Crystallization transport-controlled

It is usually believed that the growth of dendritic crystals is controlled by a bulk diffusion-controlled process which is defined as a process controlled by a transportation of solute species by diffusion from the bulk of aqueous solution to the growing crystals (e.g., Strickland-Constable, 1968 Liu et al., 1976). The appearances of feather- and star-like dendritic shapes indicate that the concentrations of pertinent species (e.g., Ba +, SO ) in the solution are highest at the corners of crystals. The rectangular (orthorhombic) crystal forms are generated where the concentrations of solute species are approximately the same for all surfaces but it cannot be homogeneous when the consumption rate of solute is faster than the supply rate by diffusion (Nielsen, 1958). 

We define the linear growth rate Vg as the linear velocity of displacement of a crystal face relative to some fixed point in the crystal. vg may be known as a function of c and c , derived from the theory of transport control, and as a function of c and cs as well, derived from the theory of surface control. Then c may be eliminated by equating the two mathematical expressions... 

From the above statements it follows that it should be possible to derive the growth kinetics and calculate the growth rate of uncontaminated electrolyte crystals when the following parameters are known molecular weight, density, solubility, cation dehydration frequency, ion pair stability coefficient, and the bulk concentration of the solution (or the saturation ratio). If the growth rate is transport controlled, one shall also need the particle size. In table I we have made these calculations for 14 electrolytes of common interest. For the saturation ratio and particle size we have chosen values typical for the range where kinetic experiments have been performed (29,30). The empirical rates are given for comparison. 

Fig. 2.3 Rate-limiting steps in mineral dissolution (a) transport-controlled, (b) surface reaction-controlled, and (c) mixed transport and surface reaction control. Concentration (C) versus distance (r) from a crystal surface for three rate-controUing processes, where is the saturation concentration and is the concentration in an infinitely diluted solution. Reprinted from Sparks DL (1988) Kinetics of soil chemical processes. Academic Press New York 210 pp. Copyright 2005 with permission of Elsevier...

Dendritic. In electrodeposited films, dendritic grains result from mass-transport-controlled growth, and the individual crystals may vary in shape. 

Nucleation is necessary for the new phase to form, and is often the most difficult step. Because the new phase and old phase have the same composition, mass transport is not necessary. However, for very rapid interface reaction rate, heat transport may play a role. The growth rate may be controlled either by interface reaction or heat transport. Because diffusivity of heat is much greater than chemical diffusivity, crystal growth controlled by heat transport is expected to be much more rapid than crystal growth controlled by mass transport. For vaporization of liquid (e.g., water vapor) in air, because the gas phase is already present (air), nucleation is not necessary except for vaporization (bubbling) beginning in the interior. Similarly, for ice melting (ice water) in nature, nucleation does not seem to be difficult. 

Figure 1-11 Interface- or transport-controlled crystal growth 51...

There are basically three, rate-limiting mechanisms for mineral dissolution assuming a fixed degree of undersaturation. They are (1) transport of solute away from the dissolved crystal or transport-controlled kinetics. 

The transport of the growth unit(s) from the bulk solution through the hydrodynamic boundary layer to a region adjacent to the adsorption layer of the crystal. This is often referred to as bulk transport-controlled, volume diffusion-controlled, or simply transport-controlled. 

If the calculated value of is equal to the measured intracrystalline lifetime, Tinira, the rate of molecular exchange between different crystals is controlled by the intracrystalline self-diffusion as the rate-limiting process. Any increase of Timn, in comparison with Tf,j L indicates the existence of transport resistances different from intracrystalline mass transport. Under the conditions of TD NMR one has A r. > Antra, thus these resistances can only be brought about by sur ce barriers. The ratio Timra/Tfn L represents, therefore, a direct measure of the influence of surface barriers on molecular transport. 

The so-called dry gel method [20-22] is another alternative for membrane synthesis where vapors containing templates (i.e., amines) and water are employed to crystallize silica or silica-alumina layers previously deposited onto the support (vapor-phase transport method) or where steam is used to crystallize silica and template or sUica-alumina and template dry layers previously deposited onto the support (steam-assisted crystallization). Using this approach, the reactant consumption is clearly diminished, an important issue for scale-up purposes on the other hand, synthesis time is delayed due to transport-controlled phenomena. 

In order to undergo a redox process, the reactant must be present within the electrode-reaction layer, in an amount limited by the rate of mass transport of Yg, to the electrode surface. In electrolyte media, four types of mass-transport control, namely convection, diffusion, adsorption and chemical-reaction kinetics, must be considered. The details of the voltammetric procedure, e.g., whether the solution is stirred or quiet, tell whether convection is possible. In a quiet solution, the maximum currents of simple electrode processes may be governed by diffusion. Adsorption of either reactant or product on the electrode may complicate the electrode process and, unless adsorption, crystallization or related surface effects are being studied, it is to be avoided, typically... 

Alternative processes. In the case of a transport-controlled process, the overall reduction rate can be significantly increased if a continuous renewal of the gas in contact with the solid oxide is readily achieved, i.e., when the oxide particles are not in close contact with each other, as, for example, in a fluidized bed, or a laminar flow reactor, where the reaction is virtually instantaneous. Under these conditions, however, the change in the water removal rate will have a strong impact on the powder properties, in particular on the average grain size. It is very unlikely that under such conditions the production of coarse powder (i.e., the formation of small W single crystals of size >10 pm) can be successful. Thus the inherent ability of the powder bed to retard the water vapor within the layer and to build up locally humid conditions is an important aspect in industrial powder manufacture. 

Fig. 62. Crystal-electrode geometry, notation, and boundary conditions for mass transport-controlled reaction of H+ at the crystal surface.

Fig. 66. The variation in shielding factor with time for the reduction of H+ in an aqueous polymaleic acid system of concentration 7.63 mM in monomer units. The flow rate employed is 1.64 x 10 2cm3 s 1. A theoretical shielding factor of0.333 is predicted on the basis of the crystal electrode geometry, for mass transport-controlled reaction of H+ at the crystal surface. Data from the authors laboratory.

For PEIMs (n = odds), crystallization at temperatures above the liquid crystalline transition may result in spherulitic morphology development. Kinetics analysis of these PEIMs indicates that the phase transition is mainly determined by the number of methyl units in the spacers. The liquid crystallization formation can be either transport-controlled or nucleation-controlled, depending on how far apart the liquid crystalline transition is from the glass transition temperature of the polymer. As all of these PEIMs form a monoclinic system, the lifetime of mesophase may be very short (in seconds). Therefore, kinetics analysis may involve two stages mesophase formation followed by true crystallization (Table 3.2). 

Dislocations must also be taken into consideration as possible high mobility paths for particles. The density Pe> 2ind the spatial arrangement of dislocations are very difficult to control. Thus, even for single crystals, transport coefficients can be structure-sensitive at temperatures less than about half the melting temperature where volume diffusion no longer predominates. 

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