Nucleation critical supersaturation

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. ... 

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

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. 

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... 

Another theory that could account for the effect of supersaturation on contact nucleation is based on the view that nuclei formed cover a range of sizes that includes the critical nucleus. Since only the nuclei larger than the critical nucleus are stable, the relationship of the size of the critical nucleus to supersaturation reflects the dependence of contact nucleation on supersaturation. This concept, which has been referred to as a survival theory, seems to have been refuted by measurements of the sizes of crystals formed by collisions. These sizes are much larger than the critical nucleus, and the survival theory would have little influence on the number of nuclei that survive. 

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. 

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... 

For the onset of homogeneous in addition to heterogeneous nucleation, the supersaturation ratio must be high. Walton compiled estimates of the critical size of cluster for several substances as shown in Table 8-1. Serious difficulties in making estimates of this type arise from the dearth of knowledge about the surface tension of soUds with extremely small particles and the assumption that the particles are spheres with equivalent surface sites. 

As indicated in Section 8-1, at a critical supersaturation concentration and after a suitable induction period, clusters of critical size form in a solution to constitute nucleation. In a simple way, this critical size can be calculated for a given temperature from Equation (7-41) rearranged to... 

Nucleation of new particles occurs through the formation of low-vapor-pressure products of gas-phase reactions. The production of this material is generally accompanied by its condensation on existing aerosol particles. In some instances nucleation can be initiated where there is rapid production of new condensable material together with low concentration of aerosol particles and thus low existing aerosol surface area. This combination results in the supersaturation of the condensable vapor, which may build up sufficiently to overcome the free-energy barrier (critical supersaturation) associated with new particle production, thereby initiating nucleation. 

The growth of calcite crystals to form speleothems is a delicately balanced process depending on the degree of supersaturation of the water and its total concentration of dissolved carbonates. Waters dripping onto speleothems require supersaturations on the order of Sk = +0.5 in order to overcome nucleation barriers (where Sic is the saturation index defined in the textbooks cited above). However, the critical supersaturation for 2-dimensional nucleation and the continued growth of a single ciystal is only slightly... 

Fig. 7 Dependence of the critical supersaturation for nucleation on solubility of nitrofurantoin in (a) formic acid, (b) formic acid water (4 1), (c) formic acid ethanol (2 1), (d) formic acid dioxane (2 1), (e) formic acid methanol (2 1), (f) formic acid water (2 1). (From Ref. °l)...

The crucial parameters governing AGn are the interfacial energy of the nucleus, AGsurface, and the supersaturation of the solution. As the rate of nucleation, J, is related to AGn, these parameters also influence Jn- The rate of nucleation is slow until a critical supersaturation is achieved, after which it rises rapidly. 

The mechanism of bubble formation by nucleation requires supersaturation of the dissolved gas [11-13] and a nucleus radius greater than the critical [7], The main sources of heterogeneous nucleation are usually surface irregularities capable of containing entrapped gas, e.g. pits and scratches. The bubbles typically develop over the electrode surface, grow in size until they reach a break-off diameter and subsequently detach into the electrolyte. After detachment, some residual gas remains at the nucleation site and another bubble will form at the same place [2,13,14], In most two-phase flow simulations [15-19], it is assumed that bubbles detach with a constant diameter, although from experiments [20,21] it is know that electrochemically formed bubbles show a size distribution. 

The particle size corresponding to the maximum in p/p, does not in general corre.spond to a critical size at which nucleation takes place. Development of a more complete theory of nucleation by ions will require the use of fluctuation theory, introduced at the end of this chapter for equilibrium systems and in Chapter 10 for homogeneous nucleation in supersaturated vapors. 

The goal of an experiment is to set up a critical chamber state—that is, a state that just produces nucleation at some height in the chamber where the vapor is critically supersaturated and droplets are visible. This occurs when the temperature difference across the chamber has been increased to the point where a rain of drops forms at an approximately constant height. Drop formation in this way must be distinguished from condensation on ions generated by cosmic rays passing through the chamber. An electrical field is applied to sweep out such ions which appear as a trail of drops. 

La Mer s qualitative interpretation of sulfur particle formation mechanism can be understood from the diagram shown in Fig. IV-13. The concentration of the molecularly dispersed sulfur formed in the above reaction slowly increases until critical supersaturation is reached. At that point nucleation and the formation of solid phase embryos take place. 

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