## STRIPE PATTERNS IN A MODEL For Block Copolymers (2010)

Citations: | 4 - 2 self |

### BibTeX

@MISC{Peletier10stripepatterns,

author = {Mark A. Peletier and Marco Veneroni},

title = {STRIPE PATTERNS IN A MODEL For Block Copolymers},

year = {2010}

}

### OpenURL

### Abstract

We consider a pattern-forming system in two space dimensions defined by an energy G ". The functional G " models strong phase separation in AB diblock copolymer melts, and patterns are represented by f0; 1g-valued functions; the values 0 and 1 correspond to the A and B phases. The parameter " is the ratio between the intrinsic, material length-scale and the scale of the domain. We show that in the limit " ! 0 any sequence u " of patterns with uniformly bounded energy G "ðu"Þ becomes stripe-like: the pattern becomes locally one-dimensional and resembles a periodic stripe pattern of periodicity Oð"Þ. 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 " 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 jru "j combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier and R€oger, 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.