Saturation ratio nucleation rate

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... 

Secondary nucleation results from the presence of solute particles in solution. Recent reviews [16,17] have classified secondary nucleation into three categories apparent, true, euid contact. Apparent secondary nucleation refers to the small fragments washed from the surface of seeds when they are introduced into the crystallizer. True secondary nucleation occurs due simply to the presence of solute particles in solution. Contact secondary nucleation occurs when a growing particle contacts the walls of the container, the stirrer, the pump impeller, or other particles, producing new nuclei. A review of contact nucleation, frequently the most significant nucleation mechanism, is presented by Garside and Davey [18], who give empirical evidence that the rate of contact nucleation depends on stirrer rotation rate (RPM), particle mass density, Mj>, and saturation ratio. 

Several modifications of this nucleation equation have been suggested by Zeldovitch [24] that allow Figure 7.5 [25] to be drawn, which shows the nucleation rate as a fimction of the saturation ratio and the number of atoms in the critical embiyo (see Section 6.3 for details). The nucleation rate is veiy low for small values of the saturation ratio. But, at a critical saturation ratio, the nucleation rate increases drastically and saturates at a maximum rate corresponding to the condensation rate, times the molecular density, PJk T. 

The saturation ratio, S = p /p, where p is the same aspsA in (11.1). Replacing p by p in (11.19) gives the forward rate constant at saturation, pj. Then, the forward rate constant under nucleation conditions (S > 1) can be written as... 

Equation (11.47) is the classical homogeneous nucleation theory expression for the nucleation rate. It is the expression that we will use to predict nucleation rates of different pure substances as a function of saturation ratio. 

To maintain the nucleation rate at a constant value J it is necessary that the saturation ratio of vapor be maintained at a constant value.3 Without outside reinforcement of the vapor concentration, the saturation ratio will eventually fall as a result of depletion of vapor molecules to form stable nuclei. The case in which the vapor concentration is augmented by a source of fresh vapor is referred to as nucleation with a continuously reinforced vapor. 

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... 

FIGURE 11.5 Upward thermal diffusion cloud chamber. Typical cloud chamber profiles of total gas-phase density, temperature, partial pressure p (n-nonane in this example), equilibrium vapor pressure p saturation ratio 5, and nucleation rate J, as a function of dimensionless chamber height at T = 308.4 K, S = 6.3, and total p = 108.5 kPa. (Reprinted with permission from Katz, J. L., Fisk, J. A. and Chakarov, V. M. Condensation of a Supersaturated vapor IX. Nucleation of ions, J. Chem. Phys. 101. Copyright 1994 American Institute of Physics.)... 

FIGURE 11.6 Fast expansion chamber. Initially undersaiurated (5< 1) at po, 7b for a time interval of typically 1 ms supercritically supersaturated (S > S, ), where Sm, is the saturation ratio corresponding to a threshold nucleation rate, at pmm, Tmin finally still supersaturated, but subcritical (1 < S < Scri<) to permii droplet growth only. Sketch at right indicates the distribution of clusters with size at the three stages of operation. 

Two effects are observed in homogeneous nucleation experiments for all substances. First, the nucleation rate is always a steep function of saturation ratio S. The second feature common to all systems is that the critical saturation ratio Sc decreases as T increases, and J increases as T increases at constant S. Also, critical nuclei become smaller as 5 increases and as T increases. 

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. 

The traditional method of studying gas-liquid nucleation involves the use of a cloud chamber. In such a chamber the saturation ratio 5 is changed until, at a given temperature, droplet formation is observable. Because once clusters reach the critical size for nucleation, subsequent droplet growth is rapid, the rate of formation of macroscopically observable droplets is assumed to be that of formation of critical nuclei. In such a device it is difficult to measure the actual rate of nucleation because the nucleation rate changes so rapidly with S. J is very small for S values below a critical saturation ratio 5, and very large for S > 5,. Thus what is actually measured is the value of 5, defined rather arbitrarily by the point where the rate of appearance of droplets is 1 cm s . ... 

We present here only one example of experimentally measured nucleation rates, those for n-butanol. Viisanen and Strey (1994) measured homogeneous nucleation rates of n-butanol in argon in an expansion chamber as a function of saturation ratio in the temperature range 225 to 265 K. In this temperature range the equilibrium vapor pressure and surface tension of n-butanol are known with sufficient accuracy that a quantitative comparison of the observed nucleation rates with those predicted by classical theory could be performed. Figure 10.7 shows the measured nucleation rates (the data points) and the predictions of the classical theory (solid lines). Nucleation rates ranging from about 10 to 10 cm s were measured. To check for possible influence of the carrier gas, measurements were also carried out with helium and xenon as carrier gases. The authors did not ob-... 

---