Diffusive crystal growth diffusion-controlled

Early investigators (N3, N4) assumed that crystal growth was controlled by solute diffusion and that the solution contacting the crystal surface was saturated. Berthoud (B4) was the first to suggest that the rate of the surface reaction must be taken into account. Valeton (VI) later added further support to this theory, and the following equations were developed for rate of crystal growth ... 

The solubility concentration is c which is smaller than the bulk concentration c. The total concentration drop Ac = c - c is split up into two contributions. The first part (c - Cj) within the concentration boundary layer is the driving force for diSusion and convection whereas the second part (cj - c ) in the very thin layer where the integration step takes place is effective for this step. The index I means Interface. In the case of growth completely controlled by diffusion and convection, (cj - c ) (c - Cj) or (Cj - c )/(c - Cj) l is valid. Contrary to this with the ratio (c - Cj)/ cj - c ) l crystal growth is controlled by the integration step. The molar flux density h directed toward the crystal srrrface is... 

In most industrial crystallizers growth is controlled by both diffusion and the BCF integration step. As a rule the molar flux density n is described by the following equation ... 

Crystal morphology describes both the size and the shape (habit) of a crystal. Crystal polyhedra develop according to the growth rates of their surfaces. Crystal growth is controlled by kinetic effects such as diffusion and heat transfer in crystallizing solutions or melts as well as by supramolecular recognition processes at crystal surfaces. 

The kinetics of crystal growth has been much studied Refs. 98-102 are representative. Often there is a time lag before crystallization starts, whose parametric dependence may be indicative of the nucleation mechanism. The crystal growth that follows may be controlled by diffusion or by surface or solution chemistry (see also Section XVI-2C). 

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. 

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). 

Nielsen, A.E. (1959b) The kinetics of crystal growth in barium sulphate precipitation. 111. Mixed surface reaction and diffusion-controlled rate of growth. Acta Chem. Scand., 13, 1680-1686. 

The dynamics of upd reactions have also been examined by STM. The formation of the ordered copper/sulfate layer [354] and copper chloride layer [355] on Au(lll) was examined in a dilute solution of Cu where the reaction was under diffusion control so that growth proceeded on a time scale compatible with STM measurements [354]. In another study, the importance of step density on nucleation was examined and the voltammetric and chronoamperometric response for Cu upd on vicinal Au(lll) was shown to be a sensitive function of the crystal miscut, as... 

Figure 1-11 Concentration profile for (a) crystal growth controlled by interface reaction (the concentration profile is flat and does not change with time), (b) diffusive crystal growth with t2 = 4fi and = 4t2 (the profile is an error function and propagates according to (c) convective crystal growth (the profile is an exponential function and does not change with time), and (d) crystal growth controlled by both interface reaction and diffusion (both the interface concentration and the length of the profile vary).

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. 

To use this method to obtain diffusivity, the dissolution must be diffusion controlled. The diffusion aspect was discussed in Section 3.5.5.1, and the heterogeneous reaction aspect is discussed later. The melt growth distance (L, which differs from the crystal dissolution distance by the factor of the density ratio of crystal to melt) may be expressed as (Equation 3-115d)... 

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. 

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