Diagram technique for finding of vertex functions in the Landau theory of heteropolymer liquids

The problem on finding the coefficients of the Landau free energy expansion into the power series of parameter of order has been considered for solutions and melts of linear heteropolymers whose molecules comprise several types monomeric units arranged stochastically. The presence of such a quenched structural disorder places this problem outside the framework of the traditional statistical physics inviting for its solution special approaches. One of them, based on the replica concept and actively engaged in theoretical physics of disordered systems, has been invoked in this paper to derive expressions for the vertex functions in the Landau theory of heteropolymer liquids. An algorithm has been formulated which permits one resorting to the simple diagram technique to write down expressions for these functions of any order in terms of the statistical characteristics of chemical quenched structure of polymer molecules. Explicit expressions for the contributions to the Landau free energy up to the fourth degree of order parameters for polymer systems with an arbitrary structural disorder have been presented to illustrate this general algorithm. Its potentialities have been also exemplified for the melt of random m-component copolymer where exact analytical formulas for these contributions up to n=6 at an arbitrary m have been derived for the first time. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2005