Directed assembly of block polymers is rapidly becoming a viable strategy for lithographic patterning of nanoscopic features. One of the key attributes of directed assembly is that an underlying chemical or topographic substrate pattern used to direct assembly need not exhibit a direct correspondence with the sought after block polymer morphology, and past work has largely relied on trial-and-error approaches to design appropriate patterns. In this work, a computational evolutionary strategy is proposed to solve this optimization problem. By combining the Cahn–Hilliard equation, which is used to find the equilibrium morphology, and the covariance-matrix evolutionary strategy, which is used to optimize the combined outcome of particular substrate–copolymer combinations, we arrive at an efficient method for design of substrates leading to non-trivial, desirable outcomes.
Left: schematic illustration showing self-assembly on a patterned substrate with pattern rectification and density multiplication. Right: experi- mental results3 on PS-PMMA diblocks showing the effects of directed self- assembly. Block copolymer's natural periodicity: L0 = 27 nm.
2-D phase diagram at varying values of A and f, for alpha = 0.02. Morphologies are obtained by evolving the CH equation for 2E8 steps from a random initial configuration (dt = 0.01). The grid size: 100 X 100.
Evolutionary results of the “I” pattern. (a) Evolution of the objective function; (b) the target morphology. (c) The optimal morphology and the spot positions. Parameters: A = 1.3, alpha = 0.002, and N = 50.
Evolutionary pattern design for copolymer directed self-assembly
Jian Qin, Gurdaman S. Khaira, Yongrui Su, Grant P. Garner, Marc Miskin, Heinrich M. Jaeger, and Juan J. de Pablo, Soft Matter 9, 11467–11472 (2103). DOI: 10.1039/C3SM51971F