Chiral edge fluctuations of colloidal membranes

We study edge fluctuations of a flat colloidal membrane comprised of a monolayer of aligned filamentous viruses. Experiments reveal that a peak in the spectrum of the in-plane edge fluctuations arises for sufficiently strong virus chirality. Accounting for internal liquid crystalline degrees of freedom by the length, curvature, and geodesic torsion of the edge, we calculate the spectrum of the edge fluctuations. The theory quantitatively describes the experimental data, demonstrating that chirality couples in-plane and out-of-plane edge fluctuations to produce the peak.

Figure 1

(a) Schematic of a colloidal membrane (b) Semi-infinite surface with a helix-like boundary. (c) Same shape viewed from above, looking down the $z$ axis at the $x\text{−}y$ plane.

Figure 2

(a) Normalized power spectrum $s_q^2\equiv q^2\langle u_q{u}_{-q}\rangle$ in which $c^{*}$ (in units of $k_{B}T$) and $\gamma$ (in units of $k_{B}T/\mu \mathrm{m}$) are fit to the data, with $\kappa /\overline{\kappa }\to \infty$, $B=100k_{\mathrm{B}}T\mu \mathrm{m}$, and $\overline{\kappa }=50k_{\mathrm{B}}T$. The value of $B$ used was found by fitting the $T=60{}^{\circ }\mathrm{C}$ data to the achiral formula (2). (b) Values of $c^{*}$ obtained from fitting (left vertical axis), and values of $\gamma$ obtained from fitting (right vertical axis).

Figure 3

Phase diagram for flat and rippled edges of chiral membranes as a function of Gaussian curvature modulus $\overline{\kappa }$ and chiral modulus $c^{*}$ for various values of line tension $\gamma$, $B=100k_{\mathrm{B}}T\mu \mathrm{m}$, and $\kappa =150k_{\mathrm{B}}T$. The flat edge is stable in the shaded area.