## Abstract

Crystallization of concentrated colloidal suspensions was studied in real space with laser scanning confocal microscopy. Direct imaging in three dimensions allowed identification and observation of both nucleation and growth of crystalline regions, providing an experimental measure of properties of the nucleating crystallites. By following their evolution, we identified critical nuclei, determined nucleation rates, and measured the average surface tension of the crystal-liquid interface. The structure of the nuclei was the same as the bulk solid phase, random hexagonal close-packed, and their average shape was rather nonspherical, with rough rather than faceted surfaces.

The study of the structure, growth, and properties of crystals is one of the most important areas of solid state physics (1). Although a great deal is known about the behavior of bulk crystals, considerably less is known about their earliest stage, when they nucleate and grow from a liquid. Homogeneous nucleation occurs when small crystalline regions form from structural fluctuations in a liquid cooled below its freezing point. The growth of these regions depends on a competition between a decrease in bulk energy, which favors growth, and an increase in surface energy, which favors shrinkage. The smallest crystals continually form by fluctuations but then typically shrink away because of the high surface energy. Growth becomes energetically favorable only when the crystallites reach a critical size. The competition between the surface energy and bulk energy is reflected in the free energy for a spherical crystallite(1)where *r* is the radius, γ is the free energy of the crystal-liquid interface per unit area, Δμ is the difference between the liquid and solid chemical potentials, and *n* is the number density of particles in the crystallite (2). The size of the critical nucleus is *r*
_{c} = 2γ/(Δμ*n*), corresponding to the maximum of Δ*G* (Eq. 1). Despite its crucial role, very little is known about this nucleation process, primarily because of the difficulty of directly observing the nuclei in real space.

No experiment has ever directly measured the size of the critical nucleus. It cannot be determined theoretically—even the surface tension is unknown. Furthermore, an additional important yet unresolved question involves the internal structure of critical nuclei, because nucleation need not occur via the stable bulk phase (3). General arguments based on symmetry suggest that nucleation may proceed by a body-centered cubic (bcc) structure as an intermediate phase, with the final equilibrium solid phase having a different symmetry (4). Computer simulations of a Lennard-Jones system found critical nuclei with a face-centered cubic (fcc) core and a predominantly bcc surface layer (5). However, light-scattering measurements from hard spheres (6–8), which provide an average over crystalline regions of all sizes, suggest that nucleation occurs through the random hexagonal close-packed (rhcp) phase, a random mixture of hexagonal close-packed (hcp) and fcc-like stackings of planes with hexagonal order, and this is supported by recent computer simulations (9). The important questions about the structure and size of the nucleating crystallites can best be resolved by direct experimental observation in real space.

The small length and time scales that characterize homogeneous crystal nucleation have to date precluded this direct real-space observation of critical nuclei in atomic or molecular systems. By contrast, however, colloidal systems can be studied directly, because their larger size and concomitant slower time scale makes them much more experimentally accessible. Colloidal particles can serve as good models for atomic or molecular materials; they show an analogous phase diagram (10), and the volume exclusion that results in crystallization of colloidal particles also plays a dominant role in traditional crystallization (2). Concentrated suspensions of hard-sphere colloids crystallize because of entropy (11): The free volume for ordered spheres (maximum packing volume fraction φ = 0.74) is greater than that of disordered spheres (maximum packing φ ≈ 0.64), and thus, the entropy is higher in the crystalline state. The phase behavior is determined solely by the volume fraction φ, with freezing occurring at φ_{f}
^{HS} = 0.494 and melting occurring at φ_{m}
^{HS} = 0.545 (HS, hard sphere) (12). Weakly charged colloids behave in essentially the same manner, but the boundaries of the phase diagram are shifted to slightly lower φ values (11). Colloidal (6–8, 13–16) and protein (17) systems have been valuable for studying crystal nucleation, but these studies did not directly visualize the critical nuclei and thus were unable to directly observe their properties.

We report on three-dimensional real-space imaging of the nucleation and growth of nearly hard-sphere colloidal crystals. We observed the formation of the smallest nuclei, which shrink far more frequently than they grow; by measuring their size distribution, we were able to directly determine the surface tension of a colloidal crystal. We were also able to determine the size of the critical nuclei by observing when the probability to grow exceeds the probability to shrink. We found that the structure of critical nuclei is rhcp, the same as that of the bulk. The critical nuclei have rough surfaces, and their average shape is ellipsoidal.

We used poly(methyl methacrylate) spheres sterically stabilized with poly-12-hydroxystearic acid (18, 19), with a radius *a* = 1.26 μm and a polydispersity of <5%. They were suspended in a mixture of decahydronaphthalene and cyclohexylbromide, which closely matches both the refractive index and the density of the particles. To make the particles visible with fluorescence microscopy, we dyed them with rhodamine, which gives the particles a small charge, resulting in freezing at φ_{f} = 0.38 and melting at φ_{m} = 0.42 (20). We used a fast laser scanning confocal microscope, which allowed us to measure the three-dimensional arrangement of colloidal particles in suspension (21, 22). We observed a 58 μm by 55 μm by 20 μm volume, containing ∼4000 particles. The total volume of our samples was ∼50 μl, and they contained a short piece of paramagnetic wire, which we agitated with a magnet to shear melt the sample before observation. Heterogeneous nucleation at the walls of the sample chamber was suppressed by fusing colloidal particles of a smaller size onto the walls before adding the colloidal samples. Further, we focused at least 8 and typically 15 particle diameters from the walls to ensure that we studied bulk homogeneous nucleation.

We observed the growth of crystallites by extracting the particle centers from the raw images with an accuracy of ∼0.04 μm (23). From these particle positions, crystalline regions were identified with an algorithm that finds ordered regions, independent of any particular crystal structure (5,24). It relies on the assumptions that the neighbors of a particle in a crystal lattice are arranged in a particular orientation about the particle and that this orientation of neighbors is the same for nearby particles. Quantitatively, this is achieved by calculating local bond-order parameters (25) that are based on spherical harmonics *Y _{lm}
* and are a measure of the orientation of the neighbors around a particle. Two adjacent particles with similar orientation of their neighbors have similar bond-order parameters and are therefore said to be joined by a crystal-like “bond” (26). Even in the disordered liquid, crystal-like bonds are not uncommon; thus, only particles with eight or more crystal-like bonds are defined as being crystal-like. All particles in a crystallite with perfect fcc, hcp, rhcp, or bcc structure are correctly recognized, whereas only an insignificant number of particles in the liquid are found to be crystal-like.

At the beginning of an experiment, samples started in the metastable liquid state, but because of random structural fluctuations, subcritical nuclei of crystal-like particles were present. This is shown in two early-time snapshots (Fig. 1, A and B), where we represent crystal-like particles as red spheres and liquid-like particles as blue spheres, shown with a reduced diameter to improve visibility. Typically, these subcritical nuclei contained no more than 20 particles and shrank to reduce their surface energy. After a strongly φ-dependent period of time, critical nuclei formed and rapidly grew into large postcritical crystallites (Fig. 1, C and D). By following the time evolution of many crystallites, we determined the size dependence of the probabilities *p*
_{g} and*p*
_{s} with which crystallites grow or shrink (27). Because *p*
_{g} =*p*
_{s} at the critical size, we plot the difference*p*
_{g} – *p*
_{s} as a function of crystallite radius and particle number *M* in Fig. 2 for a sample with φ = 0.47. We found an abrupt change from negative to positive values of*p*
_{g} – *p*
_{s} (28), allowing us to identify the critical size, which is 60 <*M* < 160, in good agreement with recent computer simulations (9). This corresponds to*r*
_{c} ≈ 6.2*a*, assuming a spherical nucleus. The volume fraction of the nuclei is larger than the φ value of the fluid; above coexistence, the difference is Δφ = 0.012 ± 0.003, independent of φ, where Δφ increases slightly for *M* > 100. We can understand this Δφ value as resulting from the higher osmotic pressure exerted by the fluid on the nuclei (16), whereas in the coexistence regime, Δφ must reflect the evolution of φ to the higher value, ultimately attained by the crystallites, where Δφ = φ_{m} − φ_{f}. The nucleation rate densities were slower than 5 mm^{−3}s^{−1} for φ < 0.45, as well as for φ > 0.53. Values of the order of 10 mm^{−3} s^{−1} were found for 0.45 < φ < 0.53. However, for 0.47 < φ < 0.53, the average size of the nuclei began to grow immediately after shear melting; thus, there was little time for the sample to equilibrate after shear melting, and we were not able to observe the formation process of critical nuclei entirely. Our nucleation rate densities are of the same magnitude as values obtained by small-angle light scattering from hard spheres (29, 30) but are two orders of magnitude larger than those obtained from Bragg scattering (31).

The direct imaging afforded by confocal microscopy enabled us to determine the structure and shape of individual nuclei. In Fig. 3, A through C, we show a crystallite that is slightly larger than the critical size. Again, the red spheres represent crystal-like particles, and the blue ones depict particles in the liquid state that are on the surface of the crystallite and have at least one crystal-like bond with a red particle. The cores of the crystalline nuclei are formed from hexagonal layers, as shown by the cut through the center of this crystallite (Fig. 3D), and their surfaces are more disordered. The interface between the highly ordered cores and the amorphous liquid exterior is not sharp; this decrease in order from the interior to the exterior is in agreement with computer simulations of a Lennard-Jones system (5) and density functional calculations for hard spheres (32).

To determine the crystal structure of the ordered cores, we used the rotational invariants *q*
_{4}(*i*),*q*
_{6}(*i*), and*w*
_{6}(*i*) (33), which are quantitative measures for the local order around particle *i*, and distinguished between fcc, hcp, bcc, and liquid (5, 9,24). Because of thermal fluctuations, even homogeneous crystals have a distribution of these parameters. Nevertheless, these distributions are distinct for every crystal structure, as shown by the calculations (34) in Fig. 4A. These are compared to measurements of experimental crystalline nuclei, shown by a black curve in Fig. 4B. The blue curve is a least squares fit with the distributions from Fig. 4A, and the fit parameters are summarized in Table 1. During the nucleation process, bcc order is completely absent; in particular, as shown in Fig. 4, the*q*
_{4} distribution for bcc is markedly different from the measured distribution. Instead, the main contributions to the structure of the experimental crystalline nuclei are due to fcc, hcp, and liquid. The mixture of fcc and hcp is the signature of rhcp structure and is consistent with the stacked hexagonal planes in Fig. 3D. The liquid contribution in the fit is mainly due to the surface of the crystallites. These results are in agreement with those expected for hard spheres, which also crystallize in the rhcp structure (7), confirming that the critical nuclei are formed only from the bulk solid phase, rather than forming from a different metastable solid phase.

In classical nucleation theory (2), it is assumed that the nuclei have a spherical shape due to surface tension. To the best of our knowledge, this assumption has never been experimentally verified. As shown by the example in Fig. 3, our data indicate that the crystallite surfaces were quite rough, suggesting that the surface tension is low. Consistent with this, the crystallite shapes appeared rather nonspherical. To confirm this, we approximated each nucleus with an ellipsoid and found that the average nucleus is best described by an ellipsoid whose major axes have the ratio 1:1.8:3.2. Nuclei containing >50 particles were found to be somewhat less anisotropic than smaller ones, but on average, the ratio of the shortest and longest ellipsoid axes was never higher than 0.65 ± 0.15 for nuclei of all sizes. This suggests that the effects of the nonspherical shape could be included in any future modification of the theory. Furthermore, we found that the radius of gyration *r*
_{g} of crystallites was related to the number of particles *M* within each crystallite as *M*(*r*
_{g}) ∝*r*
_{g}
^{df}with the fractal dimension *d*
_{f} = 2.35 ± 0.15 for all values of φ; the fractal behavior presumably reflects the roughness of their surfaces.

The interfacial tension between the crystal and fluid phases is a key parameter that controls the nucleation process, yet γ is difficult to calculate (1, 2) and to measure experimentally, but with our data, we can directly measure γ by examining the statistics of the smallest nuclei. For *r* ≪*r*
_{c}, the surface term in Eq. 1 dominates the free energy of the crystallites, and consequently, we expect the number of crystallites to be *N*(*A*) ∝ exp[–*A*γ/(*k*
_{B}
*T*)], where *A* is the surface area, which we approximate by an ellipsoid. In Fig. 5, we show that this proportionality does indeed hold, allowing us to obtain values for the surface tension. As shown in the inset, γ ≈ 0.027*k*
_{B}
*T*/*a*
^{2}and may decrease slightly with increasing φ values. This value of γ is in reasonable agreement with density functional calculations for hard spheres and Lennard-Jones systems (32). By contrast, an older density functional study (35) and computer simulations of hard spheres (9, 36) gave results approximately four times as large. Our measurement of a low value of γ is consistent with the observed rough surfaces of the crystallites; this may reflect the effects of the softer potential due to the weak charges of our particles. Approximating the critical nucleus as an ellipsoid, with *M*
_{c} ≈ 110, we obtain*A*
_{c} = 880 μm^{2}, Δμ ≈ 0.13*k*
_{B}
*T*, and Δ*G*(*A*
_{c}) ≈ 7.4*k*
_{B}
*T*.

The direct observation of the structure, dynamics, and evolution of small crystalline nuclei allowed us to characterize the key processes of nucleation and growth of colloidal crystals and to quantitatively determine the important parameters that control this process. These results also provide crucial experimental input to help guide any future refinements of theories for nucleation and growth of colloidal crystals. Moreover, the ability to obtain three-dimensional movies of the dynamics of colloidal solids will allow the direct observation and study of other important phenomena, such as defect motion, effects of impurities, and crystallization of colloidal alloys.