Nucleation-growth-overlap model

FIGURE 8.4 Schematic illustration of film formation (A) and dissolution (B) on the substrate according to the nucleation-growth-overlap model. 

The number of sites participating in the formation of Sn02 is N0 = 3.0 x 1010 cm-2. This value is similar to those usually found for active sites in the case of metals [67]. As the diagnostic criteria for the instantaneous nucleation-growth-overlap model was fulfilled and physicochemical data obtained for No is physically plausible, we may conclude that instant nucleation with lateral growth of the film occurs. Taking into account that the atomic density of the substrate is of the order of 1015 cm2 [67], we conclude that the density of the active sites on the surface can severely limit the nucleation process. 

The fact that the experimental data fit both models, the dissolution-precipitation and the instantaneous nucleation-growth-overlap models, confirms the assertion made at the beginning of this chapter, that is, some models are complementary to each other. 

Miiller-Calandra, Srinivasan-Gileadi, and instantaneous nucleation-growth overlap models... 

Diagnostic Parameters Criteria for the Different Growth Models Nucleation-Growth-Overlap (I) [39] Miiller-Calandra (II) [43,44] and Srinivasan-Gileadi (III) [47] under Potentiodynamic Conditions... 

The incorporation of discreet nucleation events into models for the current density has been reviewed by Scharifker et al. [111]. The current density is found by integrating the current over a large number of nucleation sites whose distribution and growth rates depend on the electrochemical potential field and the substrate properties. The process is non-local because the presence of one nucleus affects the controlling field and influences production or growth of other nuclei. It is deterministic because microscopic variables such as the density of nuclei and their rate of formation are incorporated as parameters rather than stochastic variables. Various approaches have been taken to determine the macroscopic current density to overlapping diffusion fields of distributed nuclei under potentiostatic control. 

In addition to these two conceptual models, there are several modified or composite models that have been developed in an attempt to better explain experimental observations. The more recent models are usually based on the concept of defect accumulation, but address the heterogeneous nature of the amorphization process, as has been observed in many semiconductors and ceramic materials by electron spin resonance (Dennis and Hale 1976) and HRTEM (Headley et al. 1981, Miller and Ewing 1992, Wang 1998). These models include the cascade-overlap model (Gibbons 1972, Weber 2000), defect complex overlap model (Pedraza 1986), nucleation and growth model (Campisano et al. 1993, Boise 1998) or models that involve a combination of these mechanisms. 

The existing models describing electrochemical phase formation involving both adsorption and a nucleation/growth process have been recently modified. A generalization of the Avrami s overlap formula for two or more competitive irreversibly growing phases has been proposed [89]. 

The Kolmogoroff-Avrami formalism can be used without restrictions in the case of 3D, 2D and ID crystallization in three-, two- and onedimensional space, respectively [5.19]. However, the method cannot be applied directly to the nucleation, growth and coalescence of 3D clusters on a plane substrate. The reason is that such clusters caimot grow in the direction perpendicular to the substrate and therefore the spread of the 3D deposit is not random in space [5.53]. Since the formulation of a rigorous theoretical model encounters principle difficulties, here we do not consider this complex case of mass electrocrystaUization. However, theoretical treatment of the nucleation, growth and overlap of circular cones, hemispheres and three-dimensional clusters with more complex geometrical forms can be found in [5.29, 5.53-5.61],... 

A comparison between equations (7.8)-(7.9) and (7.12)-(7.13) shows that the exponential terms in Eqs. (7.12) and (7.13) represent the overlap effect (overlap correction). This new model of simultaneous nucleation and growth of nuclei, Eqs. 

Support for the nucleation and growth model was provided by electron microscopy which showed that nucleation was not confined to surfaces, but occurred at lines of internal dislocation. Overlap, following growth, resulted in the formation of approximately cylindrical particles of product. The presence of gaseous hydrogen exerted [51] only a small influence on the kinetics of decomposition. 

Kinetic runs in step b in Fig. 8c started with a very fast reduction of approximately e per molecule, after which a slow reductioh took place, yielding sigmoidal reduction curves. This, indicates that reduction of Co2+ to Co° is controlled by the formation and slow growth of reduction nuclei of metallic cobalt on. the surface of the reduced phase in step a (nucleation model). Initially, the reduction rate increases because of the growth of nuclei already formed and the appearance of new ones. At a certain point the reduction nuclei start to overlap at the inflection point, the interface of. the oxidized and reduced phases and the reduction rate both begin to decrease. Reduction of this type is described by the Avrami-Erofeev equation (118)... 

It was often found that, contrary to the theoretical prediction, the value of n is noninteger (Avrami 1939). The Avrami model is based on several assumptions, such as constancy in shape of the growing crystal, constant rate of radial growth, lack of induction time, uniqueness of the nucleation mode, complete crystallinity of the sample, random distribution of nuclei, constant value of radial density, primary nucleation process (no secondary nucleation), and absence of overlap between the growing crystallization fronts. These assumptions are often not met in polymer (blend) crystallization. Also, erroneous determination of the zero time and an overestimation of the enthalpy of fusion of the polymer at a given time can lead to noninteger values for n (Grenier and Prud homme 1980). 

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