Nucleation Rate Calculation for the Phase Transition of Diblock Copolymers under Stochastic Cahn--Hilliard Dynamics

TLDR: The string method is devised to compute the minimal energy path of nucleation events and the gentlest ascent dynamics to locate the saddle point on the path in Fourier space and derive the nucleation rate formula in the infinite-dimensional case and prove the convergence under numerical discretizations.

Abstract: We focus on the nucleation rate calculation for diblock copolymers by studying the two-dimensional stochastic Cahn--Hilliard dynamics with a Landau--Brazovskii energy functional. To do this, we devise the string method to compute the minimal energy path of nucleation events and the gentlest ascent dynamics to locate the saddle point on the path in Fourier space. Both methods are combined with the semi-implicit spectral method and hence are very effective. We derive the nucleation rate formula in the infinite-dimensional case and prove the convergence under numerical discretizations. The computation of the determinant ratio is also discussed for obtaining the rate. The algorithm is successfully applied to investigate the nucleation from the lamellar phase to the cylinder phase in the mean field theory for diblock copolymer melts. The comparison with projected stochastic Allen--Cahn dynamics is also discussed.