STRIPE PATTERNS IN A MODEL FOR BLOCK COPOLYMERS

We consider a pattern-forming system in two space dimensions defined by an energy G(epsilon). The functional G(epsilon) 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 epsilon is the ratio between the intrinsic, material length-scale and the scale of the domain Omega. We show that in the limit epsilon -> 0 any sequence u(epsilon) of patterns with uniformly bounded energy G(epsilon)(u(epsilon)) becomes stripe-like: the pattern becomes locally one-dimensional and resembles a periodic stripe pattern of periodicity O(epsilon). 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 G(epsilon) Gamma-converges to a limit functional G(0). This limit functional is defined on fields of rank-one projections, which represent the local direction of the stripe pattern. The functional G(0) is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L-2-norm of the divergence of the projection field, or equivalently the L-2-norm of the curvature of the field. At the level of patterns the converging objects are the jump measures vertical bar del u(epsilon)vertical bar combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier and Roger, Arch. Rational Mech. Anal. 193 ( 2009) 475-537, provides the initial estimate and leads to weak measure-function pair convergence. We obtain strong convergence by exploiting the non-intersection property of the jump set.