Undulation instabilities in cholesteric liquid crystals induced by anchoring transitions

We use 4-cyano-4’-pentylbiphenyl (5CB, Kingston Chemicals Limited) doped with 2.8% wt (S)-4-cyano-4’-(2-methylbutyl)biphenyl (CB15, EMD Performance Materials) to obtain a CLC with a pitch of $\sim$ 5 $\mu$m. The pitch is determined using a Grandjean-Cano wedge cell Grandjean (1921); Cano (1968). Briefly, two glass slides are coated with a unidirectionally rubbed polyvinyl alcohol (PVA, Sigma-Aldrich) film to give planar anchoring. The slides are then clamped and glued with rubbing directions parallel to one another and a Mylar spacer on one side of the cell to create the wedge angle. The distance between disclination lines in the cell is measured and used, along with the angle, to determine the pitch.

CLC shells are produced using a glass capillary microfluidic device to generate double emulsions, similar to methods described in past works Shah et al. (2008); Tran et al. (2017); Lopez-Leon et al. (2012); Urbanski et al. (2017). Briefly, three different fluid phases are used to generate shells of CLC: an inner aqueous phase, the middle CLC phase, and an outer aqueous phase. The tapered circular capillary of the inner water phase is fitted into a tapered square capillary for the middle CLC phase, which is subsequently fitted into a larger circular capillary for injection of the outer aqueous phase. The middle, square capillary is treated with octadecyltrimethoxysilane (OTS, Sigma-Aldrich) to improve CLC wetting of the capillary. Three syringe pumps (Harvard Apparatus) are used to control the flow rates of each phases, with 40 mL/hr, 500 $\mu$L/hr, and 400 $\mu$L/hr as the flow rates corresponding to the outer, middle and inner phases. Both aqueous phases have 1% wt polyvinyl alcohol (PVA, 87-89% hydrolyzed, average $M_{w}=13,000-23,000$) to stabilize the emulsions. After the double emulsions are collected from the microfluidic device, the shells are left to settle in the vial and equilibrate to the planar anchoring conditions induced by PVA and water.

We use sodium dodecyl sulfate as the surfactant (SDS, Sigma-Aldrich) in varying concentrations to induce homeotropic anchoring Tran et al. (2017); Lopez-Leon et al. (2012); Noh et al. (2015). For homeotropic anchoring of the inner shell surface, SDS is included in the inner aqueous phase during the microfluidic production of CLC double emulsions. To induce homeotropic anchoring on the outer surface of the CLC shell, the double emulsions are pipetted into vials containing aqueous solutions of 1% wt PVA, 0.1 M NaCl, and SDS with concentrations ranging from 7-10 mM Tran et al. (2017). The shells are either viewed immediately to observe the planar to homeotropic anchoring transition or can be left overnight to obtain relaxed shells with homeotropic anchoring as an initial condition. To induce the homeotropic to planar anchoring transition, these homeotropic shells are pipetted into another solution with only 1% wt PVA and 0.1 M NaCl, without SDS. After introducing shells to the appropriate outer aqueous phase for the desired anchoring transition, the sample vial is gently mixed before pipetting into a viewing chamber (Grace Bio-Labs). An upright microscope in transmission mode fitted with crossed polarizers (Zeiss AxioImager M1m) and a high-resolution color camera (Zeiss AxioCam HRc) is used to capture polarized micrographs of shells.

Sodium chloride (NaCl, Fisher Scientific) is added to aqueous phases, ranging in concentration from 0.1 to 1 M, in order to increase the SDS interfacial density Brake and Abbott (2002). In some samples, a higher concentration of NaCl in the inner aqueous phase compared to the outer phase can trigger osmotic swelling, in which water permeates through the CLC shell towards the inner aqueous droplet, increasing the inner droplet volume and thinning the shell over time Tu and Lee (2012); Darmon et al. (2016a); Tran et al. (2017). However, we found the osmotic swelling of CLC shells to occur over tens of hours, while stripe transformations from the introduction or removal of SDS occur over a span of tens of minutes. Therefore, we do not expect osmotic swelling to have an effect on the initial pattern formation that is the focus of this work. Indeed, the time and the manner of pattern formation for shells with and without osmotic swelling is comparable.

The numerical simulations model the CLC molecular orientation via the Landau-de Gennes free energy where a symmetric, traceless $3\times 3$ matrix $Q_{\alpha\beta}(\mathbf{x})$ represents the nematic orientation order parameter and is defined at each site $\mathbf{x}$ of a cubic lattice, with indices $\alpha,\beta=x,y,z$ indicating the Cartesian directions Ravnik and Žumer (2009); Sussman and Beller (2019). The free energy in the bulk regions of the liquid crystal reads

with summation over repeated indices implied. The constants $A$, $B$, and $C$ are set by the thermodynamic properties of the liquid crystal. Note that the scalar order parameter is given in terms of these parameters as $S_{0}=(-B+\sqrt{B^{2}-24AC})/6C$. We also have an elastic component in the bulk which incorporates the energy penalties associated with splay, twist, and bend distortions:

where $\epsilon_{\alpha\beta\gamma}$ is the Levi-Cevita symbol and $\partial_{\alpha}\equiv\frac{\partial}{\partial x_{\alpha}}$ are the partial derivatives along the three spatial directions. When we square the two terms in the first line of Eq. (2), we are left with a sum over the remaining free indices $\alpha$ and $\beta$. More details are given in, e.g., Fukuda and Žumer (2010). We include a splay-bend coefficient $L_{24}$ that is a total derivative term, but will be important in our system since we will be interested in interfacial phenomena.

At interfaces, we also have an anchoring energy that we model using a Rapini-Papoular surface potential Rapini and Papoular (1969). For homeotropic anchoring strength $W_{0}$ and planar anchoring strength $W_{1}$, the total interface free energy reads:

where $Q^{\parallel}_{\alpha\beta}=3S_{0}(\hat{s}_{\alpha}\hat{s}_{\beta}-\delta_{\alpha\beta}/2)$ is the orientation tensor constructed from the interface’s surface normal vector $\hat{\mathbf{s}}$ and where $\bar{Q}_{\alpha\beta}=Q_{\alpha\beta}+S_{0}\delta_{\alpha\beta}/2$. The tensor $\bar{Q}^{\perp}_{\alpha\beta}$ is the projection of $\bar{Q}_{\alpha\beta}$ onto the plane of the interface, so $\bar{Q}^{\perp}_{\alpha\beta}=(\delta_{\alpha\beta}-s_{\alpha}s_{\delta})\bar{Q}_{\delta\gamma}(\delta_{\gamma\beta}-s_{\gamma}s_{\beta})$.

This free energy is then minimized over a computational domain using a conjugate gradient method in the ALGLIB package (http://www.alglib.net/). Note that the minimization here does not reflect the actual dynamics of a liquid crystalline system. In reality, the free energy would be minimized in a manner consistent with liquid crystal hydrodynamics. Nevertheless, we can obtain a qualitative idea about the evolution of various patterns by monitoring the states of the system during the minimization procedure from some specified initial condition while tracking any long-lived transient states. Often, the minimization procedure will end within local free energy minima (metastable states) which are also of interest.

In the experiments presented here, we consider 5CB mixed with a chiral dopant as our cholesteric liquid crystal. Therefore, we make use of some standard values for this compound in our simulations Ravnik and Žumer (2009). We set $A=-1$, as this can be set simply by choice of free energy units. We then utilized a one-constant approximation with $B=-12.33$, $C=10.06$, and $L_{1}=L_{2}=2.32$ (Fig. 6), consistent with the parameters of 5CB and a lattice spacing $\Delta x\approx 4.5~{}\mathrm{nm}$ close to the nematic correlation length (size of nematic defect core). These particular choices were not important for determining the qualitative features of the simulation, including the formation of the undulation instability that is the focus of this work. For the other shell simulations in Figs. 3 and 5, we used the two-constant approximation with $B=-1.091$, $C=0.6016$, $L_{1}=0.003805$, and $L_{2}=2L_{24}=0.01141$. These latter two constants were chosen to correspond to $K_{1}=K_{3}\approx 10~{}\mathrm{pN}$ and $K_{2}=5~{}\mathrm{pN}$ for the characteristic splay, bend, and twist elastic constants, respectively, for 5CB Karat and Madhusudana (1977). The choice of constants yields a large lattice spacing $\Delta x\approx 30~{}\mathrm{nm}$. Note that the bulk free energy constants $B$ and $C$ are not so important in this case because we simulate deep in the nematic phase (low temperature) with a large $\Delta x$ so that all the lattice sites retain the equilibrium value $S_{0}$ of the scalar order parameter. The minimization procedure essentially only minimizes the elastic component as the bulk remains at a uniaxial minimum. For the simulations with a deformable CLC shell interface in Fig. 8, we tuned the cholesteric to an isotropic-nematic phase transition by setting $A=5.284$, $B=-11.2612$, and $C=0.88889$, with the elastic constants $L_{1}=L_{2}=2L_{24}=0.928$. The various corresponding dimensions of the shell are given in the figure captions in SI units.

We begin first with the planar to homeotropic anchoring transition, featured in Fig. 2. Homeotropic anchoring is induced on the outer shell surface experimentally through the addition of a surfactant (SDS, see Sec. II) to the outer aqueous phase. The onset of the stripe instability is shown in Fig. 2. Note that the undulated stripes do not conform to a discernible pattern, but run in different directions along the shell surface. The nematic defects in the planar to homeotropic transition do not appear to play a large role in ordering the undulating stripes, although the stripes can be observed terminating at the director defects, highlighted in Fig. 2-iv.

We turn to simulations to elucidate the bulk CLC layer arrangements that bring about the undulating stripe patterns. The stripe behavior is simulated by taking a planar CLC shell configuration as the initial condition [Fig. 3(a), left] and minimizing the free energy under homeotropic anchoring conditions. Large stripes are spawned at early stages of the transition [Fig. 3(b)], reminiscent of those seen in experiments. Cross sections of the shell reveal the source of the large stripes to be undulation of the concentric CLC layers, evident from comparing the cross sections in Fig. 3(a) and (b). During the minimization, homeotropic anchoring increases most significantly at the director defect, the area of highest distortion [red region in Fig. 3(c)]. Interestingly, far from the director defect, the CLC layers are not as undulated as they are near the defect, evident in the cross section of Fig. 3(c). That undulations are most pronounced near the director defect suggests that pitch axis reorientation occurs first at these local regions of disorder before propagating to the rest of the system. Eventually, the typical stripe pattern of homeotropic CLC shells develops on the surface, forming focal conic domains and indicating that the pitch axis completes its reorientation and becomes parallel to the shell surfaces. The simulation results further show that the distortions due to the changing anchoring are confined to regions closest to the shell interfaces, with the most bent and distorted cholesteric layers near the surface [Fig. 3(d)].

A similar stripe instability also occurs for a homeotropic to planar anchoring transition. However, in contrast to the reverse transition, the onset of the stripe instability induced by imposing planar anchoring appears to be sensitive to the initial configuration of the system. We examine first the transition for an equilibrated, homeotropic CLC shell before turning to the transition for metastable, homeotropic states with the characteristic focal-conic-like domains decorating the CLC shell surface.

A CLC shell left to equilibrate in a solution with 7 mM SDS, 1% wt PVA, and 0.1 M NaCl for a month is shown in Fig. 4(i), first presented in Tran et al. (2017). It has two focal conic domains at it poles, with the pole in the thinnest region of the shell highlighted in Fig. 4(i). Transferring the shell to an aqueous solution without SDS, results in slow removal of surfactant from the interface and a reduction in the homeotropic anchoring strength. The double spirals of the polar focal conic domains unwind first, resulting in stripes of a larger periodicity at the poles, seen in Fig. 4(ii). As the stripes unwind, they also widen to accommodate greater regions of planar anchoring at the interface. At some point during the anchoring transition, stripes near the thinnest region of the shell begin to undulate with a periodicity of twice the pitch, shown in Fig. 4(iii) & (iv). The undulations extend along the stripes as the transition continues. Near the end of the transition to planar anchoring, most of the stripes have been removed from the system, with a few undulations remaining. One of the excess defects near the focal conic domain in Fig. 4(i) becomes the topologically required director defect in Fig. 4(v).

The absence of undulating stripes during the majority of the anchoring transition is captured through simulations, plotted in Fig. 5, where an equilibrated, homeotropically-anchored cholesteric shell relaxes to a planar-anchored state. The starting condition, shown in Fig. 5(a), has two characteristic double-spirals at both poles that correspond to the topologically-required, focal-conic-like domains where the pitch axis has a defect. During relaxation, these double spirals unwind as the pitch axis tilts toward the shell interior. We see a top view of the unwinding in Fig. 5(b). Meanwhile, the layers on the interior of the shell begin to undulate, shown on the left of Fig. 5(c). The layers near the equator undulate in opposite directions and pinch off, creating another pitch defect that eventually becomes the topologically-required nematic defect in the planar-anchored shell. Such a mechanism was conjectured in Tran et al. (2017), but is observed here explicitly in the simulation. Eventually, the pitch axis points radially, and we obtain a state close to the equilibrated planar configuration, seen on the right of Fig. 5(c). We can discern from the insets in Fig. 5(c) that one of the nematic defects (red spot on the outer surface) sits near the shell equator. Faint stripes are still observable near the director defect, shown through the inset of Fig. 5(c), right. Generally, a secondary stripe instability does not materialize, reminiscent of the experiment featured in Fig. 4, where stripes are capable of becoming wider and more planar without undulating in the majority of the system.

However, it is possible to observe the undulated stripe instability in simulations by using a thinner, smaller shell and slightly altering the initial equilibrium, homeotropic configuration, as shown in Fig. 6. Using a uniform CLC ground state with a vertical pitch axis as the initial shell configuration creates distortion of the polar focal conic domains, shown on the left of Fig. 6(a). The distorted focal conic domains are regions where deformation of the CLC is easiest, observed also in our prior work Tran et al. (2017). With tilting of the pitch axis beginning at these regions near the poles, the centers of the stretched double spirals are also where the undulations first appear. Under the anchoring transition, the stretched double spirals unwind and widen only in the double spiral region, leaving behind an undulated, planar stripe, shown in Fig. 6(b). The simulation is then trapped in this configuration, indicating a metastable state. Cross sections on the right of Fig. 6(b) reveal that the undulated stripes are a direct result of CLC layer undulations. The left panel of Fig. 6(b) further reveals that the stripe undulation appears on both the inner and outer surfaces of the shell. These modulations of the director, perpendicular to the initial stripes, mimic what is seen at the former focal conic domain of the experiment featured in Fig. 4i-ii Tran et al. (2017). As the shell adapts to planar anchoring in Fig. 4, the double spiral region also develops undulated stripes, matching the simulated behavior of the distorted focal conic domains in Fig. 6.

Curiously, undulating stripes in the homeotropic to planar anchoring transition are more pervasive in experimental systems where the initial homeotropic state is meta-stable and have many focal conic domains, exemplified in Fig. 1-i and Fig.7-i. As the surfactant is washed off, the stripes widen to accommodate larger regions of planar anchoring. When the stripes reach a width of approximately twice the pitch $P_{0}$, the stripe interior becomes filled in with secondary, arch-like stripes of a periodicity $\sim 2P_{0}$ that run along primary stripes, covering the entire shell surface (Fig. 1-iii-iv). Primary stripes being filled in with undulating, secondary stripes also matches the simulation results of Fig. 6. However, it remains unclear why stripes in metastable configurations more readily undulate than those in equilibrium configurations. The initial, seeded planar anchoring at the surface of the simulation in Fig. 6 allows for the undulated stripes to emerge during the anchoring transition. This implies that larger, local regions of planar anchoring may be necessary in experiments for secondary, undulated stripes to appear. This could possibly occur in experiments from the segregation of surfactants into regions of favorable anchoring, reported in Tran et al. (2018). Shell thickness heterogeneity in experiments may additionally play a role, as the local undulations of Fig. 4 occur near the thinnest region of the shell. Indeed, the simulated shell of Fig. 6, exhibiting pronounced, undulated stripes, is smaller and thinner than that of Fig. 5.

Despite slight differences between homeotropic and planar anchoring transitions, the emergence of undulating stripes with changes in anchoring is a general phenomenon, demonstrated through the above experiments and simulations of CLC shells. Simulation results reveal that the stripe instability is a consequence of CLC layer undulation, resulting from the pitch axis tilting to accommodate the new anchoring conditions. In the following section, we formulate a simple model through the lens of the Helfrich-Hurault instability to describe how incompatible anchoring can trigger undulations in the cholesteric bulk.