Research Article

Using chiral tactoids as optical probes to study the aggregation behavior of chromonics

Karthik Nayani, Jinxin Fu, Rui Chang, Jung Ok Park, and Mohan Srinivasarao
PNAS April 11, 2017 114 (15) 3826-3831; first published March 23, 2017; https://doi.org/10.1073/pnas.1614620114

  1. Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved February 21, 2017 (received for review September 1, 2016)

Significance

Confined liquid crystals occupy a sweet spot in their continued relevance to a host of fundamental studies as well as being exploited for many technological applications. We report on interesting phenomenology in a particularly exciting class of liquid crystals called chromonics by observing the director configurations in tactoids as the phase boundary is traversed. Unique chiral structures in chromonic tactoids are rationalized by appealing to the variation of the aggregate lengths as the concentration and temperature change. We arrive at an interesting conclusion that higher concentrations have shorter aggregates at the nematic–biphasic transition temperature. Our study opens up pathways to exploit this unique class of water-soluble liquid crystals for a host of potential applications while tuning their concentration and temperature.

Abstract

Tactoids are nuclei of an orientationally ordered nematic phase that emerge upon cooling the isotropic phase. In addition to providing a natural setting for exploring chromonics under confinement, we show that tactoids can also serve as optical probes to delineate the role of temperature and concentration in the aggregation behavior of chromonics. For high concentrations, we observe the commonly reported elongated bipolar tactoids. As the concentration is lowered, breaking of achiral symmetry in the director configuration is observed with a predominance of twisted bipolar tactoids. On further reduction of concentration, a remarkable transformation of the director configuration occurs, wherein it conforms to a unique splay-minimizing configuration. Based on a simple model, we arrive at an interesting result that lower concentrations have longer aggregates at the same reduced temperature. Hence, the splay deformation that scales linearly with the aggregate length becomes prohibitive for lower concentrations and is relieved via twist and bend deformations in this unique configuration. Raman scattering measurements of the order parameters independently verify the trend in aggregate lengths and provide a physical picture of the nematic–biphasic transition.

When an isotropic phase of a lyotropic system undergoes a phase transition to an ordered nematic phase, the pathway is usually mediated through spindle-shaped droplets called tactoids. Observation and analysis of tactoids have been an integral part of the investigations on liquid crystals, including some of the earliest experiments that motivated the seminal theory of Onsager (1⇓–3). Tactoids also provide a natural setting to study nematics under confinement (4⇓–6). Hence, they are an attractive setting as a testbed for fundamental research as well as being relevant to technological applications. This has driven the experimental investigation of tactoids in a host of materials, including dispersions of viruses (1, 3), proteins (7), inorganic platelets (8), and lyotropic chromonic liquid crystals (LCLCs) (4).

The director configuration in the tactoids is dictated by the individual contribution of the elastic constants, splay (K11), twist (K22), and bend (K33) to the Frank free energy (4, 9). Historically, tactoids of lyotropic nematic liquid crystals have been found to have an elongated shape and adopt a bipolar configuration with the director following the meridional lines (1, 8, 9). Only recently has a twisted-bipolar director configuration been experimentally realized for a LCLC system when it was crowded with polyethylene glycol (PEG) (4). Twisted-bipolar tactoids of cellulose nanocrystals have also been demonstrated recently (10). The emergence of chirality in achiral liquid crystals has fueled the curiosity of scientists ever since their discovery (11, 12). There have been a host of technological applications that take advantage of chiral configurations of confined liquid crystals (13, 14). In this work we exploit the emergence of a new chiral tactoidal structure to address fundamental questions pertaining to the aggregation behavior and the physics behind the nematic–biphasic transition of LCLCs.

LCLCs are made up of plank-like molecules with a polyaromatic core and polar peripheral groups (15⇓⇓–18). The π−π interactions result in the stacking of the constituent molecules (19). The aggregation process is believed to be isodesmic; that is, addition or removal of molecules to a column is always associated with the same energy cost (20⇓⇓⇓–24). In addition to applications of technological relevance (14, 25, 26), LCLCs provide a fertile playground for fundamental studies. Of particular interest are studies dealing with confinement of these materials. The twist elastic constant for LCLCs is much lower than splay, bend, and saddle-splay elastic constants (27, 28). This results in the emergence of spontaneous twist when LCLCs are confined to different geometries. Twisted structures have been reported previously in disodium cromoglycate (DSCG) tactoids (4), in Sunset Yellow FCF (SSY) droplets (29), and for LCLCs confined to cylinders (30, 31). Another interesting aspect of LCLCs is that the length distribution of the constituent units is polydisperse and is determined both by temperature and concentration (18, 28, 32). A consequence of the polydisperse size distribution of LCLCs is a broad biphasic regime where the nematic and isotropic phases coexist (32). However, there are several fundamental questions pertaining to LCLCs including their aggregation behavior, elasticity, and anchoring that need to be rigorously addressed.

In this work, we exploit tactoids for studying the effect of confinement on LCLCs for a range of temperatures and concentrations, as well as to uncover the underlying physics pertaining to the aggregation and phase transitional behavior of LCLCs. We show that the director configurations in the confined tactoids can be exploited as an optical guide to delineate the role played by concentration and temperature in the aggregation of LCLCs. For high concentrations we observe the well-understood bipolar tactoids. However, upon lowering the concentration we observe a mirror-symmetry–breaking transition and the emergence of twisted-bipolar tactoids. We find that, for the same reduced temperature, T/TBI (TBI is the biphasic–isotropic transition temperature for the given concentration), the twist angle of the twisted-bipolar tactoids increases as the concentration is lowered. Concurrently, for lower concentrations, we observe the emergence of a unique director configuration for tactoids, wherein the director is oriented concentrically and is free of singular defects. This surprising finding is rationalized by using a simple model to calculate the average length and length distribution of the aggregates. We conclude that, at the same reduced temperature, lower concentrations on average have longer aggregates. Consequently, the splay deformation in the bipolar and twisted-bipolar tactoids becomes prohibitive with increasing length of the aggregate (33). Hence, the director avoids the energetically costly splay close to the boojums by conforming to the unique configuration.

The aforementioned observations also bring to light rich physics pertaining to the nematic–biphasic transition of LCLCs (34). We use Raman scattering measurements of the order parameter close to the nematic–biphasic transition as an independent probe to verify the trend in the variation of aggregate lengths with concentration that was surmised from the observation of LCLC tactoids. Remarkably, the measured order parameters at TNB (TNB is the nematic–biphasic transition temperature) are independent of concentration. We provide a simple model incorporating the volume fraction and average aggregate length to help understand the nematic–biphasic transition for LCLCs.

Results and Discussion

Polarized Optical Microscopy.

Historically, tactoids of lyotropic liquid crystals have always conformed to the bipolar director configuration (1, 7, 8, 35). In this configuration the director follows the meridional lines with the surface defects (or boojums) being located at the poles. Fig. 1A is an experimental image of bipolar tactoids imaged under crossed polarizers. Note that the central part of the tactoid is completely dark when its long axis is parallel to a polarizer. This is a characteristic feature of bipolar tactoids in the absence of twist. This feature can be appreciated by comparing the experimental texture of bipolar tactoids in Fig. 1A with the schematic in Fig. S1A. The sample corresponding to Fig. 1A is 1.1 M SSY observed at 59.5 °C (T/TBI= 0.99). When the concentration is lowered to 1.0 M (observation temperature 42.8 °C, T/TBI= 0.99) we see that the bipolar tactoids develop a twist. This can be inferred via the transmitted light intensity under crossed polarizers in the central region of the tactoids in Fig. 1B, even when the long axis of the tactoid is nearly parallel to the polarizer. The twisted director configuration acts as a waveguide for polarized light and results in a transmitted intensity even under crossed polarizers (4). Fig. 1B, Inset shows the extinction of transmitted light when the polarizers are uncrossed, which serves as additional confirmation of the twisted structure (4, 30). A schematic of a twisted-bipolar tactoid is provided in Fig. S1B. We also observe a few tactoids with a texture that is distinct from that of bipolar and twisted-bipolar tactoids in Fig. 1B. As the concentration is lowered to 0.9 M (observation temperature 29.6 °C, T/TBI=0.99), the number fraction of the tactoids with the unique configuration was found to increase. This can be observed clearly in Fig. 1C.

Fig. 1.
Fig. 1.

Cross-polarized microscopy images of SSY tactoids at different concentrations: (A) 1.1 M, bipolar tactoids; (B) 1.0 M, twisted-bipolar tactoids with a small fraction of tactoids with the unique director configuration (Inset shows the extinction in the center of a twisted-bipolar tactoid when the polarizers are uncrossed; magnification: 40×); (C) 0.9 M, increasing number fraction of tactoids with the unique director configuration; and (D) 1.0-M sample with 0.5 wt% PEG.

Fig. S1.
Fig. S1.

Schematics of director configurations of bipolar and twisted-bipolar tactoids. (A and B) Cross-section view of a bipolar tactoid (A) and a twisted-bipolar tactoid (B). The color indicates the angle between the local director and the cross-section plane of the tactoids.

To understand the physics behind the phenomenology pertaining to the configurational transformations in the tactoids, we first quantify the concentration dependence of the twist angle of the twisted-bipolar tactoids at the same reduced temperature, using wave-guiding experiments (29, 36). The twist angle is defined as the angle the director on the surface of the tactoid makes with the axis of symmetry (29). We observe that, at the same reduced temperature (T/TBI=0.99), lower concentrations have a larger twist angle. Starting with purely bipolar tactoids (twist angle = 0°) the twist angle increases to about 135° for a 0.8-M sample. The data for all of the concentrations are plotted in Fig. S2. It has been shown previously in the context of LCLC droplets that the equilibrium twist angle of twisted-bipolar droplets is determined by the balance of offsetting the energetically costly splay deformation at the expense of twist and bend deformations (29). Those results show that a large fraction of the elastic deformation cost in bipolar and twisted-bipolar droplets is associated with the splay deformation close to the surface defects (29). We perform similar numerical calculations, but by fixing the ratio of bend/twist while allowing the value of splay elastic constant to vary. This is shown in Fig. S3. The plot verifies the prediction that increasing cost of splay results in an increasing twist angle of the twisted-bipolar tactoids. The twist angle data and the numerical calculations suggest that, for a given reduced temperature, the relative cost of splay (K11/K33 and K11/K22) increases as the concentration is lowered.

Fig. S2.
Fig. S2.

Concentration dependence of the twist angle of twisted-bipolar tactoids and number fraction of the tactoids with the unique director configuration. Shown is the fraction of tactoids with the unique configuration (blue triangles) and the twist angle of twisted bipolar tactoids (red circles) as a function of SSY concentration. Over 100 tactoids were sampled to determine the fraction of the tactoids with unique director configuration for each concentration. Twist angle measurements are the average of about five tactoids for each data point and were measured at T/TBI=0.99.

Fig. S3.
Fig. S3.

Numerically calculated elastic free energy as a function of twist angle in twisted-bipolar tactoids. The ratio of the twist/bend (K22/K33) elastic constant is fixed to be 0.1. The relative splay elastic constant (K11) varies from 0.5 to 2.5. Note that the twist angle corresponding to the minimum in the elastic free energy increases with the increase of K11. a.u., arbitrary units.

The elastic constants associated with the bulk deformations of a liquid crystal whose constituents are semiflexible aggregates have contrasting scaling with the aggregate length, L. The splay elastic constant scales linearly with the aggregate length (K11∼ϕL) (28, 33), where ϕ is the volume fraction of the aggregates. In contrast, in the context of the bend deformation, the semiflexible aggregates are free to conform to the director orientation and hence the bend elastic constant is dictated only by the persistence length, λp, of the aggregates (K33∼ϕλp). The twist elastic constant is also only a function of the persistence length and scales as K22∼(ϕλp)1/3 (37). Persistence length is a molecular property that is a weak function of temperature and concentration (38). On the other hand the aggregate length L is quite sensitive to changes in temperature and concentration (20). Naturally, longer aggregates result in higher relative cost of splay. From the scaling of the elastic constants and the trend in the twist angle of twisted-bipolar tactoids, it is reasonable to deduce that, at the same reduced temperature, lower concentrations have longer aggregates.

We also gather from our experiments that the fraction of tactoids with the unique director configuration increases as the concentration is lowered. This is also quantified in Fig. S2. We postulate that this phenomenon is also a consequence of the relative increase in the splay cost as the concentration is lowered. We test this idea by increasing the cost of splay deformation of LCLCs while keeping the relative cost of bend and twist mostly unchanged. We achieve this with the addition of a small amount of PEG to SSY solutions. PEG is a widely used condensing agent. The use of PEG in influencing the aggregation behavior of LCLCs is well documented (4, 39). PEG remains in the isotropic part of the solution and exerts osmotic pressure on the nematic region, resulting in the elongation of the aggregates. In essence, addition of PEG to a sample results in the increase of aggregate length, consequently resulting in the increase of splay, which scales linearly with aggregate length. Twist and bend on the other hand scale only with the persistence length and hence remain mostly unchanged.

Fig. 1D is a cross-polarized image of a 1.0-M sample of 0.5 wt% PEG. Compared with a 1.0 M PEG-free sample (Fig. 1B), Fig. 1D clearly has a significantly higher fraction of tactoids adopting a unique director configuration. We tabulate the fraction of tactoids adopting the unique director configuration upon the addition of PEG for different concentrations in Table S1. The trend is consistent with the idea that longer aggregates lead to a greater fraction of tactoids with the unique director configuration. Not surprisingly, the twist angle of the twisted-bipolar tactoids also increases upon the addition of PEG. The measured twist angles of the twisted-bipolar tactoids in Fig. 1 B (1.0 M) and D (1.0 M with PEG) are about 18° and 105°, respectively. We note that PEG also increases the concentration of SSY aggregates (39); however, the scaling of both bend and splay is linear in concentration. Hence, we argue that the phenomenon observed upon the addition of PEG is largely due to changes to the aggregate lengths. For this reason, other factors such as concentration gradients of SSY within the tactoids might not play a major role in the phenomenology observed compared with the role played by the changes to the aggregate length.

View this table:
Table S1.

The fraction of tactoids with the unique director configuration when SSY is doped with PEG

Dichroism.

After establishing the phenomenology pertaining to the tactoidal configuration changes as a function of concentration, we uncover more information regarding the unique director configuration in the tactoids by performing linear dichroism studies. The radial symmetry of the texture under crossed polarizers (Figs. 1 and 2A) indicates that the director is oriented in either a radial or a concentric fashion with a defect/escaped core at the center of symmetry. We note that the images presented in Figs. 1 and 2 are top views of the configuration with the direction of light propagation being parallel to the defect/escaped core. Comparing the regions of light extinction and light transmission in Fig. 2 B and C, we can rule out the possibility of a radial configuration. This can be ascertained by observing the intensity at the periphery of the droplet. In both instances the dark regions close to the periphery of the droplet, where the light is extinguished, are along the polarizer. This is contrary to the expectation if the droplet structure was radial. The molecular plane of SSY molecules is perpendicular to the director orientation (40). Hence, in a radial droplet, the extinction at the periphery should be perpendicular to the polarizer direction. The dichroism data suggest that the director is oriented concentrically as opposed to being radial. The simple concentric configuration involves an azimuthally oriented director with a line defect running through the center (29). However, the formation of the line defect can be avoided through a twist deformation resulting in an escaped configuration. The swirl in the dichroism images is indicative of escape of the director configuration via a twist deformation.

Fig. 2.
Fig. 2.

Cross-polarized, dichroism, bright-field, and transition images of SSY tactoids (A) under crossed polarizers, (B) under monochromatic illumination (551 nm) with vertical polarization, (C) under monochromatic illumination (551 nm) with horizontal polarization, and (D) under bright field. (E) The transition from twisted bipolar to the unique director configuration. The corresponding solution is 0.97 M SSY doped with 0.5 wt% PEG. (F) A schematic of the unique director configuration.

Further information regarding the unique configuration can be garnered by the bright-field images shown in Fig. 2D. The surprising finding here is the lack of any indication of the presence of surface defects (boojums) that are a feature of the bipolar and twisted-bipolar configurations. Boojums, when imaged under a bright-field microscope, can be readily distinguished due to the strong scattering off the defect core. The core of the defect has a different refractive index in comparison with the nematic phase. This results in strong scattering of light close to the defect. A combination of cross-polarized and bright-field images can be used to identify boojums. One such example is shown in Fig. S4 A and B, where scattering off the boojums of a twisted-bipolar tactoid is readily obvious. However, in contrast, when the tactoids with the unique configuration are observed under bright-filed microscopy, no scattering is apparent. The plane of observation of all of the images discussed in Figs. 1 and 2 is the equatorial midplane. Bright-field images of the side view of the unique configuration and other top-view images for which the focal plane is varied are shown in Fig. S5.

Fig. S4.
Fig. S4.

Combination of bright-field and crossed-polarized microscopy to identify boojums in twisted-bipolar and columnar tactoids. Shown are microscopy images of tactoids. (A and B) The polarized optical image and bright-field image of a twisted-bipolar tactoid. Two singular boojums can be clearly observed on each pole. (C and D) The polarized optical image and bright-field image of tactoids in columnar phase. The tactoids exhibit concentric structure with a disclination line in the center. The singular line defects are clearly observed. The columnar tactoids are observed with 1.04 M SSY doped with 3 wt% PEG.

Fig. S5.
Fig. S5.

Side view and variation of the focal plane for top views of the bright-field images of tactoids with the unique director configuration. Shown are microscopy images of tactoids. (A and B) The cross-polarized image and bright-field image of tactoids with the unique configuration that have nucleated on the side wall of a square capillary. Singular defects cannot be discerned in the bright-field image of the side view. The plane of focus is the midplane. The corresponding SSY concentration is 0.88 M. (C) The top view of a tactoid with the unique director configuration viewed under cross-polarizers. (D1–D7) The bright-field images of the same tactoid with varying plane of focus. The focal plane is changed from the top surface of the capillary to the bottom of the tactoid from D1 to D7. The focal plane is lowered every 2μm by each image. The radius of the tactoid is 12μm. We find no evidence of a singular defect in any focal plane of the top-view images as well as the midplane of the side-view image. (Magnification: 50× in C and D1-D7.)

The difference in the defect textures of the two configurations is made more apparent by examining the transition of a twisted-bipolar tactoid to one with the unique configuration. This is shown with a sequence of images in Fig. 2E. The sample under observation is a 0.97-M SSY solution with 0.5 wt% PEG. The tactoid initially has a twisted-bipolar configuration, and the boojums can be readily identified. As the configuration transitions, we see that the boojums are forced toward the center, while trying to maintain maximum separation via tracing a spiral (clockwise in this instance). Finally, from the last two snapshots we infer that the singular defects are replaced by a nonsingular core of the unique director configuration. Movie S1 shows the transition. Further, upon addition of 3 wt% PEG to a 1.04-M SSY solution, the tactoids enter the columnar phase and adopt a concentric configuration. Scattering off the defects in the bright-field image for this case is again readily obvious. This is shown in Fig. S4D.

Identifying the exact director configuration of the nonsingular core is beyond our experimental capability. However, we surmise that the reason for the preference of the nonsingular core as opposed to the boojums is again related to the cost of splay associated with the director field around the boojums, where most of the free-energy cost of deformation is concentrated (29). We hypothesize that the director avoids the formation of the boojums with a nonsingular core that violates the anchoring in a small region by pointing axially. Finally, the measured twist angle of the tactoids with the unique configuration is always 90°, independent of the concentration. These measurements were performed on tactoids that happen to nucleate on the side wall of a square capillary. The data are presented in Fig. S6. A schematic of the proposed configuration is provided in Fig. 2F. Movie S2 shows the schematic rotating 90° to help visualize the configuration at different orientations. Discussion regarding the shape of tactoidal droplets is provided in Fig. S7 and Table S2.

Fig. S6.
Fig. S6.

(A) Cross-polarized image of a tactoid with the unique configuration that has nucleated on the side wall of a square capillary. Note that the direction of light propagation is perpendicular to the escape direction. The corresponding SSY concentration is 0.96 M. (B) A schematic of the unique configuration observed from a side view. (C and D) The twist angle measurement of the tactoids with the unique configuration. (C) The maximum (red triangles) and the minimum (blue circles) transmitted intensity at every polarizer angle when the analyzer is rotated through 180°. (D) The transmitted intensity at every analyzer angle when the polarizer is fixed at a horizontal position.

Fig. S7.
Fig. S7.

The director configuration does not depend on the size of the tactoid. Shown are polarized optical microscopy images of tactoids with different sizes. Tactoids of significantly different sizes exhibit both textures. (Left) The diameter of the tactoids is around 25μm; (Right) the diameter of the tactoids is around 120μm. Both the twisted-bipolar and the unique director configurations are observed irrespective of the tactoid size.

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Table S2.

Aspect ratio of bipolar and twisted-bipolar tactoids as a function of SSY concentration

We conclude from the experiments that the phenomenology related to the emergence of the unique tactoidal configuration and its director arrangement is a consequence of the increasing relative cost of splay deformation. The trends in the twist angle measured for twisted-bipolar tactoids and the number fraction of the tactoids with the unique director configuration also conform with the expectation of increasing relative cost of splay. The experiments suggest that, for the same reduced temperature, splay gets costlier in comparison with twist and bend as the concentration is lowered. We postulate that this is a consequence of increasing aggregate lengths. Next, we use a simple model to estimate the length distribution of SSY aggregates to test this idea.

Length Distribution of SSY Aggregates.

The transition temperatures, TNB and TBI, increase with increasing concentration. However, the physics behind the role of temperature and concentration in affecting the length distribution and arrangement of the aggregates are not readily obvious. We estimate the length distribution of the aggregates at a given temperature, using a model based on the law of mass action (20, 41). The model assumes that the aggregation process is isodesmic. A commonly accepted value of aggregation energy (E) is about 7.25 kBT, as estimated from several experiments (18, 20, 32, 42) and molecular dynamic simulations (43). The relative volume fraction XN of aggregates containing N molecules is XN=N(X1exp(E/kBT))Nexp(−E/kBT), where X1=(1+2ϕexp(E/kBT))−1+4ϕ⁡exp(E/(kBT)2ϕexp(2E/kBT) is the relative volume fraction of individual molecules (17, 20), kB is the Boltzmann constant, and T is the absolute temperature. The volume fraction, ϕ, can be calculated from the SSY concentration using ϕ=cM/(cM+ρ) (20, 32), where Mw = 452.36 g/mol is the molar mass of SSY molecular, c is the concentration in molars, and ρ= 1.4 g/cm3 is the density of SSY aggregates (20, 44). The concentration of SSY inside the tactoid is ascertained by the tie lines in the experimental phase diagram (20).

Using this model, we estimate the length of the aggregates at the same reduced temperature (T/TBI) for different concentrations. The relative volume fraction of aggregates XN/ϕ is plotted as a function of the number of molecules in the aggregate (N) in Fig. 3. There are several striking aspects to this plot. We can immediately note that there is a systematic shift in the peak position of the plot as the concentration is varied. Higher concentrations have lower aggregate length on average. Another feature that can be readily observed in the plot is that the relative volume fraction of long aggregates (N> 30) increases as the concentration is lowered. These two observations have a significant consequence and serve as the physical basis for explaining the remarkable changes in the director configuration of the tactoids.

Fig. 3.
Fig. 3.

Simulation of size distribution of SSY aggregates. The plot provides the relative volume fraction of aggregates with different lengths (number of molecules) for different concentrations at the same reduced temperature.

From Fig. 3, we note again that for lower concentrations, the aggregates at the same reduced temperature are longer. Further, the relative volume fraction of the longer aggregates (N> 30) also increases as the concentration is lowered. Both these factors accentuate the relative cost of splay for lower concentrations. The model qualitatively explains the increase in the twist angle of twisted-bipolar tactoids and the emergence and increasing number fraction of the tactoids with the unique director configuration, as the concentration is lowered.

We independently verify the conclusion from the observations on the tactoidal droplets by measuring the order parameters close to the nematic–biphasic transition of SSY. Volume fraction and the aggregate length (aspect ratio) are the two parameters that define the order parameter when using theoretical models on the lines of Onsager (2). Motivated by this we perform Raman scattering measurements to determine the order parameter of SSY close to the transition temperature (TNB) for a range of concentrations.

Raman Measurements.

Polarized Raman intensities, I∥ and I⊥, are measured in two conditions, which represent the scattered intensities when the polarizer and analyzer are parallel and crossed, respectively. The experimentally obtained depolarization ratio I⊥/I∥(θ) is fitted to the theoretical expression (40, 45). More experimental details are provided in Fig. S8.

Fig. S8.
Fig. S8.

Polarized Raman spectra of SSY and the measured order parameters at various concentrations and temperatures, as well as the reduced order parameter (<P2> and <P4>) as a function of T/TNB. (A) Polarized Raman spectra of SSY in monodomain at 0° and 90°. I∥ corresponds to the polarization component of the Raman signal that is parallel to the laser excitation and I⊥ corresponds to the Raman signal perpendicular to the excitation. (A, Inset) The SSY director orientation in a rectangular capillary. (B) Raman peak intensities at 1,596 cm−1 measured with 360° full rotation of the capillary.

The variation of <P2> and <P4> (second and fourth moments of the orientational distribution function) as a function of temperature and concentration are shown in Fig. 4 A and B. Order parameters for SSY increase with concentration at a given temperature. This is a unique property of LCLCs. Although the quantitative values of <P2> and <P4> as a function of the temperature and concentration are informative (40), we can gain more physical insight by normalizing the temperature axis with nematic–biphasic transition temperature TNB. This can clearly help in discerning and separating the roles played by concentration and temperature. Fig. 4 C and D shows the plots of the order parameters as a function of the reduced temperature for different concentrations. It is evident from these plots that <P2> and <P4> are the same for every concentration at T=TNB. Further, the <P2> and <P4> dependence on the reduced temperature T/TNB is independent of concentration for SSY. We infer that the role played by concentration is to alter the TNB.

Fig. 4.
Fig. 4.

Order parameters measurement for various concentrations and temperatures, as well as the reduced order parameter as a function of T/TNB. (A and B) <P2> and <P4> obtained for four concentrations as a function of temperature. (C and D) Replotted order parameters <P2> and <P4> as a function of reduced temperature T/TNB for a range of concentrations.

To explain the concentration independence of order parameters, we start with the idea that the two key factors determining the orientational order are volume fraction ϕ and aspect ratio L/D (2). The average aggregate length L is proportional to the average number of molecules in an aggregate, which we estimate in Fig. 3. The diameter D, which is about the width of one SSY molecule, is the same for all of the aggregates. For hard rods, the aspect ratio L/D is a constant and determines the critical volume fraction beyond which the system undergoes a transition from the isotropic to the nematic phase (2). However, in a LCLC system, the length distribution is polydisperse and depends both on concentration and on temperature. Despite this complication, we provide a simple picture to understand the nematic–biphasic transition by examining the product ϕ<N> for all of the samples (N∼L). The weighted-average number of molecules in an aggregate <N>=∑N=1∞N⋅XN/ϕ is calculated from the length distribution of each concentration in Fig. 3. As we can see from Table 1, at TNB the product of ϕ<N> is nearly the same for all samples.

View this table:
Table 1.

The weighted-average aggregate length <N> and the product ϕ<N> for different concentrations

We note that LCLCs, in general, do not conform to Onsager’s predictions (2, 32). However, quite surprisingly, we find that the model incorporating the volume fraction and aggregate length explains the order parameter data reasonably well. This is at the heart of Onsager’s theory, wherein the order parameter can be expressed as S≈ 1−3/α, where α∼ 4/π(ϕL/D)2−45/8+o((ϕL/D)−2)).

We note that not only is the order parameter a constant at TNB for all concentrations, but also we can capture the physical picture using the simple argument that ϕ<L>/D is nearly a constant at TNB. The physical picture then corresponds to higher concentrations having shorter aggregates but a larger number of them, whereas lower concentrations have longer but fewer aggregates at the transition temperature. A more comprehensive theory can capture the details regarding the influence of temperature and polydispersity of the aggregates in affecting the nature of the phase transition (34, 46⇓–48), but we believe the essential physical picture is qualitatively captured by the simple model we provide.

Conclusions

We use tactoids as a means to uncover rich physics pertaining to the nematic–biphasic transition of LCLCs. In the process, we also find several interesting phenomenologies pertaining to tactoids, such as breaking of achiral symmetry of the director configuration as well as the emergence of a unique chiral director configuration.

Bipolar tactoids prevailed in our experiments when higher concentrations were observed optically. However, upon lowering the concentration we observed a breaking of achiral symmetry and a predominance of twisted bipolar tactoids. On further reduction of concentration, the tactoids assume a unique director profile that was never previously observed. Concurrently, the twist angle of the twisted-bipolar tactoids also increased. We conclude that the observed phenomena are a consequence of the increasing relative cost of splay deformation (at the same reduced temperature) as the concentration is lowered. An analysis of the aggregate length distribution at the same reduced temperature for different concentrations reveals that lower concentrations on average have longer aggregates. The contrasting scaling of splay elastic constant with the aggregate length in relation to the scaling of bend and twist elastic constants explains the experimental results of the tactoids qualitatively. The dramatic director configurational changes that we see then are a direct result of the changes to aggregate size. Changes to aggregate lengths can also be induced using crowding agents and other impurities like ions (4, 32). These reasons make the tactoidal droplets an extremely good candidate for sensor applications. The insights gained from optically observing tactoidal droplets were extended to understand the underlying physics of the nematic–biphasic transition. Order parameters <P2> and <P4> of SSY LCLC were measured close to the transition temperature and they were found to be independent of the concentration. We find that the product of volume fraction and the estimated average length of aggregates is almost a constant at TNB. This provides an explanation for the invariance of the order parameter with concentration. The optical observation of tactoidal droplets and the order parameter measurements then lay the basis for the physical picture we put forward, which corresponds to higher concentrations having shorter aggregates but a larger number of them, whereas lower concentrations have longer but fewer aggregates at the transition temperature.

Materials and Methods

SSY was purchased from Sigma-Aldrich and TCI America with a purity of 90%. Experimental details of the microscopy and Raman spectroscopy are provided in SI Materials and Methods.

SI Numerical Calculation of the Elastic Free Energy of Twisted-Bipolar Tactoids

The elastic free energy of a deformation can be expressed as F=1/2∫dV[K11(∇⋅𝐧)2+K22(𝐧⋅∇×𝐧)2+K33(𝐧×∇×𝐧)2], where n is the director, and K11, K22, and K33 are the Frank elastic constants associated with splay, twist, and bend bulk deformations. F is numerically calculated using the integral function of Mathematica. As the aspect ratio of the twisted-bipolar tactoids is close to 1 (data shown below), we approximate them to be spherical and adopt the director field of the twisted-bipolar droplet model in ref. 29,ntb={−zρcos(α0ρ1−z2)z2ρ2+(1−z2)2,sin(α0ρ1−z2),(1−z2)cos(α0ρ1−z2)z2ρ2+(1−z2)2},where ρ is the radius in cylindrical coordinates, z is the axis is along the bipolar axis, and α0 is the angle that the director on the surface of the tactoids makes with the z axis. Because the free energy of twist is divergent at the boojums (z=1), the integral is truncated at zc=0.999.

SI Twist Angle Measurement of the Tactoids with the Unique Director Configuration

Fig. S6A shows the side-view images of SSY tactoids with the unique configuration in a square capillary. We measured the twist angle using wave-guiding experiments. When the Mauguin limit is satisfied, the polarization of incident light is guided by the twist deformation. By rotating polarizer and analyzer, the center of the tactoid is driven to darkness when the polarizer angle is parallel (or perpendicular) to the director at the bottom of the tactoid and the analyzer is perpendicular (or parallel) to the director on the top surface. To find the minimum intensity, we rotated the polarizer every 10° and at every polarizer orientation, the analyzer was also rotated every 10°. In Fig. S6C, the minimum transmitted intensities at the center of the tactoids occur when the polarizer is horizontal. When the polarizer is fixed in horizontal orientation, the transmitted intensity with the rotation of analyzer is plotted in Fig. S6D. The minimum transmitted intensity corresponds to the vertical analyzer. This combination indicates a 90° (or 0°, which corresponds to bipolar configuration; the bipolar configuration can be ruled out from top views of the 0.88-M sample) twist of director from the center to the surface. The twist angle measured for the tactoids with the unique director configuration was independent of the concentration and always 90°. The concentration of this particular sample was 0.88 M, corresponding to the fraction of escape concentric tactoids of 0.45. The twist angle of the coexisting twisted-bipolar tactoids at this concentration is about 60°.

SI Quantifying the Shape of Bipolar and Twisted-Bipolar Tactoids

We quantify the shape of the twisted-bipolar tactoids as a function of concentration in Table S2. The cross-sections of the escaped-concentric tactoids are almost perfectly circular.

The aspect ratio is defined as the ratio of the diameter along the axis of symmetry (line joining the boojums) vs. the largest diameter along the axis perpendicular to the axis of symmetry. We see that there is no clear trend. It should be noted that in addition to elastic forces, interfacial tension needs to be accounted for when trying to understand shape. We note that the observation temperatures for the various concentrations vary from 10° to 70°. The interfacial tension may vary in this range and has to be taken into account for understanding the shape changes.

SI Raman Measurements

Polarized Raman intensities, I∥ and I⊥, are measured in two conditions, which represent the scattered intensities when the polarizer and analyzer are parallel and crossed, respectively. The experimentally obtained depolarization ratio I⊥/I∥(θ) is consequently fitted using the equationR(θ)=(r−1)2[56+40<P2>+(9−105cos4θ)<P4>]56(8r2+4r+3)−40(4r2−r−3)(1+3cos2θ)<P2>+3(r−1)2(9+20cos2θ+35cos4θ)<P4>,where <P2> and <P4> are the second and fourth moments of the orientational distribution function, and r is the ratio of differential polarizability. Measurements are performed for concentrations ranging from 0.95 to 1.15 M.

SI Influence of Confining Surfaces

The capillaries used in the experiments are devoid of any chemical treatment. We obtained qualitatively similar results when using cells made with untreated cover-glass slides (manufacturer VWR). The confining surfaces do not bias the director in any way and overall random orientations were observed for the bipolar and twisted-bipolar tactoids in both cases.

SI Materials and Methods

Sample Preparation.

SSY was purchased from Sigma-Aldrich and TCI America with a purity of 90%. Further purification was carried out by dissolving it in deionized water and adding ethanol, causing the SSY to precipitate. We then filtered the isolate and dried the powder in a vacuum oven. The purified SSY was then dissolved in deionized water to make SSY solutions. All rectangular glass capillaries were purchased from Vitrocom. Without further treatment, the capillaries were filled with isotropic SSY solutions by capillary action and then placed on a glass slide. The ends were sealed with epoxy glue to prevent the water evaporation. The capillaries used for tactoidal experiments are 100μm×1,000μm. For Raman experiments the capillary dimensions are 20μm×200μm. The dimension of the square capillary is 100μm.

Polarized Optical Microscopy.

The sample slide was placed on a Linkam T95-PE heat stage with a temperature control accuracy of 0.1°C. The heat stage was mounted on the 360° rotation stage of the Leica DMRX microscope. The low-magnification images were obtained with a 10× (N.A. = 0.3) Leica objective and the high magnification images were obtained with a 50× (N.A. = 1.4) Leica objective.

Depolarized Raman Microscopy.

A 785-nm diode laser of 100 mW total power (measured after objective) was focused into the monodomain region by a 50× (N.A. = 0.5) Leica objective. Two rotatable Glan–Thompson prisms were placed in both the excitation and emission optical paths so the desired polarization direction can be chosen independently. The spectra were obtained with a Kaiser RamanRxn system.

Acknowledgments

K.N. acknowledges funding from RBI fellowship.

Footnotes

  • ↵1K.N. and J.F. contributed equally to this work.

  • ↵2To whom correspondence should be addressed. Email: mohan{at}mse.gatech.edu.

  • Author contributions: K.N., J.O.P., and M.S. designed research; K.N., J.F., and R.C. performed research; K.N., J.F., and R.C. analyzed data; and K.N., J.F., R.C., J.O.P., and M.S. wrote the paper.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission.

  • This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614620114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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Unusual behavior of chiral chromonic tactoids
Karthik Nayani, Jinxin Fu, Rui Chang, Jung Ok Park, Mohan Srinivasarao
Proceedings of the National Academy of Sciences Apr 2017, 114 (15) 3826-3831; DOI: 10.1073/pnas.1614620114
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Proceedings of the National Academy of Sciences: 114 (15)
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