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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 $λ_{GS}$ depends on the segregation strength $α^{-1}$ and it is know since long time that in one spatial dimension $λ_{GS} \simeq α^{-1/3}$. Here we study in details the dynamics of the system in 1D for different initial conditions and by varying $α$ by five orders of magnitude. We find that, depending on the initial state, the final configuration may have a wavelength $λ_{D}$ with $λ_{min}(α)<λ_{D}<λ_{max}(α)$, where $λ_{min} \approx \ln (1/α)$ and $λ_{max}\approx α^{-1/2}$. In particular, if the initial state is homogeneous, the system exhibits a logarithmic coarsening process which arrests whenever $λ_{D}\approxλ_{min}$.