Effects of time-periodic forcing in a Cahn-Hilliard model for Langmuir-Blodgett transfer

The influence of a temporal forcing on the pattern formation in Langmuir-Blodgett transfer is studied employing a generalized Cahn-Hilliard model. The occurring frequency-locking effects allow for controlling the pattern formation process. In the case of one-dimensional (i.e., stripe) patterns one finds various synchronization phenomena such as entrainment between the distance of deposited stripes and the forcing frequency. In two dimensions, the temporal forcing gives rise to the formation of intricate complex patterns such as vertical stripes, oblique stripes, and lattice structures. Remarkably, it is possible to influence the system in the spatial direction perpendicular to the forcing direction leading to synchronization in two spatial dimensions.

Figure 1

Sketch of the experimental setup of the Langmuir-Blodgett (LB) transfer. A solid substrate is pulled out of a liquid bath covered by a monolayer of surfactants at constant density. The latter is ensured by the use of movable barriers limiting the bath surface. On the liquid surface the surfactant is in the liquid-expanded (LE) phase. The layer may undergo substrate-mediated condensation in the vicinity of the three-phase contact line and is then deposited onto the substrate (partly) in the liquid-condensed (LC) phase.

Figure 2

Snapshot of a time-periodic solution of (5) in the 1D case obtained by direct numerical integration for a domain of size $L$. The stripe period $l$ is defined as the distance between two subsequent rising flanks whereas the width of the stripes of LC phase $l_{\mathrm{LC}}$ is the distance between the two zero crossings that bracket the region with $c\left(x\right)>0$. The domain size is increased for larger $l$ while maintaining the discretization in order to fully capture the stripes. The black arrow denotes the transfer direction.

Figure 3

Shown is the dependence of mean stripe distance $\overline{l}$ on mean transfer velocity $\overline{v}$ for horizontal stripe patterns in the 1D case. Time-periodic forcing, i.e., Eqs. (5, 6, 7, 8, 9) with forcing frequency $\omega =0.002$ and amplitude is $A=0.1$ (red crosses) is compared to the case of constant transfer velocity $v=\overline{v}$ (blue crosses). The linear behavior of $\overline{l}$ [Eq. (12)] in the case of complete synchronization is shown for $W=1$ (solid black line) and $W=2$ (dashed black line). Dotted vertical lines denote the limits of the natural patterning range.

Figure 4

The dependence of the synchronization order $W$ on the forcing frequency $\omega$ for horizontal stripe patterns in the 1D case with $A=0.1$ and $\overline{v}=0.06$ shows a typical devil's staircase. Plateaus for 1:1 (blue), 2:3 (green), and 1:2 (black) synchronization are particularly highlighted.

Figure 5

Shown is the dependence of the synchronization order $W$ on the mean transfer velocity $\overline{v}$ and the forcing frequency $\omega$ at fixed $A=0.1$ for horizontal stripe patterns in the 1D case. Top and bottom panel provide different views of the same data. Several plateaus at low order $W$ ranging from 1:2 to 4:1 are distinctively colored. The full patterning range in terms of $\overline{v}$ is covered, i.e., beyond the ends of the black lines, homogeneous transfer occurs. A video showing the plot from different perspectives is provided in the Supplemental Material [66].

Figure 6

Shown are Arnold tongues in the dependence of the synchronization order $W$ on the mean transfer velocity $\overline{v}$ and the forcing amplitude $A$ at fixed $\omega =0.0014$ for horizontal stripe patterns in the 1D case. Each point represents a time-periodic solution. The synchronization order $W$ is indicated by the color scheme that is identical to the one in Fig. 5 for $W>1$. Here $W<1$ does not occur.

Figure 7

A $2:3$ synchronized horizontal stripe patterns in the 1D case as solutions of (5, 6, 7, 8, 9) with $A=0.1$, $\omega =0.007$, and $\overline{v}=0.06$. (a) A snapshot of the solution is presented. The $l_i$ (red) denote the four individually occurring stripe distances and $l_f$ is the forced stripe distance for $1:1$ synchronization [cf. (12)]. (b) Time evolution of $l_i$. (c) Fourth return map of the individual stripe distances $l_i$, indicating that the solution is 4-periodic.

Figure 8

(a) Time evolution of the stripe distance $l_i$ for horizontal stripe patterns in the 1D case with $A=0.1$, $\omega =0.003$, and $\overline{v}=0.06$ (b) and its fourth return map. The solution exhibits prime periodicity 4 and $W=4:3$.

Figure 9

Shown is the dependence of the average duty cycle $D$ on the mean transfer velocity $\overline{v}$ for horizontal stripe patterns in the 1D case in the forced case with $A=0.1$ and $\omega =0.002$ (red) in comparison to the unforced case (blue).

Figure 10

Examples of two-dimensional solutions of (5) in the case of constant transfer velocity ($A=0$). The transfer direction is denoted by the respective white arrow. (a) Vertical stripes arising at $\overline{v}=0.026$ after an initial transient has died out. (b) Horizontal stripes at $\overline{v}=0.04$.

Figure 11

Examples of snapshots of 2D solutions of (5, 6, 7, 8, 9) in the plane spanned by mean transfer velocity $\overline{v}$ (left to right) and forcing amplitude $A$ (bottom to top) at $\omega =0.002$. Shown are the states at end of simulation, namely at time $t_{\mathrm{end}}=10\tau$, where $\tau$ is the time needed for the substrate to be advected by one domain size $L$. This limit is increased for transversally inhomogeneous patterns. The initial condition is a simple meniscus with small amplitude white noise as in (9). The transfer direction is from bottom to top. Note that to capture the most interesting solutions, the shown $\overline{v}$ intervals are not equidistant.

Figure 12

Array of snapshots of 2D solutions of (5, 6, 7, 8, 9) in the plane spanned by mean transfer velocity $\overline{v}$ (left to right) and forcing amplitude $A$ (bottom to top) at $\omega =0.002$. Shown are the states at end of simulation, namely at time $t_{\mathrm{end}}=10\tau$, where $\tau$ is the time needed for the substrate to be advected by one domain size $L$. This limit is increased for transversally inhomogeneous patterns. For each run, the final solution of the previous run is used as initial condition whereby the very first initial condition is a vertically striped pattern. The transfer direction is from bottom to top. Here, the shown $\overline{v}$ intervals are equidistant.

Figure 13

(a) Branched pattern in the case of time-periodic forcing with $A=0.1$, $\omega =0.002$, $\overline{v}=0.026$. Blue and red domains represent the LE and LC phase, respectively. (b) AFM image of a branched pattern obtained for LB transfer at constant transfer velocity [32]. Dark brown and yellow domains represent LE and LC phase, respectively.

Figure 14

(a) Transversally modulated vertical stripes found in the case of time-periodic forcing with $A=0.2$, $\omega =0.002$, and $\overline{v}=0.0244$. (b) Fourier transform $\mathcal{F}\left[c\right]$ of the pattern in the plane spanned by synchronization orders $W_y$ and $W_x$. The synchronization order can be directly related to a wave number according to Eq. (12). Each mode consists of one pair of opposite wave vectors, e.g., $(W_y,W_x)=(0.75,-0.5)$ and $(W_y,W_x)=(-0.75,0.5)$ are one mode. The zero mode originates from the nonvanishing mean concentration of the pattern.