Chemical Leslie effect in a chiral $\text{smectic-}{C}^{★}$ film: Nonsingular target patterns

We analyze experimentally and theoretically the winding and unwinding of the $\vec{c}$ director in a chiral $\text{smectic-}{C}^{★}$ film crossed by an ethanol flow. This leads to a target pattern under crossed polarizers when the $+1$ defect imposed by the boundary conditions is pinned on the edge of the film. We show that the target is deformed at the center of the film when it is subjected to a flow of ethanol because of the presence of two recirculation vortices of chemohydrodynamical origin. This deformation and the two vortices disappear during the unwinding of the target when the ethanol flow is stopped. This unambiguously shows that the target deformation is due to the vortices and not to the elastic anisotropy. These two points are confirmed theoretically. An estimate of the two so-called chemomechanical and chemohydrodynamical Leslie coefficients is also derived from this study.

Figure 1

Structure of the Smectic ${\mathrm{C}}^{★}$ phase (a) and definition of the $\vec{n}$ and $\vec{c}$ directors (b).

Figure 2

Oven used in our experiments.

Figure 3

Scheme of the device used to produce the air-ethanol mixture.

Figure 4

Diagram of the optical device employed to visualize the film and its texture.

Figure 5

Reflected intensity as a function of the orientation of the $\vec{c}$ director when the polarizer and the analyzer are crossed (a) and slightly uncrossed by $5{}^{\circ }$ (b).

Figure 6

The three configurations frequently encountered. (a) With a $+1$ defect at the edge of the film; (b) with two 1/2 defects at the edge of the film; (c) with a $+1$ defect in the center of the film. Photos taken between polarizer and analyzer slightly uncrossed.

Figure 7

Film thickness as a function of the number of layers.

Figure 8

Film reflectivity as a function of its thickness given in number of layers for two different wavelengths.

Figure 9

Sequence of snapshots showing the winding of the $\vec{c}$ director under a flow of ethanol. At time $t=0$ a 50% ethanol flow is imposed to the film; (a) $t=9\mathrm{s}$; (b) $t=12.1\mathrm{s}$; (c) $t=13.2\mathrm{s}$; (d) $t=14.1\mathrm{s}$; (e) $t=14.8\mathrm{s}$; (f) $t=17.4\mathrm{s}$. The red arrows drawn on snapshots (a) and (d) show the direction of the flows. The blue arrows on snapshot (b) show the sense of rotation of the director. The film is 10 layers thick.

Figure 10

Time course of the phase $\varphi$ in the center of the film for a percentage of ethanol vapor of 50%. The slope of the dashed line gives the initial rotation frequency $f$. This curve has been obtained from the images shown in Fig. 9.

Figure 11

Rotation frequency as a function of the percentage of ethanol vapor. These measurements have been performed at the beginning of the winding with the film shown in Fig. 9. The solid line is the best linear fit of the experimental points ignoring the first two. The error bars correspond to variations observed over several measurements.

Figure 12

Experimental evidence of a flow in the center of the target during the winding of the $\vec{c}$ director. (a–c) The circles and the arrows indicate the position and the direction of the particle at different times: (a) $t=0$ s, (b) $t=1.8\mathrm{s}$; (c) $t=3.5\mathrm{s}$. The film is 10 layers thick and the percentage of ethanol vapor is 50%. (d) Trajectory followed by the particle during one turn (about 20 s here).

Figure 13

Target pattern observed at equilibrium. The film is 10 layers thick. The percentage of ethanol vapor injected under the film is 50%. (b) Phase profile measured along a radius of this film. The solid red line is the best parabolic fit.

Figure 14

Images of the target pattern obtained with different percentages of ethanol vapor. From top to bottom, the percentage is 10%, 20%, 50%, and 100%.

Figure 15

Number of turns at equilibrium as a function of the percentage of ethanol vapor for the 10-layers-thick film of Fig. 13. The error bars correspond to the variations observed by making several measurements. The solid line corresponds to the best linear fit.

Figure 16

Number of turns at equilibrium as a function of the film thickness for films of 0.5 mm in diameter. The percentage of ethanol vapor is 50%. The error bars have been calculated from the variations observed with different films. The lines correspond to the best fits with the functions given on the graph.

Figure 17

Angle $\varphi$ in the center of the film as a function of time measured during the phase unwinding when the flow of ethanol is cut off. The solid line is the best fit with an exponential law. The inset shows the target pattern observed during the phase unwinding. The film is 20 layers thick.

Figure 18

(a) Velocity profile measured along a radius during the phase unwinding. Each color corresponds to a different tracer. For each tracer, the different points with the same color correspond to different times. The solid line is a fit with a law in $r-r^3$. (b) Maximal velocity as a function of time measured during the phase unwinding and its fit with an exponential law.

Figure 19

Solution of Eq. (26) for a final winding of five turns. (a) Winding angle in the center as a function of time scaled using $1/f$ as the characteristic time. (b) Radial angle profiles at different times. Each colored curve corresponds to a point of the same color in panel (a). The dashed curve corresponds to the equilibrium curve obtained when $t\to \infty$. (c) Aspect of the target between slightly uncrossed polarizers when $n=5$. The local intensity is calculated by taking $I={sin}^2\varphi$.

Figure 20

Target pattern at equilibrium calculated for $n=5$ and two different values of the elastic anisotropy: (a) $\varepsilon =0.7$; (b) $\varepsilon =0.95$.

Figure 21

Number of turns at equilibrium divided by $n$ as a function of the elastic anisotropy when the flows are neglected. The curve has been plotted for $n=5$ but it should be emphasized that this curve is almost independent of $n$.

Figure 22

Angle $\varphi$ in the center of the film scaled by ${\varphi }_{\text{max}}=2\pi n_{\text{max}}$ as a function of the dimensionless time when the flows are nglected. Simulations were performed with $n=5$ for different values of the elastic anisotropy $\varepsilon$. Angle ${\varphi }_{\text{max}}$ corresponds to the final value of $\varphi$ during the winding. $n_{\text{max}}$ is given in Fig. 21.

Figure 23

Half-winding time as a function of the elastic anisotropy when the flows are neglected. These results have been obtained by taking $n=5$ but it should be emphasized that they depend very little on this parameter.

Figure 24

Target patterns and corresponding velocity fields in the stationary regime when $n=5, a_4=1, \varepsilon =0$, and $X=2.5$ (a, b) and $X=-2.5$ (c, d). Only the central part of the film, where the flows are the most important, is shown in panels (b) and (d), with the texture in background.

Figure 25

Number of turns scaled to $n$ in the center of the film (a) and maximal value of the norm of the velocity (b) as a function of $X$ for $n=5$ and different values of $a_4$ when the stationary regime is reached. In these simulations, $\varepsilon =0$.

Figure 26

Number of turns in the center of the film scaled to $n$ (a) or to $n_{\text{max}}$ (b) as a function of the dimensionless time. In these simulations, $n=5, \varepsilon =0$, and $a_4=1$.

Figure 27

Simulation of the complex trajectory of the particles. (a)–(d) The red dot shows the position of the particle at different times and the corresponding velocity field. (e) Trajectory of the particle during a full rotation of the director in the center of the film.

Figure 28

Angle $\varphi$ in the center of the film scaled by ${\varphi }_{\text{max}}$ as a function of the dimensionless time when the backflow is neglected. Simulations were performed with $n=5$ for different values of the elastic anisotropy $\varepsilon$. Angle ${\varphi }_{\text{max}}=2\pi n_{\text{max}}$ corresponds to the initial value of $\varphi$ at the start of the unwinding. $n_{\text{max}}$ is given in Fig. 21.

Figure 29

Half-unwinding time as a function of the elastic anisotropy when the backflow is neglected. These results have been obtained by taking $n=5$ but it should be emphasized that they depend very little on this value.

Figure 30

(a) Normalized phase as a function of the dimensionless time calculated when $\vec{v}=0$ (dashed line) and $\vec{v}\ne 0$ (solid line) by taking $a_4=1$ and $\varepsilon =0$. At time $t=0, \varphi ={\varphi }_{\text{max}}=10\pi$ (five turns); (b) velocity field when $\varphi ={\mathrm{\Phi }}_{\text{max}}/2=5\pi$ with the target pattern in background.

Figure 31

Phase profiles analytically calculated at $t=0$ when $\varphi =10\pi$ at the center of the film for different values of $\overline{\beta }$. The elastic anisotropy is neglected ($\varepsilon =0$). The profiles $\varphi (r,t)$ at time $t$ are obtained by multiplying these profiles by $e^{-t/\tau }$.

Figure 32

(a) Velocity profiles analytically calculated at $t=0$ when $\varphi =10\pi$ at the center of the film for different values of $\overline{\beta }$. The profiles $v_{\theta }(r,t)$ at time $t$ are obtained by multiplying these profiles by $e^{-t/\tau }$. The elastic anisotropy is neglected ($\varepsilon =0$). (b) Maximal velocity at time $t=0$ reached at $r\sim 0.6$ as a function of $\overline{\beta }$.

Figure 33

Half-unwinding time as a function of the dimensionless viscosity $\overline{\beta }$ when $\varepsilon =0$.