Dynamic density functional theory for microphase separation kinetics of block copolymer melts

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Abstract
In this paper, we describe a numerical method for the calculation of collective diffusion relaxation mechanisms in quenched block copolymer melts. The method entails the repeated calculation of two opposing fields—an external potential field U, conjugate to the density field ρ, and an energetic interaction field E. The external field is calculated by numerical inversion of the density functionals and the energetic interaction field is calculated directly by integration over the density field. When the two fields are balanced U=E, we recover the self-consistent field solutions; when the two fields are off balance, the spatial gradient of E–U is the thermodynamic force which drives the collective diffusion. We introduce a simple local coupling approximation for the Onsager kinetic coefficients of short freely jointed chains in weakly ordered systems. Fluctuations are added by incorporation of a random Langevin force in the diffusion equation. Numerical results of decomposition in symmetric and asymmetric diblock copolymer melts indicate that the method is capable of describing extremely slow defect annihilation relaxation modes. We find that in the nonlinear regime, the density patterns evolve to metastable states, in which isolated defects separate relatively well-ordered crystalline microdomains. These final states are typical for many industrial applications of incompletely relaxed copolymer melts.