Critical supersaturation

It appears from our analysis that the limit of solubility CJ in small particles does not coincide with the equilibrium composition Cp after the separation. This difference between the limiting mean mole fraction of component B in an initially saturated alloy (or solubility - concentration corresponding to the separation criterion) and optimal (or equiUbrium) concentration in the parent phase after the separation was called critical supersaturation [65, 66). Here, the difference AC = CJ — Cp is the critical supersaturation. The effect of critical supersaturation is that the separation is possible only (at some fixed temperature and size) if the supersaturation AC = Co — Cp is larger than AC. If the supersaturation AC AC, then 

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The figures in the table show clearly how rapidly / increases with x, and it is generally sufficient to define the critical supersaturation pressure such that In / is some arbitrary value such as unity. 

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]. 

Because of the large surface tension of liquid mercury, extremely large supersaturation ratios are needed for nucleation to occur at a measurable rate. Calculate rc and ric at 400 K assuming that the critical supersaturation is x = 40,000. Take the surface tension of mercury to be 486.5 ergs/cm. ... 

Calculate what the critical supersaturation ratio should be for water if the frequency factor in Eq. IX-10 were indeed too low by a factor of 10 . Alternatively, taking the observed value of the critical supersaturation ratio as 4.2, what value for the surface tension of water would the corrected theory give ... 

Since the critical supersaturation ratio for homogeneous nucleation is typically greater than 3, it is not often reached in process equipment. 

Jons. Amelin [Theory of Fog Condensation, Israel Program for Scientific Translations, Jerusalem, (1967)] reports that ordinary air contains even higher concentrations of ions. These ions also reduce the required critical supersaturation, but by only about 10 to 20 percent, unless multiple charges are present. 

Although surface-cooled types of MSMPR crystalhzers are available, most users prefer crystallizers employing vaporization of solvents or of refrigerants. The primary reason for this preference is that heat transferred through the critical supersaturating step is through a boil-ing-hquid-gas surface, avoiding the troublesome solid deposits that can form on a metal heat-transfer surface. 

The most important "message" of this chapter is that there is a critical supersaturation that must be exceeded before homogeneous nucleation can occur. The background given is an essential preparation for the introduction of heterogeneous nucleation. 

This observation is in accordance with the phenomena of the crystallization in the resolution operation mentioned above in the following points. There are no clear, definite critical supersaturations above which nucleation of D-threonine occurs. Ohtsuki (2), however, reported supersolubility curve for this system, who gave the value of the supersaturation width At=7 C at 50 C. Their definition of the metastablllty was that no nucleation of the enantiomer other than seeded one was observed for two hours of resolution experiments. According to this definition, the supersolubility can be determined to lie somewhere between At=8 and 5 C from the present experimental data, this being in agreement with his result. If the crystallization proceeds further, however, D-threonine crystals may start to crystallize from the solution even if the initial supersaturation is 5 C. In this sense it is no longer the metastablllty limit. 

As we have already seen, the critical supersaturation Sc. corresponding to the peak of the Kohler curve depends on a number of parameters unique to the aerosol particle. Thus, at a given supersaturation some particles will form cloud droplets and some will not. As a result, the total number of CCN will vary with the supersaturation used in the CCN measurement. This is illustrated in Fig. 14.39, which shows the concentration of CCN measured in Antarctica as a function of the percentage supersaturation for CCN that grow into droplets larger than 0.3 and 0.5 gm, respectively (Saxena, 1996). This particular set of measurements... 

Calculate the critical radius /y and critical supersaturation Sc. for activation into a cloud droplet of a 10-l5-g NaCl particle. Assume the surface tension is 72 dyn cm-1 and the liquid density is that of water. 

Figure 7. Experimental values of the critical supersaturation, ATC, as a function of saturation temperature in In-rich In-P solutions. (Reproduced with permission from reference 72. Copyright 1986 Elsevier.)...

Table 14-22 shows typical experimental values of taken from the work of Russel [. Chem. Phys., 50, 1809 (1969)]. Since the critical supersaturation ratio for homogeneous nucleation is typically greater... 

In the same samples, a second absorption feature was detected that is associated with the dopant ions themselves. These ligand-field transitions allow distinction among various octahedral and tetrahedral Co2+ species and are discussed in more detail in Section III.C. The three distinct spectra observed in Fig. 4(b) correspond to octahedral precursor (initial spectrum), tetrahedral surface-bound Co2+ (broad intermediate spectrum), and tetrahedral substitutional Co2+ in ZnO (intense structured spectrum). Plotting the tetrahedral substitutional Co2+ absorption intensity as a function of added base yields the data shown as triangles in Fig. 4(b). Again, no change in Co2+ absorption is observed until sufficient base is added to reach critical supersaturation of the precursors, after which base addition causes the conversion of solvated octahedral Co2+ into tetrahedral Co2+ substitutionally doped into ZnO. Importantly, a plot of the substitutional Co2+ absorption intensity versus added base shows the same nucleation point but does not show any jump in intensity that would correspond with the jump in ZnO intensity. Instead, extrapolation of the tetrahedral Co2+ intensities to zero shows intersection at the base concentration where ZnO first nucleates, demonstrating the need for crystalline ZnO to be... 

The droplet current / calculated by nucleation models represents a limit of initial new phase production. The initiation of condensed phase takes place rapidly once a critical supersaturation is achieved in a vapor. The phase change occurs in seconds or less, normally limited only by vapor diffusion to the surface. In many circumstances, we are concerned with the evolution of the particle size distribution well after the formation of new particles or the addition of new condensate to nuclei. When the growth or evaporation of particles is limited by vapor diffusion or molecular transport, the growth law is expressed in terms of vapor flux equation, given by Maxwell s theory, or... 

Equation (46) shows that the nucleation rate is an exponential function of the supersaturation. Hence it is expected that J will be negligible until a certain critical supersaturation is achieved after which homogeneous nucleation will be extremely fast. 

The effect of solution concentration on nucleation rate is shown qualitatively in Fig. 9. At low levels of supersaturation, the rate is essentially zero but, as concentration is increased, a fairly well defined critical supersaturation is reached (point 1), beyond which nucleation rate rises steeply (curve 1-2). Point 1 may be regarded as the threshold of the labile region. Data from a series of such curves at different temperatures establish the locus of points at which nucleation starts, i.e., the Miers supersolubility curve discussed in Section II. 

P/P )]2- As pointed out by LaMer (LI), this term dominates the rate expression, making possible the prediction of critical supersaturation ratios within 10%, despite a hundredfold error in estimating the frequency factor. Close agreement between theory and experiment in the condensation of various vapors is demonstrated by the data of Volmer and Flood (V8) and discussed by Pound (P3). 

This expression is of the same general form as Eq. (8) and the exponential term corresponds to that derived above from Eq. (7). According to Becker and Doering, then, the frequency factor is proportional to the square of the supersaturation ratio. As pointed out by Pound (P3), the Becker-Doering equation closely predicts critical supersaturation values in the condensation of various liquids. For example, for water vapor at 275°K, the calculated ratio p/p = 4.2 agrees exactly with experimental results. 

Most nucleation is in practice likely to be heterogeneous nucleation induced by solid impurite surfaces other than the solute. Nucleation on a foreign surface has a lower surface energy, which leads to a lower critical supersaturation. The rate of heterogeneous nucleation is the same form as that describing homcgeneous nucleation in equation... 

At supersaturations less than the critical supersaturation ratio for surface nucleation, surface —1-5, layer growth has been experimentally... 

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