Avrami theorem

In a real system there will be several clusters growing simultaneously. At first the clusters are separated, but as they grow, they meet and begin to coalesce (see Fig. 10.5), which complicates the growth law. For the case of circular growth considered here, the Avrami theorem [4]... 

Growth of isolated nuclei at an electrode surface is eventually limited when they start to coalesce due to their number and size and the size of the electrode area. Analysis of the overlap problem can be performed by use of the Avrami theorem [152] and leads to maxima in the current—time curves at constant potential. Potentiostatic conditions are convenient for the study of these phenomena because electrochemical rate coefficients and surface concentration conditions are well controlled. 

S(/) can be described by the well-known Avrami theorem [3.317], supposing multiple nucleation on a quasi-homogeneous substrate surface with a sufficient density of nuclei, statistically local distribution of nuclei, and overlapping of growing 2D islands ... 

The same strategy was applied in the derivation of rate equations for w-step nucleation according to a power law (cf. Eq. (21)) [133, 134], the combination of nucleation laws with anisotropic growth regimes [153], as well as truncated nucleation due to time-dependent concentration gradients of monomers [136]. MC simulations verified that the Avrami theorem is valid for instantaneous [184], progressive [185], and n-step nucleation according to a power law [184-187]. 

As previously considered with respect to the three-dimensional nucleation situation, the expression for the current transient corresponding to multiple two-dimensional nucleation requires consideration of the coalescence of growth centers, which diminishes the edge length available for attachment of new atoms or molecules, and produces a decay of the current. By means of the Avrami theorem (see Sect. 5.3.4.1), the following expressions are obtained for the limiting cases of instantaneous and progressive two-dimensional nucleation ... 

Fig. 9.15 - The overlap problem. The Avrami theorem (Equation (9.36)) relates the true surface area, S, to the nominal extended area . The figure illustrates how the overlap of growth centres shown in (a) gives rise to the real area corresponding to (b) and the extended area shown in (c).

As 5ex becomes very large, S tends towards 1, i.e. the surface is completely covered. Equation (9.36) is known as the Avrami theorem. The charge density associated with this notional extended area is given by... 

The effect of the exponential term introduced by the Avrami theorem is to introduce a maximum into the current transient, and at long times the current approaches zero. 

Note that we consider unit area in the general case S and Sex denote fractional coverage. At short times Sex 1, the clusters do not touch, and S Sex- At long times Sex -> oo and S - 1, and the whole surface is covered by a monolayer. For a proof of Avrami s theorem we refer to his original paper [4] (see also Problem 2). 

Here we derive Avrami s theorem for a simple case [4]. Consider an area A that is partially covered by N circles each of area a, where a A. The circles overlap so that the area that is actually covered is smaller than the extended area Na. Show that the probability that a particular point in not covered by any circle is ... 

However, nuclei do not grow independently of each other and their interactions results in overlap of the diffusion fields around them. Overlap may be accounted for using Avrami s theorem, [13]... 

---