Dynamics versus energetics in phase separation

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 $\lambda_{D}\approx\lambda_{min}$.