A simple and explicit numerical method for the phase-field model for diblock copolymer melts

In this paper, we present a simple and explicit finite difference method for the phase-field model for diblock copolymer melts. A diblock copolymer is a polymer consisting of two types of different monomers bonded covalently to each other to form a single copolymer chain. When the temperature is below the critical temperature, the copolymer melt exhibits microphase separation. The mathematical model is derived from a total free energy functional which contains kinetic, gradient, double well, and long-range nonlocal potentials. The Saul'yev-type scheme based on a linearly stabilized convex splitting method is used for the discretizations. The proposed method is simple and computationally efficient because the scheme is explicit and it does not require any iterative procedures. The proposed scheme not only overcomes the severe time step restriction for the explicit scheme but also works well for the simulations of lamellar and hex-cylinder structures which are characteristic morphologies for diblock copolymer melts after phase separation. Furthermore, the proposed method can be easily applied to the simulations in complex computational domains. We present various numerical tests to demonstrate the performance of the proposed scheme.