Rheology of twist-grain-boundary-$A$ liquid crystals

We report studies on the rheological properties of a liquid crystalline analog of Abrikosov phase in type-II superconductors known as twist-grain-boundary-$A$ (${\mathrm{TGB}}_A$) phase. The ${\mathrm{TGB}}_A$ phase shows a large apparent yield stress compared to the cholesteric (${\mathrm{N}}^{*}$) phase. The storage modulus ($G^{{}^{\prime }}$) of the ${\mathrm{TGB}}_A$ phase is significantly larger than the loss modulus ($G^{{}^{\prime \prime }}$). The dynamic relaxation measurements indicate a solid-like behavior of ${\mathrm{N}}^{*}$, ${\mathrm{TGB}}_A$, and smectic-$C$${}^{*}$ phases. The complex shear modulus of the ${\mathrm{TGB}}_A$ phase exhibits a power-law behavior $G^{*}$($\omega$) $\sim$ ${\omega }^{\alpha }$ with $\alpha \simeq 0.5$. The relative amplitude of $G^{{}^{\prime }}$ and $G^{{}^{\prime \prime }}$ at various temperatures indicate that the enhanced elasticity of ${\mathrm{TGB}}_A$ phase is due to the structural defects.

Figure 1

Schematic representation of structure of the ${\mathrm{TGB}}_A$ phase. The distances between two dislocations and two grain boundaries are $l_d$ and $l_b$, respectively.

Figure 2

Textures observed in polarizing optical microscope (POM) at different temperatures (a) ${110}^{\circ } \mathrm{C}$ (${\mathrm{N}}^{*}$) (b) ${73}^{\circ } \mathrm{C}$ (${\mathrm{TGB}}_A$), and (c) ${55}^{\circ } \mathrm{C}$ ($\mathrm{Sm}C$${}^{*}$). A square grid pattern characteristics of ${\mathrm{UTGB}}_{C^{*}}$ appears between ${\mathrm{TGB}}_A$ and $\mathrm{Sm}C$${}^{*}$ is also shown in the box. The physical appearance of the sample while trying to pull it up by a glass rod at corresponding temperatures are also shown.

Figure 3

Shear-rate-dependent shear stress (${\sigma }_{\text{eff}}$) at different temperatures ${\mathrm{N}}^{*}$: [${110}^{\circ } \mathrm{C}$ (squares)], near ${\mathrm{N}}^{*}$-${\mathrm{TGB}}_A$ transition: [${81}^{\circ } \mathrm{C}$ (circles)], ${\mathrm{TGB}}_A$: [${73}^{\circ } \mathrm{C}$ (uptriangles)], ${\mathrm{TGB}}_A$: [${68}^{\circ } \mathrm{C}$ (downtriangles)], $\mathrm{Sm}C$${}^{*}$: [${57}^{\circ } \mathrm{C}$ (diamonds)].

Figure 4

The strain amplitude dependence of the storage $G^{{}^{\prime }}$(solid symbols) and the loss $G^{{}^{\prime \prime }}$(open symbols) moduli at different temperatures. ${110}^{\circ } \mathrm{C}$ (hexagons) [${\mathrm{N}}^{*}$], ${73}^{\circ } \mathrm{C}$ (diamonds) [${\mathrm{TGB}}_A$], ${68}^{\circ } \mathrm{C}$ (circles) [${\mathrm{TGB}}_A$], and ${59}^{\circ } \mathrm{C}$ (squares) [$\mathrm{Sm}C$${}^{*}$] at $\omega =1$ rad/s.

Figure 5

The frequency dependence of (a) storage $G^{{}^{\prime }}$(solid symbols) and (b) loss $G^{{}^{\prime \prime }}$ (open symbols) moduli at three representative temperatures in all the phases. The solid lines are theoretical fits to Eq. (1).

Figure 6

(a) The exponent $\alpha$ characterizing the power-law contribution to $G^{{}^{\prime }}$$\left(\omega \right)$(solid squares) and $G^{{}^{\prime \prime }}$$\left(\omega \right)$ (open squares) as a function of temperature. (Inset) The inset displays the fit parameters $G_o$ and $\beta$ obtained from $G^{{}^{\prime }}\left(\omega \right)$ and $G^{{}^{\prime \prime }}\left(\omega \right)$ data. Diamonds (solid and open) and circles (solid and open) are obtained from $G^{{}^{\prime }}$$\left(\omega \right)$ and $G^{{}^{\prime \prime }}\left(\omega \right)$ data, respectively. (b) Variation of $G_o$ with reduced temperature $\chi$. The solid line showing the best-fit result $G_o\approx {\chi }^{\gamma }$, with $\gamma =1$.