Asymptotically periodic L^2 minimizers in strongly segregating diblock copolymers

TL;DR

This paper introduces a one-dimensional model for strongly segregating diblock copolymers that predicts asymptotically periodic L^2 minimizers with sharp interfaces, aligning with experimental microstructures at low temperatures.

Contribution

The paper develops a novel one-dimensional model incorporating delta correction and nonlocal interactions, revealing unexpected periodic minimizers and phase transition phenomena in diblock copolymers.

Findings

Periodic minimizers with sharp interfaces are consistent with experiments.

The model exhibits a phase transition with Maxwell's equal area rule.

The model does not predict the universal critical surface tension exponent.

Abstract

Using the delta correction to the standard free energy \cite{bc} in the elastic setting with a quadratic foundation term and some parameters, we introduce a one dimension only model for strong segregation in diblock copolymers, whose sharp interface periodic microstructure is consistent with experiment in low temperatures. The Green's function pattern forming nonlocality is the same as in the Ohta-Kawasaki model. Thus we complete the statement in [31,p.349]: ``The detailed analysis of this model will be given elsewhere. Our preliminary results indicate that the new model exhibits periodic minimizers with sharp interfaces.'' We stress that the result is unexpected, as the functional is not well posed, moreover the instabilities in $L^2$ typically occur only along continuous nondifferentiable ``hairs''. We also improve the derivation done by van der Waals and use it and the above to show the existence of a phase transition with Maxwell's equal area rule. However, this model does not predict the universal critical surface tension exponent, conjectured to be 11/9. Actually, the range $(1.2,1.36)$ has been reported in experiments [21,p. 360]. By simply taking a constant kernel, this exponent is 2. This is the experimentally ($ \pm 0.1$) verified tricritical exponent, found e.g., at the consolute $0.9$ K point in mixtures of ${}^3$He and ${}^4$He. Thus there is a third unseen phase at the phase transition point.