# Morphology selection kinetics of crystallization in a sphere

## Abstract

Crystallization under geometrical confinement is of fundamental importance in condensed matter physics, biophysics and material science. Even the influence of the simplest geometry, a sphere, on crystallization remains far from well understood, thereby making morphology control of the final superstructures challenging. Here, we employ charged colloids encapsulated in an emulsion droplet as a model system to access the crystallization kinetics at the single-particle level. We find rapid formation of ‘skin’ layers with an icosahedral arrangement of defects under the geometrical frustration effect, followed by interior ordering and slow ripening. The final morphologies are determined by dynamical interplay between the system-independent skin layer formation and the system-dependent structural transformation towards the most stable solid far from the surface. We reveal the crucial role of kinetics in morphological selection under a geometrical constraint, besides the thermodynamics, which may shed new light on the structural design of nanoscale crystals.

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## Data availability

Data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are available for this paper.

## Code availability

The computer codes used in this paper are available from the corresponding author upon reasonable request.

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## Acknowledgements

This work is supported by the National Natural Science Foundation of China (grant nos. 11774059, 11734014, 11504052 and 16Z103010253) and the Shanghai Rising Star programme (grant no. 16QA1400600). H. Tanaka acknowledges Grants-in-Aid for Scientific Research (A) (JP18H03675) and Specially Promoted Research (JP25000002 and JP20H05619) from the Japan Society for the Promotion of Science (JSPS). Z.Y. thanks the support from the Student Innovation Center at Shanghai Jiao Tong University.

## Author information

Authors

### Contributions

Y.C. performed the experiments and Z.Y. carried out the numerical simulations. All authors contributed to the data analysis. P.T. and H. Tanaka supervised the project and wrote the manuscript.

### Corresponding authors

Correspondence to Hajime Tanaka or Peng Tan.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Confocal raw images and the rotational motion of emulsion droplets.

a, ICO type, FCC type and BCC type final structure corresponding to Fig. 1 of the main text. Images are captured by the 40X objective with 1.3 NA (Numeric Aperture) and 0.24 mm working distance. The image distortions caused by the drift is small in the x-z plane at a z scanning speed 10μm/s. b, Typical rotational drift as a function of r/R0 during the ICO-type structure formation, which corresponds to Supplementary Movie 1. c, The rotation drift as a function of r/R0 in the FCC-type system, which corresponds to Supplementary Movie 2. The droplets rotate as a whole very slowly, probably due to the flow of the surrounding fluid.

### Extended Data Fig. 2 Experimental details and histograms of the final structures at various κσ.

a, Illustration of the experimental procedure of the sample preparations. Initial κσ of each samples is controlled by vibrating CHB solvent with an ultrasonicator. The equilibrium κσ of each sample, which produces the three final structures shown in the main text, is adjusted by the volume ratio between water and glycerol. The condition R0/a and the colloid volume fraction ϕ, where we find the three final structures, is illustrated in b for global icosahedral organizations, in c for single-crystal FCC core organizations and in d for single-crystal BCC core organizations. e, Histogram for κσ ~ 3.2 with a volume ratio of water:glycerol=2.5:7.5. The ICO-type final structure dominates in a range of geometric constraint 6.3 < R0/a < 17.8. f, Histogram for κσ ~ 3.0 with a volume ratio of water:glycerol=3:7. The ICO-type final structure dominates in a range 6.7 < R0/a < 17.6. Small amounts of FCC-type appear in a range of 10.0 < R0/a < 14.7. g, Histogram for κσ ~ 2.5 with a volume ratio of water:glycerol=4:6. FCC single crystal is the most favorable structure in a range of 9.6 < R0/a < 22.7. No complete ICO is observed. h, Histogram for κσ ~ 2.2 with a volume ratio of water:glycerol=1:1. BCC single crystal and polycrystal dominate.

### Extended Data Fig. 3 Structural order identification and distortions in the system.

a, Illustration of the identified BCC solids using the q4q8 bond orientational order parameter map and the bond orientational order diagram. b, Illustration of the identified HCP and FCC solids using the same approach as in a. c, $${\overline{q}}_{4}$$-$${\overline{q}}_{6}$$ distribution map of the initial crystallization taking place in the curved shell layers. The colours represents the scaled distance to the sphere center, r/R0. d, The radial distribution function of BCC solid at κσ ~ 3.0 and the real-space illustration of the local structure. The distorted BCC 110 plane is close to a hexagonal layer. The blue and green spheres represent the upper and lower layers, respectively. e, f, The radial distribution function of FCC and HCP solid at κσ ~ 3.0 and the real-space illustration of the local structures. We note that the distortion is small.

### Extended Data Fig. 4 Lattice mismatch in systems with and without image-charge effect.

a, Lattice-constant mismatch from the droplet centre to the surface as a function of r/R0 for the three morphologies in our emulsion systems. A particle layer (S0) is tightly bound to the surface due to the image-charge effect. Large-lattice mismatch is observed between S0 and the first shell layer (L1). b, Lattice-constant mismatch measured in a round glass capillary tube with a diameter of 50 μm, in which the image-charge effect is negligibly small.

### Extended Data Fig. 5 Change of the crystalline structure and lattice-constant mismatch caused by the long-range interaction in simulations.

a, Configuration (cross-section) of FCC crystals under geometric constraint at various κσ. Particles are propelled to the surface as the decrease of κσ. b, Configuration (cross-sectional view) of BCC crystals under geometric constraint at various κσ. We find a similar trend of the surface densification. c, Lattice-constant mismatch from the centre to surface as a function of r/R0 at various κσ for FCC crystal under spherical constraint in simulations. Decrease of κσ causes the propagation of the lattice mismatch from S0 to the inside. d, Lattice-constant mismatch for FCC crystal under spherical constraint in simulations. We find a similar trend as in FCC system.

### Extended Data Fig. 6 Macky and anti-Macky organizations in our systems.

a, Dependence of the number ratio of particles with HCP-type and FCC-type local orders, nHCP/nFCC, on r/R0. b, Illustration of the HCP layer rightly covering the icosahedral core in our system. c, Illustration of the icosahedral core in our system. It is composed of FCC tetrahedron grains, HCP twinning boundaries, and 5-fold disclination chains.

### Extended Data Fig. 7 The BCC-to-RHCP conversion.

The sliding motion of BCC 110 plane mainly occurs along a relatively flat plane, which creates FCC and HCP layers perpendicular to the surface at κσ ~ 3.0.

### Extended Data Fig. 8 Layering of the shell caused by the surface.

The shell and surface have a similar spherical geometry independent of their lattice-constant mismatch during the whole crystallization process, as shown by the cross-section view of six different samples.

### Extended Data Fig. 9 Lattice mismatch and structural correspondence between the surface and shell.

a, Illustration of strong anchoring of the surface defects with defective structures on the shell when the lattice-constant mismatch is small (6.5%). b-d, Illustration of the reduction of structural order correspondences between the shell and surface as the lattice-constant mismatch becomes more significant. As shown in b-d, a larger sphere has a weaker structural order correspondence between the shell and surface.

### Extended Data Fig. 10 Competing of 5-fold twinning with single-twinning in FCC system.

The core of the configuration has only one 5-fold twinning axis. a, Top view. b, Side view.

## Supplementary information

### Supplementary Video 1

Cross-section view of structural evolutions towards ICO type at the nucleation and growth stage. The red spheres indicate FCC solid, the orange spheres indicate HCP solid, the blue spheres indicate BCC solid, and the small green spheres are liquid particles.

### Supplementary Video 2

Cross-section view of structural evolutions towards FCC type at the nucleation and growth stage. The red spheres indicate FCC solid, the orange spheres indicate HCP solid, the blue spheres indicate BCC solid, and the small green spheres are liquid particles.

## Source data

### Source Data Fig. 1

3D configurations of colloidal particles.

### Source Data Fig. 2

Time sequential configurations.

### Source Data Fig. 3

3D configurations of colloidal particles.

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Chen, Y., Yao, Z., Tang, S. et al. Morphology selection kinetics of crystallization in a sphere. Nat. Phys. (2020). https://doi.org/10.1038/s41567-020-0991-9