^{1}, S. I. Kuchanov

^{2}and L. I. Manevitch

^{3,a)}

### Abstract

Rather general mean field theory of heteropolymer liquids developed earlier reduces the problem of the phase diagram construction to the determination of extremals of the free energy functional. These should be subsequently analyzed for their local and global stability. Tackling of this problem traditionally involves the examination of the behavior of the solutions of a set of nonlinear algebraic and partial differential equations at various values of the control parameters. Besides, the necessity arises here to construct in space of these parameters the lines where a polymer system loses the thermodynamic stability. To overcome mathematical difficulties encountered we employed a complex approach that combines analytical and numerical methods. A two-step procedure constitutes the essence of such an approach. First, the bifurcation analysis is invoked to find the asymptotics of the extremals in the vicinity of bifurcation points. Then these asymptotics are used as an initial approximation for the numerical continuation of specific lines, where the stability loss occurs, into regions of the parametric space far removed from bifurcation values. We realized this approach for the melt of linear binary copolymers of various chemical structure with macromolecules having a pattern of arrangement of monomeric units describable by a Markov chain. Bifurcation and phase diagrams for some of these copolymers have been constructed within a wide range of temperatures and volume fractions of a polymer.

The authors gratefully acknowledge the financial support by CRDF (Grant No. RC-2-2398-MO-02).

I. INTRODUCTION

II. PHASE DIAGRAM

III. THE BIFURCATION ANALYSIS

IV. BIFURCATION OF NONHOMOGENEOUS STRUCTURE FROM HOMOGENEOUS SOLUTIONS

V. BIFURCATION IN THE SPINODAL POINT

VI. ANALYSIS OF THE STABILITY

VII. NUMERICAL CALCULATIONS

### Key Topics

- Bifurcations
- 46.0
- Copolymers
- 29.0
- Eigenvalues
- 22.0
- Polymers
- 17.0
- Phase diagrams
- 16.0

## Figures

Phase diagram for system (1). The curves (1), (2) are spinodal and binodal, respectively [for changing from 0 to 0.4 they are trivial ones, for , curve , nontrivial spinodal], is the critical point (marked by 엯), is the Lifshitz point (marked by ×).

Phase diagram for system (1). The curves (1), (2) are spinodal and binodal, respectively [for changing from 0 to 0.4 they are trivial ones, for , curve , nontrivial spinodal], is the critical point (marked by 엯), is the Lifshitz point (marked by ×).

The spatial distribution for for the system (1) bold curve, for ; next curve, for and value , corresponding to binodal; dotted line, for . The curves are plotted in the scale .

The spatial distribution for for the system (1) bold curve, for ; next curve, for and value , corresponding to binodal; dotted line, for . The curves are plotted in the scale .

Phase diagram for system (2). The curve (1), trivial spinodal, the curve (2) is nontrivial spinodal, (3) is the trivial binodal, is the Lifshitz point (marked by ×), is the critical point (marked by 엯).

Phase diagram for system (2). The curve (1), trivial spinodal, the curve (2) is nontrivial spinodal, (3) is the trivial binodal, is the Lifshitz point (marked by ×), is the critical point (marked by 엯).

The bifurcation diagram for the system (2) at . The curve (1) is the unstable nonhomogenous branch, (2) is the stable one, (3) the stable part of the branch appeared transcritically from the spinodal point, (4) is the unstable part of the same branch, (5) is the stable homogeneous nontrivial branch rigidly arising in couple with branch (4).

The bifurcation diagram for the system (2) at . The curve (1) is the unstable nonhomogenous branch, (2) is the stable one, (3) the stable part of the branch appeared transcritically from the spinodal point, (4) is the unstable part of the same branch, (5) is the stable homogeneous nontrivial branch rigidly arising in couple with branch (4).

The bifurcation diagram at for the system (3).

The bifurcation diagram at for the system (3).

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