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:

There is expanding industry for production of bulk single crystals
for subsequent usage in electronics or for gems in jewelry. An
optimization of the technology presumes increasing productivity
that is equivalent in this case to finding a maximum rate of
crystal growth securing in the same time a quality of the crystal that means
keeping concentration of defects on atomic and higher scales on acceptable
level. In ranking by influence on the product output the kinetics of crystal
growth stays comparable very close here to the thermo and mass transportation
phenomena. Developer of the process would be greatly benefiting
from the knowledge of how much lower temperature on the border
of the crystal could be set without deviating from layerbylayer
mechanism of growth that provides the best possible quality of the crystal.

For mass produced construction materials like metals and alloys, fine
ceramic, Portland and highalumina cements and for many other materials in
industry the crystalline content of the product is strongly depend
on timetemperature conditions of treatment of the raw mixture
materials. Taking into account that different crystalline phases
can have mismatched rates of their spreading out (crystal growth
rates could be and mostly are different for them at the same
temperature), the knowledge how atomic structure of crystals
influences kinetic of crystallization could help in choosing
strategy for experiment planning in development of the best quality products.

There is necessity to avoid crystals formation during production of
materials with amorphous atomic structures like commonly used glasses or more
exotic materials like fast frozen amorphous metals. The knowledge
of growth rate of crystals in initial melts as a function of
temperature can provide guideline to defining minimum cooling
rate that is nevertheless fast enough to avoid formation of the
crystals in glass forming melts.

Formation of thin films on monocrystal substrates in vacuum by physical
or vapor deposition demands knowing the rate of crystal growth at given
temperature. To obtain crystalline film the velocity of material deposition
should be at least slower compare to the crystal growth rate.

A petrography analysis of samples in geology or found in meteorite rocks
permits to obtain their crystalline content. Additional information about
relative growth rate of these different crystalline phases helps to deceiver
a history of physical conditions of their places of origin.
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:

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.

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
twodimensional 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 twodimensional 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 twodimensional
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 semismooth surface.
An expanded model of layer by layer growth allows formation of secondary and
higher order twodimensional 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 first stage consists in development of the
LeoMonteCrystal software application for numerical simulation
of crystal growth that can reliable substitute for physical experiment
for most wide range of initial parameters and producing large
enough set of data that covers most of practically interesting
situations.

The second stage consists in finding of adequate method for approximation
of found set of experimental data with formal series of polynomial
like fitting formulas or even better by semianalytical formulas
with coefficients derived from mathematical constants. Achieving of such
approximation had required a development a state of art algorithms of
multivariate approximation by flexible constructed multivariate polynomial
formulas and fitting by free style constructed formulas these are
incorporated in our software
LeoDataAnalysis.
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