Effects of flexoelectricity and weak anchoring on a Freedericksz transition cell

We consider a mathematical model that consists of a nematic liquid crystal layer sandwiched between two parallel bounding plates, across which an external field is applied. We investigate how the number and type of solutions for the director orientation within the layer change as the field strength, anchoring conditions, and material properties of the nematic liquid crystal layer vary. In particular, we focus on how the inclusion of flexoelectric effects alters the Freedericksz and saturation thresholds.

Figure 1

Sketch showing the setup and summarizing the key parameters in dimensional coordinates.

Figure 2

Evolution of director field in time for ${\alpha }_{\{0,1\}}=\pi /2$, with (a) ${\mathcal{A}}_{\{0,1\}}=1000$ (strong anchoring), $\mathcal{F}=5$, and $\mathcal{D}=25$, (b) ${\mathcal{A}}_{\{0,1\}}=5$ (weak anchoring), $\mathcal{F}=5$, and $\mathcal{D}=25$ and (c) ${\mathcal{A}}_{\{0,1\}}=5$, $\mathcal{F}=-5$, and $\mathcal{D}=25$. All figures show the director evolution in dimensionless time when no electric field is applied $\mathcal{D}=\mathcal{F}=0$ (black solid lines) and when an electric field of strength $\mathcal{F}=\pm 5$ and $\mathcal{D}=25$ is applied (red dotted lines).

Figure 3

Director solution $\theta \left(z\right)$ for different values of the material parameter $\mathrm{{\rm Y}}$ while keeping $\mathcal{D}=10$. $\mathrm{{\rm Y}}=0$ corresponds to a NLC layer with no flexoelectric contribution. $\mathrm{{\rm Y}}=1$ corresponds to the director configuration shown in Fig. 2.

Figure 4

Bifurcation diagram showing ${\left|\right|\theta \left(z\right)\left|\right|}_2$ vs $\mathcal{F}$ with ${\mathcal{A}}_0={\mathcal{A}}_1=5$ and ${\alpha }_0={\alpha }_1=\pi /2$ for different $\mathrm{{\rm Y}}$, obtained using continuation in $\mathcal{F}$. ${\mathcal{F}}_f$ denotes the Freedericksz threshold and ${\mathcal{F}}_s$ denotes the saturation threshold with the arrows pointing where the thresholds occur for each $\mathrm{{\rm Y}}$.

Figure 5

Bifurcation diagram showing ${\left|\right|\theta \left(z\right)\left|\right|}_2$ vs $\mathcal{F}$ with $\mathrm{{\rm Y}}=1$ for different anchoring strengths: ${\mathcal{A}}_0={\mathcal{A}}_1=0.1,1,5,10,20,1000$ obtained using continuation in $\mathcal{F}$. The portion of the diagram where ${\left|\right|\theta \left|\right|}_2=\pi /2$ represents the horizontal state $[{\theta }_h\left(z\right)=\pi /2]$, while ${\left|\right|\theta \left|\right|}_2=0$ represents the vertical state $[{\theta }_v\left(z\right)=0]$. The intermediate portion (slowly decaying as $\left|\mathcal{F}\right|$ increases) represents the nontrivial solution ${\theta }_n\left(z\right)$ found numerically.

Figure 6

Bifurcation diagram showing ${\left|\right|\theta \left(z\right)\left|\right|}_2$ vs $\mathcal{F}$ with $\mathrm{{\rm Y}}=1$ for ${\mathcal{A}}_0=10.0$ and $1\le {\mathcal{A}}_1\le 12$ using continuation in $\mathcal{F}$.

Figure 7

Bifurcation diagram showing ${\left|\right|\theta \left(z\right)\left|\right|}_2$ vs $\mathcal{F}$ with $\mathrm{{\rm Y}}=1$ for ${\mathcal{A}}_0={\mathcal{A}}_1=5$ and ${\alpha }_0=0$, ${\alpha }_1=\pi /2$, obtained using continuation in $\mathcal{F}$.

Figure 8

Bifurcation diagram showing ${\left|\right|\theta \left(z\right)\left|\right|}_2$ vs $\mathcal{F}$ with $\mathrm{{\rm Y}}=1$ for ${\mathcal{A}}_0={\mathcal{A}}_1=5$ and ${\alpha }_0=\pi /2$, ${\alpha }_1=\pi /2-\psi$ ($\psi =0.1$), obtained using forward continuation in $\mathcal{F}$. Inset located at the upper left corner shows the director configuration obtained when $\mathcal{F}=12.5$. Inset located at the lower right corner shows a zoom of the bifurcation diagram, to clarify the behavior in the range $12\le \mathcal{F}\le 13$ region.

Figure 9

Bifurcation diagram showing ${\left|\right|\theta \left(z\right)\left|\right|}_2$ plotted vs $\mathcal{F}$ with $\mathrm{{\rm Y}}=1$ for (a) ${\alpha }_0=\psi$ and ${\alpha }_1=\pi /2$ and (b) ${\alpha }_0=0$, ${\alpha }_1=\pi /2-\psi$ (with $\psi =0.1$). Anchoring strengths are set to ${\mathcal{A}}_0={\mathcal{A}}_1=5$. Black solid and red dashed curves are obtained using forward continuation in $\left|\mathcal{F}\right|$ while the green dotted curve is obtained using reverse continuation in $\left|\mathcal{F}\right|$.

Figure 10

Bifurcation diagram showing ${\left|\right|\theta \left(z\right)\left|\right|}_2$ plotted vs $\mathcal{F}$ with $\mathrm{{\rm Y}}=1$ for ${\alpha }_0=0$ and ${\alpha }_1=\pi /3$. Anchoring strengths are set to ${\mathcal{A}}_0={\mathcal{A}}_1=5$. Black solid and red dashed curves are obtained using forward continuation in $\left|\mathcal{F}\right|$ while the green dotted curve is obtained using reverse continuation in $\left|\mathcal{F}\right|$.