Formulation of the project for numerical experiment on crystallization.

by Dr. Leonid Sakharov

Formulation of the project for numerical experiment of crystal growth to find universal formulas for rate of crystal growth, roughness of its surface and concentration of hole like defects.

A crystallization phenomena manifests itself in a long list of natural and technological processes. Crystals are solid materials with atoms located in regular position defined by long distance symmetry. The structure of crystals gives each of them special and often unique properties essential for material culture of civilization. The rate of transformation of atomic structure motive of matter from liquid or gas state into crystals plays important role in series of industrial productions of variety construction and other industrial materials as well for research how earth minerals are formed in geology and same for space bodies in cosmology.

The presented project for numerical simulation of crystallization is aimed to obtain data of growth rate, roughness of crystal surface and also concentration of hole like defects for as broad as possible ranges of process parameters namely: thermodynamic potential of solidification (supercooling), surface energy and geometry of molecules, using series of numerical experiments and then finding the practically applicable way to fit these results by direct calculable analytical formulas. These formulas can be used then for optimization of technological processes involved crystallization phenomena as well for better understanding conditions of natural formation of crystal structures in nature.

Here are far from complete list of occurrences when rate a of movement of border of growing crystal plays essential role:

In the spite of comprehensive general understanding of the mechanism of crystal growth on molecular level there are only two special cases derived from theoretical models permitting to obtain analytical equations applicable for practical purposes. First case is when supercooling is so large or/and surface energy is so small that influence of the surface energy is negligible in comparison to thermodynamic potential of solidification. Second case is for extremely slow rate of growth near point of phase equilibrium when relation between surface energy and thermodynamic potential is reversed, surface energy has much larger influence on growth mechanism than thermodynamic potential that could be virtually as close to zero as environmental conditions of growth can allow.

These two special cases can be characterized by the following models of molecules behavior on boundary between crystalline and feeding phases:

  1. Continues growth occurs when surface of the crystal is so rough that there are no preferential places on it to incorporate new molecule. In such case the crystal growth can be presented in terms of thermal activated reaction and analytical formula can be derived without any special mathematical complications. The growth by such mechanism could happen in some practical circumstances especially for crystals with low surface energy under high supercooling. A solidification of fast cooled metal melts could be presented as an example. Continues growth usually is recognized by no edge like shape of formed crystals such as dendrites or spheres.
  2. Layer by layer growth. This model postulates that on absolute smooth surface of the crystal the seeds for upper layer appear randomly by itself or with help of lattice dislocation and then these two-dimensional islands grow radiantly filling up next layer; and the process repeats itself indefinitely. The main condition for the model to be valid is that time for filling a layer must be so small comparably to the average time between appearances of two-dimensional nuclei so no additional seed island is likely to appear during the stage of layer spreading. That will eliminate a situation of forming secondary and higher order nuclei on the primary one. Such model is the most possible true presentation of forming precious gems these can grow so slow that it takes centuries. They are not called precious for ordinarily circumstances of creation.

    It is worth mentioning that frequency of forming new two-dimensional nuclei and its radiant spreading rate are phenomena of the same nature which are derived from the same parameters. This makes a situation, when average distances between two primary nuclei are larger than the size of crystal itself, to be a quit rare, almost exotic occasion. That means that contrary to some variations of the model for most cases one should avoid of painting erroneous picture that once one random nucleus appears it can exist there alone spreading along whole surface of the crystal; then after considerable time when no nuclei exists while next one will appear again and spread along repeating the sequence indefinitely. Actually for most situations there is some measurable concentration of nuclei on surface. The condition of applicability of such model for producing analytical mathematical expression for crystal growth is the demand that concentration of primary two dimensional nuclei on smooth surface should be much larger than concentration of secondary ones, these are defined as such which are appeared on primary nuclei during spreading phase. Actually for meticulousness sake one can say that both stages of two dimensional nucleation of primary seeds and their spreading happen at the same time but rate of secondary nucleation is negligibly small.

A problem at hands is that vast majority of significant for practice and science situations involving crystal growth in nature and in industrial processes are happening outside these two extreme sets of circumstances when process are very close to or very far from phase equilibrium. For intermediate cases in between there are no reliable and applicable formula that can connect rate of crystal surface growth in given crystallographic direction for broad range of possible sets of parameters such as supercooling, surface energy and geometric shape of molecules (structure elements of crystal lattice). The root of the problem lies not in the lack of physical understanding of fundamental processes during crystal growth but exclusively on the side of mathematical complexity for their depiction in context of molecules interaction on semi-smooth surface.

An expanded model of layer by layer growth allows formation of secondary and higher order two-dimensional nuclei. Previously described mechanisms are special cases of this more general one. As soon no analytical formula is likely to be obtained for such model the logical approach is to collect large amount of experimental data for most wide ranges of physically possible values of initial parameters and approximate them with appropriate formula.

There are often overwhelming instrumental difficulties and prohibited level of time and instrumental resources for obtaining even one experimental result via direct observation of crystal growth. It is not realistic approach to collect needed volume of experimental data by physical experimentation. Nevertheless such direct quantitative observations like high temperature microscopy technique could be used for verification of suitability of found fitting formulas at least on a conceptual level. Thus a numerical simulation of crystal growth has happened to be the only realistic approach for collecting large enough set of experimental data.

The project is divided on two main stages:

The project is successfully accomplished and results are available for confidential release for negotiated compensation.

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Oct. 18, 2017; 10:46 EST

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