LeoCrystal

Introduction.

LeoCrystal is a software program that performs numerical Monte Carlo modeling of reaction on the surface of the growing crystal for illustrative educational and research purposes. With the help of this program you can estimate influence of effective geometrical sizes of structure elements of the crystal and thermodynamic parameters of the process on the topology and kinetic of the crystallization. As soon a process of crystal growth is presented itself in practically all major technology processes an in depth understanding of complexity of it is essential for professional research in different area of applied science.
Particular interest it could be for the nonotechnology investigations permitting off site optimizing design on the molecule scale level.

The creative presentation of the surface includes a three-dimensional perspective projection and stereo view. It makes this software a great illustrative toll for teaching of the concept of phase transition for the students of very broad ranges of initial background.

Theoretical background.

Very moderate scientific background is needed to understand a material in this section. Preliminary runs of the program will make a great help to easy accommodate basic concepts.

The crystal growth is a phase transition process with sharp border between it and initial feeding phase like a liquid, gas or plasma. The structure element (molecule) of the crystal could be determined as a minimum part of it when a reaction of incorporating its in the crystal will effect with changing energy of the whole system that will be equivalent to the condensation energy of the corresponding mass of crystal. In simple words the molecule is the minimum part of the crystal that behaves as a whole crystal. The principal difference of behavior of molecules in liquids is based on the principle of the long order in the crystals structure. Each molecule has exact position relatively to the other in crystal.

In the every moment of time one molecule on the surface of a crystal have two options (taking diffusion out from consideration): to get out to initial matter or stay incorporated as a part of crystal. The same choice is true for the molecule outside of crystal in direct closeness to its surface. It can be incorporated into body of crystal or stay outside. The model implemented here is based on calculating these probabilities for each position on the surface of the crystal and comparing with the random number to decide what one of three possible events will happen: stepping ahead or back or just staying on its place.

Probabilities of transition of molecule from and in crystal is calculated on the base of model of thermal activation reaction. It presumes that molecules perform thermal vibrations and could sometimes occasionally with cooperation of neighboring molecules accept enough energy E_a - (activation barrier), to overcome energy barrier to transform into another phase. Backward transition will demand to overcome activation barrier with a different tall. The difference between these barriers is indeed a movement force of the reaction that determined its direction from state with higher energy to the deeper down along the energy scale. The crystallization is such kind of process and it cause the exposition of the energy in form of heat that also named as changing of enthalpy DH. When the temperature is constant during the process (heat is conducting away) the difference between activation energies can be calculated by the formula:

E_{a out} - E_{a in} = DG = DH*DT/To, (1)

where E_{a out} - E_{a in} are activation energies to get molecule outside of the crystal and incorporate in its body correspondingly, DG - is so named changing of so named Gibbs potential, DH - changing enthalpy per one molecule, <DT = To-T, To - temperature of equilibrium for given conditions, T - temperature.

The frequency of passing through activation barrier can by calculated with help of Arrenius equation:

g = g_o exp(-E_a/kT), (2)

where g_o - thermal frequency of molecule vibration, k = 1.38* 10^-23 J/grad (Bolzman constant). In case if all position on the surface of the crystal is energy equivalent the rate of growth can be calculated from Turnbal's formula:

V_c = a C_o (g_in - g_out) = a C_o g_o exp(-E_{a in}/kT)*(1 - exp(- DG/kT), (3)

where g_in and g_out frequencies to go in or out of crystal for any given molecule on the surface , a - height of the molecule in the growth direction, C_o concentration of the molecules in direct distance from the surface. The probabilities of these events to occur at the small period of time t can be expressed with

p_in = g_in * t and p_out = g_ou * t. (4)

The problem with the crystal growth is that position of molecules on the surface is not equivalent. They have different number of neighbors on various energy states.

One of the formal way to deal with this situation that implemented in this model is to consider that at the elementary process of coming out off the crystal for one molecule is accompanies with changing of effective value of surface square in contact with initial matter. And as soon the surface has special exceed of energy surface one can add corresponding additive to the expression for the activation energy for jumping out of crystal. Using formula (1) we can write:

E_out = E_{a in} + DG' + DS*s, (5)

where DS - is changing surface as a result of one molecule going out of a crystal. It can be calculated on the base of formula

DS = S_m - 2* S_n, (6)

where S_m - full surface of molecule and S_n - surface in direct contact with neighbors.

Now we have all the necessary initial parameters to present basic formulas for modeling crystal growth behavior:

p_out = p_in* exp(- DG/kT +(S_ms/kT)(1-S_n/S_m)) (7)

t = p_in / (g_o exp(-E_a/kT)). (8)

Most interesting conclusion from these equations is that there are only two initial parameters

DG` = DG/kT, (9)

and

Ds` = S_ms/kT, (10)

these along with sizes molecule in the three dimensions are determined all properties of the crystal growth behavior.

Input of initial parameters of the crystal growth.

In the introduction chapter we presented the initial parameters that are necessary to perform crystal growth simulation. They can be set up with help of tab control.

That control could be also used independently as a calculator of the major thermodynamic parameters. Just print out new value in corresponding edit box and press ENTER (recommended) or come to other tab or begin new simulation process for updating it and all others related. Note that program permits to change calculated parameters forcing a resetting some of the initial ones as result of updating. Major idea is that after successful updating all other values need to correspond each others.

The program will not permit to enter unreasonable values, setting it to minimum or maximum number from allowed range in a case.

Note also that for calculation of the Ds` surface of the molecule is calculated by formula:

S_m = 2 K_s *(ab+bc+ac), (11)

where a ,b ,c are sizes of the molecule along three dimensions (a in growth direction), and K_s expresses not 100% effective filling with molecules accessible volume around it. In the program it set up as 0.9.

for calculation of radius of critical two dimensional nuclei (that has minimum size that a able to future growth)

r_n =DG/s, (12)

and for rate by this mechanism:

V_n = exp(-DG_n/kT)*V_c, (13)

where DG_n - changing energy of critical two dimensional nuclei that could be calculated by formula:

DG_n = p a s²/ DG (14)

There is one more broad discussed mechanism of the crystal growth - so named spiral dislocation growth in which one presumes that on surface there is always presented same step that cause a permanent existence on its place a peak of the two dimensional spiral with radius at its center equal to radius of critical nuclei. Velocity of crystallization by this mechanism is

V_n = 2*(a/ r_n )*V_c, (15)

These theoretical models represent major issues of the crystal growth at same specific sets of conditions. In fact there are numerous cases that not fit these models and need to run simulation model to realize all complexity of the process. Most critical point that compromised mentioned above theories is here the fact that time for appearance of new two dimensional nuclei is comparable with time of filling up one layer of the crystal in contradict of basis presumption of the theories. The modeling process is solving this problem.

Results of simulation process.

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Results of simulation process are presenting on the main screen that displayed simulation field with the several option of presentation and results.

There are two main way of presentation of the result of simulation:

1. A chart reflects time history of average changing of height of the molecules on the surface, its roughness that is calculated as a standard deviation from the average position and growth rate as difference average positions per time.
2. Numeric date at the given moment for the mentioned above values plus information about value of time step between two shots in the loop of model process.

A presentation of the surface.

There are four ways of presentation of the surface of the crystal:

The point of view menu

offers several options to observe a surface of growing crystal:

1. Gray map of the surface.
   

The difference in heights of the surface is represented with different colors of corresponding rectangles. The deeper the darker.

The specific simulation area can be chosen for detailed look with selecting it with mouse at right button down till release. Zoom out/in is available with double click in area of simulation.

This way of presentation is recommended. It permits fastest performance of the program. If running long term simulation fastest result could be reached if only several molecules are displayed.

2. 3D look at perspective projection.



The natural view of the surface. The point of view could be changed with the keyboard arrows buttons alone and together Ctrl/Alt:

3. Stereo blue/red image.

The same as previous except that with blue/red glasses its possible to resolute two images for left and right eyes observing stereo image.

4. Two stereo images for each of two eyes separately.

Main menu options:

New.
Begin new simulation process for given set of parameters.

Run.
Resume paused simulation for new or changed set of parameters. Only changing of thermodynamic parameters is permitted for this option.

Stop.
Pausing of simulation.

Save.
Saving current situation for further use. Format of the file is internal for this application and can not be freely used without specific knowledge.

Load.
Reading previous performed simulation.
In the section Help of Parameters Tab there is control permitting to load one of the three character cases.

Quit.
Close application.

Recommendations and troubleshooting.

1. The LeoCrystal is oriented to the two main customer groups: to the students studying material science, chemical engineering and related topics and professionals in needed to renew a process of the crystal growth. It can be used for optimizing of existing and developing novel technologies processes that includes crystallization. In particular it could be very useful for determination set of condition in which crystals of perfect shape are growing. The additional useful tools such as automatically creation of growth rate as temperature function; finding set of unknown parameters from primary principals set of parameters that is best fit to experimental data; modification algorithm for simulation itself and other could be available.
2. The gray map style of presentation is standard for simulation. 3D presentation is just supporting tools for better understanding of the processes. Do try to avoid to use them in case of big enough sizes of the field of simulation specially in accompany with set of parameters that are course very rough surface. The time for creating 3D image could be unbearably long as if the program crashed. Just reveal the numeric value using gray map view option and repeat it to get 3D image for the simulation area with smaller sizes. The 30*30 usually performs well enough to any systems. The actual performance of the program directly depends on power of your system. 3D acceleration card is very helpful.

Additional literature.

There are number of the books and articles devoted to the crystal growth including the international journal with the same name. The list presenting bellow is not complete in any sense and could be consider only as source of inspiration for the developing the program and as an initial point for further research for the user.

1. Sakharov L.G. The comparison of temperature dependence of linear crystal growth rate in silicate melt with theoretical models in wide interval of supercooling. Saint Petersburg, 1995, 19 p.
2. Turnbull D., Cohen M.H. Crystallization kinetics and glass formation. - In. Modern aspects of the vitreous state. New Yourk, Butterworth, 1960, p. 47-50.
3. Cahn J.W., Hilling W.B., Sears G.W. The molecular mechanism of crystallization. - Acta Mettalurgica, 1964, v.12, N12, 11421-1439.
4. Gilmer G.H, jackson K.A. Computer simulation of crystal growth. - In. Current topics in materials science. v.2, 1977, North-Holland, Amst., N.Y., p. 77-114.
5. Sarma S.D. Numerical studies of epitaxial kinetics: What can computer simulation tell us about nonequilibrium crystal growth? - J. Vac.Sci.technol. A8(3), may/Jun, 1990, pp. 2714-2726.