Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term

We are primarily concerned with the variational problem with long-range int eraction. This functional represents the Gibbs free energy of the microphas e separation of diblock copolymer melts. The critical points of this variat ional problem can be regarded as the thermodynamic equilibrium state of the phase separation phenomenon. Experimentally it is well-known in the dibloc k copolymer problem that the final equilibrium state prefers periodic struc tures such as lamellar, column, spherical, double-diamond geometries and so on. We are interested in the characterization of the periodic structure of the global minimizer of the functional (corresponding to the strong segreg ation limit). In this paper we completely determine the principal part of t he asymptotic expansion of the period with respect to epsilon (interfacial thickness), namely, we estimate the higher order error term of the period w ith respect to epsilon in a mathematically rigorous way in one space dimens ion. Moreover, we decide clearly the dependency of the constant of proporti on upon the ratio of the length of two homopolymers and upon the quench dep th. In the last section, we study the time evolution of the system. We firs t study the linear stability of spatially homogeneous steady state and deri ve the most unstable wavelength, if it is unstable. This is related to spin odal decomposition. Then, we numerically investigate the time evolution equ ation (the gradient flow of the free energy), and see that the free energy has many local minimizers and the system have some kind of sensitivity abou t initial data. (C) 1999 American Institute of Physics. [S1054-1500(99)0210 2-3].