Role of initial conditions in the decay of spatially periodic patterns in a nematic liquid crystal

The decay of stripe patterns in planarly aligned nematic liquid crystals has been studied experimentally and theoretically. The initial patterns have been generated by the electrohydrodynamic instability and a light diffraction technique has been used to monitor their decay. In our experiments different decay rates have been observed as a function of the pattern wave number. According to our theoretical analysis they belong to a spectrum of decay modes and are individually selected in dependence on the initial conditions. Additional insight has emerged from a refined physical optical description of the diffraction intensity. The results compare well with experiments, which include also controlled modifications of the initial conditions to assess different decay modes.

Figure 1

(Color online) (a) Critical voltage $U_c$ and (b) dimensionless components $q_c^{\prime }=q_{c}d∕\pi$ (open squares) and $p_c^{\prime }=p_{c}d∕\pi$ (open triangles) of the critical wave vector for a Phase 5 cell $(d=9.2\mathrm{\mu }\mathrm{m})$ as function of frequency. Measurements (symbols) are compared with theory (solid lines).

Figure 2

(Color online) Schematic sketch of the light diffraction geometry.

Figure 3

(Color online) Variation of the weights $w_k$ of various decay eigenmodes (plotted with different line styles) with respect to the dimensionless wave number square ${q^{\prime }}^2$ for (a) $k=1,\dots ,4$ and for (b) $k=5,\dots ,8$, using a $\sin \left[\frac{\pi }{d}\right(z+\frac{d}{2}\left)\right]$ initial director profile. The weights $w_k$ calculated from the actual EC director profile are plotted as symbols for a few frequencies in the conductive (${q^{\prime }}^2<6$, squares for $k=1$, up triangles for $k=2$) as well as in the dielectric regime $({q^{\prime }}^2>8)$ (circles for $k=3$ and down triangles for $k=4$).

Figure 4

(Color online) Variation of the diffraction efficiency of the first ($c_1^{opt}$, solid line), the second ($c_2^{opt}$, dashed line) and the third ($c_3^{opt}$, dotted line) decay eigenmodes as a function of the dimensionless wavenumber square ${q^{\prime }}^2$.

Figure 5

(Color online) Contribution of modes to the diffraction intensity $w_k{c}_k^{opt}$ (plotted with different line styles) with respect to the dimensionless wave number square ${q^{\prime }}^2$ for (a) $k=1,\dots ,4$ and for (b) $k=5,\dots ,8$, assuming a $\sin \left[\frac{\pi }{d}\right(z+\frac{d}{2}\left)\right]$ initial director profile.

Figure 6

(Color online) Dimensionless decay rates ${\mu }_f^{\prime }$ obtained by a single exponential fit versus ${q^{\prime }}^2$ for the start of the decay. Squares and triangles indicate decay of conductive and dielectric rolls, respectively. The lines of different style depict the ${\mu }_1^{\prime },\dots ,{\mu }_7^{\prime }$ branches of the dispersion relation.

Figure 7

(Color online) Dimensionless decay rates ${\mu }_f^{\prime }$ obtained by a single-exponential fit versus the time $t_a$ elapsed between switching off the voltage and start of the fit (a) in the conductive regime (${q^{\prime }}^2=5.62$, $\overline{k}=2$, ${\mu }_2^{\prime }=31.28$, ${\mu }_1^{\prime }=22.42$), (b) in the dielectric regime (${q^{\prime }}^2=18.80$, $\overline{k}=5$, ${\mu }_5^{\prime }=147.64$, ${\mu }_4^{\prime }=121.23$, ${\mu }_3^{\prime }=94.27$, ${\mu }_2^{\prime }=84.10$, ${\mu }_1^{\prime }=73.21$).

Figure 8

(Color online) Temporal evolution of the normalized decay rates ${\overline{{\mu }^{\prime }}}_f\left(t_a\right)$ obtained by a single exponential fit for ${q^{\prime }}^2=1.88$ (solid squares), ${q^{\prime }}^2=5.62$ (solid up triangles), ${q^{\prime }}^2=9.02$ (open up triangles), ${q^{\prime }}^2=15.10$ (open circles), and ${q^{\prime }}^2=18.80$ (solid stars).

Figure 9

(Color online) Variation of $C_q$ with respect to the dimensionless wave number square ${q^{\prime }}^2$.

Figure 10

(Color online) Temporal evolution of the light intensity of the first-order diffraction fringe following the shutdown of the applied voltage in a $28\text{−}\mathrm{\mu }\mathrm{m}$ thick cell of Phase 5A. Curves with different line styles correspond to different values of the period $\mathrm{\Delta }{t}_s$ during which an increased amplitude excitation (${\epsilon }_p=0.18$ instead of $\epsilon =0.02$ ) was used.

Figure 11

(Color online) Theoretical $\left({\mu }_k^{\prime }\right)$ and measured $\left({\mu }_{expt}^{\prime }\right)$ values of the dimensionless decay rate of the director versus dimensionless ${q^{\prime }}^2$ for (a) the conductive mode of Phase 5A, (b) the dielectric mode of Phase 5. Lines of various styles correspond to the eight lowest branches of the dispersion relation. Solid triangles are the data measured at sinusoidal excitation with $\mathrm{\Delta }{t}_s=0$; open squares are the data obtained at square-wave excitation with $\mathrm{\Delta }{t}_s=0.2\mathrm{s}$.