Dynamics versus energetics in phase separation

TL;DR

This paper investigates the dynamics of phase separation in one-dimensional diblock copolymer melts, revealing how initial conditions and segregation strength influence the resulting pattern wavelength and coarsening behavior.

Contribution

It provides a detailed analysis of how initial states and parameter variations affect pattern formation and wavelength selection in 1D phase separation dynamics.

Findings

Final pattern wavelength depends on initial conditions and parameters.

Logarithmic coarsening process observed in homogeneous initial states.

Wavelength bounds are related to segregation strength, with specific asymptotic behaviors.

Abstract

Phase separation may be driven by the minimization of a suitable free energy ${\cal F}$. This is the case, e.g., for diblock copolimer melts, where ${\cal F}$ is minimized by a steady periodic pattern whose wavelength $\lambda_{GS}$ depends on the segregation strength $\alpha^{-1}$ and it is know since long time that in one spatial dimension $\lambda_{GS} \simeq \alpha^{-1/3}$. Here we study in details the dynamics of the system in 1D for different initial conditions and by varying $\alpha$ by five orders of magnitude. We find that, depending on the initial state, the final configuration may have a wavelength $\lambda_{D}$ with $\lambda_{min}(\alpha)<\lambda_{D}<\lambda_{max}(\alpha)$, where $\lambda_{min} \approx \ln (1/\alpha)$ and $\lambda_{max}\approx \alpha^{-1/2}$. In particular, if the initial state is homogeneous, the system exhibits a logarithmic coarsening process which arrests whenever $\lambda_{D}\approx\lambda_{min}$.