Growth rate diffusion-controlled

Stage III Nonlinear growth rate (Diffusion controlled regime)... 

If the growth rate is controlled by both interface reaction and diffusion (Figure 1-1 Id), then (i) the concentration profile is not flat, (ii) the interface concentration changes with time toward the saturation concentration, (iii) the diffusion profile propagates into the melt, and (iv) the growth rate is not constant, nor does it obey the parabolic growth law. 

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. 

Growth of Concretions In sedimentary rocks we commonly find concretions, that is, material formed by deposition of a precipitate, such as calcite or siderite, around a nucleus of some particular mineral grain or fossil. The origin of most concretions is not known but their often-spherical shape with concentric internal structure suggests diffusion as an important factor affecting growth. The rate of growth, if diffusion controlled, is readily amenable to mathematical... 

If the integration reaction is very fast or the rate corrstant oo, the growth rate is controlled by the diffusive/convective transport of elementary units. With (c-Cj) = (c- c ) = Ac the molar flux density vahd for small rates is given by... 

Fig. X - Growth, of diffusion-controlled structure, showing form of structure (A), and radia.1 distribution of (B), diffusive flux (C), and rate of dissolution reactions (D).

The account of the formal derivation of kinetic expressions for the reactions of solids given in Sect. 3 first discusses those types of behaviour which usually generate three-dimensional nuclei. Such product particles may often be directly observed. Quantitative measurements of rates of nucleation and growth may even be possible, thus providing valuable supplementary evidence for the analysis of kinetic data. Thereafter, attention is directed to expressions based on the existence of diffuse nuclei or involving diffusion control such nuclei are not susceptible to quantitative... 

Kinetic expressions for appropriate models of nucleation and diffusion-controlled growth processes can be developed by the methods described in Sect. 3.1, with the necessary modification that, here, interface advance obeys the parabolic law [i.e. is proportional to (Dt),/2]. (This contrasts with the linear rate of interface advance characteristic of decomposition reactions.) Such an analysis has been provided by Hulbert [77], who considers the possibilities that nucleation is (i) instantaneous (0 = 0), (ii) constant (0 = 1) and (iii) deceleratory (0 < 0 < 1), for nuclei which grow in one, two or three dimensions (X = 1, 2 or 3, respectively). All expressions found are of the general form... 

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. 

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