On the diffraction of acoustic waves by a quarter-plane

This paper follows the work of A.V. Shanin on diffraction by an ideal quarter-plane. Shanin's theory, based on embedding formulae, the acoustic uniqueness theorem and spherical edge Green's functions, leads to three modified Smyshlyaev formulae, which partially solve the far-field problem of scattering of an incident plane wave by a quarter-plane in the Dirichlet case. In this paper, we present similar formulae in the Neumann case, and describe a numerical method allowing a fast computation of the diffraction coefficient using Shanin's third modified Smyshlyaev formula. The method requires knowledge of the eigenvalues of the Laplace-Beltrami operator on the unit sphere with a cut, and we also describe a way of computing these eigenvalues. Numerical results are given for different directions of incident plane wave in the Dirichlet and the Neumann cases, emphasising the superiority of the third modified Smyshlyaev formula over the other two. © 2011 Elsevier B.V.

Hierarchical orientation of crystallinity by block-copolymer patterning and alignment in an electric field

Electron and hole conducting 10-nm-wide polymer morphologies hold great promise for organic electro-optical devices such as solar cells and light emitting diodes. The self-assembly of block-copolymers (BCPs) is often viewed as an efficient way to generate such materials. Here, a functional block copolymer that contains perylene bismide (PBI) side chains which can crystallize via π-π stacking to form an electron conducting microphase is patterned harnessing hierarchical electrohydrodynamic lithography (HEHL). HEHL film destabilization creates a hierarchical structure with three distinct length scales: (1) micrometer-sized polymer pillars, containing (2) a 10-nm BCP microphase morphology that is aligned perpendicular to the substrate surface and (3) on a molecular length scale (0.35-3 nm) PBI π-π-stacks traverse the HEHL-generated plugs in a continuous fashion. The good control over BCP and PBI alignment inside the generated vertical microstructures gives rise to liquid-crystal-like optical dichroism of the HEHL patterned films, and improves the electron conductivity across the film by 3 orders of magnitude. © 2013 American Chemical Society.