Stripe patterns in a model for block polymers

Rapport

Peletier, M.A. & Veneroni, M. (2009). Stripe patterns in a model for block polymers. (arXiv.org [math.AP], No. 0902.2611). arXiv.org, 56 pp. Lees meer: Medialink/Full text

Abstract

 

We consider a pattern-forming system in two space dimensions defined by an energy

Ge. The functional Ge models strong phase separation in AB diblock copolymer melts, and patterns are represented by {0, 1}-valued functions; the values 0 and 1 correspond to the A and B phases. The parameter e is the ratio between the intrinsic, material length scale and the scale of the domain . We show that in the limit e ¿ 0 any sequence u" of patterns with uniformly bounded energy Ge(ue) becomes stripe-like: the pattern becomes locally one-dimensional and resembles a periodic stripe pattern of periodicity O(e). In the limit the stripes become uniform in width and increasingly straight.

Our results are formulated as a convergence theorem, which states that the functional

Ge Gamma-converges to a limit functional G0. This limit functional is defined on fields of rank-one projections, which represent the local direction of the stripe pattern. The functional G0 is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L2-norm of the divergence of the projection field, or equivalently the L2-norm of the curvature of the field. At the level of patterns the converging objects are the jump measures |¿ue| combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier & Röger, Archive for Rational Mechanics and Analysis, to appear, provides the initial estimate and leads to weak measure-functionpair convergence. We obtain strong convergence by exploiting the non-intersection property of the jump set.