the case in which one diblock copolymer ensemble B(aP) characterized by a divariate distribution function is dissolved in one solvent A. The phase equilibrium conditions for this case are analogous to these for random copolymers. Thus, we have... 

---