Controlled nanochannel lattice formation utilizing prepatterned substrates

Solid substrates can be endued with self-organized regular stripe patterns of nanoscopic length scale by Langmuir-Blodgett transfer of organic monolayers. Here we consider the effect of periodically prepatterned substrates on this process of pattern formation. It leads to a time periodic forcing of the oscillatory behavior at the meniscus. Utilizing higher-order synchronization with this forcing, complex periodic patterns of predefined wavelength can be created. The dependence of the synchronization on the amplitude and the wavelength of the wetting contrast is investigated in one and two spatial dimensions, and the resulting patterns are discussed. Furthermore, the effect of prepatterned substrates on the pattern selection process is investigated.

Figure 1

Schematic drawing of a surfactant-covered meniscus on a prepatterned substrate. The height profile $h(x,y,t)$ indicates the film thickness at location $(x,y)$ and time $t$, whereas $\gamma (x,y,t)$ describes the surfactant density at the surface above $(x,y)$.

Figure 2

Spacetime plots of solutions on homogeneous substrates ($\rho =0$) at different transfer velocities, obtained from one-dimensional simulations: (a) $V=0.385$, (b) $V=0.500$, (c) $V=0.700$. Only the monolayer density $\Gamma$ is shown. Blue (dark) indicates the liquid expanded phase; yellow (bright) corresponds to liquid condensed parts of the monolayer.

Figure 3

Patterns of equal wavelength, ${\lambda }_0\approx 200$, obtained from one-dimensional simulations at different velocities $V$ on a homogeneous substrate: (a) $V=0.4$ and (b) $V=0.795$. The red line (solid line) corresponds to monolayer density $\Gamma$, and the blue line (dashed line) represents the height profile $H$. The patterns are almost inverse to each other. The solid vertical lines indicate the location $X_f$ where new domains are perpetually created.

Figure 4

Wavenumber $k_0$ of patterns transferred onto homogeneous substrates against transfer velocity $V$, as obtained from one-dimensional computations. The curve is nonmonotonous, so that the same $k_0$ can be obtained for high and for low $V$. The origins of the Arnold tongues in Fig. 7 can be determined from the intersection points of the curve with horizontals at the corresponding $n$:$m$ ratio. This is shown here for 1:1 and 3:2 synchronization with $L_p=200$.

Figure 5

Synchronization plateaus obtained from one-dimensional simulations of transfer onto a prepatterned substrate with $L_p=400$ and $\rho =0.006$.

Figure 6

Staircase obtained from one-dimensional simulations using $L_p=400$ and $\rho =0.006$, after elimination of the transfer velocity in favor of the natural frequency scale $\Omega /{\omega }_0$. Red crosses indicate patterns obtained for velocities higher than $0.52$; black squares correspond to lower $V$.

Figure 7

Arnold tongues resulting from one-dimensional computations for prepatterned substrate with (a) $L_p=200$, (b) $L_p=300$, and (c) $L_p=400$. The dashed lines show how the Arnold tongues are expected to go to $\rho =0$. The vertical solid line indicates the velocity corresponding to the maximum of the intrinsic frequency ${\omega }_0$.

Figure 8

Patterns in the one-dimensional system, obtained during transfer with five different velocities on a prepatterned substrate with $L_p=400$, $\rho =0.005$. The velocities in (a) and (c–f) are chosen from different $n$:$m$ synchronization regions, while (b) shows a pattern obtained at a velocity from between the right $3$:$1$ and $2$:$1$ regions [compare Fig. 7a].

Figure 9

Two-dimensional patterns obtained on a prepatterned substrate with $L_p=400$ for two different transfer velocities $V$. The height information corresponds to the film profile $H$, and the monolayer density $\Gamma$ is color coded.

Figure 10

Two-dimensional solutions in the velocity range of the parallel-perpendicular orientation transition for various wetting contrasts $\rho$.