Parallel and Perpendicular Lamellae on Corrugated Surfaces

Abstract

We consider the relative stability of parallel and perpendicular lamellar layers on corrugated surfaces. The model can be applied to smectic phases of liquid crystals, to lamellar phases of shortchain amphiphiles and to lamellar phases of long-chain block copolymers. The corrugated surface is modelled by having a single q-mode lateral corrugation of a certain height. The lamellae deform close to the surface as a result of chemical interaction with it. The competition between the energetic cost of elastic deformations and the gain in surface energy determines whether parallel or perpendicular lamellar orientation (with respect to the surface) is preferred. Our main results are summarized in two phase diagrams, each exhibiting a transition line from the parallel to perpendicular orientations. The phase diagrams depend on the three system parameters: the lamellar natural periodicity, and the periodicity and amplitude of surface corrugations. For a fixed lamellar periodicity (or polymer chain length), the parallel orientation is preferred as the amplitude of surface corrugation decreases and/or its periodicity increases. Namely, for surfaces having small corrugations centered at long wavelengths. For a fixed corrugation periodicity, the parallel orientation is preferred for small corrugation amplitude and/or large lamellae periodicity. Our results are in agreement with recent experimental results carried out on thin block copolymer films of PS-PMMA (polystyrene-polymethylmethacrylate) in the lamellar phase, and in contact with several corrugated surfaces.

Spontaneous Lamellar Alignment in Thickness‐Modulated Block Copolymer Films

“…[31,32] The typical values of the two elastic constants in PS/PMMA system with a period d 0 ¼ 50 nm is K 1 % 100 pN and B % 10 5 J m À3 . [32] The ratio of the two elastic constant introduces an intrinsic material parameter having length l ¼ ffiffiffiffiffiffiffiffiffiffi ffi K 1 =B p on the order of a lamellar period d 0 .…”

“…[31,32] The typical values of the two elastic constants in PS/PMMA system with a period d 0 ¼ 50 nm is K 1 % 100 pN and B % 10 5 J m À3 . [32] The ratio of the two elastic constant introduces an intrinsic material parameter having length l ¼ ffiffiffiffiffiffiffiffiffiffi ffi K 1 =B p on the order of a lamellar period d 0 . When the system is distorted over distances much larger that l (for example, for distortion regions far away from the disclination cores), the lamellae tend to preserve their equidistance; [33] the dilation term can then be disregarded and only the splay curvature term is retained for consideration.…”

Patterning with block copolymer thin films

“…The lamellae must pay an elastic deformation penalty for conforming to a sufficiently irregular substrate and will in some situations rearrange to stack perpendicularly from the surface [183]. Depending on the surface roughness amplitude and periodicity, phase transitions can occur between the two orientations.…”

Defect-Free Perpendicular Diblock Copolymer Films: The Synergy Effect of Surface Topography and Chemistry

“…On the theoretical side [33][34][35][36][37][38], previous works have been mainly devoted to study the transition between parallel to perpendicular lamellar phases in presence of a substrate. Most of these studies have been done in two dimensions and cannot address the important experimental issues related to inplane alignment, which intrinsically requires a 3D study.…”