Curvature as a Guiding Field for Patterns in Thin Block Copolymer Films

Experimental data on thin films of cylinder-forming block copolymers (BC)—free-standing BC membranes as well as supported BC films—strongly suggest that the local orientation of the BC patterns is coupled to the geometry in which the patterns are embedded. We analyze this phenomenon using general symmetry considerations and numerical self-consistent field studies of curved BC films in cylindrical geometry. The stability of the films against curvature-induced dewetting is also analyzed. In good agreement with experiments, we find that the BC cylinders tend to align along the direction of curvature at high curvatures. At low curvatures, we identify a transition from perpendicular to parallel alignment in supported films, which is absent in free-standing membranes. Hence both experiments and theory show that curvature can be used to manipulate and align BC patterns.

Figure 1

(a) Phase-height AFM image of a free-standing thin film (image size: $2.6\mu \mathrm{m}\times 2.6\mu \mathrm{m}$). (b) Local mean curvature for the membrane shape. Here the vectors $u_{1,2}$ indicate the directions of the principal curvatures and $\mathbf{N}$ is the normal vector to the membrane surface (image size: $1.8\mu \mathrm{m}\times 1.8\mu \mathrm{m}$, region indicated by dashed lines in panel a; $H_{\max }=-H_{\min }=3.84\times {10}^{-3}{\mathrm{nm}}^{-1}$). (c) Local orientation of the director field $\alpha$ of the pattern with regard to the $x$ axis. (d) Histogram showing the local distribution of angles $\theta =\alpha -\beta$ between $\alpha$ and the local orientation of the membrane wrinkles $\beta$ (see also Fig. S2 in the Supplemental Material [32]).

Figure 2

Top panels: 3D AFM phase-height images of the BC thin film on a curved substrate after annealing at $T=373\mathrm{K}$. Panels (a) and (b) show the pattern configuration after 90 min (image size: $2.0\mu \mathrm{m}\times 1.5\mu \mathrm{m}$) and 3.5 h of thermal annealing (image size: $1.0\mu \mathrm{m}\times 1.25\mu \mathrm{m}$), respectively. Height scale: 80 nm from crest to valley, $H_{\max }^2=6.25{\mu \mathrm{m}}^{-2}$. The presence of a dislocation and $+1/2$ disclinations has been emphasized with a rectangle and circles, respectively. Bottom panels: (c) Local orientation of the smectic pattern (color map indicated at the bottom). (d) Histograms showing the distribution of angles $\theta$ between the local cylinder orientation and the direction of curvature at two different annealing times.

Figure 3

(a) Schematic representation of curvature radius $R_m$ in free-standing membranes. (b) Density profiles from SCFT for the parallel (left) and perpendicular (right) configurations $C_{\parallel }$ and $C_{\perp }$ at $R_m=9Rg$. (c) Free energy per area as a function of inverse curvature radius $R_g/R_m$ for the $C_{||}$ and $C_{\perp }$ configurations. Inset shows the free energy shift per area as a function of angle $\theta$ between the cylinders and the direction of curvature relative to the $C_{\perp }$ configuration ($\theta =0$) at $R_m=50R_g$.

Figure 4

(a),(b) Schematic representation of supported thin films (green) on curved substrates (gray). (c,d) Free energy difference per area $\mathrm{\Delta }F=(F_{||}-F_{\perp })/A$ of supported thin films with parallel ($C_{||}$) vs perpendicular ($C_{\perp }$) orientation, versus curvature, for two sets of surface interaction energies I (top) and II (bottom) as described in the main text. Blue shaded regions highlight curvature regimes where the parallel orientation is more favorable ($\mathrm{\Delta }F<0$). (e),(f) Corresponding curves for the free energy per area (red) and thickness (green) in the perpendicular configuration $C_{\perp }$.