Crystallization from seed

The videos demonstrates the freezing (solidification) process of a small seed placed into an undercooled melt. The undercooled melt starts to solidify on the seed surface. The solidification process is accomplished by latent heat release. As a result, the melt starts heating up near the solid surface. These processes result in so called dendritic growth. The black line corresponds to the liquid/solid interface. The color codes the temperature (Red - hot, blue - cold).

The local growth rate is defined by heat transport efficiency. This explains the formation of dendritic structure because the heat withdrawal is efficient near sharp parts. The orientation of the side branches is defined by the crystal properties such as growth kinetics and surface energy. Here you can find specific cases corresponding to the so-called 4-fold and 6-fold symmetry meaning the existence of four or six preferred growth directions.

Dendritic growth from seed in 2D with 4-fold symmetry

Dendritic growth from seed in 2D with 6-fold symmetry

Comparison of the dendritic growth process for two seeds with different symmetries in 2D

Dendritic growth from seed in 3D with 6-fold symmetry

This symetry is a 3D analog of 4-fold symmetry in 2D.

Dendritic growth from seed in 3D with 14-fold symmetry

This symetry is a 3D analog of 6-fold symmetry in 2D.

Theory of solidification

The crystallization in pure materials is described by the movement of the interface and by the temperature field. This system is modeled by the so-called Stefan problem. A system consists of two phases, solid and liquid. They are separated by a sharp interface, \(\Upsilon\), which moves according to the local conditions. These two phases are described using different material parameters, e.g. heat capacities \(c_{ps}\), \(c_{pl}\) and heat conductivities \(K_s\), \(K_l\), where indices \(s\) and \(l\) stand for \(solid\) and \(liquid\), respectively. The phase-transition is defined through the melting-point temperature \(T_m\) and latent heat \(L\). The classic sharp-interface model of solidification in a pure material may be written as:

\[ c_{ps}\frac{\partial T_s}{\partial t} = \nabla[K_s\nabla T_s],\] \[ c_{pl}\frac{\partial T_l}{\partial t} = \nabla[K_l\nabla T_l],\] \[ L V = K_s\frac{\partial T_s}{\partial n}\vert_{x\in\Upsilon} - K_l\frac{\partial T_l}{\partial n}\vert_{x\in\Upsilon}, \] \[T_s\vert_{x\in\Upsilon} = T_l\vert_{x\in\Upsilon} = T_m- V/\mu - \Gamma \kappa,\]

where \(T_s\), \(T_l\) are temperature fields in solid and liquid phases, respectively, and \(V\) is a local growth velocity. \(\mu\) and \(\Gamma\) are kinetic and Gibbs-Thomson coefficients, respectively. \(\kappa\) is the curvature of the interface \(\Upsilon\). \(\vec n\) is a unit vector normal to \(\Upsilon\) oriented from the \(solid\) to the \(liquid\) phase. The first two equations describe heat transport in solids and liquids. Eq. XX defines heat transport through the interface and the release of the latent heat during solidification. The last equation governs the movement of the interface, taking into account the kinetic and Gibbs-Thomson effect.