European Journal of Applied Mathematics

Table of Contents - April 2012 - Volume 23, Issue 02  

Papers

Characterising the disordered state of block copolymers: Bifurcations of localised states and self-replication dynamics

KARL B. GLASNERa1

a1 Department of Mathematics and Program in Applied Mathematics, University of Arizona, Tucson, AZ 85721, USA4 email: kglasner@math.arizona.edu

Abstract

Above the spinodal temperature for micro-phase separation in block co-polymers, asymmetric mixtures can exhibit random heterogeneous structure. This behaviour is similar to the sub-critical regime of many pattern-forming models. In particular, there is a rich set of localised patterns and associated dynamics. This paper clarifies the nature of the bifurcation diagram of localised solutions in a density functional model of A−B diblock mixtures. The existence of saddle-node bifurcations is described, which explains both the threshold for heterogeneous disordered behaviour as well the onset of pattern propagation. A procedure to generate more complex equilibria by attaching individual structures leads to an interwoven set of solution curves. This results in a global description of the bifurcation diagram from which dynamics, in particular self-replication behaviour, can be explained.

(Received March 10 2011)

(Revised November 08 2011)

(Accepted November 09 2011)

(Online publication December 21 2011)

Key words:

  • Co-polymer;
  • Order–disorder transition;
  • Self-replication