Crystal growth, diffusion process

A low entropy system gets closed, and, according to a process which can be quantitatively described, goes to states of increasing entropy (radioactive decay, racemization, crystal growth, diffusion). 

Jackson, K.A., Kinetic Processes - Crystal Growth, Diffusion and Phase Transitions in Materials, Wiley-VCH, 2004. 

Crystal growth is a diffusion and integration process, modified by the effect of the solid surfaces on which it occurs (Figure 5.3). Solute molecules/ions reach the growing faces of a crystal by diffusion through the liquid phase. At the surface, they must become organized into the space lattice through an... 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

We have so far assumed that the atoms deposited from the vapor phase or from dilute solution strike randomly and balHstically on the crystal surface. However, the material to be crystallized would normally be transported through another medium. Even if this is achieved by hydrodynamic convection, it must nevertheless overcome the last displacement for incorporation by a random diffusion process. Therefore, diffusion of material (as well as of heat) is the most important transport mechanism during crystal growth. An exception, to some extent, is molecular beam epitaxy (MBE) (see [3,12-14] and [15-19]) where the atoms may arrive non-thermalized at supersonic speeds on the crystal surface. But again, after their deposition, surface diffusion then comes into play. 

A normal diffusion process, however, runs at a finite concentration of particles different from zero. In this situation it was found [101] that a fractal character (73) of the resulting structure is restricted to an interval a < R < if), where d is the diffusion length (67). Larger clusters have a constant density on a length scale larger than They are no longer fractal there. These observations have various consequences for crystal growth, and will be discussed in the next section. 

Dehydration reactions are typically both endothermic and reversible. Reported kinetic characteristics for water release show various a—time relationships and rate control has been ascribed to either interface reactions or to diffusion processes. Where water elimination occurs at an interface, this may be characterized by (i) rapid, and perhaps complete, initial nucleation on some or all surfaces [212,213], followed by advance of the coherent interface thus generated, (ii) nucleation at specific surface sites [208], perhaps maintained during reaction [426], followed by growth or (iii) (exceptionally) water elimination at existing crystal surfaces without growth [62]. 

Dynamics of Crystal Growth hi the preceding section we illustrated the use of a lattice Monte Carlo method related to the study of equilibrium properties. The KMC and DMC method discussed above was applied to the study of dynamic electrochemical nucleation and growth phenomena, where two types of processes were considered adsorption of an adatom on the surface and its diffusion in different environments. 

The electrode reaction can involve the formation of a new phase ( e.g. electro-deposition processes used in galvanizing metals). The formation of a new phase is a multi-stage process since it requires a first nucleation step followed by crystal growth (in which atoms must diffuse through the solid phase to then become located in the appropriate site of the crystal lattice). 

Clearly this is a very interesting problem and of great practical relevance, very well suited to Monte Carlo simulation. At the same time, simulations of such problems have just only begun. In the context of crystal growth kinetics, models where evaporation-condensation processes compete with surface diffusion processes have occasionally been considered before . But many related processes can be envisaged which have not yet been studied at all. 

The initial theoretical treatment of these mechanisms of deposition was given by Lorenz (31-34). The initial experimental studies on surface diffusion were published by Mehl and Bockris (35, 38). Conway and Bockris (36, 40) calculated activation energies for the ion-transfer process at various surface sites. The simulation of crystal growth with surface diffusion was discussed by Gilmer and Bennema (43). 

On the left-hand side of Fig. 1 are the various mineralforming components in their complexed and uncom-plexed forms. They can directly bind to the crystal growth sites or they can combine to form the crystallization monomers (half-filled squares). These processes can be blocked competitively by the presence of other substances that form nonproductive complexes, thereby depleting the concentrations of precursors through mass action. The diagram also shows a second phase that helps to explain the nature of oriented diffusion and subsequent adsorption of the monomers. This so-called dou-... 

If the boundary motion is controlled by an independent process, then the boundary motion velocity is independent of diffusion. This can happen if the magma is gradually cooling and crystal growth rate is controlled both by temperature change and mass diffusion. This problem does not have a name. In this case, u depends on time or may be constant. If the dependence of u on time is known, the problem can also be solved. The Stefan problem and the constant-w problem are covered below. 

One-dimensional crystal growth at constant growth rate The crystal growth rate may be controlled by factors other than the diffusion process itself. In such a case, the growth rate may be constant. Assume constant D and uniform initial melt. The diffusion problem can be described by the following set of equations ... 

Ghiorso M.S. (1987a) Chemical mass transfer in magmatic processes, III crystal growth, chemical diffusion and thermal diffusion in multicomponent silicate melts. Contrib. Mineral. Petrol. 96, 291-313. 

Crystallizing particles arriving at the surface will diffuse onto the surface (surface diffusion). As this occurs, some may return to the ambient phase, while some will be caught at kinks or steps (see Section 3.6) on the surface and will be incorporated into the crystal. When these particles are incorporated into the crystal, the solvent component will be dissociated. This process is called desolvation. In solution growth, this process will determine the growth rate. At certain points in these processes, it is necessary to overcome the energy barriers required to climb the respective steps (Fig. 3.5). 

The overall process of metal deposition and crystal growth involves several steps. One is the diffusion of ions in the solution to the metal surface. Another is the cathodic deposition step, i.e., the removal of the ion across the interfacial region to land somewhere on a terrace on the metal surface. 

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