Elastic response of colloidal smectic liquid crystals: Insights from microscopic theory

Elongated colloidal rods at sufficient packing conditions are known to form stable lamellar or smectic phases. Using a simplified volume-exclusion model, we propose a generic equation of state for hard-rod smectics that is robust against simulation results and is independent of the rod aspect ratio. We then extend our theory by exploring the elastic properties of a hard-rod smectic, including the layer compressibility ($B$) and bending modulus ($K_1$). By introducing weak backbone flexibility we are able to compare our predictions with experimental results on smectics of filamentous virus rods ($\mathit{fd}$) and find quantitative agreement between the smectic layer spacing, the out-of-plane fluctuation strength, as well as the smectic penetration length $\lambda =\sqrt{K_1/B}$. We demonstrate that the layer bending modulus is dominated by director splay and depends sensitively on lamellar out-of-plane fluctuations that we account for on the single-rod level. We find that the ratio between the smectic penetration length and the lamellar spacing is about two orders of magnitude smaller than typical values reported for thermotropic smectics. We attribute this to the fact that colloidal smectics are considerably softer in terms of layer compression than their thermotropic counterparts while the cost of layer bending is of comparable magnitude.

Figure 1

Left: Sketch of a smectic-A phase of colloidal rods of length $L$ with layer spacing ${\mathrm{\Delta }}_S$. Right: Illustration of the lamellar fluid model. Each lamella is composed of strongly aligned rods with centers of mass (red dots) roughly confined to a narrow slab of width ${\xi }^{-1}$ as indicated in blue.

Figure 2

Equation of state of a smectic-A phase of hard rods. Plotted is the osmotic pressure $P_S$ (in units $k_{B}T$ per rod volume) versus the packing fraction $\varphi$. The prediction from the theoretical model is generic and does not depend on the rod aspect ratio $\ell =L/D$. Simulation results are quoted from McGrother et al. [17] ($L/D=3.2,4,5$) and Bolhuis et al. [18] ($L/D=40$).

Figure 3

Layer compressibility modulus $B$ of a hard-rod smectic as a function of the overall packing fraction $\varphi$. The modulus is expressed in units $k_{B}T/v_r$ in terms of the rod volume $v_r=\frac{\pi }{4}L{D}^2$.

Figure 4

Two principal effects of layer bending; (top) a pure bend deformation of the smectic layer keeping a uniform (splayless) director field and (bottom) a splay deformation of the director field without layer bending. The lamellar midplane is indicated in blue.

Figure 5

(a) Penetration length $\lambda =\sqrt{K_1/B}$ of a hard-rod smectic in units of the rod length $L$ versus the rod packing fraction $\varphi$. The penetration length is shape-independent and does not depend on the rod aspect ratio. (b) Same as a function of the smectic layer spacing ${\overline{\mathrm{\Delta }}}_S$ (in units rod length $L$).

Figure 6

Theory-experiment comparison based on data for $\mathit{fd}$ virus rods in a smectic organization. (a) Lamellar spacing versus rod packing fraction $\varphi$ renormalized to the value at nematic-smectic coexistence ${\varphi }_{\mathrm{NS}}$. The interdigitation depth is fixed at $\overline{\zeta }=0.5$. Shown are results for two different $\mathit{fd}$ strands: M13KE (${\ell }_p/L=3$) and the much stiffer Y21M (${\ell }_p/L=10$). (b) Smectic potential for a number of different interdigitation depths $\overline{\zeta }=\zeta /L$ and ${\ell }_p=250$. Experimental data are based on M13KE.