Arrested coalescence drives helical coiling and networking of filamentous smectic condensates

Abstract

Liquid-liquid crystal phase separation (LLCPS) occurs in many industrial and biological settings. To date the states of the separated condensed liquid crystals have been found to be nematic, columnar, or smectic phases. Interestingly, when smectic phases condense out of the liquid, they can form filamentous condensates that spontaneously assemble into sparse networks with rich life-like dynamics. Here, we study the underlying process of filament linking and conformational changes that mediates formation of these unique networks. Microscopy reveals that new linkages between filaments are initiated by an adhesive interaction between straight filaments; the filaments snap into contact and then rapidly wind into helical coils, despite the absence of molecular chirality or transitions between mesophases. Using polarized optical microscopy, theoretical modeling, and simulation, we show that filament linking into ribbon structures is driven by arrested coalescence that depends on both interfacial energy minimization and the constraints of smectic order. The linked filaments spontaneously coil into double helices to reduce interfacial area and smectic distortion, thus driving compaction into networks. We propose a microstructure consistent with this interpretation, which quantitatively predicts the extent of arrested coalescence. In total, these findings suggest a generic pathway for network formation in liquid crystals that provides insight about the formation of condensate networks in other engineered or biological materials.

Figures (13)

• Figure 1: Filament coiling mediates networking in Hele-Shaw cells.A 12OCB and Squalane mixture within a 20 $\muup$m gap Hele-Shaw cell is cooled into the binodal to induce demixing as previously described morimitsu2024spontaneous. B Schematic energy landscape of metastable filament transformations to form unknown intermediate structures before forming flat drops. C Schematic of network formation, which occurs when filaments above threshold density collpase and form new linkages, which coarsen to aggregate nodes composed of flat drops. D Imaging on CMOS (ThorLabs) under crossed polarizers shows birefringence of smectic condensate early in network formation at 5$\times$ magnification before (i) and after (ii) a collapse event. E Zoomed view of collapse event (20$\times$ magnification) shows nascent linkage forming between initially disconnected filamentous masses. Lower panel shows zoomed insets in red and yellow. Free filaments meet at Y-junctions (white arrows) forming bright contact regions that zipper filaments together (orange arrows) and subsequently form an interdigitated, ribbed structure (purple arrows).
• Figure 2: Filaments link by partially coalescing and winding into helical coils.A Representative linkage process captured by high speed camera (1000 fps; Phantom VEO 1010) under 63$\times$ magnification. Phase contrast imaging shows the scattering interface of the condensate as bright. B Rendering of two initially-distinct filaments, colored green and blue to visually distinguish, zippering to form ribbon structure and winding to form coil structure. C Phase contrast imaging of coiled regions compacting into a supercoiled aggregate and coarsening into flat drops.
• Figure 3: Helical winding dynamics of isolated filaments in microdroplets.A Microdroplets are formed by Rayleigh-Plateau breakup of our binary fluid within an immiscible refractive-index-matched aqueous fluid (Methods). B Sufficiently small microdroplets exhibit the nucleation and growth of single filaments, which grow and evolve in isolation. Larger microdroplets can contain multiple condensate droplets, which sometimes assemble into "proto-networks" (Fig. \ref{['SIfig:microdroplet']}). C Helical coils form in isolation, but only when filaments self-contact. D High speed imaging shows dynamics of coil winding following formation of ribbon contact. E Schematic of ribbon zippering and coil winding process.
• Figure 4: Ribbons form by partial coalescence of filaments. A Birefringence pattern of ribbon (orange arrows) zippering from two free filaments (pink arrows). B Closeup shows two dark axial bands when viewing the major ribbon diameter and a single dark axial band when viewing the minor diameter, in agreement with the simulated birefringence for the proposed ribbon microstructure. C Rendering of free filaments and their arrested coalescence as a ribbon. Filaments colored arbitrarily. Inset shows schematic cross-section with definition of major diameter $d_\text{R1}$ (orange arrows), minor diameter $d_\text{R2}$ (green arrows), and nondimensional coalescence $\Omega_\text{R}$. D Arrested coalescence measured from ribbon $d_\text{R1}$ and free filament radii $R_\text{F}$ for ribbons whose zippering is captured on video. Error bars reflect one standard deviation over five manual measurements. Black line and grey shaded region indicate average and standard deviation $\Omega_\text{R}=0.37\pm0.07$. Lines show theoretical prediction for $\Omega_\text{R}$ for different values of $\gamma$, which converge to be indistinguishable from the black line for $\gamma\gg K/R_\text{F}$. E Proposed smectic microstructure within ribbon. Constant layer spacing requires two parabolic domain walls (purple) separating domains of radially-bent (colored) and straight (uncolored) smectic layers. Cross sectional contour $s$ shown in pink. F Modeled free energy density during coalescence for $K=20~\text{pN}$, $\gamma=0.3~\text{mN/m}$, and $R_\text{F}=1~\muup\text{m}$. Helfrich energy decreases monotonically and cannot arrest coalescence. Interfacial energy exhibits geometrically-determined minimum, which dominates arrested coalescence. Minima of this model plotted for several values of $\gamma$ in panel $\textbf{D}$ (colored lines). Both energies taken in reference to $\Omega_\text{R}=0$, which is that of the uncoalesced straight filaments, $\bar{F}_\text{I,F}=2\gamma / R_\text{F}$ and $\bar{F}_\text{H,F}= \left(K/\left(2R_\text{F}^2\right)\right)\text{ln}\left(R_\text{F}/R_\text{C}\right)$, where the defect core size $R_\text{C}\approx 0.1~\text{nm}$ is estimated from the molecular layer spacing morimitsu2024spontaneous. G Geometric-minimum $\Omega_\text{R}^\text{Theory}\approx0.374$ (red dot) set by competition between increasing $l$ and decreasing $s$ during coalescence. Minimum is independent of $R_\text{F}$.
• Figure 5: Coils increase contact with consistent extent of partial coalescence. A Rendering of a helical coil with definitions of pitch $P$, pitch angle $\theta_p$, and helix radius $R_\text{H}$. Filaments colored green and blue arbitrarily; blue filament is displayed translucent to make helical backbone visible (black line). B Helical radius. Black line indicates the condition for maximally-tight coils $R_\text{H}=R_\text{F}$. C Helix pitch $P$ increases linearly with $R_\text{F}$. D Extent of partial coalescence measured from $R_\text{F}$, $R_\text{H}$, and $P$ gives $\Omega_\text{C}=0.17\pm0.06$E Schematic of proposed internal mesophase structure. Orange line shows contour running between filament centers; our proposed structure treats this contour as a straight chord for simplicity. F Double-helical coil bent by background active flows displaying birefringence at different angles with respect to crossed-polarizers ($0\degree$--$90\degree$). Zoom inset shows segments with helix axis at roughly $45\degree$ and $0\degree$ offset from polarizer for comparison to simulated birefringence for our proposed internal structure. G Free energy density computed by equation \ref{['eq:energy']} shows that coils at $\Omega_\text{C}=0.17$ are favorable to ribbons at $\Omega_\text{R}=0.37$ for the range of observed $R_\text{F}$ (SI §\ref{['sec:ribbonEnergetics']} and §\ref{['sec:coilEnergetics']}).