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Crystal growth.
Process of crystal growth is a most fundamental phenomena related to material science and in the same time one of the most dramatically ignored. There are many reasons for leak of focus on the direct experimental studies of crystal growth in general and in specific on its kinetic as a function of parameters of environment such a temperature, concentrations of components in feeding phase (gas or liquid). Numerical presentation of such data are very rare and for broad range of temperatures could be count by fingers. One of the most comprehensive study of crystal growth kinetic in silicate melts was done by lead developer of LeoStatistic. There were several years works with high temperature microscope observing a temperature dependence of normal crystal growth of crystals with different structures in melts of primary system CaOAl_{2}O_{3}SiO_{2} with additional oxides of alkali and alkaline earth metals. The principal scheme of most advanced installation for high temperature microscopy of kinetic of crystal growth are shown here: Platinum wire (foil) with a small hole in which a drop of melt is placed can be heated by electrical current up to about 1700 C. The close area of melt to metal the higher temperature in it. Near the melting temperature it's possible to leave only one small seed crystal in the center of the hole. After sharp drop of temperature of the heater the seed crystal in the drop is growing all around. The normal growth rate of the crystal can be measured. The normal rate of the crystal are varying in very broad range that compensate in same way a inaccuracy of rate measurements based on variation of their shapes along with temperature changes. One of the best in sense of accuracy and accomplishment temperature dependence of normal crystal growth for is for crystallization of triclinic Anorthite, CaAl2 Si2 O8 ) from its own melt:
What is the best analytical representation of this dependence? In general words we have to find coefficients of the theoretical formula these have to be in agreement with independently found from other kind of experimentations. For crystal growth a commonly accepted model of thermal activation reaction. 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. Paraphrasing the molecule is the minimum part of the crystal that behaves as a whole crystal. The main 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 all others 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. Molecules perform thermal vibrations and could sometimes occasionally with cooperation of neighboring molecules accept enough extra energy to jump back and force of activation barrier. 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 }= ΔG = ΔH*ΔT/T_{o} where E_{a out}  E_{a in} are activation energies to get molecule outside of the crystal and incorporate in its body correspondingly, ΔG  is so named changing of so named Gibbs potential, ΔT = T_{o}T; T_{o}  temperature of equilibrium for given conditions, T  temperature. The frequency of passing through activation barrier can by calculated with help of Arrenius equation:
The frequency of passing through activation barrier can by calculated with help of Arrenius equation: γ = γ_{o} exp(E_{a}/kT) where γ_{o}  thermal frequency of molecule vibration, k = 1.38* 10^{23 } J/K (Boltzmann 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 } (γ_{in }  γ_{out}) = a C_{o }γ_{o} exp(E_{a in}/kT)*(1  exp( ΔG/kT) and replacing ΔG with its enthalpy presentation: V_{c } = a C_{o }γ_{o} exp(E_{a in}/kT)*(1  exp( ΔH*ΔT/kTT_{o}) where γ_{in }and γ_{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. If we transform previous formula taking logarithm of right and left side and considering that 1  exp(x) ~ x, replace for more reasonable grade in constant Boltzmann with Molar Gas Constant R=8.31e J/(K*mol), estimate value of ΔH ~ 8.1*10^{5} J/mol and its melting temperature T_{o}=1823 K we will get ln(V)_{ } = K_{o}  E_{a}/(8.23*T) + ln(1.1e^{5}*(1823T)/T) where K_{o}= ln(a C_{o }γ_{o}). Let's approximate experimental data with this formula. Go to tab "Statistics" click button "Custom formulas" and modify fitting formula as it shown on the picture:
The formula on the screen can be copied and pasted with the help of "Copy formulas on the screen" in "Result" tab: ln(V/((8.1e5/(8.31*1823))*(1823x)/x)) = +22.1* 1.0 +2.68e+5* 1/(8.31*T) Calculated coefficients gives value of E_{a}= 268 kJ/mol and a*C_{o}*γ_{o}= exp(22.1) = 3.96 10^{9} mcm/s. Presuming that activation energy in oxide melts are related to breaking oxygenoxygen bond it's reasonable to expect that E_{a} will be close to bond energy. Indeed average value of bond energy 242 kJ/mol is in very good agreement with found coefficient. For crystallization in its own melt value of concentration C_{o}=1.0. The size of molecule multiplied by own frequency of thermal vibration can be close to rate of sound velocity in the condensed matter that is near 10^{9} mcm/s  very close to found value. One thing is definitely not nice  in general bad fitting of the curve with experimental data especially in area close to melting point. Depend of specific mechanism of crystal growth meaning spiral or two dimensional nuclei model in the Turnbal`s formula could be additional member (ΔT/T)^{n} where n  is 1 for spiral mechanism and about 4 in case of two dimensional nucleation. Next picture is shown result of approximation with this addition:
ln(V/((8.1e5/(8.31*1823))*(1823x)/x)) = +39* 1.0 +4.5e+5* 1/(8.31*T) +1.3*ln((1823T)/T) Yes. The matching of experimental data with theoretical curve is much better.
The value of coefficient n=1.3 is quite reasonable. But value of
activation energy is nearly doubled to 450 kJ/mol. And
a*C Really I don't know the answer. My feeling is that the reason why addition of (ΔT/T)^{n}^{ }is helping to get nice fitting but spoils the approximated coefficients is that the area of applicability of it is much narrower then all domain of supercooling where data is existed. In smaller temperatures it has to move to 1.0. If you want to experiment with software visually simulated what happen on the surface of the grown crystal try LeoCrystal. 
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