Crystallization --> Growth

### Models of growth of crystal surface.

by Leonid Sakharov

The model of thermally activated crystal growth as described in the article leaves aside the very important feature of crystal growth phenomena namely the influence of surface energy.

A difference in levels of free energy between molecule situated at the surface of the crystal incorporated in it and if belonged to amorphous phase is designated by addition of component that reflect on change surface of crystal in case of molecule transformation:

ΔG = ΔμVm + σΔSm                              (1),

where Δμ - change of chemical potential in crystalline and amorphous phase, σ - surface energy of border between crystal and amorphous phase, Vm - volume of molecule, ΔSm - change of surface of crystal as result of transformation of molecule. The change of surface if molecule come out of crystal can be expressed from part of molecule surface that is in direct contact with neighbors in each of phases:

ΔSm = Sm - 2Scr                                                  (2),

where Sm - surface of whole molecule, Scr - surface of the part of molecule in direct contact with other molecules in crystalline phase.

Should all molecules on the crystal surface have exact half of their outer area belonged to crystal there should be no reason to incorporate into formulas surface energy. In reality it is geometrical impossibility. The general solution in form of simple formulas for growth rate of crystal surface that incorporated surface energy seems like matter of impossibility at least in term like mathematical theorem. There main problem is that one of the parameters of solution is the roughness of the growing crystal surface but there are no independent principles to help to define it other way to direct computer simulation.

A following figure shows a typical snapshot of positions of molecules on the crystal surface produced by computer simulation:

Thermodynamically the most advantageous configuration of molecules is the flat, completely filled layer. After one layer is completely filled to start completion of the next layer a formation of island called two dimensional nucleus has to happen. On the larger size analogous to formation of activation complex situation has to happen. The same as for activation complex a change of free energy to form two dimensional nucleus has to be positive that is thermodynamically prohibited in sense of direction of system transformation on the macro scale. The trick is that two dimensional nucleus is small enough to be an fluctuation, a result of chaotic elementary moved molecules on  a smooth surface in and out of crystalline state. A separate article is devoted to detail discussion of mechanism of two-dimensional nuclei growth.

Other than fluctuation phenomena an existence of permanent edge on the surface of growing crystal called spiral (screw) dislocation. The idea is that such dislocation can be permanent substitute to random formation of overcritical nuclei. If some deviation of ideal structure during the growth is happen like isomorphic substitution of atoms or forming of hole inside there could be permanent edge on the smooth surface. More details about dislocation induced crystal growth are in separate article.

The main premise laid under two-dimensional  and dislocation mechanism crustal growth is that layer by layer growth is happened. It means that as a rule until one layer is completely filled next one is not begins to growth. As obviously can be seen from the picture above this condition is more exception than a rule. The broad attention to layer by layer mechanism is due its most importance for big bulk crystals productions when technological parameters of process are specially set to reach as perfect crystals as possible. But actually a number of important applications in industry and science when layer by layer model is not realized are quite big. In such case crystals are formed in variety of formed from nice snowflakes, dendrites, spheres, needles and so many other shapes that makes difficult to find adequate names for all of them. The situation two stage reaction with step one includes incorporation molecule into crystal on the surface itself and step two that introduces transportation of material to surface of crystal from feeding neighborhood and exhaust heat and admixture out of crystal.

Submitted at Mar. 4, 2010; 17:29