Nonequilibrium orientational patterns in two-component Langmuir monolayers

A model of a phase-separating two-component Langmuir monolayer in the presence of a photoinduced reaction interconverting two components is formulated. An interplay between phase separation, orientational ordering, and reaction is found to lead to a variety of nonequilibrium self-organized patterns, both stationary and traveling. Examples of the patterns, observed in numerical simulations, include flowing droplets, traveling stripes, wave sources, and vortex defects.

Figure 1

Schematic illustration of the variables used to describe the tilted elongated molecules in the monolayer.

Figure 2

Concentration (left panel) and orientation (right panel) fields in the equilibrium pattern representing an array of splay defects for $\Lambda =2$, $K=0.25$, ${\pi }_0=\alpha =0$, $\beta =4$, ${\chi }_2=1$, and ${\chi }_0=1$ in the absence of reaction $\left(\kappa =0\right)$. The gray color is used in the right panel to display the local tilt $a$ of the molecules. In both panels, darker regions correspond to higher values of the displayed variables.

Figure 3

Equilibrium stripe pattern for $\Lambda =2$, $K=0.25$, ${\pi }_0=\alpha =0$, $\beta =4$, ${\chi }_2=1$, and ${\chi }_0=2$, in the absence of reaction $\left(\kappa =0\right)$.

Figure 4

Profiles of composition, tilt, and azimuth angle along the cross section of the stripe pattern indicated by a line segment in the left panel of Fig. 3. The azimuthal angle is measured with respect to the positive direction of the $x$ axis.

Figure 5

Profiles of composition, tilt, and azimuth angle along a cross section of the equilibrium solitonlike pattern with sharp wall domains found for ${\chi }_0=8$ and the same other parameters as in Figs. 2, 3.

Figure 6

Phase diagrams of the considered model for the parameter values ${\chi }_2=0.0052$, ${\chi }_0=2.27$, $\alpha =1.5$, and $\beta =2$ in the limit $K\to 0$ (a) without splay coupling and (b) with weak splay coupling $\left(\Lambda =0.01\right)$. The diagram (b) is obtained by the linear stability analysis with respect to the perturbations with $q_y=0$. Dashed lines in this diagram show the boundaries of the respective regions in the absence of splay coupling (a). Other notations are explained in the text.

Figure 7

A snapshot of concentration and orientation fields in a pattern of traveling waves that interact with orientational defects; $\Lambda =0$, $K=0.001$, ${\pi }_0=1.5$, and $\kappa =1$. The waves propagate from the upper-left to the lower-right corner of the figure. See also video (Ref. 26, video Fig7.mgp).

Figure 8

A snapshot of concentration and orientation fields in a pattern of flowing droplets; $\Lambda =K=0.01$, ${\pi }_0=1.25$, and $\kappa =0.5$. See also Ref. 26, video Fig8.mpg.

Figure 9

A snapshot of concentration and orientation fields in a pattern of traveling stripes; $\Lambda =K=0.01$, ${\pi }_0=1.25$, and $\kappa =1$. See also Ref. 26, video Fig9.mpg.

Figure 10

Concentration and tilt profiles in the cross sections of (a) flowing droplets and (b) traveling stripes, shown in Figs. 8, 9. The respective patterns and the orientations of lines used to make the cross sections are displayed in the insets. For the droplet, both the profiles along the lines parallel (“para”) and perpendicular (“perp”) to the motion direction are presented. The patterns move from left to right.

Figure 11

A snapshot of concentration and orientation fields in a pattern representing a linear wave source; $\lambda =0.01$, $K=0.01$, $\kappa =1$, and ${\pi }_0=1.25$. To obtain this pattern special initial conditions ($\phi =\pi$ for $x<L∕2$ and $\phi =0$ for $x>L∕2$) were taken. The wave structures are generated at the central vertical line and propagate to the side. See also Ref. 26, video Fig11.mpg.

Figure 12

A snapshot of concentration and orientation fields in a stationary array of splay defects; $\Lambda =1$, $K=0.1$, $\kappa =1$, and ${\pi }_0=1.25$.

Figure 13

A snapshot of concentration and orientation fields in the patterns of traveling stripes that involves azimuth reorientation; $\Lambda =1.5$, $K=0.1$, $\kappa =3.5$, and ${\pi }_0=3.7$. See also Ref. 26, video Fig13.mpg.

Figure 14

Space-time diagram showing the temporal variation of the azimuth angle along the central vertical cross section in Fig. 13. Here, the gray-scale coding is such that $\phi =0$ is shown as white and $\phi =\pi$ is shown as black. The azimuth angle is defined in such a way that $\phi =0$ corresponds to the propagation direction of the stripes.

Figure 15

Composition, tilt, and azimuth profiles for a cross section perpendicular to the traveling stripes in Fig. 13. The arrow indicates the direction of motion of the traveling structures.