# Crystal Growth

## Introduction

Crystal growth is a typical phenomenon which appears during the solidification of materials: A structurally disordered (or hardly ordered) phase (liquid/gaseous) transforms into a structurally ordered crystalline phase. On the microscopic lengthscale the growing solid phase can develop beautiful patterns during such a process. The most popular example of such a pattern is probably the snowflake but similar patterns can be found in many materials.

The animation on the right shows a two-dimensional snowflake-like single crystal (dark grey) growing into the surrounding liquid (light grey). However, this structure has a fourfold symmetry and thus a lower symmetry than a snowflake. Structures like this are typical for metals.

## Modeling

The microscopic modeling of crystal growth is generally a difficult challenge due to the complex solid structures that usually arise. The things shown here were done with a so-called sharp-interface model, which is the classical thermodynamically rigorous way of modeling. This sharp-interface model consists of three parts:

### Part 1: Transport equations

In the simplest case, only heat controls the growth process and the heat is only transported by the mechanism of diffusion. This is described by the diffusion on the right  equations for a temperature $T$ with a diffusion constant $D_l$ in the liquid phase and a diffusion constant $D_s$ in the solid phase:

$$\frac{\partial T}{\partial t}=D_l\Delta T$$

$$\frac{\partial T}{\partial t}=D_s\Delta T$$

Here, $\Delta$ is the Laplace operator.

In general, however, advection occurs as an additional transport mechanism and apart from the temperature the chemical concentration also controls the process.

### Part 2: Thermodynamical equilibrium at the phase interface

The interface between the two phases poses a boundary for the transport equations in each of the phases. The boundary value for the temperature is determined by the assumption of thermodynamical equilibrium along the boundary, i.e. the interface. Hence, the temperature $T_I$ is equal to the melting temperature $T_m$:

$$T_I=T_m$$

However, this expression of thermodynamical equilibrium is too simple and incomplete because it neglects the existence of an interface tension between the two phases. The final boundary equation is:

$$T_I=T_m-T_m\frac{\gamma}{L}\kappa$$

Here, $\gamma$ is the interface tension, $L$ is the latent heat per volume and $\kappa$ is the local curvature of the interface. The interface tension $\gamma$ is usually anisotropic, i.e. it depends on the direction.

### Part 3: Conservation at the phase interface

To close the system of equations for this moving boundary problem an additional equation is necessary. In this case it relates the movement of the phase interface to the temperature gradients at the interface and thus states the conservation of energy:

$$L v_n = \left(D_sc_{p,s}\vec{\nabla}T|_s-D_lc_{p,l}\vec{\nabla}T|_l\right)\vec{n}$$

vn is the local growth velocity of the interface in the local normal direction n and cp,s and cp,l are the specific heat of the solid and the liquid phase, respectively.  $\vec{\nabla}$ is the nabla operator.

The computationally most difficult part with sharp-interface models is to keep track of the phase interface and to resolve its curvature sufficiently also in complex patterns.

## The Dendritic Morphology

If the dynamics of the atomistic processes at the solid-liquid interface depends on the direction in space (i.e. is anisotropic) the solid structure develops a so-called dendritic morphology.

The animation on the right side shows the growth of a single dendritic arm. (The "camera" moves along with the tip of the growing structure.) The characteristic feature of a dendrite is that the solid structure grows mainly in preferred directions.

## The Seaweed Morphology

If the dynamics of the atomistic processes at the solid-liquid interface depends only weakly or not at all on the direction in space (i.e. is isotropic) the solid structure develops a so-called seaweed morphology.

The animation on the right shows the growth of such a seaweed structure. The growth of this structure is obviously more isotropic than the dendritic growth. A characteristic feature of this morphology is the tendency to develop a lot of gaps in the solid structure (the so-called tip-splitting instability).

## Technical Details

The animations were created with the following procedure:

• The computer simulation created a pile of single files each containing a snapshot of the growing structure. The data were not in a special format.
• A little self-written hack created JPEG pictures out of the snapshots.
• The convert tool from the software suite ImageMagick was used to create finally a GIF animation out of all the JPEG pictures.

If you have serious questions: jurgk[at]pks.mpg(dot)de

## References

• about the model, its numerical implementation and the morphologies:
T. Ihle and H. Müller-Krumbhaar: Fractal and compact growth morphologies in phase transitions with diffusion transport. Phys. Rev. E, 49(4):2972, 1994