Rate constants for nucleation

A kinetic model for single-phase polymerizations— that is, reactions where because of the similarity of structure the polymer grows as a solid-state solution in the monomer crystal without phase separation—has been proposed by Baughman [294] to explain the experimental behavior observed in the temperature- or light-induced polymerization of substimted diacetylenes R—C=C—C=C—R. The basic feature of the model is that the rate constant for nucleation is assumed to depend on the fraction of converted monomer x(f) and is not constant like it is assumed in the Avrami model discussed above. The rate of the solid-state polymerization is given by... 

Note that k+ is the forward rate constant for nucleation. If one chooses a particular value of ci/co (e.g., one-half), then the left-hand side of the equation will remain constant, and one may measure the time, t, required to... 

Differential thermal analysis, DTA, was also carried out to determine the optimum nucleation temperature according to the method reported by Marotta et al. who concluded that, if samples are held for the same time tn, at each heat-treatment temperature Tn, then In 1 (kinetic rate constant for nucleation) is proportional to (1/Tp) - (1/Tp ), where Tp and Tp are, respectively the crystallization exotherm temperatures obtained with and without a nucleation hold. Plotting (1/Tp) - (1/Tp ) against nucleation hold temperature gives a bell shaped curve, with the optimum nucleation... 

The parameter A, in turn, is inversely proportional to the rate constants for nucleation, nuclei and chains growth (Kgm, Kgs, Kgi) respectively. Thus, the analysis of non-isothermal reduction of nickel from NiO allows determination of the possible mechanism for this process, which is interpreted as quasi-chain reaction accompanied by the nuclei growth with equal probability of the size distribution. The proposed kinetic analysis can be used for other thermally activated processes from unstable precursors such as oxalates, metal carbonyls etc. 

This so-called linear nucleation law (k = rate constant for nucleation) presupposes an equal a priori probability of nucleation at all inner and outer surfaces, as well as a slow rate of nucleation. However, if the time required for the thermal decomposition of the crystal is large compared to 1/k, then will become approximately equal to A after only a short period at the beginning of the reaction. Thai is, nucleation will be complete almost immediately. 

Table 3 Values of the rate constants for nucleation and growth of Au particles in aqueous solution during the redox reaction between chloride gold(III) complex ions and ascorbic acid. Temperature 25°C...

The primary crystallization process is characterized by three parameters. These are the rate of radial growth of the spherulite, G, the time constant for nucleation, t , and the time constant for the primary crystallization process, Tc, which is determined from the Avrami equation. All three parameters seem to depend on the stereoregularity of the polymer, but the nucleation rate seems to depend most strongly. 

By assuming f iist-order kinetic processes, Tomellini et al. formulated the rates of each of the four steps in the kinetic scheme as a function of the surface densities of the transforming species, with /y, n, K and as rate constants for the corresponding steps. The equations for diamond nucleation density and rate were then derived by solving these coupled kinetic equations. For small values of surface coverage (S(t) 1) during the entire nucleation process, the solution reduces to... 

The net flux J is the nucleation rate, and (11.11) is a direct expression for the nucleation rate as a function of the forward and reverse rate constants for the clusters. 

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