Abstract
Facile geometricstructural response of liquid crystalline colloids to external fields enables many technological advances. However, the relaxation mechanisms for liquid crystalline colloids under mobile boundaries remain still unexplored. Here, by combining experiments, numerical simulations and theory, we describe the shape and structural relaxation of colloidal liquid crystalline microdroplets, called tactoids, where amyloid fibrils and cellulose nanocrystals are used as model systems. We show that tactoids shape relaxation bears a universal single exponential decay signature and derive an analytic expression to predict this out of equilibrium process, which is governed by liquid crystalline anisotropic and isotropic contributions. The tactoids structural relaxation shows fundamentally different paths, with first and secondorder exponential decays, depending on the existence of splay/bend/twist orientation structures in the ground state. Our findings offer a comprehensive understanding on dynamic confinement effects in liquid crystalline colloidal systems and may set unexplored directions in the development of novel responsive materials.
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Introduction
Colloidal liquid crystals are a class of soft matter formed when shapeanisotropic nanoparticles are dispersed in a an isotropic fluid^{1}. When confined to a finite volume, rodlike colloidal particles selforganize into various structures that are set by a delicate balance between anisotropic viscoelastic and surface properties^{2,3,4}. The subtle balance between these contributions results in facile response to external fields^{1,5} such as mechanical, flow, electric, and magnetic, giving rise to many opportunities and technological applications. Examples include displays, spatial light modulator and tunable filters in medical devices and optics, liquid crystal biosensors for rapid diagnostics, and new functional material such as artificial muscles exploiting liquid crystalline anisotropic physical properties^{6,7,8}. However, facile responsiveness to external fields (and disturbance) makes the colloidal liquid crystals very fragile to be studied experimentally under dynamical conditions^{5,9}. In particular, relaxation of liquid crystalline droplets under mobile confinement is still poorly understood despite its central importance in a variety of phenomena in condensed matter physics. This includes particle packing^{10}, selfassembly^{11}, and relaxation of colloidal liquids^{12} with implications in the field of active nematic, e.g., living liquid crystals^{13}, where the understanding of the hydrodynamics of the liquid crystals is critical^{14}.
Liquid crystalline droplets, known as tactoids, are a particularly significant example of colloidal liquid crystals, since they consist of microconfined liquid crystalline colloids with a selfselected shape/structure stemming out from the thermodynamicallydriven phenomena from which they emerge, i.e., spontaneous liquid–liquid crystalline phase separation^{15,16,17,18,19,20,21,22,23}. In stark contrast with spherical liquid crystalline emulsions, achieved commonly by emulsifying liquid crystals in another immiscible liquid (like water in oil)^{24,25}, tactoids hold spindlelike, prolate, or oblate shapes with different nematiccholesteric internal structures^{2,3,4,15,16,17,18,19,20,21,22,26}, as a consequence of the coupling between the vanishingly small interfacial tension, the surface anchoring at the interface, the chirality of colloids and the anisotropic elastic properties^{3,4}. These features make tactoids a very unique system with peculiar viscoelastic^{27,28,29} and boundary^{3,4} properties, thus adding theoretical challenges to the experimental ones when describing these complex colloidal systems under dynamical conditions. For instance, recent experiments suggest that the boundary has a significant impact on the local structure of colloids^{30,31,32,33} and on the equilibration pathways of structural relaxation of colloidal systems^{34}; yet such understanding mainly comes from the examination of colloidal systems with static boundary conditions^{30,31,32,33,34,35}. Moreover, one of the challenges of the current study is to disentangle the rate of selfassembly of the liquid crystalline tactoids from shape relaxation, providing insights on the kinetics of selfassembled complex colloidal systems.
Here we report the shape and structural relaxation dynamics of colloidal liquid crystalline tactoids. We use βlactoglobulin amyloid fibrils and cellulose nanocrystals as model rodlike colloidal liquid crystalline systems. We disentangle shape and structural relaxation and show—by integrated experimental and numerical measurements— that the shape relaxation of the tactoids follows a single exponential decay that depends on the material isotropic and anisotropic properties and the size of the droplets. We develop a theoretical model to predict the shape of the tactoids out of the equilibrium state, by considering the Hamiltonian of the tactoids in presence of an external flow field. We also show that the structural relaxation of the tactoids follows different paths depending on the colloidal mesogens configurations at the ground state; homogenous and bipolar tactoids relax through a firstorder exponential decay whereas cholesteric ones follow a secondorder exponential decay. We use direct experimental measurements of the order parameter, supported by direct numerical simulation (DNS) of a complete structurecomposition multiscale model, to discuss the nature of the structural relaxation of the liquid crystalline droplets and how it relates to the splay, bend and twist structures of tactoids at relaxed state. Our results offer original insights on the structural organization of colloidal suspensions out of equilibrium, under dynamic confined boundaries and evolving shapes.
Results
Relaxation of different classes of liquid crystalline droplets
The classical approach to study relaxation of droplets involves using the fourroll mill geometry developed originally by Taylor^{36}. This approach, however, is not applicable to tactoids due to their labile nature, making prohibitive isolating a single tactoid in such a geometry. Thus, in our experiments we take advantage of a microfluidic system with contractionabrupt expansions design^{37,38}, allowing to elongate tactoids with different volumes and then let the elongated tactoids relax to the equilibrium state in the abrupt expansion zone (the details on the microfluidic channel are provided in Methods and Supplementary Note 1). To be able to form the tactoids in a microfluidic chip, we prepared the suspension of the liquid crystalline with a concentration that is set within the isotropic–nematic coexistence region. After injection of the suspension into the microfluidic system, tactoids with various volumes are formed inside the channel following nucleation and growth path. Series of experiments with βlactoglobulin amyloid fibril and cellulose nanocrystals liquid crystalline droplets were performed and analyzed under crossed polarizers and LC (liquid crystal)PolScope device, allowing capturing not only the shape of the tactoids but also their internal structure (Fig. 1), see Supplementary Movies 1–3. Regardless of the tactoids volume, all the tactoids at the initial state are extended and the director field inside the tactoids is aligned parallel to the long axis of the tactoid, known as homogenous configuration^{2,4}. During the relaxation, an initially extended droplet with volume V (≈r^{2}R with R the major and r the minor axes of tactoids) ~10^{2} µm^{3} undergoes the shape relaxation while its structure remains unchanged as homogenous configuration (Fig. 1a). As explained in detail later, while its configuration remains unchanged as homogenous, the tactoid still undergoes structural relaxation. A tactoid with a larger volume, ~10^{3} µm^{3}, shows relaxation on both its shape and structure where the director field changes from homogenous structure to bipolar with a director field that smoothly follows the tactoid interface (Fig. 1b). For a tactoid with volume ~10^{4} µm^{3} as shown in Fig. 1c, while its shape relaxes to a nearly—yet not perfectly—spherical shape, the structural relaxation takes place while changing the director field from homogenous to cholesteric configuration, that is easily distinguishable from its characteristic striped texture.
We additionally captured both the shape and structural relaxation of the tactoids using DNS (see Supplementary Movies 4–6), and found good agreement with our experimental results as shown in Fig. 1. Details of the DNS are given in the Methods section.
Shape relaxation of liquid crystalline droplets
To characterize the underpinning physics of the relaxation of the liquid crystalline droplets, we first analyze the shape of the tactoids during the relaxation. We measured the long axis of tactoids at different time t and quantified the relaxation behavior, as suggested previously for homogenous droplets^{39}, by \({{{{{\mathcal{R}}}}}}=\tfrac{R\left(t\right)\,\,{R}_{{{{{{\rm{equil}}}}}}.}}{{R}_{{{{{{\rm{init}}}}}}.}\,\,{R}_{{{{{{\rm{equil}}}}}}.}}\), where R_{equil.}, R_{init.}, and R(t) are the halflength of the long axis of tactoid at equilibrium, at the initial time, and at a given time, respectively (Fig. 2a–c). The value of \({{{{{\mathcal{R}}}}}}\) is one at time zero and zero when the tactoid reaches its equilibrium shape. We observed that \({{{{{\mathcal{R}}}}}}\) for all classes of tactoids (homogenous, bipolar, and cholesteric configurations) follows essentially a single exponential decay (Fig. 2a–c). Thus, to capture the shape relaxation behavior of the tactoids, we use \({{{{{\mathcal{R}}}}}}\,=\,{{{{{\rm{exp}}}}}}(t/{\tau }_{{{{{{\rm{s}}}}}}})\), where the τ_{s} is the characteristic shape relaxation time of the tactoids. A good agreement between the fitting, using exponential decay \({{{{{\mathcal{R}}}}}}\,=\,{{{{{\rm{exp }}}}}}(t/{\tau }_{{{{{{\rm{s}}}}}}})\), and the experimental data of the shape relaxation of the tactoids with different relaxed configurations, allows us to obtain the τ_{s} from the shape decay curve (Fig. 2a–c). Hence, a good collapse of the shape relaxation data of the tactoids with different volumes and relaxed configurations onto a single master curve is observed when time is rescaled with τ_{s} (Fig. 2d). We additionally performed experiments on the shape relaxation of the tactoids resulting from breakup events of initially extended tactoids in homogeneous configuration, and observed a single exponential decay trend similar to the initially extended tactoids relaxation behavior, see Supplementary Fig. 2.
To calculate τ_{s}, we combine theories on simple droplet relaxation, dimensional analysis, and DNS that shows an excellent agreement with our experiments, see Figs. 1, 2a–c. As elaborated in Supplementary Note 2, we find:
where τ_{s} is expressed as the sum of two contributions: the first term is the liquid crystalline anisotropic contribution (τ_{a}) induced by the presence of colloidal mesogens accounting for orientational order, gradient elasticity, anisotropic viscoelasticity, rotational dissipation, and concentration gradients while the second term is the characteristic shape relaxation time of elongated isotropic tactoids (τ_{i}). We express τ_{i}, following the wellestablished relation of the characteristic shape relaxation time of elongated isotropic droplets^{37,40,41}, as \({\tau }_{{{{{{\rm{i}}}}}}}\,=\,\frac{\beta {\mu }_{{{{{{\rm{I}}}}}}}{R}_{{{{{{\rm{equiv}}}}}}{{{{{\boldsymbol{.}}}}}}}}{\gamma }\), where γ is the interfacial tension, R_{equiv.} = ((r^{2}R)^{1/3}) is the equivalent radius of tactoids, μ_{I} the viscosity of the medium phase taken to be equal to viscosity of the isotropic phase and \(\beta \,=\,\frac{\left(2\hat{\eta }\,+\,3\right)\left(19\hat{\eta }\,+\,16\right)}{40\left(\hat{\eta }\,+\,1\right)}\), where \(\hat{\eta }\,=\,\frac{{\mu }_{{{{{{\rm{N}}}}}}}}{{\mu }_{{{{{{\rm{I}}}}}}}}\) is the ratio of viscosities of the nematic phase μ_{N} and that of the isotropic medium, μ_{I}. We obtain τ_{a}, as noted in Supplementary Note 2, based on dimensional analysis and parametric studies through our validated DNS on material properties involved in the liquid crystalline selfassembly and tactoid size. In Eq. 1, ω is the anchoring strength, which takes into account the coefficient of the concentrationorientation gradient and its coupling in the numerical simulations^{42} (Supplementary Note 2), the term ck_{B}T is thermal energy per unit volume of dispersion with c, k_{B}, and T, the number density, Boltzman constant, and temperature, respectively. The term K is the Frank elastic constant for splay and bending (assumed to be equal) and K_{2} is the Frank twist elastic constant. It should be noted that, from both theory^{43,44} and experimental measurements^{45,46,47} on different systems of rigid rodshaped liquid crystals, including filamentous colloids^{48} analogous to those studies here, the ratio of \(\frac{K}{{K}_{2}}\) in Eq. 1 is always greater than ½, and thus Eq. 1 only contains real arguments. The term ξ represents the coherence length that is an indicator of length over which longrange ordering takes place. The term \({M}_{{{\varnothing }}}\,\propto\, \frac{{{{{{\rm{ln}}}}}}\left(L/D\right)}{{{{{{{\rm{c}}}}}}\mu }_{{{{{{\rm{N}}}}}}}L}\) is the mass mobility and \({M}_{{{{{{\rm{Q}}}}}}}\,\propto\, \frac{2{{{{{\rm{ln}}}}}}\left(2L/D\right)\,\,1}{{{{{{{\rm{c}}}}}}\mu }_{{{{{{\rm{N}}}}}}}{L}^{3}}\) is the rotational mobility^{49} (∝ stands for proportionality); here L and D, respectively, the length and diameter of the rodlike mesogen, assumed to be equal to the weighted mean length of the fibrils L_{f,w} and effective diameter D_{eff.} proposed by Onsager^{50}, respectively. We list all properties of the liquid crystals used in this study in Table 1 along with the details on calculations and measurements in Supplementary Notes 3, 4, supported by refs. ^{51,52,53,54,55,56,57,58,59}. The relation for τ_{a} is formulated through a hybrid approach based on DNS results and dimensional analysis and can be turned to an equation by use of a single constant prefactor (b = 54.0). The term b is thus an aggregated value reflecting prefactors present in the proportionality terms such as mobilities. We compare the experimental results for τ_{s} with our prediction with a single fitting parameter b and find an excellent agreement (see Fig. 2e).
To test the generality of the present approach, in addition to BLG I in Table 1, we also perform experiments with one other system of amyloid fibrils (BLG II) having different length distribution compared to BLG I as a result different material properties^{60} and with sulfated cellulose nanocrystals (SCNC); see Table 1. The tactoids of both systems also illustrate single exponential decay during the shape relaxation like BLG I in Fig. 2a–c. The results for characteristic shape relaxation time of BLG II and SCNC show very good agreement with Eq. 1 prediction (Fig. 2e), suggesting that Eq. 1 is general enough to describe the relaxation behavior of most biocolloidal liquid crystalline tactoids.
Deformation of tactoids under external stresses
Having established a general picture on the dynamic shape relaxation of the tactoids, we go into modeling the deformation of the tactoids under external stresses. While this has been well documented for simple fluids^{40,61}, for liquid crystalline tactoids the physics become complex due to the energy terms associated with the internal structure of the tactoids, their anisotropic viscoelasticity and the confining boundary features. We look at the deformation of the droplet under uniaxial flow field with extension rate given \(\dot{\varepsilon }_{xx}\,{{\mbox{=}}}\,\frac{\partial {u}_{x}}{\partial x}\), where the u_{x} is the flow speed and x is the direction of the motion of the flow, although the approach is general and can be used to model tactoids deformation under any external stresses. We consider the energy gained by the tactoids under the external stresses imposed by the extensional flow field and incorporate it to freeenergy landscape of the tactoids that is well described by a scaling form of Frank–Oseen elasticity theory^{2,4}. In particular, the rate of energy \(\frac{{dE}}{{dt}}\) gained by the tactoids under any external normal stresses σ can be expressed as \(\frac{{dE}}{{dt}}\,=\,\int {{{{{\boldsymbol{\sigma }}}}}}.{{{{{{\bf{u}}}}}}}_{{{{{{\rm{i}}}}}}}\,{dS}\), where u_{i} is the displacement of the interface of the tactoids. We consider the case where the tactoid is stretched due to the stresses applied by uniaxial extensional flow and in Supplementary Note 5 we calculate \(\frac{{dE}}{{dt}}\) to be
The freeenergy landscape of the tactoids includes two energetic terms, the bulk elastic and surface free energies, associated with the tactoid at equilibrium. The total free energy of the tactoid F_{E} is described in scaling form as^{2,4}
where the first term accounts for the surface free energy of the tactoids that is due to the interfacial tension and the anchoring strength. The last two terms are the bulk elastic free energy of the tactoids where the second term accounts for splay and bending free energy and the third term is the twist elastic free energy. The term θ (=n ∙ ∇ × n with n the nematic director) is the twist term in the Frank–Oseen elasticity theory and the term q_{∞} (=2π/P_{∞} with P_{∞} the natural pitch of the system) is the chiral wave number. We measured and summarized the properties of the suspensions used in this study in Table 1 (see also Supplementary Note 4).
By energy conservation, the rate of the energy gained by the tactoids due to the normal stresses from the flow field must be equal to the rate of the energy changes in the free energy of the tactoids associated with their elastic/interfacial energy. Note that setting the two energies equal stands valid here since the process happens significantly faster than the rate at which the heat can flow out, thus separating the time scale for the transfer of the energy associated with structural changes from the time scale for heat dissipation. Additionally, as all three classes of the homogenous, bipolar, and cholesteric tactoids hold a homogenous configuration under extreme deformation, as can be seen in Fig. 1 and our recent study^{9}, we ignore the second term in Eq. 3, implying, as usual for homogeneous tactoids, that the bulk elastic energy due to the splay and bending is zero. Additionally, the third term is eliminated as, in the homogenous configuration and at constant tactoid volume, it does not change under deformation, so the rate of the energy gained by this term becomes zero. All in all, by setting Eq. 2 equal to the time derivative of Eq. 3 yields:
giving the steadystate elongated shape of the tactoids under a given extensional flow field in terms of r as a function of \(\dot{\varepsilon }\) and V. There is no analytical solution for Eq. 4, but the numerical solution along with our experimental data are presented in Fig. 3. Here, to best match experimental observations, the second term is rescaled by a prefactor of 0.14, which is fully justified by the use of a scaling form of the Frank–Oseen energy landscape. Our results suggest that the short axis of the tactoids r decreases as the extension rate increases, where r lines, corresponding to tactoids with different volumes, converge to a single universal curve at large values of extension rate (Fig. 3a). The most remarkable consequence of our analysis is that for high extension rates, r becomes independent of the volume, that is, the crosssection of the tactoid is simply ruled by extension rate, and that at identical extension rates, tactoids of different volumes V only differ by their long radius R, which is directly proportional to V. In the regime of low extension rate, the short axis of the tactoids becomes volumedependent and increases logarithmically with the increase in the volume of the tactoids (Fig. 3b), which is nicely supported by the collected experimental dataset. Equation 4 can be used to predict the R_{init.}(=V/r^{2}) as the maximum tactoids deformation that can be reached under a given extensional flow rate. Note that in Fig. 3a, b the experiments of the large volume of the tactoids (e.g., V = 30,000 μm^{3}) at the high extension rate is limited by the experimental setup. This is so as the length of the tactoids in the stretched forms becomes extremely high compared to the size of taken images, preventing to fully capture tactoids with large volume at high extension rate.
Structural relaxation of nematiccholesteric tactoids
The above treatment describes comprehensively the evolution of the confining boundaries of the tactoids, but provides no information on the evolution of their internal structure. We thus turn our attention to the study of the orientational order parameter of the director field during relaxation of the tactoids. In our experiments, we captured the relaxation of the tactoids by LCPolScope allowing us to access the retardance images of the tactoids during relaxation and analyze the order parameter S, where S = \({{{{{\mathcalligra{r}}}}}}\)/d∆n_{0} with \({{{{{\mathcalligra{r}}}}}}\) the optical retardance value, d the thickness of the sample and ∆n_{0} the birefringence corresponding to a perfectly aligned nematic phase, i.e., when the order parameter is 1^{62,63}. We measured the retardance value \({{{{{\mathcalligra{r}}}}}}\) of every pixel within the tactoids, and accordingly, the d is calculated for every pixel assuming a spindle shape for the tactoids (see Supplementary Note 6). The exact value of the ∆n_{0} is unique for a given liquid crystalline system and is often challenging to obtain experimentally, thus here we present our calculation independent form ∆n_{0}. We define \({{{{{\mathcal{S}}}}}}=\frac{S\left(t\right)\,\,{S}_{{{{{{\rm{equil}}}}}}.}}{{S}_{{{{{{\rm{init}}}}}}.}\,\,{S}_{{{{{{\rm{equil}}}}}}.}}\) to capture the structural relaxation of the tactoids, similar to the one used for the shape relaxation of the tactoid, and most importantly, fully independently of ∆n_{0}. The experimental results of \({{{{{\mathcal{S}}}}}}\) obtained for tactoids with different relaxed configurations against scaled time are shown in Fig. 4a–c. The structural relaxation of the tactoids with small volumes that hold homogenous and bipolar at equilibrium follows a firstorder exponential decay. However, for the larger extended droplets that relax to cholesteric structure, \({{{{{\mathcal{S}}}}}}\) shows nonmonotonic behavior where initially \({{{{{\mathcal{S}}}}}}\) decreases until its minimum, then it starts to increase before reaching its equilibrium, indicating a secondorder exponential decay (Fig. 4c). To illustrate such a undershoot behavior more clearly, while maintaining \({{{{{\mathcal{S}}}}}}\) as a positivelydefined object, we use in this case \({{{{{\mathcal{S}}}}}}=\frac{S\left(t\right)\,\,{S}_{{{{{{\rm{minimum}}}}}}}}{{S}_{{{{{{\rm{init}}}}}}.}\,\,{S}_{{{{{{\rm{minimum}}}}}}}}\) and show the rate of the changes in \({{{{{\mathcal{S}}}}}}\) versus the scaled time in Fig. 4d–f (note that the two definitions of \({{{{{\mathcal{S}}}}}}=\frac{S\left(t\right)\,\,{S}_{{{{{{\rm{minimum}}}}}}}}{{S}_{{{{{{\rm{init}}}}}}.}\,\,{S}_{{{{{{\rm{minimum}}}}}}}}\) and \({{{{{\mathcal{S}}}}}}=\frac{S\left(t\right)\,\,{S}_{{{{{{\rm{equil}}}}}}.}}{{S}_{{{{{{\rm{init}}}}}}.}\,\,{S}_{{{{{{\rm{equil}}}}}}.}}\) are equivalent for homogeneous and bipolar tactoids following a monotonic decay). It is clear from Fig. 4 that numerical simulations capture the firstorder exponential decay in the case of homogenous and bipolar tactoids. In the case of cholesteric tactoids, our numerical simulation results bear qualitatively similar behavior for the structural relaxation of the tactoids although simulations underestimate the equilibrium order parameter obtained experimentally for cholesteric tactoids structural relaxation (Fig. 4c). To illustrate this further, we show the rate of the changes in \({{{{{\mathcal{S}}}}}}\) versus the scaled time in Fig. 4d–f. Quantitatively, the tactoids with the homogenous and bipolar configuration at relaxed state follow \({{{{{\mathcal{S}}}}}}={{{{{\rm{exp }}}}}}(t/{\tau }_{{{{{{\rm{c}}}}}}})\), where we experimentally find the characteristic structural or configurational relaxation time τ_{c} to be 24.4 s and 31.9 s for homogenous (V = 644 µm^{3}) and bipolar (V = 2751 µm^{3}) tactoids, respectively. In contrast, the tactoid with cholesteric configuration at equilibrium (V = 16,414 µm^{3}), shows two characteristic configurational relaxation times captured by \({{{{{\mathcal{S}}}}}}={{{{{{\rm{c}}}}}}}_{1}{{{{{\rm{exp }}}}}}\left(t/{\tau }_{{{{{{\rm{c}}}}}},1}\right)\,+\,(1\,\,{{{{{\rm{c}}}}}}_{1}){{{{{\rm{exp }}}}}}(t/{\tau }_{{{{{{\rm{c}}}}}},2})\), with c_{1} a constant equal to −1.6, and τ_{c,1} and τ_{c,2} found to be 105.9 s and 333.7 s, respectively.
To interpret the physics behind these structural relaxations, we inspect the time scale related to bending, splay and twist terms in tactoids. Compared to the relaxation to homogeneous/bipolar tactoids, which involves nematic ordering with at best splay/bend relaxation, when the configuration relaxes to cholesteric tactoids, an additional twist relaxation takes place (see numerical simulation results in Supplementary Movies 4–6). This suggests that the second exponential decay associated with the cholesteric tactoids originates from the twist term. We suggest that in \({{{{{\mathcal{S}}}}}}={{{{{{\rm{c}}}}}}}_{1}{{{{{\rm{exp}}}}}}\left(t/{\tau }_{{{{{{\rm{c}}}}}},1}\right)\,+\,(1\,\,{{{{{\rm{c}}}}}}_{1}){{{{{\rm{exp }}}}}}(t/{\tau }_{{{{{{\rm{c}}}}}},2})\), where τ_{c,2} is significantly longer than τ_{c,1}, the second exponential decay τ_{c,2} originates from the chiral twist term while τ_{c,1} originates from simple nematic ordering. We base this statement on two grounds. First, we compare the lengthscales of the nematic and cholesteric ordering. We consider the length scale of the cholesteric phase as the length which is required for the phase to form a single periodic twist, that is set by the inverse wave number (i.e., the pitch) which is in the order of 10^{1} µm. In contrast, to form nematic ordering (characterized by splay and bending), the length scale is defined in the range of the length of the fibrils (mesogens) which is in the order of 10^{−1} µm. Thus, we argue that the larger twist lengthscales compared to splay/bend imply longer relaxation times for the twisting deformation. Secondly, from the experiments of the relaxation of the cholesteric tactoids using a LCPolscope, allowing us to capture the changes in the director field over time and follow the twist dynamics in the director field (see Supplementary Fig. 6), we find that change/rotation in the director field (the twist) takes place until the latest stages of the relaxation process and the twist changes are significant in the latest stage, suggesting again that the second exponential decay τ_{c,2} originates from the twist rearrangement. It is also worth mentioning that the structural relaxation time in tactoids is much lower than under static fixed boundary conditions. The structural relaxation time is of the order of hours for similar BLG and SCNC systems inside a capillary tube^{35} as opposed to tens/hundreds of seconds here. It is indeed known that boundary mobility increases the mobility of the mesogens thus promoting faster kinetics in structural relaxation^{12}.
What is the configuration of the extended tactoid with a given initial volume after relaxation? We are able to predict the relaxed configuration of the tactoids using the theoretical modeling recently developed starting from a scaling form of Frank–Oseen elasticity theory^{4}. According to this theory, the tactoids at equilibrium hold homogeneous configuration when (V/α)_{Homogenous} < (K/γω)^{3}, bipolar configuration when (K/γω)^{3} < (V/α)_{Bipolar} < [1.7γ/(K_{2}q_{∞}^{2})]^{3}, and cholesteric configuration when (V/α)_{Cholesteric} > [1.7γ/(K_{2}q_{∞}^{2})]^{3}. Approximating α equal to 3 for homogenousbipolar and 1.5 for bipolarcholesteric boundaries following ref. ^{4}, we computed these threshold values for BLG I and found V_{Homogenous} ≲ 800, 800 ≲ V_{Bipolar} ≲ 11,000, and V_{Cholesteric} ≳ 11,000 µm^{3}. This confirms that the tactoids shown in Fig. 1 follow a relaxation path until equilibrium. Thus, knowing the initial volume of the tactoids, their configuration after relaxation can be predicted simply from the scaling form of Frank–Oseen elasticity theory and physical parameters of the system such as elastic constants, interfacial energy and anchoring strength^{4}. We provide the nematiccholesteric phase diagram of the tactoids collected from the samples in a cuvette at equilibrium showing tactoids configuration as a function of the volume in Supplementary Note 7, as a further demonstration that the initially stretched tactoids reach an equilibrium configuration after relaxation.
We have presented an integrated picture based on experiments, numerical simulations and theory, allowing a disentanglement and comprehensive description of the shape and structural relaxation of initially stretched colloidal liquid crystalline droplets. We have shown that these tactoids undergo relaxation on both shape and structure, when the external flow field maintaining them in a nonequilibrium state is released. Independently of the size, the shape relaxation of the tactoids is characterized by a single exponential decay, which is explained well by taking into account both isotropic and anisotropic features of the tactoids. In contrast, the structural relaxation follows different fates, with first and secondorder exponential decays, depending on the existence of splay, bend and twist contributions in the ground state, whose relative weight depend directly on the size of the tactoid. We have discussed the fundamental physical mechanisms behind the shape and structural relaxation and highlighted their interdependence. These results bring forward our understanding of dynamic processes in liquid crystalline systems based on filamentous colloids and introduce a combined experimental, theoretical and numerical formalism which can be extended to heterogeneous complex fluids, soft matter and biological colloids in general.
Methods
Amyloid fibrils liquid crystals
βlactoglobulin was purified from whey protein following ref. ^{64} and dissolved in MilliQ water at 2 wt%. The solution was cleared from aggregates by filtering using 0.45 µm Nylon syringe filter and pH of the solution was adjusted 2 by adding HCl. Later, the solution was heated for 5 hrs over a hot plate at 90 °C. Once the amyloid fibrils were prepared, we shortened the length of fibrils using the mechanical shear force method. Two sets of the solutions were prepared, see Supplementary Note 3 for details on length and height distributions. The solutions were dialyzed for 5 days using 100 kDa MWCO Spectra/Por dialysis membrane against pH 2 MilliQ. The bath was changed every 24 h. To reach isotropic–nematic coexistence region concentration for the solution, suspensions upconcentrated using 6–8 kDa MWCO Spectra/Por 1 dialysis membrane against 6 wt% polyethylene glycol solution (mol wt: M_{r} ~ 35,000, Sigma Aldrich) in pH 2 milliQ water. The solutions were kept in the fridge until phase separation happens, allowing reporting the isotropic and nematic phases concentrations in Table 1.
Cellulose nanocrystal solution
Cellulose nanocrystal suspensions were prepared by mixing freezedried cellulose nanocrystal (FPInnovations) in MilliQ water. To make sure that cellulose nanocrystal dispersed well, the solution was ultrasonicated for 120 s. This was followed by centrifugation for 20 min at 12,000 × g to remove aggregates. SCNC solution with concentration within the isotropic–nematic coexistence region was obtained by initially mixing 2.5 wt% freezedried cellulose nanocrystal in MilliQ water.
AFM measurement
To perform the AFM measurement, a droplet of diluted suspension (0.01 wt%) was deposited on freshly cleaved mica. After 2 min, the mica was rinsed with MilliQ water and dried with air stream. The images of the sample were captured at ambient conditions with MultiMode VIII scanning probe microscope (Bruker) in tapping mode. The software FiberApp^{65} was used to analyze the images and measure the length and height distributions of fibrils.
Microscopy measurement
We performed the experiments using an optical microscope Zeiss equipped with crossed polarizers and combined with LCPolScope universal compensator. Under crossed polarizers, timeseries images at the frame rate of 12 frames per minute were taken. The microfluidic channel was placed on the microscope in a way that the tactoids long axis held 45° angle with respect to one of the crossed polarizers. This allowed unambiguous measurement of the tactoids short and long axis during relaxation. To perform the measurements, MATLAB program and ImageJ software were used. Furthermore, we used optical microscopy combined with LCPolScope universal compensator at timeseries mode capturing 3 frames per minute. LCPolScope images were used to analyze the internal structure of the tactoids. Additionally, LCPolScope produces the retardance images giving the retardance value of the image, pixel by pixel, which were used to measure the order parameter.
Microfluidic
The classical soft lithography approach was employed to make microfluidic systems^{66}. We made PDMS by mixing polydimethylsiloxane (PDMS) monomer and curing agent (Dow Corning Slygard 184) with ratio 10 to 1. The plain glass slide (Corning 2947) was used as the base plate to attach the PDMS channel.
We used microfluidic system with rectangular crosssection with width of the channel at expansion zone w_{e} = 600 µm, the width of the contraction zone w_{c} = 50 µm, and the height of the channel h = 100 µm (see Supplementary Information for the schematic of the microfluidic system).
Experimental details
We performed all the experiments at room condition. The equipment used to run experiments in microfluidics consists of a Harvard Apparatus syringe pump, 250 µl Hamilton syringe, flexible tubing with inner diameter 0.8 mm, and the needle with inner diameter 0.34 mm and outer diameter 0.64 mm.
Direct numerical simulations
Past studies^{23,28,67,68,69} have shown that the timedependent Ginzburg–Landau model can capture the spatiotemporal coupled relaxation dynamics of shape and structure. Furthermore, our approach can result in high fidelity simulations capturing dynamics of spatiotemporal liquid crystalline selfassembly including breakage, coalescence, and defect evolution. However, these are out of scope of the current study. In the present study, we applied this modeling approach and we focused on the relaxation dynamics of the shape and structure in initially extended tactoids, which are isolated in isotropic phase. The implementation has been elaborated in detail in our previous works, see refs. ^{23,28,67}. At the initial time, we consider an elongated tactoid according to the experimental observations. Thereafter, we let the elongated tactoids relax. Through relaxation, the total free energy is minimized according to the timedependent Ginzburg–Landau model, by which the excess free energy such as surface tension and elasticity are relieved. The elongated tactoid selfselects the equilibrium shape and structure through a spontaneous thermodynamicdriven relaxation. Furthermore, the matrix surrounded the tactoid under relaxation is essentially kept at the isotropic concentration. Note that, in present work, we study three prime liquid crystalline configurations; homogenous nematic, bipolar nematic, uniaxial cholesteric, which are all fully rotationally symmetric^{70}. Given this fact, to reduce computational costs, we rely on rectangular twodimension simulations which provide a good 3D description since there is no need to discriminate between point, line and ring disclinations; see Supplementary Movies 4–6.
Data availability
The data that support the findings of this study are available from the corresponding author upon request
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Acknowledgements
We thank X.Cao (ETHZ) for help in microfluidic chips fabrications and Samuel Mathews for support to maintain inhouse highperformance supercomputer. We thank Prof. Andrew de Mello (ETHZ) for granting access to his laboratory and Y.Yuan for helpful discussions. This work is supported by Sinergia grant no. CRSII5_189917 from the Swiss National Science foundation (R.M.).
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H.A. and R.M. conceived and initiated the project, designed the experiments, analyzed the data, developed theoretical modeling of the tactoids deformation, contributed to the theoretical analysis of relaxation time of the tactoids, and wrote the bulk of the paper. H.A. built the experimental apparatus and performed the experiments. M.B. contributed to the experiments and carried out the AFM measurements. S.A.K. and A.D.R. designed the simulations, analyzed the data, developed the theoretical analysis of relaxation time of the tactoids, and contributed to the writing of the paper. S.A.K. performed the simulations. A.D.R. and R.M. supervised the research. All authors discussed and edited the paper.
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Almohammadi, H., Khadem, S.A., Bagnani, M. et al. Shape and structural relaxation of colloidal tactoids. Nat Commun 13, 2778 (2022). https://doi.org/10.1038/s4146702230123y
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DOI: https://doi.org/10.1038/s4146702230123y
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