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

Abstract

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.Key words. Cahn-Hilliard equation, diblock copolymer, Landau-Brazovskii energy, string method, nucleation rate, functional determinant ratio AMS subject classification. 65C20 DOI. 10.1137/120876307 1. Introduction. In the past few decades, diblock copolymers [22,12] have been extensively investigated due to their interesting physical and chemical features as well as widespread applications in material science [13,6]. Roughly speaking, the studies of copolymers mainly focus on two kinds of issues: determining equilibrium phases and studying the dynamical behaviors. For the first issue, it has been known for a long time that the diblock copolymers admit various ordered phase separation in the microscale. Several kinds of stable ordered phases, such as the lamellar phase, cylinder phase, and spherical phase, etc., have been discovered in experiments, and lots of work has been done to determine the phase diagram [33,23]. For the second issue, people try to understand the dynamical behaviors of the diblock copolymers, for example, the interfacial motion [19,24], spinodal decomposition phenomena [28], and nucleation events between different kinds of ordered phases [31,21,5].In this paper, we study the nucleation rate calculation for diblock copolymers between the lamellar and cylinder phases with a Landau-Brazovskii energy functional, and the random fluctuation is assumed to be of stochastic Cahn-Hilliard type. In this model, it is supposed that diblock copolymer melts contain lots of chains, each of which contains monomers of types A and B linked together (see Figure 1.1). On each chain, the composition ratios of these two kinds of monomers are f A and f B , respectively, which satisfy f A + f B = 1. The order parameters φ A (r) and φ B (r) characterize

The phase field method for geometric moving interfaces and their numerical approximations

“…For example, phase field modeling of polymeric materials has received much attention in theoretical chemistry and materials science. Phase field models of diblock copolymers, such as the model due to Ohta and Kawasaki [343], have had a long history and is an active area of research [110,361,419,86,388,305,107].…”

“…Parallel algorithms and numerical simulations of the Cahn-Hilliard-Cook equation have also been reported in [463]. Stochastic Cahn-Hilliard dynamics have also been used to study nucleation in microstructure evolution in [256,304,305]. It should be noted that the added noises may not always have physical meaning and those stochastically perturbed models may not associate with sharp interface models.…”

Embedding of the saddle point of index two on the PES of the ring opening of cyclobutene

“…[23] The article also narrows down this type of curve which sometimes does not describe an RP, because it can miss the TS. Other works doing the analysis and possibly use it as an RP are due to Li et al, [24] to Samanta et al, [25] and to Zeng et al [26] Let be VðqÞ an N-dimensional surface over R N , let gradðqÞ be its gradient vector, r q VðqÞ, and let HðqÞ be the matrix of its second derivatives, the Hessian r q grad T . The reason for introduction trajectories by a "gentlest ascent" dynamics (GAD) is the purposeful search for SPs of index one, in a first step, or for SPs of a higher, fixed index, for example for "summits" of index two.…”

“…[32] Recently, an application of the GAD to Physical Chemistry is demonstrated for point defect activity on a surface. [24,33] We will apply GAD to a single molecule in the empty space. The description of the PES of an n-atomic molecule can be done in 3n26 internal coordinates.…”

“…Thus, NTs into all directions can start from here if one selects the sign of Eq. (24) according to the index of the stationary point. [16,17,52] This means that the search direction of an NT, r, is optional at SPs.…”