Liquid Crystal Ordering in Densely Packed Colloidal Suspensions of Highly Anisotropic Monolayer Nanosheets

Abstract

Monolayer nanosheets of zirconium phosphate in aqueous suspension exhibit short-range repulsion and long-range attraction, producing, at overall volume fractions larger than about half a percent, phase separation into higher-concentration liquid crystal and lower-concentration isotropic regions. At high concentrations, this phase separation takes the form of an emulsion of condensed, liquid-crystalline droplets, which anneal to form lens-shaped tactoids. These tactoids provide an opportunity to study the liquid crystal ordering of inorganic nanosheets in the limit of large shape anisotropy (diameter/thickness~400) and high packing fraction (volume fraction $\gtrsim$ 70%). The internal liquid crystal structure of the tactoids remains nematic even under conditions that would usually favor ordering into lamellar smectics. Local lamellar ordering is suggested by short-range, smectic-like layer correlations, but a full transition into a smectic phase appears to be inhibited by the nanosheet edges, which act as a perturbative population of dislocation loops in the system of layers. Under conditions of thermal equilibrium, the nanoplates organize positionally to enable bend deformation of the director, a hallmark of the nematic phase and its principal distinction from the smectic, where bend must be expelled.

1. Introduction

Colloidal dispersions of disc-shaped particles order to form liquid crystal (LC) phases, a result of the competition between the entropy of orientation of the disc planes, packing entropy, electrostatic repulsion, and long-range attraction [1,2,3]. A wide variety of inorganic platelet systems have been found to exhibit liquid crystal ordering, including gibbsite [4], niobates [5,6], titanates [7], smectite clays [8], layered double hydroxides [9,10], solid acids H₃Sb₃P₂O₁₄ [11], fluorinated, layered clay minerals [12], beidellite [13], and zirconium phosphate (Zr(HPO₄)₂·H₂O, abbreviated ZrP) [14,15]. In addition to nematic phases of rod- and disc-shaped particles, lamellar smectic phases of discs have been claimed in several different experimental systems [4,10,11,12,14], prompted by observations of multiple higher harmonic orders of X-ray scattering indicating lamellar (smectic-like) correlations and by predictions based on density functional theory of interacting hard discs [16].

Despite these efforts, there are still fundamental questions about the nature of the phase formed at such low nanosheet volume fractions. Also, the dynamics of the phase formation process are rarely discussed. Here, we explore in detail the LC ordering of colloidal suspensions of monolayer nanosheets under simultaneous conditions of extreme particle shape anisotropy and high volume fraction in a study of micron-size ZrP monolayer sheets, which exhibit long-range attraction that separates them into high-concentration LC and low-concentration isotropic (Iso) phases. High-resolution X-ray diffraction shows that the lamellar ordering remains only short-ranged, even for a diameter/thickness ratio of 400 and volume fractions up to 70% where the gap between the sheets is ~12 Å, or about half their thickness. Remarkably, at such high concentrations, the phase-separated LC spontaneously forms tactoids that are lens-shaped at equilibrium and have nearly ideal nematic internal structure. The nematic director field inside the tactoids (locally normal to the planes of the disc-like nanosheets), observed directly using freeze-fracture transmission electron microscopy, shows clearly that the colloidal nanosheets organize and shift laterally (parallel to their planes) in order to position their edges (which correspond to edge dislocation loops in a conventional lamellar packing) closer to the tactoid boundaries on average so as to enable the bend of the nematic director required by the tactoid structure. This is likely a manifestation of the fact that an enforced population of dislocation loops moving around by thermal fluctuations pushes three-dimensional smectics below the critical dimensionality for maintaining quasi-long-range ordering in the presence of thermal layer displacement fluctuations [17]. There is no evidence in the ZrP suspensions of the expulsion of director bend, a key property of thermotropic smectics [18].

2. Results and Discussion

2.1. Phase Behavior

In these experiments, we use α-ZrP discs exfoliated into monolayers using n-tetrabutylammonium hydroxide (TBA⁺OH⁻) in aqueous suspension [19,20], a process in which the TBA⁺ cations intercalate from the edges of the α-ZrP discs and diffuse inward to fully exfoliate the discs into monolayer ZrP/TBA nanosheets [21]. Each nanosheet is a ZrP/TBA complex, with a single, 6.8 Å thick ZrP layer sandwiched between two TBA⁺ monolayers, each 10 Å thick, so that each nanosheet has an effective thickness t = 26.8 Å and a diameter D = 1060 ± 150 nm (Figure S1), giving an extremely large shape anisotropy of D/t~400 [22]. A nanosheet of area A is taken to have a volume A × (26.8 Å) so that the overall volume fraction ϕ_sample of a ZrP/TBA suspension is N × A × (26.8 Å)/V, where N is the total number of nanosheets and V is the total volume of the suspension, i.e., the nanosheet volume includes the TBA monolayers.

Previous experimental studies have shown that monolayer ZrP/TBA aqueous suspensions above a certain concentration exhibit ordered phases that are birefringent and display typical LC textures [15]. Our ZrP/TBA suspensions were prepared with overall volume fractions ϕ_sample ranging from 0.3% to 30%. A typical set of suspensions is displayed in Figure S2. The suspensions were characterized by small-angle synchrotron X-ray scattering (SAXS) using a high-resolution, single-crystal diffractometer. Quasi-Bragg peaks were observed for suspension concentrations of ϕ_sample = 1.5% and higher. As the concentration increases, the quasi-Bragg scattering gets stronger and sharper, with up to four visible harmonics (Figure 1a), suggestive of lamellar ordering [11]. However, even at high concentrations (ϕ_sample = 24%), the main peak is still much broader than the resolution of the synchrotron X-ray beam (Figure 1a, inset). In addition, the widths of the harmonics increase linearly with their order, which is not consistent with the scattering from smectics, where the exponent of the power-law decay of the scattering wings depends quadratically on the diffraction order. More details are discussed below.

The location of the fundamental quasi-Bragg peak is taken to be 2π/d, where d is the center-to-center distance between adjacent nanosheets in a lamellar stack and is plotted as a function of the inverse concentration of nanosheets in Figure 1b. The steep black line is the asymptote corresponding to t/ϕ_sample, the lamellar periodicity of a hypothetical single phase that uniformly fills the entire volume, and in which the colloidal layers making up the lamellae are continuous, i.e., with no voids or gaps in between. The observed intersheet spacing is uniform and much smaller than t/ϕ_sample, indicating that there is phase separation between a low-concentration (water-rich) phase and a high-concentration (nanosheet-rich) phase, whose concentrations we denote as ϕ_low and ϕ_high, respectively. This is consistent with previous studies that found an inhomogeneous distribution of clay sheets in suspensions [12,23,24,25]. Over most of the concentration range, the sheet spacing in the nanosheet-rich phase is d = t/ϕ_high, where ϕ_high ~ 7 × ϕ_sample. This large difference between ϕ_high and the nominal ϕ_sample implies that the nanosheets have long-range attraction and short-range repulsion, stabilizing an equilibrium lamellar structure over the entire range of concentrations studied. Relatively few nanosheets remain in the water-rich phase, which is disordered and isotropic and does not contribute to the scattering. Referring to the plot in Figure 1b, we see that for this nanosheet system, at the highest concentration studied (ϕ_sample = 30%), the center-to-center nanosheet spacing is d = 28.7 Å. The thickness of the solvent layer between the sheets is 1.9 Å, meaning that the sheets are almost in contact. In this limit, the high-concentration phase has a volume fraction ϕ_high~93%, and the sheet spacing is d ~ D/300. At the lowest concentration at which we observe quasi-Bragg scattering, ϕ_high = 1.5%, and the sheet spacing d ~ 270 Å ~ D/40 is again a small fraction of D. The diffuse nature of the X-ray scattering indicates that the high-density phase is nematic with lamellar correlations at all concentrations.

The overall morphology of our ZrP samples depends on their preparation, as seen in Figure S3. When freshly mixed, the entire volume appears to be a turbid liquid crystal. However, this apparent uniform LC is a mixture of nanosheet-rich and water-rich phases, which can be partially separated by weak centrifugation. Each suspension was loaded into a cylindrical glass capillary (0.8 mm in diameter) and flame-sealed. The capillary was then swing-bucket-centrifuged at 7000× g for times ranging from a few minutes to 1 h. After this weak centrifugation, denser sediment was observed at the bottom of the capillary and clear supernatant at the top (Figure S3). We then immediately carried out SAXS measurements. No X-ray scattering was detected from the supernatant. The scattering from the liquid-crystalline sediment after centrifugation showed little change in either the Bragg peak position (open circles in Figure 1b) or the peak width as compared with the original suspension (Figure 1b inset). The apparent volume fraction of the sediment ϕ_sed, calculated based on the relative height of the sediment in the capillary after centrifugation (ϕ_sed/ϕ_sample = V_sample⁄V_sed), increases with centrifugation duration (dark pink region). But there is essentially no change in d until a threshold centrifugation duration and/or strength, above which the d spacing becomes substantially smaller (light pink region). These results suggest that the monolayer nanosheets in the original suspension phase separate into a water-rich phase and a nanosheet-rich phase in which the intersheet spacing is much smaller than t/ϕ_sample. Weak centrifugation (in the dark pink region of Figure 1b) stratifies these phases into two distinct regions without noticeably affecting their internal structure.

Confocal fluorescence microscopy, discussed next, indicates that the concentration of nanosheets in the water-rich phase is extremely low, essentially zero at the higher sample concentrations. Under this circumstance, if centrifugation achieved complete separation of the two phases and if the observed short-range, lamellar ordering corresponded to layers uniformly filled with sheets, then the dark pink region in Figure 1b would extend all the way to the black line (ϕ_sed = t/ϕ_sample,). The fact that the volume of the sedimented fraction gets smaller with no change in d indicates that centrifugation below the threshold does not compact the nanosheet-rich phase uniformly, resulting in packed ϕ_high domains embedded in ϕ_low regions. These morphologies will be discussed further below. That the entrained water-rich phase may not be completely separable by weak centrifugation is also indicated at lower concentrations, where the phase separation appears to result in a foam or gel of the two phases.

The absence of a significant concentration of nanosheets in the water-rich phase means that their osmotic pressure is relatively low, raising the question of what is determining the periodicity of the lamellae in the nanosheet-rich phase. There clearly must be attractive interactions between the plates. In order to explore this further, we added deionized water to the suspensions, either as-prepared or sedimented, and then remixed them. The addition of deionized water always had the effect of increasing d. The intersheet spacing is therefore determined not only by the characteristics of ZrP but also by the ions in the solution, principally H⁺, OH⁻, and the TBA⁺ cations. In equilibrium, some TBA⁺ cations dissociate from the nanosheets and partition at different net concentrations in the two phases. Adding more water changes this equilibrium, resulting in the dissociation of more TBA⁺ cations so that the nanosheets end up with more uncompensated charges, giving a stronger electrostatic repulsion of the nanosheets and thus a larger intersheet spacing. To verify this effect, samples with the same concentration of ZrP but different relative amounts of TBA were prepared and characterized using SAXS. The intersheet spacing became smaller with increasing TBA⁺ cation concentration, as shown in Figure 1c, in accordance with the dilution experiments.

In an effort to understand the attraction between the sheets, and given the lamellar character of the suspensions indicated by the SAXS data, we applied the DLVO theory of interacting planar sheets [2,3,26] to this system, considering a sheet interaction energy/area, U_A, given by the balance of Debye electrostatic interactions and a power law attraction from van der Waals (VdW) forces (see Supporting Notes). The electrostatic interactions are characterized by the Debye length, λ_D (Figure S4, [27]). λ_D is determined by the concentration of the TBA⁺ counter-ions in solution, c, and by the solvent pH [21], which we assume is the same everywhere, both between the sheets and in the solution. Both the electrostatic repulsion and the VdW attraction are functions of the separation of the nanosheets, δ = d − 26.8 Å. The VdW energy/area of attraction U_A(δ) ∝ -A_n/δⁿ was calculated for several power-law exponents (n = 2, 3, and 4), for each of which there is a corresponding Hamaker constant, A_n, that is determined by fitting the experimental data. We found that the only case consistent with the experiment is for n = 3 (Figure S5), corresponding to a retarded VdW attraction. In the lower range of nanosheet concentrations (ϕ_sample ≤ 4.5%), U_A exhibits a secondary minimum at spacing δ_eq, which moves to smaller values with increasing sheet and, therefore, TBA⁺ concentration, in agreement with the experimental values (Figure S6). The depth of this minimum decreases with increasing c, such that it disappears in the DLVO theory for ϕ_sample > 4.5% (Figure S7) at d ~ 90 Å. In the DLVO model, the disappearance of the secondary minimum induces a jump in δ down to the primary minimum, which in the present case would correspond to the suspension collapsing to aggregates with the sheets in contact. Such a collapse is not observed in our experiments, however. At the highest ϕ_sample (30%), the sheet spacing is eventually reduced to d = 28.7 Å, as seen in Figure 1b, and this decrease is continuous, with a well-defined spacing between the sheets at all ϕ_sample. This indicates that there is some additional repulsive interaction between the sheets at small separations inside the DLVO secondary minimum, which we attribute to the short-range ordering of the TBA⁺ counter ions at high c. Nevertheless, DLVO theory successfully describes the attraction at long distances and the dependence of sheet separation on concentration when the system is in the secondary DLVO minimum for ϕ_sample ≤ 4.5%.

2.2. Aggregation into Columnar Stacks (Figure 2) or Tactoids (Figure 3)

Confocal fluorescence microscopy (using a Nikon A1R) was used to explore the detailed structure of the monolayer ZrP/TBA suspensions. A small amount of fluorescein powder was dispersed in the suspensions by Vortex shaking. The resulting mixtures were filled into 20-micron-thick glass sandwich cells, which were then sealed with oil to prevent evaporation. Fluorescein is a water-soluble dye that shows strong, green fluorescence in the monolayer ZrP/TBA suspensions. Water-rich domains in a phase-separated system would be expected to fluoresce more strongly than the nanosheet-rich ones. In the lower-concentration suspensions, there is little contrast between the ordered domains and the isotropic background, but all the ZrP/TBA nanosheet suspensions we studied using confocal fluorescence microscopy were clearly phase-separated (Figure 2 and Figure S8). In low-concentration suspensions (ϕ_sample < 9%), the nanosheets stack in columns oriented perpendicular to the substrates (Figure 2).

Figure 2. Slice-view of a glass cell (substrates in the x-y plane) of a microphase-separated ZrP/TBA suspension (ϕ_sample = 3.0%) containing fluorescein imaged by confocal fluorescence microscopy using a 488 nm excitation laser and a 100X oil immersion objective (NA 1.45). The water-rich domains give a stronger fluorescence signal, while the nanosheet-rich domains are darker, appearing in the main image as micron-sized, circular domains. The z-x and z-y cross-sections reveal that these domains are nanosheets stacked in columns oriented preferentially perpendicular to the glass plates, forming a nematic phase with strong columnar correlations. The columns are slightly tilted in the x-z and y-z sections because of slow flow in the y-direction. This foam-like 3D structure is also shown in Figure S8.

Figure 2. Slice-view of a glass cell (substrates in the x-y plane) of a microphase-separated ZrP/TBA suspension (ϕ_sample = 3.0%) containing fluorescein imaged by confocal fluorescence microscopy using a 488 nm excitation laser and a 100X oil immersion objective (NA 1.45). The water-rich domains give a stronger fluorescence signal, while the nanosheet-rich domains are darker, appearing in the main image as micron-sized, circular domains. The z-x and z-y cross-sections reveal that these domains are nanosheets stacked in columns oriented preferentially perpendicular to the glass plates, forming a nematic phase with strong columnar correlations. The columns are slightly tilted in the x-z and y-z sections because of slow flow in the y-direction. This foam-like 3D structure is also shown in Figure S8.

The X-ray diffraction study indicates that the spacing of the nanosheets in the nematic suspensions is much smaller than what one might predict assuming a spatially homogeneous distribution of the nanosheets (Figure 1d). This implies that the nematic suspensions are form or get-like, comprising both water-rich, isotropic domains and nanosheet-rich, nematic domains. Additional experiments, presented below, indicate that this microphase separation can be divided into two modes of self-assembly, depending on concentration, distinguishable by the structure of the nanosheet assembly: a columnar nematic/Iso region with a typical structure, shown in Figure 2, where the nanosheets stack into regular arrays of columns, and a lamellar nematic tactoid/Iso region shown in Figure 3, where the nanosheets stack into lamellae and the locations of the nanosheets within each layer are random.

Figure 3. Phase separation and microstructure evolution of a high-concentration ZrP/TBA suspension. A suspension with ϕ_sample = 18% was observed over the course of 17 days in transmission in an optical microscope between crossed polarizers (first row, a1–a4), with only one polarizer (second row, b1–b4), and using a confocal fluorescence microscope to visualize a sample containing fluorescein (third row, c1–c4). (a1, b1, and c1) are taken immediately after sample preparation, while (a2, b2, and c2) are taken 3 days afterwards, (a3, b3, and c3) 10 days afterwards, and (a4, b4, and c4) 17 days afterwards. Several large tactoids were observed after about two weeks. Image (d3) shows the structure observed 3 μm above (c3) in the same cell. (e–g) Schematic illustration of the formation of aggregates and tactoids: (e) The nanosheets self-assemble into thin, small, tactoid-like aggregates a few microns across. (f) These small aggregates assemble to form larger, lens-shaped structures. (g) Further annealing of the nanosheets gives smooth, lens-shaped tactoids in which the nanoplates have nematic ordering with short-range smectic-like correlations.

Figure 3. Phase separation and microstructure evolution of a high-concentration ZrP/TBA suspension. A suspension with ϕ_sample = 18% was observed over the course of 17 days in transmission in an optical microscope between crossed polarizers (first row, a1–a4), with only one polarizer (second row, b1–b4), and using a confocal fluorescence microscope to visualize a sample containing fluorescein (third row, c1–c4). (a1, b1, and c1) are taken immediately after sample preparation, while (a2, b2, and c2) are taken 3 days afterwards, (a3, b3, and c3) 10 days afterwards, and (a4, b4, and c4) 17 days afterwards. Several large tactoids were observed after about two weeks. Image (d3) shows the structure observed 3 μm above (c3) in the same cell. (e–g) Schematic illustration of the formation of aggregates and tactoids: (e) The nanosheets self-assemble into thin, small, tactoid-like aggregates a few microns across. (f) These small aggregates assemble to form larger, lens-shaped structures. (g) Further annealing of the nanosheets gives smooth, lens-shaped tactoids in which the nanoplates have nematic ordering with short-range smectic-like correlations.

If the fresh ZrP/TBA suspensions are allowed to sediment for some time under gravity (without centrifugation), they stratify into an isotropic region on top and a dense, liquid-crystalline region at the bottom [14]. In low-concentration suspensions (ϕ_sample < 9%) with columnar nematic structures (Figure 2), even those that have been left undisturbed for more than a year, the gravity-separated regions can be re-mixed easily by shaking. In high-concentration suspensions (ϕ_sample ≥ 9%), phase separation is irreversible, and mechanical shaking does not return the suspensions to their freshly-made condition. The phase separation of a ϕ_sample = 18% sample over the course of a few weeks was observed directly under the optical microscope (Figure 3). Daily vortex-shaking for about 10 min was applied to the sample to promote the annealing process. The nanosheets self-assemble into aggregates several microns across, within each of which there is nematic ordering with short-range, smectic-like lamellar correlations (a “lamellar nematic”). These small aggregates anneal dynamically and stick to one another, forming larger, tactoid-like assemblies. As time goes on, these tactoid-like aggregates become increasingly well-ordered, their internal voids being eliminated and a larger fraction of the sheets attaining equilibrium separation. Thermal fluctuations and intermittent vortex-shaking anneal these aggregates to form large tactoids, with all the nanosheets eventually contained in the tactoids and none left in the surrounding water (Figure S9). Excess water in the tactoids is gradually expelled and the tactoids become smooth and lens-shaped (Figure 4). These large tactoids become increasingly stiff over time, making them less and less susceptible to self-adhesion. As a result, the large tactoids can remain in contact for long periods without flattening substantially, with the curvature of the surface suppressing fusion at the contact point. The absence of nanosheets in the isotropic solution indicates that the sheets are bound to the surfaces of the tactoids with energy much larger than k_BT. This makes global equilibration processes like Ostwald ripening very slow once the tactoids are formed. However, the universality of the tactoid structural theme over the whole population of tactoids, and their internal homogeneity, leaves no doubt that they are internally equilibrated. In these high-concentration suspensions, this self-assembly process could not be reversed by shaking or sonication: both actions were found rather to promote and speed up tactoid formation.

Figure 4. Lens-shaped tactoids formed in a high-concentration (ϕ_high~70%) ZrP/TBA suspension (ϕ_sample = 18%) after three weeks of aging. There was little further change in the appearance of this sample even after one year. Optical microscope images of lens-shaped tactoids (a) with only one polarizer and (b) between crossed polarizers. (c) Confocal fluorescence image of tactoids in solution containing fluorescein. The tactoid boundaries can be fitted with circular arcs (dashed red lines). (d) Freeze-fracture TEM image of part of a tactoid, showing nanosheets (some indicated with cyan lines) exhibiting a non-uniform director field and homeotropic anchoring at the LC/Iso boundary (dashed white line). Director bend (splay deformation of the nanosheets) is evident everywhere inside the tactoid, especially near the tip. (e) The nanosheet orientation along several diameters (paths 1–5 in (d)) is plotted vs. distance from the long axis of the tactoid (solid white line) as symbols. The model curves show the local tangential orientations, along the selected paths, of families of circles intersecting at the lens rim, as shown in Figure S13. (f) Comparison of nematic and smectic tactoid structures. We have seen no evidence of smectic tactoids in ZrP suspensions.

Figure 4. Lens-shaped tactoids formed in a high-concentration (ϕ_high~70%) ZrP/TBA suspension (ϕ_sample = 18%) after three weeks of aging. There was little further change in the appearance of this sample even after one year. Optical microscope images of lens-shaped tactoids (a) with only one polarizer and (b) between crossed polarizers. (c) Confocal fluorescence image of tactoids in solution containing fluorescein. The tactoid boundaries can be fitted with circular arcs (dashed red lines). (d) Freeze-fracture TEM image of part of a tactoid, showing nanosheets (some indicated with cyan lines) exhibiting a non-uniform director field and homeotropic anchoring at the LC/Iso boundary (dashed white line). Director bend (splay deformation of the nanosheets) is evident everywhere inside the tactoid, especially near the tip. (e) The nanosheet orientation along several diameters (paths 1–5 in (d)) is plotted vs. distance from the long axis of the tactoid (solid white line) as symbols. The model curves show the local tangential orientations, along the selected paths, of families of circles intersecting at the lens rim, as shown in Figure S13. (f) Comparison of nematic and smectic tactoid structures. We have seen no evidence of smectic tactoids in ZrP suspensions.

Tactoids with a variety of shapes have recently been reported in colloidal suspensions of gibbsite sheets by the Utrecht group [28,29,30]. The tactoids in our samples are lens-shaped (Figure 4 and Figure S10), showing substantial birefringence between crossed polarizers when they are viewed edge-on but essentially none when they are lying down (Figure S11). Using an optical compensator, we determined that the nanosheets have negative optical anisotropy, i.e., light with polarization parallel to the plane of the nanosheets propagates with the largest refractive index. The nanosheets are therefore oriented on average in the lens plane. Confocal fluorescence images of typical tactoids are shown in Figure 4c. The boundaries of the tactoids in these images are circular, indicating that the tactoids are lens-like with spherical caps.

The internal structure of the tactoids was characterized using freeze-fracture transmission electron microscopy (FFTEM) (Figure 4d). The individual ZrP/TBA nanosheets in the tactoids are seen to be smooth and flat (Figure S12) and oriented parallel to the surface at well-defined tactoid boundaries, indicating homeotropic anchoring of the director at the LC/Iso interface. The nanosheet orientation inside each tactoid is therefore, in general, non-uniform, displaying both splay and bend deformations, the latter forbidden in conventional lamellar/smectic phases. The orientations of the nanosheets measured relative to the mid-plane of the lens (Figure 4e) follow closely the local orientation of circles intersecting at the lens rim, indicating a nearly bi-spherical internal structure (Figure S13). The LC phase is therefore nematic rather than smectic, even at very high sheet concentration (in the suspension with ϕ_sample = 18%, SAXS measurements give ϕ_high~70% and a gap between the sheets of only 12 Å), consistent with the observation of broad SAXS peaks (Figure 1a). This is different from small domains of thermotropic smectics (which have similar layer spacing), such as bâtonnets nucleating from the isotropic phase, which remain smectic, adapting the layer structure to surface curvature by the formation of focal conic structures in the smectic domains [31].

The de Gennes length, the maximum distance over which director bend deformations occur, is given by $\lambda =\sqrt{K/B}$, where K is the Frank elastic constant and B the compressional modulus. In the tactoids, if we assume that $K ~ k_{\mathrm{B}}T$/D and $\sqrt{B} ~ k_{\mathrm{B}}T/\left[\sqrt{K}{\left(d-t\right)}^2\right]$ [32,33], we obtain $\lambda ~ {\left(d-t\right)}^2/D$~0.003 nm. This is small compared with both the lamellar spacing and the typical radius of layer curvature R observed in our tactoids, indicating that if they were smectic, the tactoid interior would feature equidistant neighboring layers, and bend deformations would be expelled. That is, the tactoid would be filled by focal conic domains, with splay preferred in the regions of high deformation near the defects, like those in Figure 4f, for example. Our observations, however, reveal no hints of this sort of structure, indicating rather that bend of the director is preferred in regions of high deformation, implying that the phase is nematic rather than smectic.

Our observations of the internal structure of lens-shaped tactoids of nanosheets indicate that the director n is perpendicular to the surface of the revolution of circular arcs about the axis (the z-axis) that bisects the line joining its two foci F₁ (−L, 0) and F₂ (L, 0) and that there is strong orientational anchoring at the outer boundary. This rotation generates a family of pairs of spheres centered on the z-axis and intersecting on a ring generated by the revolution of the two foci about the z-axis (coincident with the rim of the tactoid, Figure S13). A 3D toroidal coordinates system (ξ, η, φ) is generated by rotating the 2D bipolar coordinate system about the z-axis (see Figure 5 inset and Supporting Notes, [34]). The director n of the sheets in toroidal coordinates is (1, 0, 0). The ξ-coordinate of a point P is the angle F₁PF₂. The η-coordinate is the natural logarithm of the ratio of the distances d₁ and d₂ of the point P from the two foci, $\eta =\ln (d_1/d_2)$. The φ-coordinate is the azimuthal angle about the z-axis. The elastic free energy F_e of the director field is obtained by integrating the Oseen–Frank free energy density (see Equation S15, [35]) over the volume of the tactoid in toroidal coordinates, given by

$$F_{\mathrm{e}}={{\displaystyle \int }}_0^{2\pi }d\phi {{\displaystyle \int }}_0^{+\infty }d\eta {{\displaystyle \int }}_{\pi -\alpha }^{\pi +\alpha }{\displaystyle \frac{2L{K}_{\mathrm{S}}\sinh \eta {\sin }^2\xi +\frac{1}{2}L{K}_{\mathrm{B}}{\sinh }^3\eta }{{\left(\cosh \eta -\cos \xi \right)}^3}}d\xi$$

(1)

K_S and K_B are, respectively, the elastic constants for splay and bend deformation. The twist term is identically zero and does not contribute to the free energy, consistent with the symmetry of the tactoid and the constraint of homeotropic anchoring. Since the integral of the η-coordinate diverges as η approaches infinity, an upper cutoff value β for η is introduced. The elastic free energy may then be expressed as

$$\begin{gathered} F_{\mathrm{e}}=K_{\mathrm{S}}R\times {\displaystyle \frac{\pi }{3}}\sin \alpha {\sec }^3{\displaystyle \frac{\alpha }{2}}\left(3\sin {\displaystyle \frac{\alpha }{2}}+\sin {\displaystyle \frac{3\alpha }{2}}\right) \\ +K_{\mathrm{B}}R\times {\displaystyle \frac{\pi }{3}}\sin \alpha \left(12\alpha +{\sec }^4{\displaystyle \frac{\alpha }{2}}\sin {\displaystyle \frac{\alpha }{2}}-38\tan {\displaystyle \frac{\alpha }{2}}+12\tan {\displaystyle \frac{\alpha }{2}}\ln {\displaystyle \frac{\cosh \beta +\cos \alpha }{1+\cos \alpha }}\right) \end{gathered}$$

(2)

where R is the tactoid radius (the radius of curvature of the lens-shaped tactoid boundary) and 2α is the central angle (Figure 5 inset), so that $L=R\sin \alpha$ (see Equation S15). The last term in the bend energy diverges since β goes to infinity at the lens rim, favoring the escape of bend deformation at the lens rim as observed in the FFTEM images (Figure 4d and Figure S14). This behavior is different from what is observed in spindle-shaped tactoids formed by rod-shaped particles, where point defect boojums are formed [35]. The escape region can be considered the core of a bend disclination that runs around the rim of the tactoid. When the core dimension is small compared with the volume filled by the defect, a common situation in liquid crystals, it is generally possible to solve the Frank elastic problem in the bulk (in this case, combined with the surface structure problem; see Supporting Notes) by treating the defect as a line singularity with a certain tension, determining β.

Assuming K_S/K_B~14, a value suggested by simulations of thin, hard discs with large order parameter [26], the fitting of L(α), which is given in detail in Figure S15, gives K_B/γ ~ 1.23 µm, where γ is the surface tension. This value is consistent with the lower bound of K_S/γ > 13.5 µm calculated from the largest lens-shaped tactoid (L = 21.1 µm) observed in our experiments. As discussed above, the formation of tactoids is driven by the competition between bulk elastic energy and surface energy. In our samples, the interfaces of water-rich and ZrP-rich domains in low-concentration suspensions have arbitrary shapes, and there are no tactoids, implying that the surface tension is small, and the elastic energy dominates. Tactoids start to appear at high concentrations, where the surface energy starts to dominate. Therefore, the bare surface tension probably increases faster with concentration and order parameter than K/R.

On the other hand, the tactoid aspect ratio L/h depends on α, as $L/h=\cot \left(\alpha /2\right)$. The tactoid size and aspect ratio are therefore related by α, as shown in Figure S15 and Figure 5. The model is insensitive to the value of K_B, i.e., the L/h curve remains essentially unchanged even in the case of a vanishingly small bend elastic constant K_B, indicating that large bend deformation is allowed. Therefore, bend deformation is truly dominant in the monolayer ZrP/TBA nanosheet system. In addition, while the tactoids in our suspensions have an equilibrium lens shape, they appear to have a kinetically determined size, indicating that a kinetic phenomenon that may resemble emulsion polymerization [36] determines the size distribution of the well-formed tactoids.

Having characterized the LC/Iso phase separation and observed by FFTEM that the LC phase at high concentration is nematic, it is of interest to return to the X-ray structure factors of Figure 1a, asking whether they are consistent with nematic ordering. The ZrP nanosheets are very thin, so where the quasi-lamellar spacing is large, the form factor is nearly flat in q. At higher concentrations, the form factor reduces the higher harmonic peaks somewhat, but the lower harmonic peak shapes are not substantially influenced by this. The FWHM values plotted in Figure S16 show that the X-ray correlation length varies from 100 Å at low ϕ_sample to 400 Å at large ϕ_sample. These lengths are much smaller than the biphasic domain sizes, at least in the larger ϕ_sample preparations where they can be visualized (Figure 2, Figure 4, Figures S10 and S11). This means that the X-ray structure is intrinsic to the high-concentration phase and not determined by the domain morphology. This is also clear from the weak centrifugation results of Figure 1b: the inset shows that although the centrifugation completely changes the biphasic domain morphology by squeezing out most of the free water, the X-ray line shape and the d-spacing are not affected during this process.

The multi-peak arrays of quasi-Bragg harmonic reflections are, at first glance, suggestive of smectic ordering but, as noted above, the width of these harmonics increases approximately linearly with the diffraction order, which is not consistent with the observed behavior of smectics, where the exponent of the power-law decay of the scattering wings depends quadratically on the diffraction order. Briefly, in clay suspensions, the principal deviation from perfect lamellar ordering is not the relative mean square displacement of flexible layers growing quadratically with separation along z, as in the de Gennes/Caillé description of smectics [37,38], but rather the smooth splay in the orientation of the planes of flat layers, as described in Figure 4d–f, that is the dominant structural theme of the tactoids. In this case, the relative layer displacements grow linearly with separation, leading to the width of the harmonics increasing linearly with the order. This feature of the tactoids turns out to be characteristic of the scattering in many clay systems, including ones that do not phase separate. This phenomenon will be addressed in a future paper.

3. Conclusions

Aqueous suspensions of monolayer ZrP/TBA nanosheets with high aspect ratio phase separate into liquid crystalline, nanosheet-rich and isotropic, water-rich domains at overall volume fractions ϕ_sample > 0.5%. Optical and FFTEM observations indicate that the LC phase internal to the LC domains is nematic with strong columnar correlations at intermediate volume fraction, transitioning to nematic order with strong, short-range lamellar correlations in high-volume-fraction suspensions. Lens-shaped nematic tactoids with non-uniform director field form spontaneously in concentrated suspensions. Strong anchoring at the tactoid boundaries results in splay and bend deformation of the internal LC director field in a geometry that indicates a large energy cost of splay deformation relative to that of bend (K_S >> K_B). Weak bend elasticity, observed in simulations of suspensions of thin discs at high volume fraction [26], is entirely inconsistent with the development of smectic ordering (or preordering, for that matter), which must involve the expulsion of bend deformation because of the divergence of K_B. The nematic structure of the tactoids and the observation of diffuse lamellar X-ray scattering at all concentrations up to ϕ_high ~ 70% suggest that smectic ordering is not a feature of the phase diagram of colloidal suspensions of high aspect ratio ZrP monolayer nanosheets.