Phase Field Modeling of Dendritic Growth on Spherical Surfaces

Abstract

The spontaneous formation of a crystal phase is one of the most common and beautiful pattern formation mechanisms in nature. Different instabilities in the crystal interface may lead to the growth of ramified structures, known as dendritic crystal growth. In this work, we use a Phase Field Model and numerical simulations to study 2D dendritic growth on curved surfaces. We show how the degree of ramification of a growing nucleus is modified by the underlying curvature of the substrate.

Nanocrystal Assemblies: Current Advances and Open Problems

“…The sampling of surface energetics via kinetic Monte Carlo (KMC) to grow and dissolve atoms in the crystal lattice circumvents the first two issues mentioned. ,,,− The trade-offs of KMC are (i) not predicting full trajectories but only the most probable intermediate states of the kinetic pathway of growth, therefore lacking in a complete description of diffusion, which is key for crystallization; (ii) considering a perfect lattice, thus lacking information on strain accumulation due to the formation of defects and their displacement, or lattice mismatch when growing over a seed composed of a different metal, both related to symmetry-breaking of NC; and (iii) the consideration of a homogeneous and constant chemical potential of the solution over the entire NC surface, thus not capturing (iii.a) ligand adsorption depending on the crystalline direction of each facet, which relates to transformation between different Wulff shapes, , nor (iii.b) heat and mass transfer-limited crystallization – that relates to symmetry breaking by a inhomogeneous precursor supply to the facets of the growing NC. , Inserting such effects into the prediction of growth and dissolution rates of KMC models while keeping the low computational cost nature of the method is an important challenge to enhance the physical chemistry prediction capabilities of a framework able to handle the growth of realistic-sized NCs.…”

Higher order time-implicit bounds preserving discontinuous Galerkin discretizations for partial differential equations

“…At present, using the phase field method [78,97], the Allen-Cahn equation has been widely used to model many complicated moving interface problems, such as the process of phase separation of a binary alloy at a fixed temperature [32,46], the mixture of two incompressible fluids, phase transitions and interfacial dynamics in materials science [6,32]. In particular, special phase separations may appear on static and dynamic surfaces, such as phase separation on lipid bilayer membranes [63,128] and dendritic crystal growth on curved surfaces [93]. We will analyze in this thesis the stability and prove optimal error estimates for higher order accurate semi-implicit numerical discretizations of the Allen-Cahn equation.…”