Supersaturation nucleation rate

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]. 

The central quantity of interest in homogeneous nucleation is the nucleation rate J, which gives the number of droplets nucleated per unit volume per unit time for a given supersaturation. The free energy barrier is the dommant factor in detenuining J J depends on it exponentially. Thus, a small difference in the different model predictions for the barrier can lead to orders of magnitude differences in J. Similarly, experimental measurements of J are sensitive to the purity of the sample and to experimental conditions such as temperature. In modem field theories, J has a general fonu... 

Over 50 acidic, basic, and neutral aluminum sulfate hydrates have been reported. Only a few of these are well characterized because the exact compositions depend on conditions of precipitation from solution. Variables such as supersaturation, nucleation and crystal growth rates, occlusion, nonequilihrium conditions, and hydrolysis can each play a role ia the final composition. Commercial dry alum is likely not a single crystalline hydrate, but rather it contains significant amounts of amorphous material. 

The most important variables affecting nucleation rate are shown by equations 10 and 11 to be iaterfacial eaergy, temperature, and supersaturation. 

In order to treat crystallization systems both dynamically and continuously, a mathematical model has been developed which can correlate the nucleation rate to the level of supersaturation and/or the growth rate. Because the growth rate is more easily determined and because nucleation is sharply nonlinear in the regions normally encountered in industrial crystallization, it has been common to... 

While Eq. (18-26) has been popular among those attempting correlations between nucleation rate and supersaturation, recently it has become commoner to use a derived relationship between nucleation rate and growth rate by assuming that... 

Here it can be seen that the nucleation rate is a decreasing function of growth rate (and supersaturation). The physical explanation is believed to be the mechanical influence of the crystallizer on the growing suspension and/or the effect of Bujacian behavior. 

Schierholtz and Stevens (1975), Noor and Mersmann (1993) and Chen etal. (1996) determined nucleation rates by integrating the total crystal number formed over a period and related it to an estimate of supersaturation in the precipitation of calcium carbonate, barium carbonate and barium sulphate respectively. 

Secondary nucleation is an important particle formation process in industrial crystallizers. Secondary nucleation occurs because of the presence of existing crystals. In industrial crystallizers, existing crystals in suspension induce the formation of attrition-like smaller particles and effectively enhance the nucleation rate. This process has some similarity with attrition but differs in one important respect it occurs in the presence of a supersaturated solution. 

Figure 6.21 Nucleation rate of calcium oxalate versus supersaturation at 31 °C (Zauner and Jones, 2000a)...

The nucleation rate is plotted versus the supersaturation for different stirrer speeds in a log-log diagram (Figure 6.21). The kinetic order n in the correlating equation... 

The predicted transient supersaturation levels (and corresponding nucleation rates) are also shown in Figure 7.3. These considerations predict that the high levels in the early period of natural cooling can be avoided by controlled cooling. 

The size of crystals produced in the gas-liquid system varied from 10 to 100 pm by controlling the level of supersaturation, while the liquid-liquid system produced crystals of 5—30 pm. The wide variation of crystal size is due to the marked sensitivity of the nucleation rate on the level of supersaturation, while the impurity content is another variable that can affect the crystal formation. 

Application of Eqs. (21)-(27) to the calculations of the nucleation rates J for various alloy models revealed a number of interesting results, in particular, sharp dependence of J and embryo characteristics on the supersaturation, temperature, interaction radius, etc. These results will be described elsewhere. 

A difiiculty with this mechanism is the small nucleation rate predicted (1). Surfaces of a crystal with low vapor pressure have very few clusters and two-dimensional nucleation is almost impossible. Indeed, dislocation-free crystals can often remain in a metastable equilibrium with a supersaturated vapor for long periods of time. Nucleation can be induced by resorting to a vapor with a very large supersaturation, but this often has undesirable side effects. Instabilities in the interface shape result in a degradation of the quality and uniformity of crystalline material. 

Transfomation from a meta-stable phase, such as supersaturated solution, to a thermodynamically more favorable phase requires first the crystal nucleation of a germ of the new phase. According to the classical nucleation theory, the volume nucleation rate J (cm" sec ), describing the number of nuclei(i.e., a critical germ) formed per volume per time, is given by ... 

When this supersaturation exists, the nucleation rate will be proportional to the probability of formation of a favorable configuration of particles of the primary product. According to the Boltzmann law, this probability is determined by the work of formation of a single nucleus ... 

The rate of nucleation is dependent on the degree of supersaturation as described in section 2.4.1, and because this will always be larger for Form 1 it may be incorrectly assumed that Form I will always precipitate first. The true situation is somewhat more complicated because the critical size, activation energy and nucleation rate also depend on the solid state that is being formed [6]. It is quite feasible and a regular occurrence, that a less stable polymorph will have a higher rate of nucleation than a more stable form, as illustrated in figure 6. 

Nucleation. The energy barrier and the nucleation rate depend critically on the supersaturation. 

At a high degree of supersaturation, the nucleation rate is so high that the precipitate formed consists mostly of extremely small crystallites. Incipiently formed crystallites might be of a different polymorphous form than the final crystals. If the nucleus is smaller than a one-unit cell, the growing crystallite produced initially is most likely to be amorphous substances with a large unit cell tend to precipitate initially as an amorphous phase ("gels"). 

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