Dynamics versus energetics in phase separation

Abstract

Phase separation may be driven by the minimization of a suitable free energy $\mathcal{F}$. This is the case, e.g., for diblock copolimer melts, where $\mathcal{F}$ 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 ${\lambda }_D\approx {\lambda }_{min}$.

14 pages. To appear in JSTAT. Corrected typos

References in corpus (4)

• Nonlinear dynamics in one dimension: On a criterion for coarsening and its temporal law
• Coarsening dynamics in one dimension: The phase diffusion equation and its numerical implementation
• 1D Cahn-Hilliard equation: Ostwald ripening and modulated phase systems
• Stability of frozen waves in the Modified Cahn--Hilliard model

Cited by in corpus (2)

• Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling
• Coarsening and wavelength selection far from equilibrium: a unifying framework based on singular perturbation theory