Growing Spheres

Next we consider similar problems as the growth or etching of sohd films, but now the solid is a sphere instead of a planar film. For growth or dissoluhon of a sphere with initial radius Rai the volume is F and d V = An dR, so that the number of moles of the solid B is 

If C/is = C/ib, then aU quantities on the right side in the preceding equation are independent of time and size so the equation can be rearranged 

For reaction-limited growth of a sphere the preceding equation gives 

If the surface reaction rate coefficient is large compared to the mass transfer coefficient, k m/4) then the process is mass transfer limited. For growth this requires that 

For mass transfer around a sphere we have from Chapter 7 

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Two examples will now be given of solution of the convective diffusion problem, transport to a rotating disk as a stationary case and transport to a growing sphere as a transient case. Finally, an engineering approach will be mentioned in which the solution is expressed as a function of dimensionless quantities characterizing the properties of the system. 

Convective diffusion to a growing sphere. In the polarographic method (see Section 5.5) a dropping mercury electrode is most often used. Transport to this electrode has the character of convective diffusion, which, however, does not proceed under steady-state conditions. Convection results from growth of the electrode, producing radial motion of the solution towards the electrode surface. It will be assumed that the thickness of the diffusion layer formed around the spherical surface is much smaller than the radius of the sphere (the drop is approximated as an ideal spherical surface). The spherical surface can then be replaced by a planar surface... 

In the general case, the initial concentration of the oxidized component equals Cqx and that of the reduced component cRed. If the appropriate differential equations are used for transport of the two electroactive forms (see Eqs 2.5.3 and 2.7.16) with the corresponding diffusion coefficients, then the relationship between the concentrations of the oxidized and reduced forms at the surface of the electrode (for linear diffusion and simplified convective diffusion to a growing sphere) is given in the form... 

Together with the boundary condition (5.4.5) and relationship (5.4.6), this yields the partial differential equation (2.5.3) for linear diffusion and Eq. (2.7.16) for convective diffusion to a growing sphere, where D = D0x and = Cqx/[1 + A(D0x/T>Red)12]- As for linear diffusion, the limiting diffusion current density is given by the Cottrell equation... 

Dissolving and Growing Spheres 385 For mass-transfer-limited growth of a sphere in a stagnant fluid the radius varies with time as... 

With homogeneous nucleation, new nuclei are being formed continuously the number of nuclei and thus of growing spheres is proportional to time ... 

The Avrami equation (1), originally developed for the crystallization of metals from melt, has been applied by many researchers to the crystallization of oils and fats in order to elucidate information on their crystallization mechanism. The Avrami equation is based on the model of a growing sphere crystallizing from a melt of uniform density without impingement. The usual Avrami exponent, used to draw conclusions with respect to the crystallization mechanism of the system, is observed to be about three or four for oils and fats after rounding off to whole integers. 

Philipse (5) also assumed that fast hydrolysis created an active monomer bulk. He studied the growth of silica nuclei, already synthesized, after extra addition of different amounts of TES with static light scattering. To explain his growth curves (radius versus time), he used a diffusion-controlled particle growth in a finite bulk of monomers or subparticles. The model contained equations from classical flocculation theories. It takes into account the exhaustion of the monomer bulk and the retarding influence of an (unscreened) electrostatic repulsion between growing spheres and monomers. 

If G is constant, then the rate of volume increase is proportional to the surface area of the growing sphere, a reasonable conclusion since growth occurs at this developing surface. This assumption requires that the radial lamellae branch at a rate sufficient to maintain density and that the degree of crystallinity is constant throughout the spherulite. This infers... 

The film formed on the surface usually has an irregular structure of growing spheres of different sizes and is therefore sometimes called a cauliflower structure, e.g.. Figure 11.8. 

Witze CP, Schrock VE, Chambre PL. (1968) Flow about a growing sphere in contact with a plane wall. Int. 1. Heat Mass Transfer, 11 1637-1652. 

Evans [2] calculated the expectancy of the Poisson probability distribution for the constant propagation rate of domains and two simple nucleation modes instantaneous and spontaneous with the constant rate, F i) = B. Billon et al. [13] extended this approach to the case of time-dependent nucleation rate. According to the Evans theory, an arbitrarily chosen point A can be reached before time t by growing spheres nucleated around it in a distance r (precisely in a distance within the interval (r, r + dr)) before time t - rIG their number is equal to an integral of the nucleation rate F(t) over the time interval (0, t - r/G), multiplied by the considered volume, Artr dr. The total number of spheres occluding the point A until time t is calculated by second integration, over a distance ... 

One notes that V(xd) is equal to the volume of unimpinged spherulite expressed by Equation (7.2a) and Equation (7.2b). In derivations based on probability calculus it is also assumed, as in the extended volume approach, that a growing sphere that passes through an arbitrary point as the first one represents a real spherulite. It appears that the concept of extended volume and probability calculus yield the same result if applied to crystallization in infinite volume with the nucleation and growth rate independent of spatial coordinates. 

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