Growth supersaturation

An adaptation of the Damkochicr number (Da) is a useful concept for evaluation of mixing effects in crystallization. It is the ratio of the characteristic mixing time to its corresponding process time (nucleation induction time, crystal growth/supersaturation release time, or reaction time). Studies of these times and the resulting predicted Damkoehler number in a laboratory setting can provide evidence of possible scale-up problems. 

Supersaturation is the driving force in nucleation and growth. After dissolving the chemical species in a solvent, whether or not of a predetermined nature, the solution must be supersaturated in order to observe nucleation or growth. Supersaturation is the difference between the chemical potential of the solute molecules in the supersaturated (ji) and saturated states, respectively. For one molecule the expression of this difference is ... 

The resistance to nucleation is associated with the surface energy of forming small clusters. Once beyond a critical size, the growth proceeds with the considerable driving force due to the supersaturation or subcooling. It is the definition of this critical nucleus size that has consumed much theoretical and experimental research. We present a brief description of the classic nucleation theory along with some examples of crystal nucleation and growth studies. 

Once nuclei form in a supersaturated solution, they begin to grow by accretion and, as a result, the concentration of the remaining material drops. There is thus a competition for material between the processes of nucleation and of crystal growth. The more rapid the nucleation, the larger the number of nuclei formed before relief of the supersaturation occurs and the smaller the final crystal size. This, qualitatively, is the basis of what is known as von Weimam s law [86] ... 

In the LS analysis, an assembly of drops is considered. Growth proceeds by evaporation from drops withi < R and condensation onto drops R > R. The supersaturation e changes in time, so that e (x) becomes a sort of mean field due to all the other droplets and also implies a time-dependent critical radius. R (x) = a/[/"(l)e(x)]. One of the starting equations in the LS analysis is equation (A3.3.87) withi (x). 

Over 50 acidic, basic, and neutral aluminum sulfate hydrates have been reported. Only a few of these are well characterized because the exact compositions depend on conditions of precipitation from solution. Variables such as supersaturation, nucleation and crystal growth rates, occlusion, nonequilihrium conditions, and hydrolysis can each play a role ia the final composition. Commercial dry alum is likely not a single crystalline hydrate, but rather it contains significant amounts of amorphous material. 

Nucleation of particles in a very short time foUowed by growth without supersaturation yields monodispersed coUoidal oxide particles that resist agglomeration (9,10). A large range of coUoidal powders having controUed size and morphologies have been produced using these concepts (3,14). 

Supersaturation has been observed to affect contact nucleation, but the mechanism by which this occurs is not clear. There are data (19) that infer a direct relationship between contact nucleation and crystal growth. This relationship has been explained by showing that the effect of supersaturation on contact nucleation must consider the reduction in interfacial supersaturation due to the resistance to diffusion or convective mass transfer (20). 

Models used to describe the growth of crystals by layers call for a two-step process (/) formation of a two-dimensional nucleus on the surface and (2) spreading of the solute from the two-dimensional nucleus across the surface. The relative rates at which these two steps occur give rise to the mononuclear two-dimensional nucleation theory and the polynuclear two-dimensional nucleation theory. In the mononuclear two-dimensional nucleation theory, the surface nucleation step occurs at a finite rate, whereas the spreading across the surface is assumed to occur at an infinite rate. The reverse is tme for the polynuclear two-dimensional nucleation theory. Erom the mononuclear two-dimensional nucleation theory, growth is related to supersaturation by the equation. 

The screw dislocation theory (27), often referred to as the BCE theory (after its formulators), shows that the dependence of growth rate on supersaturation can vary from a paraboHc relationship at low supersaturation to a linear relationship at high supersaturation. In the BCE theory, growth rate is given by... 

An empirical approach can also be used to relate growth kinetics to supersaturation with a power-law function of the form... 

Both supersaturation and temperature can have different effects on the growth rates of different faces of the same crystal. Such occurrences have implications with respect to crystal habit, and these are dealt with in a later section. 

Another type of crystallizer is the Oslo-type unit shown in Figure 24. In units of this type, the object is to form a supersaturated solution in the upper chamber and then reHeve the supersaturation through growth in the lower chamber. The use of the downflow pipe in the crystallizer provides good mixing in the growth chamber. 

Growth on Foreign Nuclei As noted above, foreign nuclei are often present in abundance and permit fog formation at much lower supersaturation. For example,... 

Crystal Formation There are obviously two steps involved in the preparation of ciystal matter from a solution. The ciystals must first Form and then grow. The formation of a new sohd phase either on an inert particle in the solution or in the solution itself is called nucle-ation. The increase in size of this nucleus with a layer-by-layer addition of solute is called growth. Both nucleation and ciystal growth have supersaturation as a common driving force. Unless a solution is supersaturated, ciystals can neither form nor grow. Supersaturation refers to the quantity of solute present in solution compared with the quantity which would be present if the solution were kept for a veiy long period of time with solid phase in contac t with the solution. The latter value is the equilibrium solubility at the temperature and pressure under consideration. The supersaturation coefficient can be expressed... 

Certain qualitative fac ts in connection with supersaturation, growth, and the yield in a crystalhzation process are readily apparent. 

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