Theory of microphase separation in bidisperse chiral membranes

We present a Ginzburg-Landau theory of microphase separation in a bidisperse chiral membrane consisting of rods of opposite handedness. This model system undergoes a phase transition from an equilibrium state where the two components are completely phase separated to a state composed of microdomains of a finite size comparable to the twist penetration depth. Characterizing the phenomenology using linear stability analysis and numerical studies, we trace the origin of the discontinuous change in microdomain size that occurs during this phase transition to a competition between the cost of creating an interface and the gain in twist energy for small microdomains in which the twist penetrates deep into the center of the domain.

Figure 1

(a) Schematic of the experimental system. (b) The primary results of this work summarized in a phase diagram as a function of the smectic alignment parameter $J$ and the twist-composition coupling parameter $a$. The lines indicate the phase boundary between the microphase separated and bulk phase separated states as obtained from linear stability analysis (black solid line) and numerical integration of Eqs. (2) and (3) (green dashed line). The snapshots show configurations at steady state obtained from numerics at the indicated parameter values (circles). (c) Illustration of the evolution to steady state for two parameter sets. The left panel shows the evolution of the composition field, and the right panel shows the twist profile in the steady state. The results shown here and in the main text are for a 60:40 mixture with ${\lambda }_{\psi }=0.1$ and $q_0=0.1$.

Figure 2

(a) The largest eigenvalue $\omega \left(k\right)$ of the linear stability matrix in Eq. (4) as a function of the wave vector $k$ for indicated values of the alignment strength $J$, with $a=0.5$ and ${\lambda }_{\psi }=0.1.$ (b) Dependence of the optimal microdomain size on $J{a}^2$ obtained with three different analysis methods: the steady-state mean radius of microdomains obtained by numerical integration (upper line), the radius which minimizes the GL free energy (calculated as described in the text; middle line), and wavelength corresponding to the fastest-growing mode calculated by linear stability analysis (lower line) for ${\lambda }_{\psi }=0.1$. Throughout this paper, all lengths are given in units of the twist penetration length ${\lambda }_{\text{t}}$.

Figure 3

(a) The free energy density in a domain $f$ (solid black line) and the separate contributions $(f_{\mathrm{LC}}$, dot-dashed blue line; $f_{\psi }$, dashed red line; and $f_{\mathrm{Cross}}$, dotted pink line) are shown as a function of domain radius $R$, measured in units of ${\lambda }_{\mathrm{t}}$, for $J=30$ and $a=0.7$. The domain radius is varied from ${0.2\lambda }_{\mathrm{t}}$ to ${20\lambda }_{\mathrm{t}}$. (b) The same free energy density contributions as in (a), but focusing on the behavior at small $R$.

Figure 4

(a) The theoretical twist profile obtained from dynamical analysis is compared with the one obtained from the experiments [12]. (b) The theoretical free energy density profile as a function of the microdomain radius for the parameters $J=20$ and $a=0.7$ is compared against the experimentally determined free energy. The radius is in units of the twist penetration length observed in the experiments: ${\lambda }_t=0.48\mu \text{m}$. The theoretical free energy has also been shifted to match the experimental curve at the minimum.

Figure 5

Comparison of twist and $\psi$ profiles predicted by the quasistatic calculation (black line) with results from the numerical integration of the full model dynamics (blue dots). The profiles from the dynamics are ensemble averaged over droplets of the same size in equilibrium configurations. (a) Twist profiles for droplets with radius greater than twist penetration depth ${\lambda }_{\text{t}}$. (b) Twist profiles for droplets with radius smaller than ${\lambda }_{\text{t}}$, exhibiting a cutoff of the exponential decay inside the droplet. (c) $\psi$ profile for droplets with radius larger than ${\lambda }_{\text{t}}$. (d) $\psi$ profile for droplets with radius smaller than ${\lambda }_{\text{t}}$.

Figure 6

Dependence of twist and $\psi$ profiles on $J$ and $a$ obtained from dynamical simulations by ensemble averaging over equilibrium droplets of the same size. (a) Twist profiles for droplets with radius greater than twist penetration depth ${\lambda }_{\text{t}}$ for indicated values of $J$ and $a$. (b) Twist profiles for droplets with radius smaller than ${\lambda }_{\text{t}}$, exhibiting a cutoff of the exponential decay inside the droplet. (c) $\psi$ profile for droplets with radius larger than ${\lambda }_{\text{t}}$. (d) $\psi$ profile for droplets with radius smaller than ${\lambda }_{\text{t}}$.

Figure 7

(a) Change in $\psi$ profile as a function of ${\lambda }_{\psi }$ for droplets of the same size normalized to 1 for $J=40$ and $a=0.6$, calculated by numerical simulations. (b) Location of the boundary between micro- and bulk phase separation calculated by numerical simulations for ${\lambda }_{\psi }=0.1$ (red solid line) and ${\lambda }_{\psi }=1$ (blue dashed line).

Figure 8

Location of the boundary between micro- and bulk phase separation for $q_0=0.1$ (lower line), $q_0=0.5$ (middle line), and $q_0=0.7$ (upper line) for ${\lambda }_{\psi }=0.1$ calculated from the numerical simulations. The line is a polynomial interpolation to serve as a guide to the eye.