Homogeneous nucleation rate

Figure 1. Calculated homogeneous nucleation rate for various values of Tyg as a function of temperature.

Expressing the specific homogeneous nucleation rate constant, kh, according to (2) and substituting equation (1-4) into equation (II-6), (II-6) can finally yield a general expression for the particle nucleation rate, fit), (12,58, 9). 

In general, the rate at which critical nuclei are formed, called the homogeneous nucleation rate, is given by the form... 

From a kinetic point of view, the homogeneous nucleation rate (Nr) [57] and the growth rate (Gr) [58] can be expressed as a function of the supersaturation ratio ... 

Equation 7 shows the exponential variation of the homogeneous nucleation rate with an increasing supersaturation ratio. Equation 8, coming from the model of Strickland-Constable [59], establishes that a linear relation exists between growth rate and supersaturation ratio. The direct comparison of Nr and Gr is difficult because of their different units. However, Fig. 7, although only schematic, shows the variation of these two rates versus S, predicting that for a certain value of S the nucleation rate will become preponderant on the growth rate. Moreover, it also shows, that, in the case of preponderance... 

The melt viscosity determines the likelihood that a glass will form with rapid cooling. In highly viscous melts, the atomic mobility is reduced substantially, suppressing the homogeneous nucleation rate and crystallizability, which are dependent on the ease with which atomic rearrangement can occur. 

FIGURE 6.6 Generalized nucleation rate diagram that describes the homogeneous nucleation rate as a function of the saturation ratio. The number of ions in a critical nucleus, n. is given by equation (6.21) and A = 4j8 -y V / 27/ [feBTln(10)P. Experimental nucleation rates, , are from a BaS04 precipitation reaction. Redrawn, with permission... 

Hao, Z., Iqbal, A. and Herren, F. (1999c). l,4-Diketo-3,6-bis-4-chloro-phenyl-pyrrolopyrrole in /3 and y modification, prepd. by acid hydrolysis of soluble carbamate. Derivation and preparation by cooling. Ciba-Geigy Ltd. EP 690059 B1. [271 ] Harano, Y. and Oota, K. (1978). Measurement of crystallization of potassium bro-mate from its quiescent aqueous solution by differential scanning calorimeter. Homogeneous nucleation rate. /. Chem. Eng. J., 11, 159-61. [70]... 

Aqueous phase concentration of oligomeric radicals of degree i (mol/L) Critical degree of polymerization for homogenous nucleation Rate of propagation (volume growth per particle) (L/sec)... 

Figure 9. Temperature ranges of states of stable, superheated and supercooled water at atmospheric pressure. Stationary homogeneous nucleation rate during crystallization (1) and boiling-up (2). Inverse isothermal compressibility for stable and metastable states of water (3) in the absence of the spinodal in a supercooled liquid (3 ) and in the case of its presence according to (3 % T - the temperature of the spinodal of a superheated liquid.

Indicated in Fig. 9 are temperature ranges of supercooled, stable and superheated water at atmospherie pressure. Ibidem one can see curves representing the temperature dependenee of the logarithm of the homogeneous nucleation rate for crystallization (curve 1) and boiling-up (curve 2). The maximum rate of formation of vapor nuclei is attained at the approach of the spinodal determined by condition (3). Fig. 9 also shows how the inverse isothermal eompressibility =-v(5p/5v) changes with temperature (curve 3). An arrow shows the temperature of the spinodal of superheated water. 

Wolk, J., Strey, R. Wyslouzil, B. E. (2004) Homogeneous Nucleation Rates of Water State of the Art. In Nucleation and Atmospheric Aerosols 2004 if International Conference, Conference Proceedings (Eds. M. Kasahara, M. Kulmala) 101-114... 

TABLE 14.2 Examples of Homogeneous Nucleation Rates Observed for Different Systems... 

FIGURE 14.6 Nucleation of emulsified milk fat as a function of cooling temperature, (a) Concentration of catalytic impurities Ncat. (b) Initial heterogeneous nucleation rate J() the broken line gives a rough estimate of the homogeneous nucleation rate. (After results by P. Walstra, E. C. H. van Beresteyn. Neth. Milk Dairy J. 29 (1975) 35.)... 

The next section presents an alternate derivation of a classical homogeneous nucleation rate expression based on a somewhat different approach. The development in the next section provides a fuller appreciation of classical theory and its underlying assumptions, but the section may be skipped by those interested primarily in application of the theory. We note that several useful tables of physical properties appear in Section 11.2... 

Let us evaluate the homogeneous nucleation rate for water as a function of the saturation ratio S. To do so, we use (11.47). Table 11.4 gives the nucleation rate at 293 K for saturation ratios ranging from 2 to 10. By comparing Tables 11.1 and 11.4, the effect of temperature on i can be seen also. We see that the nucleation rate of water varies over 70... 

TABLE 11.4 Homogeneous Nucleation Rate and Critical Cluster Size for Water at T = 293 K ... 

FIGURE 11.7 Nucleation rates measured in supersaturated n-butanol vapor as a function of saturation ratio for various temperatures ranging from 225 to 265 K (Viisanen and Strey 1994). Different symbols indicate different carrier gases at 240 K. Predictions of classical nucleation theory are shown by the lines. (Reprinted with permission from Viisanen, Y., and Strey, R., V. M. Homogeneous Nucleation Rates for n-Butanol, J. Chem. Phys. 101. Copyright 1994 American Institute of Physics.)... 

A Calculate the homogeneous nucleation rate of ethanol at 298 K for saturation ratios from 2 to 7 using (10.47) and (10.74). Comment on any differences. 

Hale, B. N. (1986) Application of a scaled homogeneous nucleation rate formalism to experimental data at T[Pg.534]

Strey, R., Wagner, P. E., and Viisanen, Y. (1994) The problem of measuring homogeneous nucleation rates and molecular contents of nuclei Progress in the form of nucleation pulse measurements, J. Phys. Chem. 98, 7748-7758. 

Viisanen, Y., and Strey, R. (1994) Homogeneous nucleation rates for n-butanol, J. Chem. Phys. 101, 7835-7843. 

Nielsen (1964), developed an expression for the homogeneous nucleation rate ... 

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