Theory nucleation

In the classic nucleation theory, the free energy of forming a cluster of radius r containing n atoms or molecules is the sum of two terms  

The dynamic picture of a vapor at a pressure near is then somewhat as follows. If P is less than P , then AG for a cluster increases steadily with size, and although in principle all sizes would exist, all but the smallest would be very rare, and their numbers would be subject to random fluctuations. Similarly, there will be fluctuations in the number of embryonic nuclei of size less than rc, in the case of P greater than P . Once a nucleus reaches the critical dimension, however, a favorable fluctuation will cause it to grow indefinitely. The experimental maximum supersaturation pressure is such that a large traffic of nuclei moving past the critical size develops with the result that a fog of liquid droplets is produced. 

The final equation obtained by Becker and Doting may be written down immediately by means of the following qualitative argument. Since the flux I is taken to be the same for any size nucleus, it follows that it is related to the rate of formation of a cluster of two molecules, that is, to Z, the gas kinetic collision frequency (collisions per cubic centimeter-second). 

For the steady-state case, Z should also give the forward rate of formation or flux of critical nuclei, except that the positive free energy of their formation amounts to a free energy of activation. If one correspondingly modifies the rate Z by the term an approximate value for I results  

While Becker and Doting obtained a more complex function in place of Z, its numerical value is about equal to Z, and it turns out that the exponential term, which is the same, is the most important one. Thus the complete expression is 

FIGURE 20.5 Illustration of the all the evaporation and condensation rate constants the are required for multi-component nucleation of sulfuric acid (H2SO4) and water (H2O) including the relevant products of sulfuric acid dissociation bisulfate (HSO ), sulfate (SO ), and hydronium (H3O+). 

We will now use this expression to calculate homogeneous gas phase nucleation rates. The result we will get is known as classical nucleation theory. 

Historically, in calculating nucleation ratios, Eq. (36) was not used for the equilibrium cluster concentrations. Rather, everyone employed 

As (In S) appears in the exponential in Eq. (42) for the nucleation rate, J depends very strongly on the supersaturation S. This is illustrated in Fig. 6, where the nucleation rates calculated from Eq. (42) are shown for water at 0°C. The magnitude of the nucleation rates, expressed in the number of nuclei formed in a volume of 1 cm in 1 sec can b st be appreciated by example. Typical measuring techniques cannot detect less than 1 drop per cm. At a supersaturation of 3 the nucleation rate is 2x 10 cm sec Thus one 

2x 10 years, which is longer than the age of the universe, 2x lO years. At a supersaturation of 3.5 the nucleation rate is 10 and an event would be detected in 3 years. At 5 = 3.75 a nucleation event would occur with 1 hr, at S = 4.0 an event would occur every 0.2 sec, and at S = 4.25 a nucleation event would occur every 10 msec. 

This very rapid change in the nucleation rate with supersaturation gives rise to the apparent criticality of nucleation. Below the critical supersaturation 5, at for instance S = 3.75, an experimentalist would see very little happening, while at 5 = 4.25 copious nucleation would be observed. As a result, rather than asking What is the nucleation rate 7 at a particular supersaturation S and temperature T the experimentalist will generally ask At what critical supersaturation 5 is a nucleation rate of 7=lcm sec observed at a temperature 77 Generally then, in comparing experiment and theory critical supersaturations are compared rather than nucleation rates. 

Since the drops have a curved surface of radius r the vapor pressure P/f is higher than that of the flat liquid surface. Thus, the difference in the Gibbs energies is 

however, is not the whole energy difference. In addition, the drop has a surface tension which has to be considered. The total change in the Gibbs free energy is 

In a drop of radius r there are n = Anr /W, moles of molecules, where Vm is the molar volume of the liquid phase. Inserting leads to 

This is the change in Gibbs free energy upon condensation of a drop from a vapor phase with partial pressure P. 

Let us analyse Eq. (2.32) in more detail. For P P,f, the first term is positive and therefore AG is positive. Any drop, which is formed by randomly clustering molecules will evaporate again. No condensation can occur. For P P,-f, AG increases with increasing radius, has a maximum at the so-called critical radius r and then decreases again. At the maximum we have dAG/dr = 0, which leads to a critical radius of 

It should be noted that in this system, polymerization, in terms of formation of siloxane bonds, is a reversible process. The concentration of OH ion that promotes ionization and condensation equally promotes hydrolysis and depolymerization. 

In studying models it is found that when an oligomer such as octamer, or decamer, is the starting point for further addition of monomer, the geometry is such that by the time the original oligomer has become fully condensed to a nearly anhydrous core of SiOj surrounded by silicon atoms forming the outer hydroxylated surface of the particle, the latter must contain 40-50 silicon atoms. 

It is interesting that the formulas calculated for the composition and size of silica particles in Chapter 1 give values in this range. Thus a 48-mer would have an OH Si ratio of 0.8-0.5 by calculation, whereas on the model it appears to be about 0.7. The calculated hydroxylated diameter is 1.6 nm and from the model it appears to be about 1.3 nm. The equivalent diameter of an anhydrous SiO particle is 1.09 nm. 

The theory of homogeneous nucleation has apparently not yet been developed on a quantitative basis, but some relationships have been considered between degree of supersaturation, interfacial energy of silica to water, and the critical size of nuclei. 

A major difference between the silica-water system and other aqueous solutions of inorganic compounds is that in the case of silica a catalyzed making and breaking of siloxane bonds occurs, whereas no such requirement for a catalyst seems to be 

---

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... 

Qualitative examples abound. Perfect crystals of sodium carbonate, sulfate, or phosphate may be kept for years without efflorescing, although if scratched, they begin to do so immediately. Too strongly heated or burned lime or plaster of Paris takes up the first traces of water only with difficulty. Reactions of this type tend to be autocat-alytic. The initial rate is slow, due to the absence of the necessary linear interface, but the rate accelerates as more and more product is formed. See Refs. 147-153 for other examples. Ruckenstein [154] has discussed a kinetic model based on nucleation theory. There is certainly evidence that patches of product may be present, as in the oxidation of Mo(lOO) surfaces [155], and that surface defects are important [156]. There may be catalysis thus reaction VII-27 is catalyzed by water vapor [157]. A topotactic reaction is one where the product or products retain the external crystalline shape of the reactant crystal [158]. More often, however, there is a complicated morphology with pitting, cracking, and pore formation, as with calcium carbonate [159]. 

The resistance to nucleation is associated with the surface energy of forming small clusters. Once beyond a critical size, the growth proceeds with the considerable driving force due to the supersaturation or subcooling. It is the definition of this critical nucleus size that has consumed much theoretical and experimental research. We present a brief description of the classic nucleation theory along with some examples of crystal nucleation and growth studies. 

The classic nucleation theory is an excellent qualitative foundation for the understanding of nucleation. It is not, however, appropriate to treat small clusters as bulk materials and to ignore the sometimes significant and diffuse interface region. This was pointed out some years ago by Cahn and Hilliard [16] and is reflected in their model for interfacial tension (see Section III-2B). 

Classic nucleation theory must be modified for nucleation near a critical point. Observed supercooling and superheating far exceeds that predicted by conventional theory and McGraw and Reiss [36] pointed out that if a usually neglected excluded volume term is retained the free energy of the critical nucleus increases considerably. As noted by Derjaguin [37], a similar problem occurs in the theory of cavitation. In binary systems the composition of the nuclei will differ from that of the bulk... 

The visible crystals that develop during a crystallization procedure are built up as a result of growth either on nuclei of the material itself or surfaces of foreign material serving the same purpose. Neglecting for the moment the matter of impurities, nucleation theory provides an explanation for certain qualitative observations in the case of solutions. 

F. F. Abraham, Homogeneous Nucleation Theory, Academic, New York, 1974. 

Miich effort in recent years has been aimed at modelling nucleation at surfaces and several excellent reviews exist [20, 21 and 22]. Mean-field nucleation theory is one of these models and has a simple picture at its core. 

Models used to describe the growth of crystals by layers call for a two-step process (/) formation of a two-dimensional nucleus on the surface and (2) spreading of the solute from the two-dimensional nucleus across the surface. The relative rates at which these two steps occur give rise to the mononuclear two-dimensional nucleation theory and the polynuclear two-dimensional nucleation theory. In the mononuclear two-dimensional nucleation theory, the surface nucleation step occurs at a finite rate, whereas the spreading across the surface is assumed to occur at an infinite rate. The reverse is tme for the polynuclear two-dimensional nucleation theory. Erom the mononuclear two-dimensional nucleation theory, growth is related to supersaturation by the equation. 

Equation 16—18 can be simplified considerably by recognizing that in many systems the quantity s is much less than 1. In that case, ln(l -H 5) is approximately s. Making this substitution, the growth rate from the mononuclear two-dimensional nucleation theory becomes... 

Nucleation in solids is very similar to nucleation in liquids. Because solids usually contain high-energy defects (like dislocations, grain boundaries and surfaces) new phases usually nucleate heterogeneously homogeneous nucleation, which occurs in defect-free regions, is rare. Figure 7.5 summarises the various ways in which nucleation can take place in a typical polycrystalline solid and Problems 7.2 and 7.3 illustrate how nucleation theory can be applied to a solid-state situation. 

Langer, who was the first to introduce the ImF method in his original paper [Langer 1969] on nucleation theory called it bubble . 

Gibbs considered the change of free energy during homogeneous nucleation, which leads to the classical nucleation theory and to the Gibbs-Tliompson relationship (Mullin, 2001). 

Sawada et al. [110] and the authors of this Chapter [104,111] have proposed another theory, the bundle-like nucleation theory, for the mechanism of ECC formation. Both groups of workers suggested that crystallization under high pressure starts from partially extended-chain nucleation rather than from the folded-chain nucleation as proposed by Hikosaka [103,104]. This theory was established on the basis of the following facts ... 

The layout of this article is as follows Section 2 considers the equilibrium aspects of the crystals whilst Sects. 3 and 4 explain the growth theories, divided into nucleation and non-nucleation theories, respectively. Finally, Sect. 5 provides an overview and suggests future lines of investigation. 

We then turn to a more recent approach to the determination of i by Point [51, 52] in Sect. 3.7 explaining how it differs from that of Lauritzen and Hoffman (LH). Section 3.8 covers other proposed nucleation models and we conclude with an overview of nucleation theories and their successes and most notable shortcomings. 

There are some recent nucleation theories which do not fit in with this approach and will be discussed in Sect. 3.8. 

Although specific calculations for i and g are not made until Sect. 3.5 onwards, the mere postulate of nucleation controlled growth predicts certain qualitative features of behaviour, which we now investigate further. First the effect of the concentration of the polymer in solution is addressed - apparently the theory above fails to predict the observed concentration dependence. Several modifications of the model allow agreement to be reached. There should also be some effect of the crystal size on the observed growth rates because of the factor L in Eq. (3.17). This size dependence is not seen and we discuss the validity of the explanations to account for this defect. Next we look at twin crystals and any implications that their behaviour contain for the applicability of nucleation theories. Finally we briefly discuss the role of fluctuations in the spreading process which, as mentioned above, are neglected by the present treatment. 

The growth rate of many crystals is often observed to depend upon temperature in a manner consistent with nucleation theories. If measurements are made on growth from solutions of different concentrations then, at equivalent thicknesses, the dependence of growth rate upon concentration may be determined. Equations (3.16) and (3.17) can be used to predict the concentration dependence of this nucleation approach. 

Note, however, that a) the size of a stable nucleus depends on the model used (see Sect. 3.5.1), and b) molecular nucleation theory (Sect. 3.8.1) rules out such a mechanism. 

---