'Orientations of dendritic growth during solidification'. Interfacial Wave Theory of Pattern Formation in Solidification. On the solid-liquid interface, we can define a surface energy γ s l This instability has two causes: anisotropy in the surface energy of the liquid-solid interface and the attachment kinetics of particles to crystallographic planes when they have formed. This formation will at first grow spherically until this shape is no longer stable. the first appearance of solid growth, in the supercooled liquid. Dendrite formation ĭendrite formation starts with some nucleation, i.e. This model uses the microscopic structure of the material and uses the general understanding of nucleation to accurately predict how a dendrite will grow. 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 breakthrough lead to the microscopic solvability condition theory (MSC), however this theory still failed since although for isotropic surface tension there could not be a steady solution, it was experimentally shown that there were nearly steady solutions which the theory did not predict. This result meant that a system with a steady needle growth solution necessarily needed to have some type of anisotropic surface tension. Through this they found that the problem for isotropic surface tension had no solutions.

Microscopic Solvability Condition Ī decade later several groups of researchers went back to the Nash-Glicksman problem and focused on simplified versions of it. 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. They claimed a system would be unstable for small σ causing it to form dendrites. 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. 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.įour years later, in 1978, Langer and Müller-Krumbhaar proposed the marginal stability hypothesis (MSH). This became known as the maximum velocity principle (MVP) but was ruled out by Glicksman and Nash themselves very quickly.

However they only found an inaccurate numerical solution close to the tip of the needle and they found that under a given growth condition, the tip velocity has a unique maximum value.

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. The first dendritic patterns were discovered in palaeontology and are often mistaken for fossils because of their appearance. A simplified diagram for a smooth solid-liquid interface at the atomic level.

Dendrite gemstone