Crystallization with initially flat interface

The videos demonstrate the freezing (solidification) process of a flat crystalline surface placed into an undercooled melt. The crystalline interface triggers the solidification process on it. The latent heat release caused by the phase-transition suppresses decreases the crystal growth rate. The interplay of the latent heat release and heat withdrawal from the growth region results in so-called dendritic growth. Heat transport efficiency governs the local growth rate. The needle shape ensures an efficient heat withdraw. Therefore, formation of the dendritic structure (needle shaped crystalls) is typical for such systems. The orientation of the growing patterns is a function of the crystal properties like growth kinetics and surface energy. Here you can find specific cases corresponding to the so-called 4-fold (2D) and 6-fold (3D).

Dendritic growth and flat surface instability in 2D

Dendritic growth and flat surface instability in 2D. Crystallographic directions rotated \(10^o\)

Dendritic growth and flat surface instability in 3D

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.