Curvature elasticity of smectic-C liquid crystals and formation of stripe domains along thickness gradients in menisci of free-standing films

Smectic liquid crystals with a layering order of rodlike molecules can be drawn in the form of free standing films across holes. Extensive experimental studies have shown that smectic-C (SmC) liquid crystals (LCs) with tilted molecules form periodic stripes in the thinner parts of the meniscus, which persist over a range of temperatures above the transition of the bulk medium to the SmA phase in which the tilt angle is zero. The prevailing theoretical models cannot account for all the experimental observations. We propose a model in which we argue that the negative curvature of the surface of the meniscus results in an energy cost when the molecules tilt at the surface. The energy can be reduced by exploiting the allowed $(\nabla ·\mathbf{k})(\nabla ·\mathbf{c})$ deformation which couples the divergence of k, the unit vector along the layer normal, with that of c, the projection of the tilted molecular director on the layer plane. We propose a structure with periodic bending of layers with opposite curvatures, in which the c-vector field itself has a continuous deformation. Calculations based on the theoretical model can qualitatively account for all the experimental observations. It is suggested that detailed measurements on the stripes may be useful for getting good estimates of a few curvature elastic constants of SmC LCs.

Figure 1

Schematic diagram showing the layered structure of smectic-C liquid crystals. The director n is an apolar vector along the average orientation direction of the tilted rodlike molecules. Its projection on the layer plane is a polar vector c, which is assumed to be a unit vector. k along the layer normal is also a unit vector. Note that the symmetry of SmC is preserved when both k and c change sign.

Figure 2

Schematic diagram of the meniscus region of a free standing smectic film. Only the left section of a film taken between two parallel walls (shaded region) is shown. The thickness $h$ of the meniscus decreases as one moves away from the wall along X, reaching a flat part with thickness $h_F$ at $X=X_{\max }$. Edge dislocations located at the midplane of the meniscus mediate the variation in $h$. The Burgers vector of the dislocation (shown by the symbol T rotated by 90°) is equal to the layer thickness $d$ near the thinner parts and several times $d$ in the thicker parts. The surfaces of the meniscus have a circular shape with a radius $R_M\sim 100\mu \mathrm{m}$. As the sample is cooled towards SmA to SmC transition temperature ($T_{\mathrm{AC}}$), the director in the layers near the surface develops a tilt angle, and the corresponding c vector can be expected to develop a bend distortion by following the surface profile, shown by the dashed lines near the surfaces. It is argued in the text that this actually costs a positive energy, which is reduced by the formation of stripe distortions of the thinner parts of the structure with periodicity along Y. Δ is the small width of a slice with thickness $h$ used in the theoretical analysis.

Figure 3

(a) Schematic diagram of the proposed model of the stripe domains in the thinner regions of the meniscus of a free standing film of SmC liquid crystal. Side view along X of a slice as shown in Fig. 2, which is also along the ζ axis of the local cylindrical coordinate system, and pointed into the plane of the diagram. In the central V-shaped section, the layers bend in circular arcs of angular extent α around the center at O. The layers in the neighboring sections bend around symmetrically located centers above the uppermost layer, producing inverted V-shaped sections. The spatial distortion of the tilted director n field is shown in a middle layer. The thicker ends of the triangular symbols signify projections towards the reader. Both the layers and the n fields vary smoothly across the neighboring sections. The radius of the nearest layer from O is $R_i=\mathrm{OD}$, and that of the upper-most layer $\mathrm{OA}=R_i+h$. The spatial period of the stripe pattern is $w=\mathrm{AB}+\mathrm{BC}$, and the amplitude of the surface modulation is given by $a$. (b) Top view showing the c-vector field of the stripe domains in a slice of width Δ. Note that both the c field and the corresponding k field as seen in (a) have opposite divergences in neighboring sections.

Figure 4

(a) Variation of the stripe width $w$ with the height $h$ of the meniscus in the SmC phase. The linear dependence is in accord with experimental results [3, 16]. (b) Variation of the amplitude $a$ of the periodic surface profile of the stripes, as a function of the stripe width $w$. The relative magnitude of $a$ is a few times larger than the experimental value, but the linear dependence reflects the experimental trend [3, 16].

Figure 5

Schematic diagram of a section of stripe above the SmC to SmA transition temperature $T_{\mathrm{AC}}$. The shaded areas of thickness δ at the top and bottom surfaces are assumed to have a nonzero tilt order, while the rest of the medium is in the SmA phase without any tilt order. As described in the text, in order to reduce the positive elastic energy of bending of the layers, the center $O_A$ is pushed far away from the surface, increasing $R_i\left(=O_{A}L\right)$ to tens of micrometers.