Steady state nucleation

Using the fluxing technique, Lau and Kui [33] determined that the critical cooling rate for forming a 7-mm diameter bulk amorphous Pd4QNi4()P2o cylinder was 0.75 K/sec. From this value, they estimated that the steady-state nucleation frequency was on the order of lO" m s. On the other hand, Drehman and Greer [34] estimated that the steady state nucleation frequency at 590 K is 10 m" s, which is also the maximum... 

The prediction of transformation diagrams after Bhadeshia (1982). Later work by Bhadeshia (1982) noted that the approach of Kirkaldy et al. (1978) could not predict the appearance of the bay in the experimentally observed TTT diagrams of many steels, and he proposed that the onset of transformation was governed by nucleation. He considered that the time period before the onset of a detectable amount of isodiermal transformation, r, could be reasonably defined as the incubation period, r necessary to establish a steady-state nucleation rate. The following expression for r, was then utilised... 

Transient Nucleation If a liquid is cooled continuously, the liquid structure at a given temperature may not be the equilibrium structure at the temperature. Hence, the cluster distribution may not be the steady-state distribution. Depending on the cooling rate, a liquid cooled rapidly from 2000 to 1000 K may have a liquid structure that corresponds to that at 1200 K and would only slowly relax to the structure at 1000 K. Therefore, Equation 4-9 would not be applicable and the transient effect must be taken into account. Nonetheless, in light of the fact that even the steady-state nucleation theory is still inaccurate by many orders of magnitude, transient nucleation is not discussed further. 

Stage II is the quasi-steady-state nucleation regime. During this period, the distribution of clusters has built up into a quasi-steady state and stable nuclei are being produced at a constant rate. 

Non-Steady-State Nucleation The Incubation Time. Although in principle, non-steady-state nucleation in single-component systems can be analyzed by solving the time-dependent nucleation equation (Eq. 19.10) under appropriate initial and boundary conditions, no exact solutions employing this approach have been obtained. Instead, various approximate solution have been derived, several of which have been reviewed by Christian [3]. Of particular interest is the incubation time described in Fig. 19.1. During this period, clusters will grow from some initial distribution, usually essentially free of nuclei, to a final steady-state distribution as illustrated in Fig. 19.5. 

Single-Component System with Isotropic Interfaces and No Strain Energy. This relatively simple case could, for example, correspond to the nucleation of a pure solid in a liquid during solidification. For steady-state nucleation, Eq. 19.16 applies with AQC given by Eq. 19.4 and it is necessary only to develop an expression for /3C. In a condensed system, atoms generally must execute a thermally activated jump over a... 

Two-Component System with Isotropic Interfaces and Strain Energy Present. An example of this case is the solid-state precipitation of a 5-rich (i phase in an A-rich a-phase matrix. For steady-state nucleation, Eq. 19.16 again applies. However, for a generalized ellipsoidal nucleus, the expression for AQ will have the form of Eq. 19.28. Also, /3 must be replaced by an effective frequency, as discussed in Section 19.1.2. 

Solution. Important assumptions include that the interfacial free energy is isotropic, that elastic strain energy is unimportant, and that the nucleation rates mentioned are for steady-state nucleation. The critical barrier to nucleation, AQe, can be calculated for the 0.3 atomic fraction B alloy using the tangent-to-curve construction on the curves in Fig. 19.18b to provide the value Aga = —9 x 107 Jm-3 for the chemical driving force for this supersaturation at 800 K. AQc is given for a spherical critical nucleus by... 

The molecular model of amphiphile bilayers can also be used for describing the process of hole nucleation by the classical nucleation scheme [408,409] as resulting from a series of bimolecular reactions characterised by the nucleation rate J (s 1) which is the frequency with which the / -sized nucleus holes become supemucleus holes of size / +1. For steady-state nucleation, J is known to be [408,409]... 

When the supemucleus holes grow fast enough, the results of probability considerations of steady state nucleation [412] can be used to show that probability P of bilayer rupturing until time i > 0 and the bilayer mean lifetime rare given by [402,403]... 

Theoretical approaches to nucleation go back almost 80 years to the development of Classical Nucleation Theory (CNT) by Volmer and Weber, Becker and Doring and Zeldovich [9,10,17-20]. CNT is an approximate nucleation model based on continuum thermodynamics, which views nucleation embryos as tiny liquid drops of molecular dimension. In CNT, the steady-state nucleation rate /, can be written in the form / a where jS, is the monomer condensation... 

When conditions for homc eneous nucleation are first created, an induction time or delay time is required before the steady state nucleation rate, Jq> established. The nucleation rate has the following transient behavior [9] ... 

Fig. 4.8 shows experimental plots of the number of nuclei Znuc vs. time (pulse duration) at different overvoltages (pulse amplitudes) in the electrodeposition of mercury on platinum [4.37]. The steady state nucleation rate (dZnuc/dt = /= const.) is... 

From the linear parts of the Znuc/f curves of Fig. 4.8, the steady state nucleation rate,/, for different overvoltages can be evaluated. Fig. 4.9a represents a typical In /vs. 1/177 P plot for the system of Fig. 4.8. The analysis of the data according to eq. (4.32) shows that, in the overpotential interval of 84 - 106 mV studied, the nucleation energy... 

Figure 4.9 Experimental steady state nucleation rate values calculated from the data of Figure 4.8 (a) in a In/vs. 1/1 1 plot and (b) in a In/vs. I /1 plot [4.46], The most probable straight line in Fig. 4.9a is represented as a dashed line in Fig. 4.9b.

In this section we review the essential parameters required to describe homogeneous steady-state nucleation. Extensive discussions of the ideas presented in this section are given in the excellent book by Abraham on nucleation theory and the fine review article of Burton. In what follows we sketch the important ideas used to determine homogeneous steady-state nucleation rates from equilibrium thermodynamic information. Since the kinetic results are approximate, our purpose is to outline the key assumptions introduced rather than giving a complete formal development. The assumptions used to extract nucleation rates from thermodynamic information have been critiqued by others. The rates determined from classical nucleation theory can be expected to be qualitative in some cases and quantitative when the assumptions... 

A more general tactic is to design for negligibly slow crystallization rates. The rate for the homogeneous, steady-state nucleation rate, /, of a single-component liquid (or glass) can be expressed by,69... 

Assuming that the kinetic pre-factor, Jo, in the expression for the steady-state nucleation rate J... 

Figure 9.4 (a) Steady state nucleation rate as a function of temperature for a glass close to the Li20-2Si02 composition, (b) Reflection optical micrograph of a Ba0-2Si02 glass after heat treatment. 

S > 1 and a constant net growth rate of 7, Nj. At saturation (S = 1) all 7i+1 /2 = 0, whereas at steady-state nucleation conditions all 7,+i/2 = 7. There is a third distribution that we will not explicitly introduce until the next section. It is the hypothetical, equilibrium distribution of clusters corresponding to S > 1. Thus it corresponds to all 7i+1/2 = 0, but 5 > 1. Because of the constraint of zero flux, this third distribution is called the constrained equilibrium distribution, Nf. We will distinguish this distribution by a superscript e. 

The induction time, tmj, is the sum of the time needed for reaching steady-state nucleation, t the nucleation time, t and the time required for the critical nucleus to grow to a detectable size, tg. Thus,... 

If the nucleation time is much greater than the growth time, the induction time is inversely proportional to the steady-state nucleation rate and... 

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. 

Further on we will solve Eq. (4) to obtain the steady state nucleation rate, but first a number of observations must be made. 

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