Structural parameters of twist-bend nematics and splay-bend nematics in Dozov's theory

This paper presents the results of numerical calculations revealing how the structural parameters (i.e., the pitch $p_{\mathrm{TB}}$, the spatial period $p_{\mathrm{SB}}$, and the tilt angle ${\theta }_{\mathrm{TB}}$ or ${\theta }_{\mathrm{SB}}$) of twist-bend nematics ($N_{\mathrm{TB}}$) and splay-bend nematics ($N_{\mathrm{SB}}$) depend on the values of elastic constants in Dozov's theory [I. Dozov, Europhys. Lett. 56, 247 (2001)]. Alternative formulas for $p_{\mathrm{TB}}$, ${\theta }_{\mathrm{TB}}$, $p_{\mathrm{SB}}$, and ${\theta }_{\mathrm{SB}}$ have been derived and it has been proved that they give more accurate results than the expressions proposed by Dozov. Although the determination of the fourth-order elastic constants $C_1$, $C_2$, and $C_3$ is not feasible in a simple way, the order of magnitude of the sum $C_1+C_2$ has been easily estimated and is equal to ${10}^{-31}\mathrm{J}\mathrm{m}$. Moreover, the numerical calculations have shown that twist-bend nematics can exist even when $K_{11}$ is smaller than $2K_{22}$ and thus Dozov's criterion $K_{11}>2K_{22}$ for the stability of the $N_{\mathrm{TB}}$ phase is not strictly satisfied.

Figure 1

Pitch $p_{\mathrm{TB}}$ (a) and tilt angle ${\theta }_{\mathrm{TB}}$ (b) of twist-bend nematics as a function of the fourth-order elastic constant $C_1$. The curves correspond to the results obtained for different values of the fourth-order elastic constant $C_2$ which are given in the graphs in ${10}^{-31}\mathrm{J}\mathrm{m}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $K_{33}=-0.5\mathrm{pN}$.

Figure 2

Pitch $p_{\mathrm{TB}}$ (a) and tilt angle ${\theta }_{\mathrm{TB}}$ (b) of twist-bend nematics as a function of the twist elastic constant $K_{22}$. The curves correspond to the results obtained for different values of the fourth-order elastic constant $C_1$ which are given in the graphs in ${10}^{-31}\mathrm{J}\mathrm{m}$. The other material parameters are as follows: $K_{33}=-0.5\mathrm{pN}$, $C_2=2\times {10}^{-31}\mathrm{J}\mathrm{m}$.

Figure 3

Pitch $p_{\mathrm{TB}}$ (a) and tilt angle ${\theta }_{\mathrm{TB}}$ (b) of twist-bend nematics as a function of the bend elastic constant $K_{33}$. The solid lines correspond to the results $p_{\mathrm{TB}}^n$ and ${\theta }_{\mathrm{TB}}^n$ obtained by means of numerical calculations. The dashed curves have been plotted on the basis of values $p_{\mathrm{TB}}^f$ and ${\theta }_{\mathrm{TB}}^f$ calculated from Eqs. (4) and (5). The insets present how the percentage differences $\mathrm{\Delta }{p}_{\mathrm{TB}}=\left[\left(p_{\mathrm{TB}}^f-p_{\mathrm{TB}}^n\right)/p_{\mathrm{TB}}^n\right]\times 100\%$ and $\mathrm{\Delta }{\theta }_{\mathrm{TB}}=\left[\left({\theta }_{\mathrm{TB}}^f-{\theta }_{\mathrm{TB}}^n\right)/{\theta }_{\mathrm{TB}}^n\right]\times 100\%$ depend on $K_{33}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $C_1=C_2=2\times {10}^{-31}\mathrm{J}\mathrm{m}$. In (a) the solid curve and the dashed one almost overlap.

Figure 4

Spatial period $p_{\mathrm{SB}}$ (a) and tilt angle ${\theta }_{\mathrm{SB}}$ (b) of splay-bend nematics as a function of the splay elastic constant $K_{11}$. The curves correspond to the results obtained for different values of the fourth-order elastic constant $C_1$ which are given in the graphs in ${10}^{-31}\mathrm{J}\mathrm{m}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $K_{33}=-0.5\mathrm{pN}$, $C_2=2\times {10}^{-31}\mathrm{J}\mathrm{m}$, $C_3=0$. In (b) the curves overlap.

Figure 5

Spatial period $p_{\mathrm{SB}}$ (a) and tilt angle ${\theta }_{\mathrm{SB}}$ (b) of splay-bend nematics as a function of the splay elastic constant $K_{11}$. The curves correspond to the results obtained for different values of the fourth-order elastic constant $C_2$ which are given in the graphs in ${10}^{-31}\mathrm{J}\mathrm{m}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $K_{33}=-0.5\mathrm{pN}$, $C_1=2\times {10}^{-31}\mathrm{J}\mathrm{m}$, $C_3=0$. In (b) the curves overlap.

Figure 6

Spatial period $p_{\mathrm{SB}}$ (a) and tilt angle ${\theta }_{\mathrm{SB}}$ (b) of splay-bend nematics as a function of the splay elastic constant $K_{11}$. The curves correspond to the results obtained for different values of the fourth-order elastic constant $C_1$ which are given in the graphs in ${10}^{-31}\mathrm{J}\mathrm{m}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $K_{33}=-0.5\mathrm{pN}$, $C_2=C_3=2\times {10}^{-31}\mathrm{J}\mathrm{m}$. The inset in (b) shows the enlargement of the region bounded by the dashed rectangle.

Figure 7

Spatial period $p_{\mathrm{SB}}$ (a) and tilt angle ${\theta }_{\mathrm{SB}}$ (b) of splay-bend nematics as a function of the splay elastic constant $K_{11}$. The curves correspond to the results obtained for different values of the fourth-order elastic constant $C_2$ which are given in the graphs in ${10}^{-31}\mathrm{J}\mathrm{m}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $K_{33}=-0.5\mathrm{pN}$, $C_1=C_3=2\times {10}^{-31}\mathrm{J}\mathrm{m}$. The inset in (b) shows the enlargement of the region bounded by the dashed rectangle.

Figure 8

Spatial period $p_{\mathrm{SB}}$ (a) and tilt angle ${\theta }_{\mathrm{SB}}$ (b) of splay-bend nematics as a function of the splay elastic constant $K_{11}$. The curves correspond to the results obtained for different values of the bend elastic constant $K_{33}$ which are given in the graphs in $\mathrm{pN}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $C_1=C_2=2\times {10}^{-31}\mathrm{J}\mathrm{m}$, $C_3=0$.

Figure 9

Spatial period $p_{\mathrm{SB}}$ (a) and tilt angle ${\theta }_{\mathrm{SB}}$ (b) of splay-bend nematics as a function of the splay elastic constant $K_{11}$. The curves correspond to the results obtained for different values of the fourth-order elastic constant $C_3$ which are given in the graphs in ${10}^{-31}\mathrm{J}\mathrm{m}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $K_{33}=-0.5\mathrm{pN}$, $C_1=C_2=2\times {10}^{-31}\mathrm{J}\mathrm{m}$. The inset in (b) shows the enlargement of the region bounded by the dashed rectangle.

Figure 10

Spatial period $p_{\mathrm{SB}}$ (a) and tilt angle ${\theta }_{\mathrm{SB}}$ (b) of splay-bend nematics as a function of the splay elastic constant $K_{11}$. The solid lines correspond to the results $p_{\mathrm{SB}}^n$ and ${\theta }_{\mathrm{SB}}^n$ obtained by means of numerical calculations. The dashed curves have been plotted on the basis of values $p_{\mathrm{SB}}^f$ and ${\theta }_{\mathrm{SB}}^f$ calculated from Eqs. (6) and (7). The insets present how the percentage differences $\mathrm{\Delta }{p}_{\mathrm{SB}}=\left[\left(p_{\mathrm{SB}}^f-p_{\mathrm{SB}}^n\right)/p_{\mathrm{SB}}^n\right]\times 100\%$ and $\mathrm{\Delta }{\theta }_{\mathrm{SB}}=\left[\left({\theta }_{\mathrm{SB}}^f-{\theta }_{\mathrm{SB}}^n\right)/{\theta }_{\mathrm{SB}}^n\right]\times 100\%$ depend on $K_{11}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $K_{33}=-0.1\mathrm{pN}$, $C_1=C_2=2\times {10}^{-31}\mathrm{J}\mathrm{m}$, $C_3=0$.

Figure 11

Critical value $K_{11}^c$ as a function of the twist elastic constant $K_{22}$. The solid line corresponds to the results obtained by means of numerical calculations. The dashed line has been plotted on the basis of values calculated from the formula $K_{11}^c=2K_{22}$ given by Dozov. The other material parameters are as follows: $K_{33}=-0.5\mathrm{pN}$, $C_1=C_2=2\times {10}^{-31}\mathrm{J}\mathrm{m}$, $C_3=0$.

Figure 12

Critical value $K_{11}^c$ as a function of the bend elastic constant $K_{33}$. The curves correspond to the results obtained for different values of the twist elastic constant $K_{22}$ which are given in the graph. The other material parameters are as follows: $C_1=C_2=2\times {10}^{-31}\mathrm{J}\mathrm{m}$, $C_3=0$.

Figure 13

Critical value $K_{11}^c$ as a function of the fourth-order elastic constant $C_2$. The curves correspond to the results obtained for different values of the fourth-order elastic constant $C_1$ which are given in the graph in ${10}^{-31}\mathrm{J}\mathrm{m}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $K_{33}=-0.5\mathrm{pN}$, $C_3=0$.

Figure 14

Critical value $K_{11}^c$ as a function of the fourth-order elastic constant $C_2$. The curves correspond to the results obtained for different values of the fourth-order elastic constant $C_1$ which are given in the graph in ${10}^{-31}\mathrm{J}\mathrm{m}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $K_{33}=-0.5\mathrm{pN}$, $C_3=2\times {10}^{-31}\mathrm{J}\mathrm{m}$. The inset presents the enlargement of the region bounded by the dashed rectangle.

Figure 15

Critical value $K_{11}^c$ as a function of the fourth-order elastic constant $C_3$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $K_{33}=-0.5\mathrm{pN}$, $C_1=C_2=2\times {10}^{-31}\mathrm{J}\mathrm{m}$.

Figure 16

Critical value $K_{11}^c$ as a function of the bend elastic constant $K_{33}$. The curves correspond to the results obtained for different values of the fourth-order elastic constant $C_1$ which are given in the graph in ${10}^{-31}\mathrm{J}\mathrm{m}$. The other material parameters are as follows: $K_{22}=5\mathrm{pN}$, $C_2=2\times {10}^{-31}\mathrm{J}\mathrm{m}$, $C_3=0$.