Experimental Measurement of Nucleation Rates

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

The question arises as to how closely the predictions from (11.47) and (11.74) agree.  

Derivations of these two expressions were based on similar assumptions, but these 

The traditional method of studying gas-liquid nucleation involves the use of a cloud chamber. In such a chamber the saturation ratio S is changed until, at a given temperature, 

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 .  

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The explosive onset of nucleation has made the experimental measurement of nucleation rates difficult, as measurable rates can be obtained only under a very limited range of experimental conditions. An additional difficulty has been counting the actual number of particles formed, since substantial concurrent particle coarsening often occurs (see Fig. 19.1). A common procedure has therefore been to find the driving force (which is relatively easy to quantify) that is necessary to produce... 

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... 

The natural first question to ask is whether the crystal-liquid surface free energy can be measured experimentally by some method that is independent of nucleation kinetics. In gas-liquid nucleation studies, for example, it is routine to measure the surface tension of the liquid and to use its equality with the gas-liquid surface free energy to make predictions of nucleation rates and compare them with experiment. For the liquid-solid transition, the situation is quite different, however. This is true first because the surface tension and the surface free energy are no longer strictly equal due to the possible existence of strains in the crystal. The second reason is that measurements of liquid-solid free energies or interfacial tensions are by no means simple to devise or carry out, and so are available only in certain special cases. These limited experimental data are summarized in this section. 

The presence of ions has been shown experimentally to enhance the rate of nucleation of liquid drops in a supersaturated vapor. Katz et al. (1994) showed, for example, that the nucleation rate of n-nonane, measured in an upward thermal diffusion cloud chamber, at an ion density of 16 x 106 ions cm-3, increased by a factor of 2500 over that in the absence of ions. These investigators also confirmed experimentally that the nucleation rate is directly proportional to the ion density. The phenomenon of ion-induced nucleation plays an important role in atmospheric condensation, particularly in the ionosphere. While both positive and negative ions increase the nucleation rate, many substances exhibit a preference for ions of one sign over the other. 

Not only is theoretical analysis in this field difficult, but also experimental measurements of the velocity and temperature fields surrounding a bubble are at the limit of present capabilities, because of the small length and time scales involved. This lack of information, together with the uncertainty introduced by the presence of the solid wall, make comparison of theory with experiment difficult. The status of experimental information on bubble growth rates is, therefore, briefly reviewed. Finally, a number of special topics, including bubble nucleation, are touched upon. Some general comments conclude the review. 

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-... 

Brief reference has already been made to the experimental measurement of crystal nucleation and growth rates in section 6.2.5, and a continuous MSMPR crystallizer suitable for the purpose is depicted in Figure 6.21. For cooling crystallization studies, only one feedstock storage vessel and delivery point would be needed. The duplicate feed circuits depicted in Figure 6.21 would be required if reaction crystallization, precipitation or salting-out studies are to be made. 

Until the early 80s, experimental tests of classical nucleation theory were restricted to measuring the supersaturation required to trigger an easily measurable nucleation rate, typically 1 cm sec This is known as the critical supersaturation. Using the density and surface tension of water. Equation (23) predicts an increase of seventeen orders of magnitude in the nucleation rate as a result of a 10% change in supersaturation, from 2 to 2.2. It follows that the critical supersaturation is not a sensitive probe of the accuracy of nucleation theories. It also follows from equation (23) that this constitutes an incomplete test of the theory, insofar as it does not test the actual J(S) functionality. A review by McGraw [36] is a particularly useful and comprehensive survey of experimental measurements of critical supersaturations it also proposes a useful corresponding states correlation for this quantity. 

The inverse problems discussed in Sections 6.1 and 6.2 were addressed in the absence of nucleation and growth processes. In this section we investigate inverse problems for the recovery of the kinetics of nucleation and growth from experimental measurements of the number density. It is assumed, however, that particle break-up and aggregation processes do not occur. Determination of nucleation and growth rates is of considerable practical significance since the control of particle size in crystallization and precipitation processes depends critically on such information. We will dispense with the assumption of self-similar behavior, as it is often not observed in such systems. Also, we provide here only a preliminary analysis of this problem, as it is still in the process of active investigation by Mahoney (2000). 

In panel (b) of Fig. 11 we reproduce experimental data from Rathke and coworkers [148], studying the onset of nucleation and measuring the nucleation rate in hexadecane-H CO2 mixtures at 40°C. Our crude and coarse-grained SCF calculations are able to reproduce some qualitative features of this careful, experimental study The topology of the phase diagram, i.e., the location of binodals and the triple line are similar. Most notably, the experiment observes bubble nucleation where the SCF calculations predict barriers on the order of a few tens of A T, and the line of constant nucleation barrier is closer to the binodal at higher pressure (close to the triple line) and is further inside the miscibility gap at lower pressure. 

Fig. 3. Measured crystal nucleation rates / as of function of volume fraction in a system of hard-sphere colloids. The data are taken from Ref. [5] (open circles) and Ref. [6] (filled cubes). The hues result from a two parameter fit of Eq. (4) to the experimental data. The inset shows the dimensionless nucleation rate densities plotted logarithmically versus /((f>lsp). The figure is taken from Ref [4]...

While it is possible that surface defects may be preferentially involved in initial product formation, this has not been experimentally verified for most systems of interest. Such zones of preferred reactivity would, however, be of limited significance as they would soon be covered with the coherent product layer developed by reaction proceeding at all reactant surfaces. The higher temperatures usually employed in kinetic studies of diffusion-controlled reactions do not usually permit the measurements of rates of the initial adsorption and nucleation steps. 

The models incorporate two microscopic parameters, the site density and the critical nucleus size. A fit of experimental current transients to the models allows conclusions, for example, concerning the effect of additives on nucleation rate. Fabricus et al. found by analysis of current transients that thiourea increases the nucleation density of copper deposited on glassy carbon at low concentration, but decreases it at higher concentration [112], Schmidt et al. found that Gold nucleation on pyrolytic graphite is limited by the availability of nucleation sites [113], Nucleation density and rate were found to depend on applied potential as was the critical nucleus size. Depending on concentration, critical nuclei as small as one atom have been estimated from current transient measurements. Michailova et al. found a critical nucleus of 11 atoms for copper nucleation on platinum [114], These numbers are typical, and they are comparable to the thermodynamic critical radii [86],... 

A very accurate measurement of Ccrjt would allow back-calculation of the surface energy for a given crystal. Because Ccrjt is dependent on the square of Y, such a measurement could be a very sensitive method of measuring interfacial energy at dislocation outcrops. The calculated interfacial energy from our experiments is 280+ 90 mJm- for the rhombohedral face of quartz at 300°C. Parks (10) estimated 25°C value of 360 + 30 mJm is well within the experimental error of our measurement. The best way to determine the value of Ccrjt would be to measure etch pit nucleation rate on... 

One approach which has resulted in experimental implementation is that of Randolph and co-workers f88-92 >. Using a simulation (21) Randolph and Beckman demonstrated that in a complex RTD crystallizer, the estimation of nuclei density could be used to eliminate cycling or reduce transients in the CSD. Randolph and Low (gg) experimentally attempted feedback control by manipulation of the fines dissolver flow rate and temperature in response to the estimated nuclei density. They found that manipulation of fines flow rate upset the fines measurement indicating that changes in the manipulated variable disturbed the measured variable. Partial fines dissolution resulting from manipulation of the fines dissolver temperature appeared to reduce CSD transients which were imposed upsets in the nucleation rate. In a continuation of this work Randolph et. al. < 921 used proportional control of inferred nuclei density to control an 18 liter KCl crystallizer. 

An alternative scheme, proposed by Garside et al. (16,17), uses the dynamic desupersaturation data from a batch crystallization experiment. After formulating a solute mass balance, where mass deposition due to nucleation was negligible, expressions are derived to calculate g and kg in Equation 3 explicitly. Estimates of the first and second derivatives of the transient desupersaturation curve at time zero are required. The disadvantages of this scheme are that numerical differentiation of experimental data is quite inaccurate due to measurement noise, the nucleation parameters are not estimated, and the analysis is invalid if nucleation rates are significant. Other drawbacks of both methods are that they are limited to specific model formulations, i.e., growth and nucleation rate forms and crystallizer configurations. 

FIGURE 9.30 Theoretically predicted and experimentally measured concentrations of H2S04 required for homogeneous nucleation of sulfuric acid at a rate of 1 particle cm 3 s 1 (adapted from Hoppel et al., 1994 based on theoretical calculations of Jaecker-Voirol and Mirabel (1989) and experimental data of Wyslouzil et al. (1991). 

No study has been made to discover which of the several resistances is important, but a simple rate equation can be written which states that the rate of the over-all process is some function of the extent of departure from equilibrium. The function is likely to be approximately linear in the departure, unless the intrinsic crystal growth rate or the nucleation rate is controlling, because the mass and heat transfer rates are usually linear over small ranges of temperature or pressure. The departure from equilibrium is the driving force and can be measured by either a temperature or a pressure difference. The temperature difference between that of the bulk slurry and the equilibrium vapor temperature is measured experimentally to 0.2° F. and lies in the range of 0.5° to 2° F. under normal operating conditions. 

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