Crystal Growth

1 Introduction
2 Modeling
3 The Dendritic Morphology
- 3.1 The role of symmetry
4 The Seaweed Morphology
5 Technical Details
6 References

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 above 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 following diffusion 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, $Δ$ 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, $γ$ is the interface tension, L is the latent heat per volume and $κ$ is the local curvature of the interface. The interface tension $γ$ 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:

$L \, v_n = ( D_s \, c_{p,s} \, \vec{\nabla} T|_s - D_l \, c_{p,l} \, \vec{\nabla} T|_l ) \, \vec{n} \,$

v_n is the local growth velocity of the interface in the local normal direction n and c_p,s and c_p,l are the specific heat of the solid and the liquid phase, respectively. $\nabla$ 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 above 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.

Crystal Growth

Contents

Introduction

Modeling

Part 1: Transport equations

Part 2: Thermodynamical equilibrium at the phase interface

Part 3: Conservation at the phase interface

The Dendritic Morphology

The role of symmetry

The Seaweed Morphology

Technical Details

References