Saddle-splay-induced periodic edge undulations in smectic-$A$ disks immersed in a nematic medium

We report experimental studies on the phase behavior of binary mixtures of 1″,7″-bis(4-cyanobiphenyl-4′-yl)heptane (CB7CB) and 4,4-diheptyloxyazoxybenzene, which exhibit, apart from the nematic ($N$) and twist-bend nematic ($N_{\mathrm{TB}}$) phases, the induced smectic-$A$ ($\mathrm{Sm}-A$) phase for weight fraction of CB7CB between 0.05 and 0.70. In planar nematic layers, the $N_{\mathrm{TB}}$ phase separates as droplets of tactoidlike planform; the chirality of droplets manifests in the optical dissimilarity between their opposite angular ends. Our main result is that, in the appropriate two phase region, $\mathrm{Sm}-A$ nuclei with positive dielectric anisotropy change over to disks immersed in the nematic above some electric field, their edges decorated by periodic bright spots, a result which was earlier reported in another binary system exhibiting the induced $\mathrm{Sm}-A$ phase [R. Pratibha and N. V. Madhusudana, Physica A 224, 9 (1996)]. We develop a simple theory for the threshold of this distortion, which is a periodic undulation of the edge of the disk, demonstrating that it arises from saddle-splay elasticity of $\mathrm{Sm}-A$, the low $\mathrm{Sm}-A–N$ interfacial tension unable to suppress the distortion. The observed increases in the number of bright spots with field, and with the radius of the disk at a given field, in both the experimental systems are also accounted for by the model. The distortion, which results in the most direct visualization of saddle splay in $\mathrm{Sm}-A$, is also exhibited by disks nucleating on surfaces treated for homeotropic anchoring.

Figure 1

Concentration $X$ (wt. % of CB7CB) temperature phase diagram determined from x-ray diffraction and polarization microscopic studies. The $N$ phase, common to all the studied mixtures, occurs over a temperature range that reaches a minimum for $X\approx 30$ and a maximum for $X\approx 70$. The $\mathrm{Sm}-C$ phase of pure HOAB is completely suppressed for $X>5$; the $\mathrm{Sm}-A$ phase is induced in the concentration range 5–70; above $X\approx 70$, only the $N_{\mathrm{TB}}$ mesophase is observed below the nematic.

Figure 2

(a) Tactoid-like $N_{\mathrm{TB}}$ drops nucleated at $64.8{}^{\circ }\mathrm{C}$ within a nematic monodomain with the director n along $y$, in a 5-μm-thick sample of the mixture $M_{70}$ ($X=70.6\mathrm{wt}.\%$). (b) The geometry of $N_{\mathrm{TB}}$ domains becomes circular in the Freedericksz reoriented state at 5 V, 1 kHz. Scale: 5 μm each subdivision.

Figure 3

x-ray measurements of the layer spacing $d$ (left ordinate) and peak intensity $I$ (right ordinate), for different compositions of the CB7CB/HOAB system in the temperature range $54-112{}^{\circ }\mathrm{C}$. No low angle scattering was observed for the mixture with 85% CB7CB. Squares and circles denote $d$ and $I$, respectively. The marginal increase in $\mathrm{Sm}-A$ spacings with decreasing temperature is a result of chain stiffening.

Figure 4

Bâtonnets of the $\mathrm{Sm}-A$ phase coexisting with the $N$ phase in a 5-μm-thick layer of $M_{52}$. Diagonally parallel polarizers. Scale: 5 μm each subdivision.

Figure 5

Circular $\mathrm{Sm}-A$ domains formed deep in the Freedericksz state of a 20-μm thick, initially planar nematic sample of $M_{50}$ at $85.5{}^{\circ }\mathrm{C}$, subjected to a field with $f=10\mathrm{kHz}$ and $U=42\mathrm{V}$ ($\approx 20U_{\mathrm{F}}$) and observed using unpolarized mercury green light. Scale: 2 μm each subdivision.

Figure 6

(a)–(i) The course of transformation of $\mathrm{Sm}-A$ bâtonnets into homeotropic disks with corrugated boundaries under the influence of increasing electric field of frequency 1 kHz, in a 5-μm-thick layer of $M_{52}$ at $86{}^{\circ }\mathrm{C}$. (j) $\mathrm{Sm}-A$ disks developing at angular locations of zigzag bâtonnets in $M_{30}$ at $108{}^{\circ }\mathrm{C}$; 15 V, 1 kHz. $P\left(45\right)-A$(135) in (a) and (b); $P$(0) in (c)–(i); and $P\left(15\right)-A$(135) in (j). $L=5\mu \mathrm{m}$.

Figure 7

Transformation of the near homeotropic state of $\mathrm{Sm}-A$ disks into focal conic defect state with reducing field of frequency 1 kHz. (a)–(e) Initially planar $M_{52}$ layer; $L=5\mu \mathrm{m}, T=85{}^{\circ }\mathrm{C}$. Scale: 2 μm each subdivision. Progressive growth of FCDs from the left edge in contact with a substrate in (f), (g) $M52$ and (h)–(j) $M30$ ($z$-stacked images). Scale: (a)–(g) 2 μm each subdivision, (h)–(j) 5 μm each subdivision.

Figure 8

Linear variation of the number of bright image spots $N_S$ due to the undulatory edge of $\mathrm{Sm}-A$ disks as a function of (a) voltage of the sine wave field of frequency 1 kHz in a 5-μm-thick $M_{52}$ sample at $85{}^{\circ }\mathrm{C}$ and (b) radius of the disk in a 20-μm- thick $M_{50}$ sample at $85.5{}^{\circ }\mathrm{C}$, subjected to a sine wave field with $U=42\mathrm{V}$ and $f=10\mathrm{kHz}$.

Figure 9

Schematic of a $\mathrm{Sm}-A$ sample with thickness $D=ml$, where $l$ is the layer spacing, and subjected to an electric field E acting along the layer-normal ($z$ axis). A vertical displacement $u_0$ at the edge of the sample (lying in the $yz$ plane) dies down along the $x$ axis. $\theta \left(x\right)$ is the angle made by the layer-normal with the $z$ axis.

Figure 10

Schematic diagram of a $\mathrm{Sm}-A$ sample with a sinusoidal undulation [given by Eq. (3)] imposed in the $yz$ plane, lying in the plane of the page. An electric field $E$ acts along the $z$ axis.

Figure 11

Schematic diagram of a circular disk made of $\mathrm{Sm}-A$ layers, with a periodic undulation of the layers with an amplitude $u_0$ at the edge $r=R$. As discussed in the text, the amplitude dies down exponentially along the radial directions towards the center, and for clarity a slower decay is depicted in the figure. There is $no$ periodicity of the radius $R$ (though the perspective may seem to indicate such a structure).

Figure 12

Different contributions to the free energy (in atto $J$) of a $\mathrm{Sm}-A$ disk, with a radius of 5 μm surrounded by isotropic phase, and under the action of an electric field $E=4.76\mathrm{V}/\mu \mathrm{m}$, as functions of ν, the number of undulations at the edge. The positive splay elastic energy [curve (a) at $\nu =45$] rapidly increases for $\nu >20$, and the dielectric energy [curve (b)] shows a small increase with ν. The interfacial energy [curve (c)] is very small, and stays close to 0 compared to the other contributions. The saddle-splay contribution [curve (d)], which is negative, sharply increases in magnitude with ν. The total free energy [curve (e)] has a minimum at $\nu =34$.

Figure 13

Schematic of a $\mathrm{Sm}-A$ disk (a) at higher voltages [as in Fig. 7, for $E=3\mathrm{V}/\mu \mathrm{m}$], with its normal along $E$; and (b) at lower voltages [as in Fig. 7, for $E=2.4\mathrm{V}/\mu \mathrm{m}$], with its normal tilted relative to $E$. The $N$ director remains strongly anchored to the $\mathrm{Sm}-A$ director at the interface.

Figure 14

The total contributions from all the three regions of the nematic LC as described in the text to the elastic energy [curve (a) at $\nu =20$], and the dielectric energy [curve (b)] added to $\mathrm{\Delta }{F}_{\mathrm{sm}}$ give rise to a minimum at $\nu =25$ [curve (c)].

Figure 15

(a) Theoretical variation of $\nu$ with $E$ in a $\mathrm{Sm}-A$ disk with $R=5\mu \mathrm{m}$ as described in the text. The essentially linear dependence reflects that seen in experiments (see Fig. 8 above and Fig. 8 in Ref. [5]). (b) The dependence of ν on the radius of the disk $R$, fixing $D=2\mu \mathrm{m}, E=4.756\mathrm{V}/\mu \mathrm{m}$ and $\kappa =200\mathrm{pN}$. The dependence is linear, as seen experimentally in both the experimental systems (see Fig. 8(b) above and Ref. [5]). This simply implies that ${\lambda }_{\phi }$ is determined mainly by the decay lengths as long as the radius $R$ of the disk is considerably larger than those lengths.

Figure 16

$\mathrm{Sm}-A$ domains formed in the homeotropic nematic phase in a 9-μm-thick layer of $M_{50}$ at $84{}^{\circ }\mathrm{C}$. Scale: 2 μm each subdivision.

Figure 17

Focal conic domains of the $\mathrm{Sm}-A$ phase formed in the homeotropic nematic phase in a 9-μm-thick layer of $M_{50}$. Scale: 5 μm each subdivision.