Equilibrium crystallization growth

At equilibrium, crystal growth and dissolving rates become equal, and the process of Ostwald ripening may now appear, in which the larger crystals grow at the expense of the smaller ones. The kinetics of the process has been studied (see Ref. 103). 

During equilibrium crystal growth from a melt a U-series parent and daughter will be incorporated according to their equilibrium partition coefficients, Dp and D respectively ... 

With the advances of the non-equilibrium crystal growth techniques, other 1I1-V magnetic semiconductors than (Ga,Mn)As and (In.Mn)As with different host semiconductors and different transition metals have appeared and the investigation of properties of these new materials are underway. 

In order to fully understand such technologically important phenomena as near-equilibrium crystal growth and homogeneous nucleation, it is necessary to have detailed information as to the microscopic properties of the equilibrium interface between a crystal and its melt. Unfortunately, there is an almost total lack of experimental data on such systems as the interface lies between two condensed phases of similar density, making study difficult. Thus, computational methods, such as computer simulation or density functional theory, go beyond their usual roles in the interpretation of experimental data and become important in determining the generic phenomenology of such systems. 

The relaxation of a thermodynamic system to an equilibrium configuration can be conveniently described by a master equation [47]. The probability of finding a system in a specific state increases by the incoming jump from adjacent states, and decreases by the outgoing jump from this state to the others. From now on we shall be specific for the lattice-gas model of crystal growth, described in the previous section. At the time t the system will be found in the state. S/ with a probability density t), and its evolution... 

For crystal growth from the vapor phase, one better chooses the transition probability appropriate to the physical situation. The adsorption occurs ballistically with its rate dependent only on the chemical potential difference Aj.1, while the desorption rate contains all the information of local conformation on the surface [35,48]. As long as the system is close to equilibrium, the specific choice of the transition probability is not of crucial importance. 

In the next section we describe a very simple model, which we shall term the crystalline model , which is taken to represent the real, complicated crystal. Some additional, more physical, properties are included in the later calculations of the well-established theories (see Sect. 3.6 and 3.7.2), however, they are treated as perturbations about this basic model, and depend upon its being a good first approximation. Then, Sect. 2.1 deals with the information which one would hope to obtain from equilibrium crystals — this includes bulk and surface properties and their relationship to a crystal s melting temperature. Even here, using only thermodynamic arguments, there is no common line of approach to the interpretation of the data, yet this fundamental problem does not appear to have received the attention it warrants. The concluding section of this chapter summarizes and contrasts some further assumptions made about the model, which then lead to the various growth theories. The details of the way in which these assumptions are applied will be dealt with in Sects. 3 and 4. 

Chemical vapor deposition processes are complex. Chemical thermodynamics, mass transfer, reaction kinetics and crystal growth all play important roles. Equilibrium thermodynamic analysis is the first step in understanding any CVD process. Thermodynamic calculations are useful in predicting limiting deposition rates and condensed phases in the systems which can deposit under the limiting equilibrium state. These calculations are made for CVD of titanium - - and tantalum diborides, but in dynamic CVD systems equilibrium is rarely achieved and kinetic factors often govern the deposition rate behavior. 

We have seen that the deposition of crystals from the vapor is much too slow to model by MD techniques. Most laboratory equipment for producing thin films involves relatively slow crystal growth processes, and is not suitable for direct simulation. Information on the stability and properties of thin films can be obtained by similar modeling techniques, however. We describe below some of our results that provide necessary data to find the equilibrium configuration of thin films at low temperatures. 

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. 

Snow crystals [4] Their macroscopic structure is different from a bulk three-dimensional ice crystal, but they are formed by homologous pair-pair interaction between water molecules and are static and in thermodynamic equilibrium. It should be noted, however, that dendritic crystal growth is a common phenomenon for metals [5-7] and polymers. The crystals grow under non-equilibrium conditions, but the final crystal is static. 

Primary crystallization occurs when chain segments from a molten polymer that is below its equilibrium melting temperature deposit themselves on the growing face of a crystallite or a nucleus. Primary crystal growth takes place in the "a and b directions, relative to the unit cell, as shown schematically in Fig. 7.8. Inevitably, either the a or b direction of growth is thermodynamically favored and lamellae tend to grow faster in one direction than the other. The crystallite thickness, i.e., the c dimension of the crystallite, remains constant for a given crystallization temperature. Crystallite thickness is proportional to the crystallization temperature. 

Lateral growth occurs in real systems but is not accounted for in the model of Flory. What allows its incorporation into these new calculations is the assignation of the chains to their most probable positions the chains continuously seek positions of equilibrium as crystallization proceeds. This means that all amorphous links have the same propensity for crystallization, which therefore tends to eliminate a distinction between lateral and longitudinal crystal growth (keep in mind that different levels of crystallinity favor one growth pattern over the other -low crystallinity favors fibrils, high crystallinity favors lamellae). 

Since the denominator falls in the range Dt to 1, concentration in the solid is closer to that of the liquid away from the interface than equilibrium fractionation would require. Again, disequilibrium partitioning during crystal growth decreases solid-liquid chemical fractionation. 

As Fig. 6.13 illustrates growth and dissolution are not symmetric with respect to the saturation state. At very high undersaturation, the rate of dissolution becomes independent of S and converges to the value of the apparent rate constant. This is why studies of dissolution far from equilibrium allow to study the influence of inhibition/ catalysis on the apparent rate constant, independently from the effect of S. The same is not true for crystal growth. 

In their pioneering work on crystal growth Burton, Cabrera and Frank predicted that on an atomic length scale the equilibrium structure of a crystal surface should exhibit a transition from a smooth state at low temperatures to... 

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