Дендритные кристаллы: от теории к современным моделям роста

Crystal that develops with a typical multi branching form

A crystal dendrite is a crystal that develops with a typical multi branching form, resembling a fractal. The name comes from the Ancient Greek word , which means "tree", since the crystal's structure resembles that of a tree. These crystals can be synthesised by using a supercooled pure liquid, however they are also quite common in nature. The most common crystals in nature exhibit dendritic growth are snowflakes and frost on windows, but many minerals and metals can also be found in dendritic structures.

The first dendritic patterns were discovered in palaeontology and are often mistaken for fossils because of their appearance. 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. 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. 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. Through this they found that the problem for isotropic surface tension had no solutions. This result meant that a system with a steady needle growth solution necessarily needed to have some type of anisotropic surface tension. 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.

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.

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. Eventually, the sides of this parabolic tip will also exhibit instabilities giving a dendrite its characteristic shape.

The 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. The experimental results indicated that at lower supercooling (up to 1.3 K), these convective effects are indeed significant. Compared to the growth in microgravity, the tip velocity during dendritic growth under normal gravity was found to be up to several times greater.