## Curving Crystals

When a material with a different set of lattice parameters is grown on the surface of a crystal of a second material, the stresses at the interface can affect the growing crystal. **Meng et al.** (p. 634) studied the growth of colloidal crystals on top of a curved water droplet. Owing to the elastic stress caused by the bending of the crystal, strong distortions occurred in the growing crystal, but, nonetheless, large single-crystalline domains with no topological defects were formed.

## Abstract

Although the effects of kinetics on crystal growth are well understood, the role of substrate curvature is not yet established. We studied rigid, two-dimensional colloidal crystals growing on spherical droplets to understand how the elastic stress induced by Gaussian curvature affects the growth pathway. In contrast to crystals grown on flat surfaces or compliant crystals on droplets, these crystals formed branched, ribbon-like domains with large voids and no topological defects. We show that this morphology minimizes the curvature-induced elastic energy. Our results illustrate the effects of curvature on the ubiquitous process of crystallization, with practical implications for nanoscale disorder-order transitions on curved manifolds, including the assembly of viral capsids, phase separation on vesicles, and crystallization of tetrahedra in three dimensions.

Since Nicolaus Steno’s pioneering work on crystal growth in the 17th century (*1*), it has been established that the shape of a crystal is a vestige of its growth pathway. Near equilibrium, crystals grown from the melt form compact, faceted structures that minimize interfacial area and energy (*2*, *3*); further from equilibrium, kinetic instabilities (*4*) permit the formation of crystals with much larger interfacial areas, such as dendrites and snowflakes (*5*).

Less well understood is the role of elastic stress, which can arise from the curvature—or lack thereof—of the space in which the crystal grows. For example, two-dimensional (2D) crystals of spheres on a spherical substrate are strained because of the incompatibility of the preferred triangular lattice packing with the Gaussian curvature of the sphere, which bends the lattice lines. Similarly, in Euclidean 3D space, the absence of curvature frustrates the crystallization of tetrahedra (*6*, *7*). Large, compliant crystals can alleviate this curvature-induced elastic stress by incorporating topological defects such as grain boundary scars (*8*, *9*) or pleats (*10*) in the ground state. But for rigid crystals on curved manifolds, for which topological defects such as dislocations have large core energies, the increase of elastic stress with crystal size can also affect the growth process, such that the ground states may be inaccessible. The effects of curvature on the growth pathways are potentially important for analogous processes involving the ordering of identical subunits in curved spaces, such as the assembly of viral capsids (*11*), filament bundle packing (*12*), self-assembly of molecular monolayers (*13*), functionalization of nanoparticles (*14*), and the growth of solid domains on vesicles (*15*–*17*).

We use confocal microscopy to examine the structures of rigid 2D colloidal crystals growing on the inside walls of highly curved spherical water droplets (Fig. 1A). The particles start in the interior of the droplet, but within a short time nearly all of them attach to the droplet surface through depletion attraction (*18*). Once at the interface, the particles attract one another through the same interaction. The short range of the attraction creates a wide coexistence region (fig. S1) between a low-density 2D fluid and a rigid, brittle crystal phase that, unlike the 2D crystals made from repulsive particles (*8*–*10*), cannot easily deform to cover the entire droplet. Because the high interfacial tension of the oil-water interface precludes distortion of the enclosing spherical droplets, and because the depletion attraction confines particles to the droplet interface, the crystals are forced to adopt the curvature of the droplet as they grow.

This constraint has a marked effect on the crystal structures observed at long durations: Whereas 2D crystals grown on flat surfaces are compact, crystals grown on spherical surfaces are composed of slender, single-crystal segments that wrap around the droplets (Fig. 1B). In droplets with higher surface coverage, the thin segments, which we call “ribbons,” join together to form branched patterns with voids and gaps between them.

If these crystals were on flat surfaces, the voids could be filled by additional particles. But the parallel transport caused by the Gaussian curvature forces the crystalline directions to be mismatched at the void borders (Fig. 2A), making it impossible to continue the crystal without introducing topological defects. Such defects are, however, absent in our system, in contrast to curved crystals made from repulsive particles (*8*–*10*). As we show by digitally unwrapping the crystal structures (Fig. 2A), each crystal is a single grain, and the only defects are vacancies.

Morphologically, these structures resemble dendritic crystals. We quantify the morphology using two metrics: the circularity, a measure of the perimeter/area ratio, and the fractal dimension, a measure of anisotropy. In contrast to the compact crystals formed on flat surfaces, crystals grown on curved surfaces have much lower circularity (Fig. 2B and fig. S2) and fractal dimension (fig. S3).

In flat space, crystals with such large interfacial energies can only result from kinetic instabilities. Here, however, such instabilities are unlikely to be the cause of the anisotropy, as we do not observe dendritic crystals forming on flat surfaces under growth conditions that are otherwise identical to those of the curved surfaces (Figs. 1B and 2B). Kinetic instabilities occur when the diffusion of particles along the crystal-fluid interface is slow relative to the growth rate. This happens when the domain size is comparable to the Mullins-Sekerka wavelength λ_{s} (*19*, *20*), which is only weakly affected by curvature (*18*). We estimate λ_{s} to be at least 100 μm (*18*), much larger than the typical width (25 μm or less) of our curved crystals. We therefore exclude kinetic instabilities as the cause of the morphology. We can also exclude kinetic effects arising from fluid-fluid coexistence, as our system is far from the metastable fluid-fluid critical point (*18*).

The remaining possibility is an elastic instability. A continuum model shows that such an instability occurs because the crystal must compress as it grows larger, owing to the Gaussian curvature of the growth surface. Consider forcing a flat, disc-shaped crystal of diameter *a* onto a sphere with radius *R*. For domain sizes comparable to the sphere radius or smaller, the change in circumference—or, equivalently, the elastic strain—scales as (*a*/*R*)^{2}, so that the net free energy change, including the elastic energy cost, to form a circular solid domain on a curved surface is
(1)(*18*, *21*), where *Y* is the 2D Young’s modulus, γ is the line tension, and Δ*f* is the chemical potential difference between the coexisting solid and fluid phases. On flat substrates (*R* → ∞), the elastic energy vanishes, and crystallites larger than a critical nucleus size *a*_{c} = 2γ/Δ*f* can grow isotropically without limit (dashed line in Fig. 3A). But on curved substrates, the elastic energy increases with domain size *a*, as measured along a geodesic. Isotropic growth becomes unfavorable beyond a critical size *a** ~ (Δ*f* /*Y*)^{1/4}*R* (solid line in Fig. 3A). This restriction on isotropic growth is the origin of the elastic instability (*22*). The predictions of this simple model are consistent with our measurements of the maximum isotropic domain size, which scales linearly with *R* (fig. S4A).

Although the crystal could grow larger by incorporating topological defects (*23*, *24*), the short range of the attraction in our system makes such defects energetically costly. Geometry dictates that some interparticle distances near a five-fold or seven-fold defect are larger than the interaction range, effectively breaking the corresponding bonds. Incorporating a five-fold defect, for example, breaks five bonds. This explains the observations in Fig. 2: The stiffness of the potential favors tearing rather than stretching to accommodate stress.

Thus, if the crystal is to continue to grow, it must do so by increasing its perimeter/area ratio. It must grow anisotropically. More specifically, the crystal should transition from a disk to a ribbon, or multiple ribbons, when its size exceeds *a**. This behavior arises because the elastic energy of a ribbon scales with the fifth power of its width *w* but only linearly with its length *l* (*21*, *25*). The net free energy change of forming a ribbon-like crystalline domain is
(2)This energy function is shown as a landscape in Fig. 3B. By maintaining a constant width (set by the curvature in the later stages of growth), the crystallite can grow to arbitrarily large lengths, limited only by the number of particles and the area available for growth. The anisotropic growth allows the crystal to avoid the size restriction imposed by elastic energy at the modest cost of a larger interfacial energy.

To test this model, we measured the dynamics of growth in single domains (Fig. 4A and movie S1). We find that a crystal first grows isotropically until it reaches a critical size, then grows anisotropically, increasing its length while maintaining a much smaller but constant width. Examination of the final crystal shapes for several hundred droplets (Fig. 4B and fig. S4B) shows that the length of the domains can grow to several times the droplet radius, while the width is restricted to a fraction of the radius of curvature. The near-constant value of *w/R* seen in Fig. 4 qualitatively agrees with our model, which predicts that the elastic stress limits the width to *w* ∝ *R*, whereas *l* can increase without penalty. By fitting the model to the observed domain widths, we extract an effective spring constant for the interparticle interaction that is consistent with theoretical and independent experimental estimates (*18*), again lending support to the model.

The observation of the final, branched crystal shapes is also consistent with our physical picture. A domain can extend its length along any of the three crystallographic axes. If it changes its growth axis, it will bend by ±60°, and if it grows in two directions at once, a branch forms. All of these growth patterns are roughly equivalent energetically, as long as the width of the growing section remains less than the critical width dictated by elasticity and curvature. Hence, we expect to see, and do see, changes in direction and branches in the domains. The branches can also arise from the merger of crystallites that have nucleated independently. The branched structures resulting from such mergers maintain their curvature-dependent width *w* in each section (see movies S2 to S4). The voids in these structures persist because the crystal directions are mismatched in adjacent sections of the crystal. In contrast, crystallites on flat substrates have no width restriction or curvature-induced mismatch and can therefore easily merge into isotropic shapes.

We conclude that anisotropic crystal growth, usually a result of a kinetic instability, can also occur by slow growth under the geometric constraint of Gaussian curvature. Our results illustrate a generic route for the growth of rigid, defect-free structures in curved spaces. This route may be particularly relevant to nanoscale substrates, where the curvature is appreciable on molecular length scales. For example, solid domains consisting of narrow stripes radiating outward from a circular core have been observed in diverse systems, such as phases on lipid vesicles (*17*) and metal coatings on nanoparticles (*14*). Our analysis suggests that the width of the stripes and the core size should be determined by the interplay among curvature, elasticity, and bulk energy. A similar interplay may affect the assembly pathways of viral capsids. Recent in vitro experiments (*26*) show that capsids can assemble following a two-step mechanism analogous to our crystallization process: The capsid proteins first attach to a substrate (an RNA molecule) and then bind together into an ordered shell. The intermediate states of this process—long a subject of speculation (*11*)—might contain voids that, like those in our curved crystals, help the capsids avoid excess elastic stress (*27*, *28*).

Similar rules may govern other crystallization and packing problems where global geometry is incompatible with local lattice packing. For example, crystallization of tetrahedra is frustrated in flat (Euclidean) 3D space (*6*). However, logs and helices of tetrahedra are prevalent in first-order phase transitions of tetrahedra from disordered to dense quasicrystalline phases (*7*). The existence of these structures, which were first described by Bernal (*29*), may reflect a similar physical principle, in which growth along one dimension allows a crystal to escape the restrictions of geometrical frustration.

## Supplementary Materials

## References and Notes

- 1.↵
- 2.↵
- 3.↵
- 4.↵
- 5.↵
- 6.↵
- 7.↵
- 8.↵
- 9.↵
- 10.↵
- 11.↵
- 12.↵
- 13.↵
- 14.↵
- 15.↵
- 16.
- 17.↵
- 18. See supplementary materials on
*Science*Online. - 19.↵
- 20.↵
- 21.↵
- 22.↵
- 23.↵
- 24.↵
- 25.↵
- 26.↵
- 27.↵
- 28.↵
- 29.↵
- 30.↵
- 31.↵
- 32.↵
- 33.
- 34.
- 35.
- 36.
- 37.
- 38.
- 39.
- 40.
- 41.
- 42.
- 43.
- 44.
- 45.
- 46.
- 47.
- 48.
- 49.
- 50.
- 51.
- 52.
- 53.
- 54.
- 55.
- 56.
- 57.
- 58.↵
**Acknowledgments:**We thank F. Spaepen for helpful discussions. Supported by the Harvard Materials Research Science and Engineering Center through NSF grant DMR-0820484.