Dendrite (crystal)
The first theory for the creation of these patterns was published by Nash and Glicksman in 1974, they used a very mathematical method and derived a non-linear integro-differential equation for a classical needle growth.
This became known as the maximum velocity principle (MVP) but was ruled out by Glicksman and Nash themselves very quickly.
In the following two years Glicksman improved the numerical methods used, but did not realise the non-linear integro-differential equation had no mathematical solutions making his results meaningless.
Four years later, in 1978, Langer and Müller-Krumbhaar proposed the marginal stability hypothesis (MSH).
This hypothesis used a stability parameter σ which depended on the thermal diffusivity, the surface tension and the radius of the tip of the dendrite.
They claimed a system would be unstable for small σ causing it to form dendrites.
At the time however Langer and Müller-Krumbhaar were unable to obtain a stability criterion for certain growth systems which lead to the MSH theory being abandoned.
A decade later several groups of researchers went back to the Nash-Glicksman problem and focused on simplified versions of it.
This result meant that a system with a steady needle growth solution necessarily needed to have some type of anisotropic surface tension.
Nowadays the best understanding for dendritic crystals comes in the form of the macroscopic continuum model which assumes that both the solid and the liquid parts of the system are continuous media and the interface is a surface.
This model uses the microscopic structure of the material and uses the general understanding of nucleation to accurately predict how a dendrite will grow.
[2] Dendrite formation starts with some nucleation, i.e. the first appearance of solid growth, in the supercooled liquid.
This instability has two causes: anisotropy in the surface energy of the solid/liquid interface and the attachment kinetics of particles to crystallographic planes when they have formed.
This curvature undercooling, the effective lowering of the melting point at the interface, sustains the spherical shape for small radii.
However, anisotropy in the surface energy implies that the interface will deform to find the energetically most favourable shape.
[3] For both above and below the critical anisotropy the Wulff construction provides a method to determine the shape of the crystal.
In principle, we can understand the deformation as an attempt by the system to minimise the area with the highest effective surface energy.
[4] Eventually, the sides of this parabolic tip will also exhibit instabilities giving a dendrite its characteristic shape.
For instance, a dendrite growing with BCC crystal structure will have a preferred growth direction along the
The table below gives an overview of preferred crystallographic directions for dendritic growth.
[3] Note that when the strain energy minimisation effect dominates over surface energy minimisation, one might find a different growth direction, such as with Cr, which has as a preferred growth direction
-type) For metals the process of forming dendrites is very similar to other crystals, but the kinetics of attachment play a much smaller role.
These pseudofossils form as naturally occurring fissures in the rock are filled by percolating mineral solutions.
A three-dimensional form of dendrite develops in fissures in quartz, forming moss agateThe Isothermal Dendritic Growth Experiment (IDGE) was a materials science solidification experiment that researchers used on Space Shuttle missions to investigate dendritic growth in an environment where the effect of gravity (convection in the liquid) could be excluded.
