Faraday waves on a nematic liquid crystal, and its coupling with Marangoni convection about the thermal phase transition

Using a linear hydrodynamic theory, we demonstrate that Faraday waves occur in liquid crystalline fluids. The use of already experimentally known material parameters of a $N$-(4-methoxybenzylidene)-4-butylaniline liquid crystal allows us to confirm and realize the predictions of this theory. It provides the critical wave number and necessary driving acceleration at instability wave onset. Additionally, these observables experience an abrupt change originated by Marangoni convection due to the temperature gradient at the isotropic-nematic phase transition temperature. Correspondingly, the Marangoni number versus temperature also shows a sharp change in the transition temperature.

Figure 1

Calculated critical parameters: $k_c$ (a) and $a_c$ (b) as a function of the MBBA nematic-isotropic transition temperature difference $\mathrm{\Delta }T=T-T_c$ of a subharmonic wave. External frequency $\omega =20\pi \mathrm{Hz}$ circles symbol $\circ$ with $\mathbf{H}\left|\right|\mathbf{k}$, triangle symbol $▿$ for $\mathbf{H}\perp \mathbf{k}$ in the nematic phase. Layer thickness used $L=0.0045\mathrm{m}$. The same properties $k_c,a_c$ in the isotropic phase are depicted with square symbol $\square$. Filled-black symbols include Marangoni convection $M_T={\gamma }_1{l}_c^{2}A/\left({\eta }_1\alpha \right)$ due to temperature gradient. Grey symbols (online red and blue cyan colors) do not include the Marangoni effect $M_T=0$. Used material data of MBBA as cited in Ref. [18].

Figure 2

The calculated Marangoni number $M_T={\gamma }_1{l}_c^{2}A/\left({\eta }_1\alpha \right)$ versus temperature difference $\mathrm{\Delta }T=T-T_c$ about the critical temperature $T_c=388\mathrm{K}$ for MMBA liquid crystal. In the nematic phase, the heat thermal diffusivity is anisotropic, being ${\alpha }_{\left|\right|}$ for its measurement along the optical molecular axis (filled-black circle symbol, red color online) and perpendicular to it, ${\alpha }_{\perp }$ shadow (cyan color online), circle symbol. The nematic liquid with depth $L=0.0045\mathrm{m}$. The isotropic case is for $\mathrm{\Delta }T=T-T_c\ge 0\mathrm{K}$. $l_c=\sqrt{\gamma /\rho g}$ is the capillary length. The temperature gradient $A$ values are tabulated in Supplemental Material [56]. Used data for ${\alpha }_{\left|\right|}, {\alpha }_{\perp }$ of MBBA as cited in Ref. [18]. Note that $\alpha$ scale as ${10}^{-7}{\mathrm{m}}^2/\mathrm{s}$.

Figure 3

Subharmonic's threshold wave number $k_c$ (a) and acceleration $a_c$ (b) versus temperature of MBBA nematic liquid with depth $L=0.0025\mathrm{m}$ and forcing frequency $\omega =20\pi \mathrm{Hz}$. Black-filled symbols include Marangoni convection $M_T={\gamma }_1{l}_c^{2}A/\left({\eta }_1\alpha \right)$ whereas shadow symbols without it (cyan and red color symbols online).

Figure 4

Subharmonic's threshold wave number $k_c$ (a) and acceleration $a_c$ (b) versus temperature of MBBA nematic liquid with depth $L=0.0045\mathrm{m}$ and forcing frequency $\omega =120\pi \mathrm{Hz}$. Black-filled symbols include Marangoni convection $M_T={\gamma }_1{l}_c^{2}A/\left({\eta }_1\alpha \right)$ whereas shadow symbols without it (cyan and red color symbols online)—same materials parameters of Fig. 1.

Figure 5

Subharmonic's threshold wave number $k_c$ (a) and acceleration $a_c$ (b) versus temperature of MBBA nematic liquid with depth $L=0.0025\mathrm{m}$ and forcing frequency $\omega =120\pi \mathrm{Hz}$. Black-filled symbols $•, \blacksquare , ▾$ include Marangoni flow $M_T={\gamma }_1{l}_c^{2}A/\left({\eta }_1\alpha \right)$ whereas shadow symbol without it (cyan and red color symbols online). Dashed lines are guides to the eyes—the same material parameters as Fig. 1.

Figure 6

Subharmonic acceleration $a_c$ versus liquid layer thickness $L$ at forcing frequency $\omega =60\pi \mathrm{Hz}$ while taking into account the temperature gradient given by the Marangoni number $M_T={\gamma }_1{l}_c^{2}A/\left({\eta }_1\alpha \right)$. Symbol $•$ is for MBBA nematic liquid at $\mathrm{\Delta }T=T-T_c=-2\mathrm{K}$, whereas the red-filled triangle corresponds to isotropic phase at $\mathrm{\Delta }T=T-T_c=2\mathrm{K}$. In the same plot, symbols $\circ$ and $▿$ results from a semi-infinite layer $L\to \infty$. Inset depicts threshold wave number $k_c$ versus $L$.