Crystallization --> Growth

### A model of thermally activated crystal growth.

by Leonid Sakharov

A model of thermally activated crystal growth is a foundation for theory of crystallization.

In the model of thermally activated crystal growth addresses to the elemental act of incorporation of molecule on the surface into crystal. In the spite (or because) simple mathematical formulation and universality of the model itself meanings of elements in it are sometimes paradoxical and demands separate discussions and definition in any specific situation.

In the model of thermally activated crystal growth the there are four type of molecules in system:

• - inside crystal;
• - inside noncrystalline phase;
• - in crystal at its surface of crystal in direct contact with noncrystalline phase;
•  - in noncrystalline phase in direct contact with crystal surface.

Fig. 1 presents a relation between energy levels of molecules in noncrystalline (liquid) phase and in crystal as well in intermediate state so called activation complex that is formed during elementary act of transition of molecule from on phase to other and back. Level of free energy inside crystal Gcr is defined by its structure and constant (if put outside of discussion influence of defects of crystal). The energy level of molecule in liquid Glq represents most possible formation of atoms in noncrystalline phase. If molecule in liquid by random thermal movement of atoms transforms its structure to crystal it has to form on the way an intermediate structure called activated complex. The energy level of the activated complex equivalent to minimum energy that is necessary to regroup atoms to be incorporated into crystal structure. Invert process of removing molecules from crystal into liquid demands a formation of the same activated complex.

In the model frequencies of the processes of incorporation of molecule  into crystal and opposite movement of molecule from crystal into noncrystalline phase are defined by modified Arrhenius equation:

γ = γ o e(-Ea in/kT)       (1),

γ = γ o e(-Ea out/kT)      (2),

where γo - frequency of thermal vibrations of atoms, T - absolute temperature, k - Boltzmann's constant.

A difference between energy levels in crystal and liquid:

 ΔG = Glq - Gcr = Ea out- Ea in = ΔH* ΔT (3), Tl

where ΔH - enthalpy of crystallization, Tl- temperature of equilibrium of growth, ΔT = Tl- T.

Taking into account that the act of transition of molecule leads to shift of crystal surface on the distance equal to size of molecule in direction perpendicular to surface the rate of crystal growth is equal:

V =d × (γ -  γ-)                                                                           (4),

where d is molecule size. Substituting frequencies for incorporating and outing a molecule by their presentations in formulas 1 and 2 the final formula have the form as following:

 V =d γo× e(-Ea in/kT) × [1 - e(- ΔH* ΔT )] (5), kTTl

The formula 5 is called also a Turnbull equation for kinetic of crystal growth.  The Turnbull formula is very good first estimation for experimentally observed bell-like temperature dependences of crystal growth initially described by Gustav Tammann.

The constant component of the Turnbull formula d×γo has dimension of velocity and with some stretch of rigidness could be directly connected to the sound velocity. For the simplistic but reasonable model of structure element as cubic harmonic oscillator with M (mass), d - effective size, k - elastic module that connects with shear module G defined by the formula G=k/d. Frequency of harmonic oscillations is  γ= 1/2π× (k/M)1/2. The velocity sound is defined by formula: Vsound = (G/ρ)1/2, where ρ - is density defined as  ρ = M/d3. After trivial substitutions frequency equation:

d ×γ =1/2π× Vsound  ≈  0.16 ×Vsound        (6)

In case if shape of molecule is better approximated not by cube with size of edges equal to d but with cuboid characterized by three sizes of its edges: a = d (direction of growth), b and c (in plane of growing surface that is perpendicular to direction of crystal growth). Defining volume of molecule as Vm = a × b × c and after simple transformation formula 6 can be transformed into very similar one with addition of shape coefficient:

d ×γ = a ×γ =1/2π× a/(bc)1/2 × Vsound  ≈  0.16 × a/(bc)1/2 ×Vsound        (7)

Important to note that formulas 6 and 7 are obtained using very rough estimations and as such has to be considered as the very first assessment. The most value of such approach is that there are nothing better to make an estimation for frequency of thermal vibrations of molecules in the crystal on the base of experimentally measurable parameters.

Activation energy Ea in in Turnbull equation has meaning of minimum change of free energy to form intermediate activation complex that is necessary to make rearrangements of atoms in liquid phase in direct contact of crystal to be incorporated into it. A very analogues of the situation is the movement in very packed crowd. If one person want to change his position or just turn around some push of other people should be required. The most gentle efforts of possible to make the way is the analogous of activation energy. Its meaning is directly connected although not identical to the activation energy that is manifesting itself in diffusion phenomenon. If in case of diffusion both start and end atom configuration have the same order pattern for case of  crystallization the unique structure of atoms must be reached in result of rearrangement. This condition could  demand for example to break more covalent bonds between atoms compare to diffusion atom moves to achieve specific atom configuration characterizing the growing crystal.

As a rule one can expect that activation energy of crystallization is no less then measured for diffusion for the given temperature and most possible is significantly large. For case of glass forming liquids like silicates with covalent bonds three dimensional net structure the process of crystallization may demand for forming activation complex it's reasonable to expect that activation energy will be close to multiple of bond energy between atoms.

An example for qualitative effect on crystal growth by variation of activation energy can be demonstrated by comparison of growth silicate and aluminum oxides. On the chart there are curves of growth rates of from temperature calculated by Turnbull formula for following values of parameters:

 Compound d×γ, cm/s Tl, K ΔH, J/mol Ea, J/mol Al2O3 0.16×105 2337 100800 1.45×105 SiO2 0.16× 105 1975 11400 2.90×105

The important to note that values of parameters above are very rough interpolation from limited experimental data. Specifically temperatures of equilibrium and enthalpy of crystallization are calculated for data of phase diagram (these are not precise too much for the temperature range). Values of activation energies are corresponds to oxygen-oxygen bond energy for Al2O3 and twice large for SiO2, that is arbitrary guess nevertheless in good  fit with common knowledge of maximum of crystal growth for these compounds. Important to note that only two times difference of activation energies for Al2O3 and SiO2 can provide hundred thousand times difference in rates of crystal growth. No wonder that silica oxide is primary material for glass in optical fibers and corundum is growing as large bulk crystals used as material for special applications.

The model of thermally activated crystal growth as described above completely ignores an effect of the surface energy (it omit also heat and material transport). The omission of obvious influence of surface shape based the premise that surface of growing crystal is absolute and uniformly rough. Such surface is supposed to have symmetrical distribution of changes of square area as result of molecules incorporated into or jumping out of crystal with average value equal zero. For such surface an influence of the surface energy does not exist. In reality an observation of facet like flat sides of growing crystals clear demonstrates that such assumption is not always true (almost never 100% true). Actually the thermodynamically the roughness of  surface is supposed to be as minimal as possible. The influence of surface energy on mechanism of growth will be discussed in more details in article "Models of growth of crystal surface".

Submitted at Mar. 4, 2010; 17:31