Growth diffusion-controlled

Fick s first and second laws describe the diffusion of solute to the surface and the difhision of solvent and other coordination ions away from the surface of a growing crystal. 

In these equations, J is the flux perpendicular to VC and V C is the Laplacian of the concentration of solute, solvent, or other coordination 

FIGURE 6.11 The energetics of crystal growth from solution (a) Movement of the solvated solute molecule etnd (b) corresponding energy changes for each transformation. Redrawn, with permission from E3well and Scheel [28]. 

These equations have the following error function solution [39]  

In the case of slow ciy stal growth, it can he assumed that this stea state (in reality, a pseudo-steady state) is set up faster than the crystal grows and 

The rate of change of the mass of a particle resulting from diffusion of vapor molecules of species A to the particle is expressed by (12.9), or the equivalent expression 

The vapor pressure of A just above the particle surface can be related to the particle mass through the Kelvin equation, (9.86), which after expressing the particle diameter as a function of its mass by 

We define the saturation ratio S = Pa/P reference growth rate. 

This selection simplifies the algebraic calculations, but it is nontheless arbitrary. The dimensionless growth rate corresponding to (12.114) and (12.116) is 

In the continuum regime / varies proportionally to while in the kinetic regime the dependence changes to p. Therefore, in the continuum regime, I varies like Dp, while in the kinetic regime like Dp. 

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We can find a good example of this diffusion-controlled growth in plain carbon steels. As we saw in the "Teaching Yourself Phase Diagrams" course, when steel is cooled below 723°C there is a driving force for the eutectoid reaction of... 

Fig. 7.3 Simplified scheme for the diffusion-controlled growth of multilayered scales on pure iron and mild steel above 570° C...

Reactions of the general type A + B -> AB may proceed by a nucleation and diffusion-controlled growth process. Welch [111] discusses one possible mechanism whereby A is accepted as solid solution into crystalline B and reacts to precipitate AB product preferentially in the vicinity of the interface with A, since the concentration is expected to be greatest here. There may be an initial induction period during solid solution formation prior to the onset of product phase precipitation. Nuclei of AB are subsequently produced at surfaces of particles of B and growth may occur with or without maintained nucleation. 

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... 

As a rule, short nucleation times are the prerequisite for monodisperse particle formation. A recent mechanistic study showed that when Pt(acac)2 is reduced by alkylalu-minium, virtually all the Pt cluster nuclei appear at the same time and have the same size [86]. The nucleation process quickly consumes enough of the metal atoms formed initially to decrease their concentration below the critical threshold. No new metal cluster nuclei are created in the subsequent diffusion-controlled growth stage. 

Nielsen, A. E. 1964. Diffusion controlled growth. Kinetics of Precipitation. Pergamon Press, Oxford, UK. p. 34. 

Fick s diffusion laws, depending on the shape of the liquid as determined by the particle surfaces (1,31-33). However, in such systems the rate of depletion of a supersaturated solution by diffusion controlled growth would be very fast, and if the rate is actually slow (measurable) the rate control is likely to be a surface process. 

A general treatment of a diffusion-controlled growth of a stoichiometric intermetallic in reaction between two two-phase alloys has been introduced by Paul et al. (2006). A reaction couple in which a layer of Co2Si is formed during inter-diffusion from its adjacent saturated phases was used as a model system. In the discussion it has been emphasized that the diffusion couple is undoubtedly one of the most efficient and versatile techniques in solid-state science it is therefore desirable to have alternative theories that enable us to deduce the highest possible amount of information from the data that are relatively easily attainable in this type of experiments. 

If the transport process is rate-determining, the rate is controlled by the diffusion coefficient of the migrating species. There are several models that describe diffusion-controlled processes. A useful model has been proposed for a reaction occurring at the interface between two solid phases A and B [290]. This model can work for both solids and compressed liquids because it doesn t take into account the crystalline environment but only the diffusion coefficient. This model was initially developed for planar interface reactions, and then it was applied by lander [291] to powdered compacts. The starting point is the so-called parabolic law, describing the bulk-diffusion-controlled growth of a product layer in a unidirectional process, occurring on a planar interface where the reaction surface remains constant ... 

Many high-pressure reactions consist of a diffusion-controlled growth where also the nucleation rate must be taken into account. Assuming a diffusion-controlled growth of the product phase from randomly distributed nuclei within reactant phase A, various mathematical models have been developed and the dependence of the nucleation rate / on time formulated. Usually a first-order kinetic law I =fNoe fi is assumed for the nucleation from an active site, where N t) = is the number of active sites at time t. Different shapes of the... 

Anticipating the discussion on acetylene polymerization [98], extensively reported in Section IV, a value of n = 0.6 has been found, which implies a linear diffusion-controlled growth where the molecular librational and translational oscillations control the approach of the monomers to the active sites (chain terminations). 

Diagnostic Relationships Between Current, Maximum Current, and Time. Scharifker and Hills (26) developed a theory that deals with the potentiostatic current transients for 3D nucleation with diffusion-controlled growth. According to this theory, the theoretical diagnostic relationship in a nondimensional form is given by... 

The influence of surface tension on the diffusion-controlled growth or... 

Electrodeposition on transparent material such as indium tin oxide (ITO) can be used for electrochromic applications [328]. Pb deposition on indium-tin oxide electrode occurs by three-dimensional nucle-ation with a diffusion-controlled growth step for instantaneous nucleation [329], and the electrode process has also been studied using electrochemical impedance spectroscopy [328]. 

Surface reaction controlled growth Diffusion-controlled growth... 

Therefore, the most efficient preparation of monodispersed particles can be achieved by regulating the feed rate of reactants (= — dC/dt) as proportional to tm in the case of diffusion-controlled growth and to t1 in the case of reaction-controlled growth, as illustrated in Fig. 4.1.10. 

Fig. 4.1.10 Maximum consumption rates of solute as a function of time for diffusion-controlled growth and reaction-controlled growth.

Diffusion controlled growth of the ApBq and ArBs layers Increasing the thickness of the ArBs layer will inevitably result in a change of its growth regime from reaction to diffusion controlled with regard to... 

Using glassy carbon they observed a three-dimensional nudeation of gallium, with diffusion-controlled growth of the nuclei. The diffusion coefficients for the Ga(III) and the Ga(I) species were 2.28 x 10 7 and 9.12 x 10-7cm2 s 1, respectively. 

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