Nucleation power

In principle, nucleation should occur for any supersaturation given enough time. The critical supersaturation ratio is often defined in terms of the condition needed to observe nucleation on a convenient time scale. As illustrated in Table IX-1, the nucleation rate changes so rapidly with degree of supersaturation that, fortunately, even a few powers of 10 error in the preexponential term make little difference. There has been some controversy surrounding the preexponential term and some detailed analyses are available [33-35]. 

Equations (4.24) and (4.29) are equivalent, except that the former assumes instantaneous nucleation at N sites per unit area while the latter assumes a nucleation rate of N per unit area per unit time. It is the presence of this latter rate which requires the power of t to be increased from 2 to 3 in this case. 

Silicone foam thus formed has an open ceU stmcture and is a relatively poor insulating material. Cell size can be controlled by the selection of fillers, which serve as bubble nucleating sites. The addition of quartz as a filler gready improves the flame retardancy of the foam char yields of >65% can be achieved. Because of its excellent dammabiUty characteristics, siUcone foam is used in building and constmction fire-stop systems and as pipe insulation in power plants. Typical physical properties of siUcone foam are Hsted in Table 10. 

Correlations of nucleation rates with crystallizer variables have been developed for a variety of systems. Although the correlations are empirical, a mechanistic hypothesis regarding nucleation can be helpful in selecting operating variables for inclusion in the model. Two examples are (/) the effect of slurry circulation rate on nucleation has been used to develop a correlation for nucleation rate based on the tip speed of the impeller (16) and (2) the scaleup of nucleation kinetics for sodium chloride crystalliza tion provided an analysis of the role of mixing and mixer characteristics in contact nucleation (17). Pubhshed kinetic correlations have been reviewed through about 1979 (18). In a later section on population balances, simple power-law expressions are used to correlate nucleation rate data and describe the effect of nucleation on crystal size distribution. 

A. D. Randolph and D. Etherton, Study of Gypsum Crystal Nucleation and Growth Rates in Simulated Flue Gas Desulfurization Eiquors, EPRI Report CS1885, Electric Power Research Institute, Palo Alto, Calif., 1981. 

Powers, H.E.C., 1963. Nucleation and early crystal growth. Industrial Chemist, 39, 351-355. 

Principles The reduction reaction is controlled essentially by the usual kinetic factors such as concentration of reactants, temperature, agitation, catalysts, etc. Where the reaction is vigorous, as, for example, when a powerful reducing agent like hydrazine is used, wasteful precipitation of A/, may occur throughout the whole plating solution followed by deposition on all exposed metallic and non-metallic surfaces which can provide favourable nucleation sites. In order to restrict deposition and aid adhesion, the selected areas are pre-sensitised after cleaning the sensitisers used are often based on noble metal salts. 

One of the more important uses of OM is the study of crystallization growth rates. K. Cermak constructed an interference microscope with which measurements can be taken to 50° (Ref 31). This app allows for study of the decompn of the solution concentrated in close proximity to the growing crystal of material such as Amm nitrate or K chlorate. In connection with this technique, Stein and Powers (Ref 30) derived equations for growth rate data which allow for correct prediction of the effects of surface nucleation, surface truncation in thin films, and truncation by neighboring spherulites... 

Concentration may also result from waterline evaporation. In higher pressure WT power boilers, it may develop as a result of high-firing cycle operations, which may lead to departure from nucleate boiling (DNB). 

Nucleation obeying a power law with constant rate of interface advance (normal growth)... 

These various distributions reflect the variations in reactivity of the reacting sites. Johnson and Kotz [444] discuss in detail the Weibull and other distributions which find application when conditions of strict randomness of the exponential distribution are not satisfied. From an empirical point of view, the power transformation is a practical and convenient method of introducing a degree of flexibility into a model. Gittus [445] has discussed some situations in which the Weibull distribution may be expected to find application, including nucleation and growth processes in alloy transformations. 

In Mampel s treatment [447] of nucleation and growth reactions, eqn. (7, n = 3) was found to be applicable to intermediate ranges of a, sometimes preceded by power law obedience and followed by a period of first-order behaviour. Transitions from obedience of one kinetic relation to another have been reported in the literature [409,458,459]. Equation (7, n = 3) is close to zero order in the early stages but becomes more strongly deceleratory when a > 0.5. 

Gamer and Hailes [462] postulated a chain branching reaction in the decomposition of mercury fulminate, since the values of n( 10—20) were larger than could be considered consistent with power law equation [eqn. (2)] obedience. If the rate of nucleation is constant (0 = 1 for the generation of a new nuclei at a large number of sites, N0) and there is a constant rate of branching of existing nuclei (ftB), the nucleation law is... 

The strongly acceleratory character of the exponential law cannot be maintained indefinitely during any real reaction. Sooner or later the consumption of reactant must result in a diminution in reaction rate. (This behaviour is analogous to the change from power law to Avrami—Erofe ev equation obedience as a consequence of overlap of compact nuclei.) To incorporate due allowance for this effect, the nucleation law may be expanded to include an initiation term (kKN0), a branching term (k N) and a termination term [ftT(a)], in which the designation is intended to emphasize that the rate of termination is a function of a, viz. 

Fig. 6. Two reaction models which result in obedience to the power law [eqn. (2), n = 2 ] at low a, or the Avrami—Erofe ev equation [eqn. (6), n = 2 ] over a more extensive range of a. In (a), there is growth of semi-circular nuclei in a thin plate of reactant in (b), there is cylindrical growth of linear internal nuclei. In both examples, rapid nucleation (0 = 0) is followed by two-dimensional growth (X = 2).

Before discussing such theories, it is appropriate to refer to features of the reaction rate coefficient, k. As pointed out in Sect. 3, this may be a compound term containing contributions from both nucleation and growth processes. Furthermore, alternative definitions may be possible, illustrated, for example, by reference to the power law a1/n = kt or a = k tn so that k = A exp(-E/RT) or k = n nAn exp(—nE/RT). Measured magnitudes of A and E will depend, therefore, on the form of rate expression used to find k. However, provided k values are expressed in the same units, the magnitude of the measured value of E is relatively insensitive to the particular rate expression used to determine those rate coefficients. In the integral forms of equations listed in Table 5, units are all (time) 1. Alternative definitions of the type... 

From microscopic measurements of the rates of nucleation and of growth of particles of barium metal product, Wischin [201] observed that the number of nuclei present increased as the third power (—2.5—3.5) of time and that the isothermal rate of radial growth of visible nuclei was constant. During the early stages of reaction, the acceleratory region of the a—time plot obeyed the power law [eqn. (2)] with 6 temperature coefficients of these processes were used by Wischin [201]... 

The kinetic observations reported by Young [721] for the same reaction show points of difference, though the mechanistic implications of these are not developed. The initial limited ( 2%) deceleratory process, which fitted the first-order equation with E = 121 kJ mole-1, is (again) attributed to the breakdown of superficial impurities and this precedes, indeed defers, the onset of the main reaction. The subsequent acceleratory process is well described by the cubic law [eqn. (2), n = 3], with E = 233 kJ mole-1, attributed to the initial formation of a constant number of lead nuclei (i.e. instantaneous nucleation) followed by three-dimensional growth (P = 0, X = 3). Deviations from strict obedience to the power law (n = 3) are attributed to an increase in the effective number of nuclei with reaction temperature, so that the magnitude of E for the interface process was 209 kJ mole-1. 

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