Derivation of the Nucleation Rate

We now have a direct expression for the nucleation rate, (11.11), with f given by (11.23). The difficulty in using this equation lies, as noted, in properly evaluating the evaporation coefficients y.  

To approach this problem consider the formation of a dimer from two monomers at saturation (S = 1) as a reversible chemical reaction  

This is not a true chemical reaction but rather a physical agglomeration of two monomers. The forward rate constant is p], and the reverse rate constant is j. The ratio Pj/y is just the equilibrium constant for the reaction. Similarly, the formation of a trimer can be written as a reversible reaction between a monomer and a dimer 

Thus the product of rate constant ratios in (11.23) is just the equilibrium constant for the formation of an i-mer from i individual molecules under saturated equilibrium conditions  

is the Gibbs free energy change for the reaction above, then (11.23) can be written as 

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Zel dovich theory — The theory determines the time dependence of the nucleation rate 7(f) = d N (f )/df and of the number N(t) of nuclei and derives a theoretical expression for the induction time T needed to establish a stationary state in the supersaturated system. The -> Zel dovich approach [i] (see also [ii]) consists in expressing the time dependence of the number Z(n,t) of the n-atomic clusters in the supersaturated parent phase by means of a partial differential equation ... 

Borensen C, Kirchner U, Scheer V, Vogt R, Zellner R (2000) Mechanism and kinetics of the reactions of NO2 or HNO3 with alumina as a mineral dust model compound. J Phys Chem A 104 5036-5045 Borys RD, Lowenthal DH, Mitchell DL (2000) The relationships among cloud microphysics, chemistry, and precipitation rate in cold mountain clouds. Atmos Environ 34 2593-2602 Bowles RK, McGraw R, Schaaf P, Senger B, Voegel JC, Reiss H. (2000) A molecular based derivation of the nucleation theorem. J Chem Phys 113 4524-4532... 

The morphologies of the copper deposits electrodeposited at an overpotential of 550 mV are cauliflower-like and dendritic ones.68 The size of the cauliflower-like particles did not change with increasing temperature, but the size of sub-particles constituting the cauliflower-like forms which decreased with increasing temperature of electrodeposition. The decrease of the size of sub-particles with increasing temperature can be explained by the well-known dependence of the nucleation rate on temperature,69 which was derived by Volmer and Weber.70... 

To determine this potentially small range of conditions, Deqaguin and FedoseevP 1 derived the ratio of the nucleation rates of diamond to graphite on the (111) diamond sur ce as a fimction of supersaturation, given by... 

Figure 9.9 Plot of the nucleation rate as a function of growth rates, both derived from plots of population densities vs. crystal size, such as in Figure 9.8. The nucleation rate is referred to as suspension density

As an illustration of this mode of progressive nucleation we consider the same time dependence of the nucleation rate as in Chapter 5.2.2 but only in the cases in which explicit expressions can be derived for the current Init) pure ions transfer, pure diffusion and pure ohmic control of the growth... 

A plot of In n versus L is a straight line whose intercept is In and whose slope is —l/Gt. (For plots on base-10 log paper, the appropriate slope correc tion must be made.) Thus, from a given product sample of known shiny density and retention time it is possible to obtain the nucleation rate and growth rate for the conditions tested if the sample satisfies the assumptions of the derivation and yields a straight hne. A number of derived relations which describe the nucleation rate, size distribution, and average properties are summarized in Table 18-5. 

The expression for the nucleation rate 5 in the compartment / is derived from the theory of primary nucleation and found to be (Mullin, 2001)... 

The present work aims to derive fully microscopic expressions for the nucleation rate J and to apply the results to realistic estimates of nucleation rates in alloys. We suppose that the state with a critical embryo obeys the local stationarity conditions (9) dFjdci — p, but is unstable, i.e. corresponds to the saddle point cf of the function ft c, = F c, — lN in the ci-space. At small 8a = c — cf we have... 

The nucleation rate onto a crystal is determined by the flux onto an ensemble of substrates. As the nuclei should be widely separated for the nucleation approach to be valid, this does not appear to be unreasonable. However, the subsequent way in which this flux is used to determine the thickness and growth rate seems somewhat inconsistent as explained below. However, a modification of the derivations would satisfy this query, and it is not likely that this will greatly affect the results. 

Johans et al. derived a model for diffusion-controlled electrodeposition at liquid-liquid interface taking into account the development of diffusion fields in both phases [91]. The current transients exhibited rising portions followed by planar diffusion-controlled decay. These features are very similar to those commonly observed in three-dimensional nucleation of metals onto solid electrodes [173-175]. The authors reduced aqueous ammonium tetrachloropalladate by butylferrocene in DCE. The experimental transients were in good agreement with the theoretical ones. The nucleation rate was considered to depend exponentially on the applied potential and a one-electron step was found to be rate determining. The results were taken to confirm the absence of preferential nucleation sites at the liquid-liquid interface. Other nucleation work at the liquid-liquid interface has described the formation of two-dimensional metallic films with rather interesting fractal shapes [176]. 

An alternative scheme, proposed by Garside et al. (16,17), uses the dynamic desupersaturation data from a batch crystallization experiment. After formulating a solute mass balance, where mass deposition due to nucleation was negligible, expressions are derived to calculate g and kg in Equation 3 explicitly. Estimates of the first and second derivatives of the transient desupersaturation curve at time zero are required. The disadvantages of this scheme are that numerical differentiation of experimental data is quite inaccurate due to measurement noise, the nucleation parameters are not estimated, and the analysis is invalid if nucleation rates are significant. Other drawbacks of both methods are that they are limited to specific model formulations, i.e., growth and nucleation rate forms and crystallizer configurations. 

The kinetics of nucleation of one-component gas hydrates in aqueous solution have been analyzed by Kashchiev and Firoozabadi (2002b). Expressions were derived for the stationary rate of hydrate nucleation,./, for heterogeneous nucleation at the solution-gas interface or on solid substrates, and also for the special case of homogeneous nucleation. Kashchiev and Firoozabadi s work on the kinetics of hydrate nucleation provides a detailed examination of the mechanisms and kinetic expressions for hydrate nucleation, which are based on classical nucleation theory. Kashchiev and Firoozabadi s (2002b) work is only briefly summarized here, and for more details the reader is referred to the original references. 

These observations can be represented as a special case of the general rate equation derived by the application of order-disorder theory to diffusionless transitions in solids.3 According to this equation, the shape of the rate curve is determined by the relative numerical values of zkp/kn and of c. The larger the factor is relative to c, the more sigmoidal the curves become. This is understandable since the propagation effect which is responsible for the autocatalytic character of the transformation becomes more noticeable when kPlkn is large and c small. Under these conditions some time elapses before a sufficient number of nucleation sites are formed then the... 

Here Ii(t - u) is the growth current of a single cluster born at time t = u, [1 - 0(u)] is the actual free surface fraction available for the nucleus formation, and J u) is the nucleation rate at time t = u. Similarly to the case of - instantaneous nucleation an expression for the current density jN(t)canbe derived accounting either for direct clusters coalescence [iii-vi] or for overlapping of planar diffusion zones within which nucleation is fully arrested [vii-xi], In the latter case,... 

J is the number of nuclei formed per unit time per unit volume, No is the number of molecules of the crystallizing phase in a unit volume, v is the frequency of atomic or molecular transport at the nucleus-liquid interface, and AG is the maximum in the Gibbs free energy change for the formation of clusters at a certain critical size, 1. The nucleation rate was initially derived for condensation in vapors, where the preexponential factor is related to the gas kinetic collision frequency. In the case of nucleation from condensed phases, the frequency factor is related to the diffusion process. The value of 1 can be obtained by minimizing the free energy function with respect to the characteristic length. 

While the classical theory of nucleation is limited by the implicit assumptions in its derivation, it successfully predicts the nucleation behavior of a system. Inspection of the equation above clearly suggests that the nucleation rate can be experimentally controlled by the following parameters molecular or ionic transport, viscosity, supersaturation, solubility, solid-liquid interfacial tension, and temperature. 

Branching nucleation forms the starting point for a derivation of the Prout-Tompkins reaction model [13] (see below). The main rate equations for nucleation are summarized in Table 3.1. and illustrated in Figure 3.1. 

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