Nucleation kinetic expression

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. 

The account of the formal derivation of kinetic expressions for the reactions of solids given in Sect. 3 first discusses those types of behaviour which usually generate three-dimensional nuclei. Such product particles may often be directly observed. Quantitative measurements of rates of nucleation and growth may even be possible, thus providing valuable supplementary evidence for the analysis of kinetic data. Thereafter, attention is directed to expressions based on the existence of diffuse nuclei or involving diffusion control such nuclei are not susceptible to quantitative... 

The Avrami—Erofe ev equation, eqn. (6), has been successfully used in kinetic analyses of many solid phase decomposition reactions examples are given in Chaps. 4 and 5. For no substance, however, has this expression been more comprehensively applied than in the decomposition of ammonium perchlorate. The value of n for the low temperature reaction of large crystals [268] is reduced at a 0.2 from 4 to 3, corresponding to the completion of nucleation. More recently, the same rate process has been the subject of a particularly detailed and rigorous re-analysis by Jacobs and Ng [452] who used a computer to optimize curve fitting. The main reaction (0.01 < a < 1.0) was well described by the exact Avrami equation, eqn. (4), and kinetic interpretation also included an examination of the rates of development and of multiplication of nuclei during the induction period (a < 0.01). The complete kinetic expressions required to describe quantitatively the overall reaction required a total of ten parameters. 

Kinetic expressions for appropriate models of nucleation and diffusion-controlled growth processes can be developed by the methods described in Sect. 3.1, with the necessary modification that, here, interface advance obeys the parabolic law [i.e. is proportional to (Dt),/2]. (This contrasts with the linear rate of interface advance characteristic of decomposition reactions.) Such an analysis has been provided by Hulbert [77], who considers the possibilities that nucleation is (i) instantaneous (0 = 0), (ii) constant (0 = 1) and (iii) deceleratory (0 < 0 < 1), for nuclei which grow in one, two or three dimensions (X = 1, 2 or 3, respectively). All expressions found are of the general form... 

Giovanoli and Briitsch [264] studied the kinetics of vacuum dehydroxylation of 7-FeO 0H(- -7 Fe203). It was not possible to demonstrate satisfactory obedience to a single kinetic expression. Microscopic examinations detected the occurrence of random nucleation over reactant surfaces and crystallographic indications of the specific structural reorganization steps, which occur at the reaction interface, are discussed. 

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. 

An alternative approach to describe nucleation from the amorphous state utilizes the glass transition temperature (Tg) concept (Williams et al. 1955 Slade and Levine 1991). Based on this approach, molecular mobility below Tg is sufficiently limited to kinetically impede nucleation for very long times. Amorphous systems, at temperatures above Tg, nucleate at a rate depending on the temperature difference above Tg. Williams et al. (1955) suggested that the rate of nucleation increases rapidly at temperatures just above Tg according to a kinetic expression given by the WLF (Williams-Landel-Ferry) equation. 

These relationships provide a sufficient foundation for kinetic analyses where the nucleation step, although the essential precursor to all subsequent changes, contributes only a single term to the overall composite kinetic expression. 

It should be noted, however, that most conditions of deposition from the vapor phase have been shown to be such that classical nucleation theory is not well-suited to describe the nucleation kinetics of diamond, since the critical nucleus size is on the order of a few atoms.P The small size of the critical nucleus makes it quite inappropriate to use the classical thermodynamic variables to describe the nucleation processes. Under such conditions, the Gibbs free-energy of the formation of a critical nucleus carmot be expressed... 

The Touchstone quartz cementation model assumes that the precipitation rate per unit nucleation surface area is described by an Arrhenius kinetic expression (Walderhaug 2000) and that the nucleation surface area for quartz overgrowths is a function of grain size... 

The kinetic expressions for growth and nucleation such as those given by Eqs. (9.5) and (9.6), and the initial and boundary conditions given by Eq. (9.9) complete the model formulation. 

The structure and interrelationship of the batch conservation equations (population, mass, and energy balances) and the nucleation and growth kinetic equations are illustrated in an information flow diagram shown in Figure 10.8. To determine the CSD in a batch crystallizer, all of the above equations must be solved simultaneously. The batch conservation equations are difficult to solve even numerically. The population balance, Eq. (10.3), is a nonlinear first-order partial differential equation, and the nucleation and growth kinetic expressions are included in Eq. (10.3) as well as in the boundary conditions. One solution method involves the introduction of moments of the CSD as defined by... 

The initial conditions of the moment equations are derived directly from the initial population density n(L,0). Equations (10.12) through (10.16), together with the nucleation and growth kinetic expressions [Eqs. (10.5) and (10.6)], can be solved numerically to give the moments of the CSD for a batch or semibatch system as a function of time. The CSD can be reconstructed from its moments by the methods described by Hulbert and Katz (1964) and Randolph and Larson (1988). 

A continuous cooling crystallizer is required to produce potassium sulphate crystals (density pc = 2660kgm, volume shape factor a = 0.7) of 750pm median size Lm at the rate Pc = 1000 kg h. On the basis of pilot-plant trials, it is expected that the crystallizer will operate with steady-state nucleation/ growth kinetics expressed (equation 9.39 with j = 1 and i = 2) as B = 4 X IO MtG m s. Assuming MSMPR conditions and a magma density Mt = 250 kg m, estimate the crystallizer volume and other relevant operating conditions. 

Note that if 0 is taken to be zero (which means the embryo is completely surrounded by the liquid) and is replaced by P[, 10.18 becomes the same as (10.14) for the homogeneous nucleation. This expression shows that the contact angle can reduce the kinetic limit of superheat. For example, the contact angle of water on common materials may vary between 0°C to 108°C which may reduce the limit of superheat by more than 20°C. 

Nucleation occurs by random aggregation and detachment of molecular growth units. From a probabilistic point of view, the cluster reaches a critical size n when both attach and detach frequencies are equal. For n > n, the clusters tend to grow with a probability P n) (Equation 10.29) that, according to the kinetic theory of nucleation, is expressed as (Curcio et al. 2008)... 

Crystallization kinetics. Expressions that describe crystal growth and nucleation rates from solution. CSD. Ciystal size distribution. 

Under a number of aspects the process resembles homogeneous nucleation and consequently the rate of block copolymer self-assembly is often approximated by the expression for homogeneous nucleation kinetics [50]. 

Figure 7 illustrates the relative rates of precipitation of y in y and y in y observed in a dendritic region of the alloy containing 22.0 % A1 aged for 48 h at 700 C. It is quite evident that the precipitation of y in y is much slower than the precipitation ofy iny. This is most likely due to the faster coarsening kinetics of y in y, but the slower nucleation kinetics ofy iny undoubtedly contribute as well. According to Calderon et al., A canbe expressed by the equation... 

The theory of nucleation in condensed phases has been well covered in recent reviews and can be briefly treated here. In this treatment we shall adopt without comment the standard thermodynamic formalism and will not consider questions such as the applicability of bulk thermodynamic quantities for describing very small systems. We shall also neglect any possible modification of the kinetic expressions due to a Lothe-Pound type treatment ( ) of the translational and rotational free energies of the embryos because any such modification should be quite small for nucleation in condensed phases. 

In Chapter 1 we considered the equilibrium properties of small clusters and derived explicit thermodynamic expressions for the work AG(r ) of nucleus formation. We hould emphasize, however, that giving us the value of the energy barrier AG n) thermodynamics do not say anything about the rate J of appearance of nuclei within the supersaturated parent phase. The reason is that this important physical quantity depends on the mechanism of nucleus formation and can be determined only by means of kinetic considerations. It is the purpose of this Chapter to present the fundamentals of the nucleation kinetics, to derive explicit expressions for the nucleation rate J and to reveal the supersaturation dependence of this quantity. In doing this we consider the nucleus formation on the assumption that the process is a set of consecutive bimolecular reactions of the type ... 

Volmer and Weber [2.14], Farkas [2.15] and Kaischew and Stranski [2.16-2.18] were the first who examined the stationary nucleation kinetics and derived theoretical expressions for the stationary nucleation rate. However, in this Chapter we shall present the results of the more rigorous treatments of Becker and Doring [2.6] and ofZeldovich [2.19] and Frenkel [2.20] who laid the foundations of the contemporary classical nucleation theory (see also [2.V-2.9] and [2.21-2.24]). For the sake of simplicity we shall neglect both the line tension effects (equations (1.42) and (1.70)) and the dependence of the specific free surface energy on the size of the clusters (equation (1.43). 

The Zeldovich-Frenkel approach to the stationary nucleation kinetics [2.19, 2.20] consists in the replacement of the Becker and Doling expression... 

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