Experimental System.

The isotropic–nematic phase separation of colloidal suspensions of fluorescently labeled sepiolite clay particles was studied by using confocal imaging of samples with concentrations in the coexistence range. The phase diagram of our system was determined before confocal analysis, and, from this, we extracted the range of concentrations of interest. Here, we used the effective bulk rod volume fraction ϕ as a measure of the concentration, which was determined from the number density of rods and the mean rod volume. Rods were taken to be cylinders of length equal to that measured with transmission electron microscopy (TEM) plus 2δ, and diameter equal to the width measured plus 2δ. Here, δ=4 nm, the length scale of steric stabilization. The number density was determined from the mass fraction and rod mass density.

Confocal images of the system were obtained with a Leica SP5 confocal microscope using a white-light laser emitting at 500 nm. Borosilicate glass capillaries with cross-sections of 1×0.1 mm were filled with rod suspensions at effective volume fractions of ϕ=0.021,0.026,0.031,0.043,0.054,0.066 and glued to microscope slides with epoxy. The samples were stirred by using a vortex mixer for 1 min before filling the capillaries. Our analysis focuses on the time evolution of the phase separation, for which we have defined t=0 as the moment when the stirring stops. We presume that the vortex mixing leaves the system in an isotropic state and that t=0 is the start of the demixing.

Computer Simulation.

We modeled the experimental system using rods with a steep repulsive interaction. We used molecular dynamics (MD) and neglected the effect of solvent-mediated hydrodynamic interactions. Significant though these are (22), here, we chose to focus our computational resources on simulating as large a system size as possible, so that the length scales and timescales may be comparable to the experiments. Our systems consisted of up to 180,000 rods with polydispersity only in length. The length distribution of our model rods is described by a Gaussian with an average rod length of L=24.48σ±9.06σ. Three different global volume fractions were explored here: ϕ=0.012 (isotropic), ϕ=0.145 (coexistence), and ϕ=0.357 (nematic). For ϕ=0.145, we found a phase coexistence with around 40% of the rods in a highly aligned nematic phase. The snapshots shown correspond to a system of 180,000 rods with a volume fraction ϕ=0.140 which corresponds to phase coexistence. Further details can be found in Materials and Methods and SI Appendix.

Phase Behavior.

We present the phase diagram of our system in Fig. 2. Here, we considered the fraction of nematic phase fnem as a function of (effective) volume fraction ϕ obtained from bulk observations of phase coexistence (Materials and Methods). The key point is that the isotropic–nematic phase coexistence was very substantially broadened due to polydispersity, as indicated in the yellow shaded region in Fig. 2. Such broadening due to polydispersity is in quantitative agreement with theoretical predictions for hard rods (21, 27). The (effective) volume fractions of the isotropic and the nematic phase in the coexistence were calculated by fitting the experimental data in Fig. 2 to a straight line. The ratio of the volume fractions at coexistence for our polydisperse system is ϕnemcoex/ϕisocoex=0.215/0.024=9.00, while that for a monodisperse system of very similar aspect ratio L/D=25 is just ϕnemcoex/ϕisocoex=0.157/0.127=1.22 (28). Thus, polydispersity massively increases the density difference between the isotropic and nematic phases.

This quantitative agreement with the theoretical phase behavior for hard rods suggests that attractions are not important, but residual attractions cannot be ruled out. However, significant attractions along the length of the rods cause bundling (29, 30), while “patchy” attractions lead to the formation of a random network of rods (30). While it is hard to be certain of the absence of attractions, since either bundling or a random network would suppress the isotropic–nematic transition, the occurrence of the transition suggests that, for our system, neither effect is dominant. We therefore conclude that any attractions are weak. This is quite different from conventional gelation in spheres, where attractions drive gelation (1, 14).

A further interesting observation concerns the shape of the nematic regions, as shown in the confocal image in Fig. 2B. Here, the contrast is due to the much higher concentration of rods in the nematic compared to the isotropic phase, and the brightness levels were set such that isotropic appeared dark. Nucleating nematic droplets were expected to be elongated in shape, approximately elliptical, but with sharp ends (i.e., tactoids) (31), as has been observed in experiments on more monodisperse systems than those we consider here (32, 33). Indeed, the morphology of the bicontinuous network was consistent with tactoid-like objects having fused together. Thus, we cannot rule out the formation of tactoids and their subsequent coalescence prior to imaging. Note also that since the rod width was subresolution, we did not obtain the local director field in our experimental data.

Spinodal Decomposition.

Allied with the observation of morphology distinct from that of (isolated) tactoids anticipated in the case of nucleation and growth, we found that the isotropic–nematic phase separation occurs in a spinodal-like fashion. Even at the shortest observation time accessible to our experiments (45 s from stopping vortex mixing) and at the weakest supersaturation (ϕ = 0.021), we never observed nucleation and growth. That is to say, we did not find nucleation of nematic regions; these had always formed prior to our shortest observation time.

Gels often exhibit a bicontinuous texture. In Fig. 3A, we show a three-dimensional (3D) rendering of regions identified as nematic. We have confirmed that the regions identified as the nematic phase indeed percolate in all three dimensions and thus concluded that the percolation requirement for gelation is met. Close inspection of data such as that rendered in Fig. 3A suggests some alignment of the nematic domains. We believe this to be related to the capillary into which the sample is flowed for imaging. Such alignment may present an opportunity to produce networks whose orientation may be controlled.

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Fig. 3.

Polydisperse hard rods exhibit the properties required for spinodal gelation: a percolating network and dynamic asymmetry. (A) A 3D rendering of nematic domains identified from a confocal microscopy image of a sample with a bulk effective volume fraction ϕ=0.043 prior to phase separation. The image corresponds to t=993 s and shows a volume with dimensions of 58×58×35 μm. Colors denote connected domains. The blue domain percolates the 3D image. (B) Intensity time-correlation functions for the isotropic (iso) and nematic (nem) phases. The correlation function c(t) is detailed in the main text and shows that the nematic phase exhibits very much slower dynamics than does the isotropic. The lines are fits to a stretched exponential, from which we extract the relaxation time for each phase. Note that here we consider a coarse-grained description of the particle dynamics, as discussed in the text. EXP, experiment. (C) Computer simulation (SIM) data for intermediate scattering functions (which characterize particle motion). Here, we distinguish the orientation in the case of the nematic between parallel and perpendicular to the director. The wave vector k=0.789 is chosen to correspond to 240 nm, which is the pixel size in the experiments. The volume fractions are ϕ=0.012 and ϕ=0.36 for the isotropic and nematic data, respectively (46).

Dynamics.

We now turn to the dynamical asymmetry between the phases, which is a necessary ingredient for viscoelastic phase separation, i.e., spinodal gelation (2). We shall see that our system exhibits a rather unusual form of dynamic asymmetry, due to the anisotropic dynamics of the nematic phase. In both our experiments and simulations, we measured the dynamics in a system of either isotropic or nematic phase. In this way, we probed the dynamics of the bulk phase, rather than domains in the gel network. To measure the dynamical behavior in our experiments, we used a time-correlation function c(t), which measured the dynamics using pixel intensities (34). Because the width of the rods was subpixel resolution (a pixel corresponds to 240 nm), our analysis gave a coarse-grained measure of the dynamics rather than probing individual rods. In the nematic phase, where the particle separation was subresolution, a number of rods contributed to each pixel.

In Fig. 3B, we fit the time-correlation function according to a stretched exponential form c(t)=c0⁡exp[−(t/τ)b], where c0 and b are constants, to obtain a measure of the structural relaxation time. From our fitting, we determined relaxation times of τiso=270 ms for the isotropic and τnem=202 s for the nematic, thus indicating a considerable degree of dynamic asymmetry of three orders of magnitude. See Materials and Methods for further details. Our experimental data suggest that this dynamical asymmetry is comparable to the differences in dynamic properties of fast and slow phases in gels made of spheres (15).

Further insight was gained from simulation data shown in Fig. 3C, which suggest that the situation is profoundly different from the case of spheres, in which the dynamics have no preferred direction. Here, we compared the intermediate scattering function F(k,t) (SI Appendix) at global volume fractions ϕ=0.012 and ϕ=0.36, which correspond to the isotropic and nematic phase, respectively. Note that, unlike in the experiments which considered a coarse-grained dynamics, here, we considered the particle dynamics, but at a similar length scale with respect to the rod dimensions. We saw that the dynamics of the nematic phase was such that the relaxation time perpendicular to the director was around 10 times longer than that along the director. At concentrations in the coexistence region, this led to a long-lived network of flowing channels, promoting the lifetime of the network. The main result here is qualitative: In both experiment and simulation, nematic and isotropic phases have strong dynamical contrast. Thus, we argue that three key ingredients for gelation are present: spinodal decomposition leading to a bicontinuous (percolating) network of nematic and isotropic domains (criteria 1 and 3) and dynamical contrast between the two phases (criterion 2). By analogy to work with spheres (14), within each phase, we expected the molecular dynamics to be a reasonable description of the real dynamics. Below, we consider the time evolution (aging) of this nonequilibrium system, criterion 4.

Not all volume fractions in the simulations mapped to those in the corresponding experiments. In particular, we found the isotropic phase at somewhat higher volume fraction than in the experiments. We note that determining effective volume fractions in experiments is a challenging task (35), not to mention that the gels formed were, of course, out of equilibrium, so we would not necessarily expect a perfect mapping. We leave the accurate determination of the equilibrium phase diagram of this system to the future, noting that the network which forms is out of equilibrium and that the experiments and simulations do follow different dynamics, which can influence the network formation (36).

Coarsening.

A further key feature of gels formed by spinodal decomposition is that they coarsen over time, and this is governed by the dynamics of the more viscous phase (2, 15, 26). Our system was no exception, and in Fig. 4, we present the coarsening behavior at different volume fractions in the coexistence region (Materials and Methods). The confocal images and 3D renderings in Fig. 4 A and B reveal structural evolution of the nematic network for ϕ=0.043 at t=149 and 993 s from the start of the experiment. To obtain a quantitative description of coarsening, we determined a length scale from the domain size. To do this, we fit h(r)=g(r)−1 with an exponential decay h(r)=A⁡exp(−r/ξ), where A is a constant and ξ is a correlation length which measures the extent of the nematic domains. Here, g(r) is the pixel-based radial distribution function (SI Appendix). Note that we have imposed a spherically symmetric length scale on a system of anisotropic particles. The length scales resulting from fitting of h(r) are shown in Fig. 4D for a range of rod volume fractions. At early times, for our deepest quench (ϕ=0.066), the initial growth rate had an exponent >1/2 (solid line in Fig. 4D). This is faster growth than for gels formed of spheres (15, 37), but at longer times and for weaker quenches, the growth rate was reduced, and for certain values was compatible with that of diffusive growth of 1/3 (dashed line in Fig. 4D). We carried out a similar analysis for the simulation data at ϕ=0.140, which is in the coexistence region (Fig. 1B). The correlation functions h(r) here were based on the correlations of the rod centers and fitted with exponentials to obtain a correlation length ξ. These lengths are plotted in Fig. 4D with the time scaled to the experiments by using the relaxation time in the isotropic phase τI to fix the timescale for both simulations and experiments. We saw very similar time evolution between the simulations and experiments. This is supportive of the idea that any residual attractions in the experimental system are unimportant; however, our choice of dynamics in the simulations may affect the rate of coarsening. Overall, the clear observation of coarsening satisfies criterion 4 for the time evolution of gels.

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Fig. 4.

Time evolution of the system. (A and B) Confocal images for a bulk volume fraction of, i.e., prior to demixing ϕ=0.043, at 149 s (A) and 993 s (B) showing coarsening of the network. (Scale bars, 10 μm.) A, Inset, and B, Inset correspond to 3D renderings of each sample where connectivity is clearly shown. The 3D volumes have dimensions 58×58×58 μm. A, Inset, and B, Inset show renderings of the full 3D image. (C) Pair correlation functions h(r) plotted at the times shown in seconds for ϕ=0.066 (thick solid lines). These are fitted with a decaying exponential (dashed lines) as described in the text. (D) Quantifying coarsening with the correlation length obtained from pair correlation h(r) fitting. The gray solid line indicates an exponent of 1/2 and the dashed line 1/3, as indicated. Bulk rod volume fractions are indicated for the experimental data.