Introduction

Modern engineering industry needs new materials with special characteristics. There are two ways to ensure required properties for the materials: physical experiment and computer modeling. Each of the approaches have some advantages and disadvantages.

Physical experiment has one advantage: it reproduces interaction of all factors. But it has a lot of disadvantages:

• power-intensive, high-priced, long-term;
• results depends on input material quality;
• depends of casual factors;
• some parameters of experiment are interdependent. Therefore the results of experiment can interpret in different ways.

As for computer modeling:
Advantages

• rapid obtaining of result;
• cheapness;
• low power consumption;
• effective management of paramrters.

Actuality of theme

Actuality of this task consists in decrease of expenditure of time and means for organization and realization of experiment.

Purpose and tasks of development and research

The purpose of development is creation of programmatic complex which allows to research the process of crystallization on the different stages.

The tasks of development and research are:

• study of physical principles and mathematical apparatus of process of crystallization;
• selection of descriptions of crystals for the construction of geometrical model of process;
• study of the existent systems for modeling of crystal growth;
• development of algorithm for the modeling of process of nucleation and crystal growth in melt;
• development of the system of effective management the parameters of experiment;
• development of intuitive user interface.

Object and subject of investigation, novelty of work

Object of investigation is a process of crystallization in liquid melt.

Subjects of investigation are geometric simulation and visualization of process of crystallization in melt.

Novelty of work: the methods of modeling of crystallization in liquid melt and methods of visualization of this process were develop.

Approbation of results

1. VI international theoretical and practical conference «Geometric simulation and computer technologies: theory, practice, education» (Ukraine, Kharkov, 21-24 April, 2009)
2. ISDM 2009 (16th International Students’ Day of Metallurgy, Czech Republic, Ostrava, 23-25 April 2009).

Review of researches and developments on the theme

Different methods are used for the modeling of crystallization.Methods are divided by two categories. First category - methods which midels macrolevel's processes. Second category - methods which models microlevel's processes [1]. One of the most popular methods - Monte Carlo. It describe changes of crystal's lattice growing in gas. But it is ineffective for the calculation of crystal growth in melt [1]. On the macrolevel process of crystallization is continuous process. Process can be described by differential equations in the partial derivative (for example, model of the fields of behavior [2] which used modification of diffusion equation). For computer simulation of nonequilibrium growth of crystals in a two-dimensional medium with a phase-separating impurity is used method of cells which characterized by numerical value [3].

Process of crystallization is researched in Bryansk state university (I.K. Kul'bovskiy, S.V. Karelin, D.A. Ilyushkin, [4]), Novosibirsk state university (A. Brednikhina, [5]), Physicotechnical institute A.F. Ioffe of the Russian academy of sciences (V.A. Lagunov, A.B. Sinani, [6]).

The method of numeral modeling is investigated in the Moscow state institute of electronic technique by V.A. Goncharov [7], Physicotechnical institute of A.F. Ioffe of the Russian academy of sciences (M.V. Bogdanov, [8]).

Now there are a few software products which models physical processes.
For example, Phun - program for the imitation of physical processes, development of the Swedish students of Umeo university [9]. Realization of method Monte Carlo is presented by the program LeoCrystal [10]. However the similar programs did not get mass distribution. It is related to their narrow orientation ([10]) or teaching purpose ([9]).

Physical bases of modeling

Properties of material (strength, wear resistance, impact toughness) depend on its structure:

• shape of particles;
• distribution of particles by size
• specific number of particles’s contacts
• specific volume of particles
• part of contact surface

Surface of slice can be described by 2d model in some conditions. 3d model is necessary for describe condition when crystal and direction of section oriented randomly. 3d scheme lets to analyse surface of samples. Visualization of the intermediate stages of crystallization must represent details of its flowing which determining forming of the desired crystalline structures.

Model building

All crystals have an ideal form (cube, triangular prism) only in case when in the process of growth they do not adjoin. If there is interaction crystals are deformed, because on the border of contiguity growth is halted.

Nucleus of a crystals characterized by value set: nucleation rate, size, location, rate of facet’s growth.
It is necessary to take into account:

• every particle’s facet growth along surface normal
• rate of facet’s growth can be different

Different rates of facet’s growth are explained by facet’s type (packing density of face). This quantity characterized by ratio of number of particles to unit of area. For example, there are three type of facets (figure 1) in cube crystal system {100}, {110}, {111}

There are two types of model: vectorial model and raster model. Vectorial model is more difficult in description, but it’s processing is faster. Raster model processed slowly, but it’s realization is easily and more corresponded with physical parameters of crystallization.

• Objects’ description is available with help of list of vertexes, edges and facets.
• Normal fixed for every facet
• Growth of object realized like extension of every facet and displacement along normal
• Line of intersection between facets calculate after every step of growth
• Line of intersection fixed. It is a boundary.

• Objects’ description is available with help of complex of cells of space which characterized by numerical value (inside – 0.5, outside – 0, boundary - 1).
• Growth of object: we look for cells with value 1 which have cell-neighbours with value 0.
• If such cells exists, we change value from 1 to 0.5, from 0 to 1.
• Cells without free neighbour (0 – cells) became fixed boundary

Implementation

Vectoral and raster algorithm realize 2d-model for the time being (visual scheme of crystal growth). This model can be used like a prototype of photo of real slice. An analysis of programmatic image will be more simple, derived characteristics will be more exact and single-valued.

Correctness of visual scheme is checked up by comparison of photo and computer generated figure

Analysis of visual scheme.
Visual scheme is processed by methods of linear analysis. Main idea of these methods:
1. We calculate number of boundary crystal-crystal and crystal-melt.
2. We staticize finding using by theorems of metallography.

Modeling results

Results of vectoral algorithm working is shown on figure 2 (you can see 5 crystal and its facets' positions during 6 steps). Results of raster algorithm working is shown in figure 3.

Figure 3 - Results of raster algorithm working.

On the figure 4 (dynamic figure) you can see process of crystal growth.

Resume and perspective

1. The realized methods allow to get the visual scheme of crystallization at limits on the form of crystals and speed of their growth.

2. It is necessary to take into account for the increase of model precision that initial forms of nucleus of a crystals can be different and it’s orientation in space is probabilistic observation.

3. An actual task is programmatic realization of synthesis of 3d scheme. It gives an opportunity to analyse the surfaces of samples.

Literature

• Muller-Krumbhaar H., Saito Yu. Crystal Growth and Solidification / Surfactant Science Series, volume 89. CRC Press – 2000. ISBN 0824703235. pp. 853-854
• Sunagawa I. Crystals: Growth, Morphology and Perfection / Cambridge University Press – 2004.
• Martiouchev L. M., Seleznev V. D. , Skopinov S. A. Computer Simulation of Nonequilibrium Growth of Crystals in a Two-Dimensional Medium with a Phase-Separating Impurity / Journal of Statistical Physics, Vol. 90, Nos. 5/6, 1998.
• Cборник статей брянского государственного технического университета. URL: http://www.tu-bryansk.ru/content/journal-18
• Моделирование роста 2-х кристаллов на подложке. URL: http://www.ict.nsc.ru/ws/YM2007/12700/Brednikhina.htm
• Статья, в которой рассмотрен метод молекулярной динамики для исследования структуры твердого тела при переходе из аморфного в кристаллическое состояние. URL:http://www.physics.wups.lviv.ua/depts/KFM/prysjan/metals/p1087_1091.pdf
• Гончаров В.А. Численное моделирование процесса выращивания полупроводниковых кристаллов из расплава методом направленной кристаллизации // Теор. основы хим. технологии. 2001. Т. 35. № 3. С. 257-264.
• Bogdanov M.V.,Ofengeim D.Kh.,Zhmakin A.I. Industrial Challenges for Numerical Simulation of Crystal Growth / CEJP 2(1) 2004 183-203.
• Описание программы, моделирующий общие закономерности физики. URL: http://my-soft-blog.net/phun-multyashnaya-fizika
• Реализация метода Монте-Карло. URL: http://www.leokrut.com/leocrystal.html
• Шаскольская М.П. Кристаллография. Учебник для втузов. М.: Высш. школа, 1976, 391 с.
• Арзамасов Б.Н., Крашенинников А.И., Пастухова Ж.П., Рахштадт А.Г. Научные основы материаловедения. Учебник для вузов. – М.:Изд-во МГТУ им. Н.Э. Баумана, 1994 г., 366 с.

Remark