Crystal growth integration

It has often been observed that the plot of ln(L) versus L results in curvature rendering the method of determining the growth rate from the slope strictly inappropriate, but ways to accommodate such deviations have also been proposed. Thus, if G = G(L) integration of equation 3.15 leads to the following expression for determining crystal growth rates (Sikdar, 1977)... 

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... 

The process of substituting elements for the silicon is called doping, while the elements are referred to as dopants. The amount of dopant that is required in practical devices is very small, ranging from about 100 dopant atoms per million silicon atoms downward to 1 per billion. Dopants are usualty added to the silicon after the crystal growth process, when an integrated circuit is being formed on the surface of the wafer. 

The exponents i and s in equations 15.13 and 15.14, referred to as the order of integration and overall crystal growth process, should not be confused with their more conventional use in chemical kinetics where they always refer to the power to which a concentration should be raised to give a factor proportional to the rate of an elementary reaction. As Mullin(3) points out, in crystallisation work, the exponent has no fundamental significance and cannot give any indication of the elemental species involved in the growth process. If i = 1 and s = 1, c, may be eliminated from equation 15.13 to give ... 

This results In a set of first-order ordinary differential equations for the dynamics of the moments. However, the population balance Is still required In the model to determine the three Integrals and no state space representation can be formed. Only for simple MSMPR (Mixed Suspension Mixed Product Removal) crystallizers with simple crystal growth behaviour, the population balance Is redundant In the model. For MSMPR crystallizers, Q =0 and hp L)=l, thus ... 

Figure 5.9. Time evolution of the radii of selected 2D spherulites from Fig. 5.8. The curves correspond to R t) obtained by integration of Eq. (5.9). Reprinted from Journal of Crystal Growth, Vol. 209, J. Caro, J. Fraxedas and A. Figueras, Thickness-dependent spherulitic growth observed in thin films of the molecular organic radical p-nitrophenyl nitronyl nitroxide, 146-158, Copyright (2000), with permission from Elsevier.

Further, substituting Eq. (12-8) into Eq. (12-3), integrating the resulting equation between t - 0 and t = tf and rearranging yields the expression for the overall crystal-growth rate coefficient as... 

At least two resistances contribute to the kinetics of crystal growth. These resistances apply to (1) integration of the crystalline unit (e.g., solute molecules) into the crystal surface (i.e., lattice), and (2) molecular diffusion or bulk transport of the unit from the surrounding solution to the crystal surface. As aspects of molecular diffusion and mass transfer are covered elsewhere, the current discussion will focus only on surface incorporation. 

Examination of Eqs. (12) to (16) reveals five distinct rate steps dissolution of B the reaction between A and B the generation of P nuclei in the liquid phase the mass transfer of dissolved P to the growing P crystals and the surface integration of the solute P into the crystal lattice (i.e., the crystal growth step). The relative importance of each of these steps can be characterized by a dimensionless number. For a reaction which is second order overall, and for nucleation and growth kinetics which can be represented by conventional power law expressions, we have... 

The use of relative size l/L permits analysis of crystal growth without detailed knowledge of the mathematical relation between growth rate and supersaturation. Integrating Eq. (36) from the lower limit W = 0 at l = 0, the expression for cumulative weight of crystals up to size l is ... 

Protein crystal growth involves the incorporation of a complex unit into an existing lattice. The growth unit usually includes the covalent polypeptide chain, water molecules that are integral components of the folded protein structure, and additional water molecules and solvent ions that may become immobilized at crystal lattice contacts. Direct inter-... 

Desolvation of the growth unit may occur anywhere in steps 2-A or the solvent may be adsorbed with the growth unit. As any of the above steps can be the rate limiting step in the crystal growth process and they are dependent on conditions such as supersaturation, temperature, additives or solvent, and the hydrodynamics of the system, crystal growth is generally divided into two main mechanisms volume-diffusion controlled or surface-integration controlled (Fig. 9). ... 

Crystal growth is volume-diffusion controlled when the diffusion of molecules from the bulk to the crystal surface is the rate-limiting step while growth is considered surface-integration controlled if the incorporation of a growth unit into the lattice is the slowest process. Many crystallization studies involving proteins. 

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