Vertex Functions for the Description of Heterophase States in the Landau Theory of Phase Transitions in Polydisperse Copolymers

Abstract

The diagram technique suggested earlier to find vertex functions for the description of homophase states in the framework of the Landau theory of phase transitions in polydisperse copolymers is extended to the description of heterophase states.

Features of the phase behavior of block copolymers related to their polydispersity

“…This degeneracy is associated with rotation or reflec tion spatial symmetry of the mesophase. Thus, each eigenvalue (10) of a hexagonal mesophase corre sponds to three eigenfunctions [28]: (11) Note that these functions sequentially pass into each other during rotation of vector r with Cartesian coor dinates x and y through an angle of . This circum stance implies the invariance of their sum with respect to these rotations such that we can choose it within the limits of accuracy of a normalization factor as a basis function in expansion (9):…”

“…The fundamental problem of the theory is to derive, in terms of the self consistent field theory, an expression for nonequilib rium free energy functional with respect to internal variables and . It is important to emphasize that, during the description of polydisperse polymers, this functional exhibits a nonlinear dependence on variables [7,11,18]. The determination of the absolute minimum of this functional at given values of external variables makes it possible to establish the thermodynamic equilibrium of the discussed system.…”

“…In this case, free energy func tional linearly depends on the volume fractions of the coexisting phases, and the free energy of each of them is described through a single replica functional in a conventional three dimensional Euclidean space [7,11,18]. The minimization of this functional, the expression for which was first given in [19,20], allows solution of the following important problems of ther modynamics of block copolymers: (i) finding a variety 10 11 12 20 21 22…”

“…There are a few publications concerned with the implementation of the procedure for determining the vertex functions that characterize the free energy of melts of statistical and block copolymers [7][8][9][10][11][12] as well as the construction of phase diagrams of melts of these copolymers in terms of the weak segregation approach [13][14][15][16][17][18].…”

“…A specific feature of this functional is that it is determined in so called replica dimensionality space 3m, where m is an arbitrary integer that signifies the number of replicas. It was shown in [11] that, during the study of the new phase nuclei, which are formed upon the loss of the thermodynamic stability of the spatially homogeneous state of a polydisperse het eropolymer, it is sufficient to be restricted only to a value of m = 1. In this case, the replica space coincides with the conventional three dimensional Euclidean space; this circumstance considerably simplifies the problem of determining the dependence of the tem perature at which the nuclei of a new phase are formed and its morphology on the parameters of the sample of the studied polydisperse block copolymer.…”