PHYSICAL REVIEW E 105, 014801 (2022) Two-dimensional crystalization on spheres: Crystals grow cracked Laureano Ortellado , Daniel A. Vega, and Leopoldo R. Gómez * Department of Physics, Universidad Nacional del Sur-IFISUR-CONICET, Av. Alem 1253, Bahía Blanca, Argentina (Received 22 September 2021; accepted 3 January 2022; published 13 January 2022) Here we study how curvature affects the structure of two-dimensional crystals growing on spheres. The mechanism of crystal growth is described by means of a Landau model in curved space that accounts for the excess of strain on crystal bonds caused by the substrate’s curvature (packing frustration). In curved space elastic energy penalization strongly dictates the geometry of growing crystals. While compact faceted crystals are observed when elastic energy contribution can be neglected, cracked crystals with ribbonlike forms appear as the main mechanisms to reduce elastic frustration for highly curved systems. DOI: 10.1103/PhysRevE.105.014801 I. INTRODUCTION The growth of a two-dimensional (2D) crystal layer on a planar substrate is dictated by the competition between an energy gain, obtained by forming a piece of the equilibrium crystal, and a line tension penalty due to the interface of the nuclei with the surroundings [1,2]. This free-energy compe- tition produces a critical size for crystal growth. Only those nuclei overcoming this critical size can grow by adding crystal particles to their surface. Sub-critical nuclei collapse by sur- face tension. In addition to this simple mean-field description of the nucleation and growth, some complexities may arise from different energy exchange mechanisms, limited parti- cle’s diffusion, and anisotropic nuclei line tension, between other effects, which can lead to a variety of dynamics and nuclei shapes [1,2]. On the contrary, it could be simply impossible to grow a perfect crystal on a curved substrate [3]. This is because the underlying curvature may induce distortions of the crystal lat- tice, increasing its strain energy (an effect known as geometric frustration) [4,5]. In this sense, depending on the underlying geometry, a crystal may need to be highly deformed to wrap on the curved surface, increasing thus the elastic energy of the lattice structure. For example, the free energy of a perfect circular crystal of size R growing on a sphere of radius a (spherical cap) has been modeled through the free energy [6,7]: F cap = 2π Rγ − π | f |R 2 + π 384 Y R 6 a 4 , (1) where Y is the two-dimensional Young’s modulus of the crys- tal,  f is the energy difference between crystal and melt, and γ is the line tension of the interface between the two regions. In this equation, the first two terms represent the com- petition between surface and line tension energies discussed above. The last term is a free-energy penalization induced by the substrate’s nonplanar geometry, which inhibits fur- * lgomez@uns.edu.ar ther growth of the nuclei beyond the equilibrium size R eq ∼ ( f /Y ) 1 4 a [7]. In a planar domain, it is also possible to frustrate a crystal under stress if the intrinsic curvature is associated with the interactions of the particles [8]. However, growing nuclei on curved surfaces could reduce the elastic frustration in two ways. One possibility is by the inclusion of topological defects while growing [9,10]. This is because topological defects can contribute to reducing elastic distortions on the lattice, allowing further propagation. The other possibility is by changing the shape of the crystal to ribbonlike or ramified structures, which has been recently observed in experiments with colloidal crystals growing in spheres [7], in Monte Carlo simulations of self-assembly of viral capsids [11], and phase field simulations [12,13]. In this work, we study the process of isothermal crystalliza- tion on spherical substrates by using a free-energy functional which takes into account the elastic stress induced by ge- ometric frustration. We show that elastic stress dictates the growth pathway. Growing crystals may change their shape and also crack, depending on the substrate’s curvature and lattice rigidity. II. MODEL In Landau’s theory of phase transitions, the free-energy functional of the system is expanded in terms of an ap- propriate order parameter (r ), which is mainly related to the underlying symmetries of the system [14]. In studies of crystallization, the complex order parameter  is commonly chosen as scalar function representing the local density and orientation of the material, and the dynamics of the phase transition can be studied through a relaxational equation of the form [15] ∂  ∂ t =−μ δF δ , (2) where μ is the mobility coefficient of the system, F depends on the details of the system studied, and δF/δ is the func- tional derivative of F in terms of the complex order para- meter . 2470-0045/2022/105(1)/014801(7) 014801-1 ©2022 American Physical Society