Nucleation problems

In this Section we use Eqs. (2)-(10) to derive several relations for the free energy F ci of a stationary noiiuniform alloy. These relations can be used to study properties of interphase and antiphase boundaries, nucleation problems, etc. 

Applications of Eqs. (17)-(19) to concrete problems show that when the fluctuative effects are important (as in the nucleation problem treated in Sec. 7), the PCA-)-f approximation can be notably more accurate than the lower ones, (17), (13) and (18), while the PCA and MFA residts usually do not greatly differ with each other. 

Schneir et. al. have avoided the nucleation problem by using the same metal (Au) for both deposition and substrate (Schneir, J. Hansma, P.K., Elings, V. Gurley, J. Wickramasinghe, K. ... 

In terms of (14), imperfections appear as a modification of the local anisotropy K r) and lead to a nucleation-field and coercivity reduction [105, 110-112], The solution of the nucleation problem is simplified by the fact that Eq. (14) has the same structure as the single-particle Schrodinger equation, J i(r) and Hc being the respective micromagnetic equivalents of V(r) and E. Consider, for example, an imperfection in form of a cubic soft inclusion of volume l) in a hard matrix. The corresponding wave functions are particle-in-a-box states, and the nucleation field is [5]... 

Within the proposed model the Kelvin equation no longer presents a paradox around the nucleation problem. Although some nuclei may appear with sub-critical size, it is by no means true for all of them. The possibility is now... 

The resulting variation of Xj, with the parameter s is shown in Fig. 3, which implies that a rapid cooperative transition occurs in the vicinity of a critical temperature P (the value of Tfor which s = 1). It is the fact that a is very small that ensures a sharp transition. If (T were taken to be 1 (which corresponds to ignoring the nucleation problem), this model would yield = s/(l + s), which corresponds to a very gradual transition. For high-molar-mass PBG, a value of a- 2 X 10 seems to be appropriate to obtain a fairly good quantitative fit to the experimental data. It should be noted that the transition is sharpest, and the model is most successful, when polypeptides with N s 1000 are used. 

Hartman P (1973) Crystal Growth An Introduction. North-Holland Pubhshing Co, Amsterdam Herring C (1951) Some theorems on the free energies of crystal strrfaces. Phys Rev 82 87-93 Hettema H, McFeaters JS (1996) The direct Monte Carlo method applied to the homogeneous nucleation problem. J Chem Phys 105 2816-2827... 

Sputtered TiW and TiN are already in extensive use as barrier layers against Si diffusion in contacts. It is therefore fortunate that these layers show also good adhesion to CVD-W (i.e. no "new" adhesion material needs to be introduced). Sputtered TiN has some drawback in that, especially with the blanket H2/WF6 chemistry, substantial initiation times (of the order of 10 minutes) can be observed [Rana et al.8, Iwasaki et al.9]. This will be exhibited by apparent lower deposition rates and thickness or uniformity control problems. The reason for the nucleation problem atop TiN is not... 

A common problem in macromolecular crystallization is inducing crystals to grow that have never previously been observed. The single major obstacle to obtaining any crystals at all is, however, ensuring the formation of stable nuclei of protein crystals. In cases where the immediate problem is growing crystals, attention must be thus directed to the nucleation problem, and any approach that can help promote nucleation should be considered. 

Nucleation arid Crystal Growth. When a new phase forms dining the reduction process, the first step is phase nucleation. If the structure of the new phase does not match the host matrix, phase nucleation problems can occur (incoherent nucleation). The... 

The model developed above serves as a convenient starting point for carrying out a dynamical analysis of the nucleation problem from the perspective of the variational principle of section 2.3.3. A nice discussion of this analysis can be found in Suo (1997). As with the two-dimensional model considered in section 2.3.3, we idealize our analysis to the case of a single particle characterized by one degree of freedom. In the present setting, we restrict our attention to spherical particles of radius r. We recall that the function which presides over our variational statement of this problem can be written genetically as... 

Obviously, crystallization of pure titaniosilicate zeolite Beta without Al could then be possible if the nucleation problem was solved. We have done this by seeding with highly active zeolite Beta seeds comprised of very small zeolite Beta crystals (typically —0.05 mm and below) showing good stability in the synthesis media (TEOS/Seed method) (ref. 12). In this way it is possible to synthesize highly crystalline zeolite Ti-Beta with Si/Al ratios well above those obtained by other synthesis procedures, for example Si/Al ratios about 1000. Additionally, as shown by XPS, the crystals obtained by this procedure consist of an inner core of... 

The effect of particle-matrix interfacial free energy is often overlooked but is particularly important in the nucleation and coarsening of internal oxides. Consider the classical nucleation problem of forming a spherical nucleus. If strain is neglected, the free energy of formation of a nucleus of radius r is given by Equation (5.31),... 

There exists an extensive literature on nucleation theory. A great diversity of problems have been discussed. They range from homogeneous gas phase nucleation,to condensation of the primodial vapor in the solar system to form meteorites, and to formation of voids in nuclear reactor materials. Several collections of review articles " as well as a book have been published recently devoted solely to nucleation problems. In these works, each author has advocated his own particular approach to nucleation theory or dealt solely with his own pet nucleation problem. 

Initially we will examine homogeneous gas phase nucleation. This is the classic test problem of nucleation theory. It is believed that condensation of a gas is the simplest nucleation problem. This problem provides a test of the conceptual usefulness of nucleation theory. The lessons learned in discovering how to solve this as yet unsolved problem will hopefully provide a guide to more complex nucleation problems. 

Then in Section 5 we will present the most complex nucleation problem that has been solved to date—formation of voids in nuclear reactor materials. This problem will demonstrate the great conceptual usefulness of nucleation theory even if there is so little information available about the system of interest that there is no possibility of obtaining numerically correct nucleation rates. 

In this section we will use a slightly different trick to solve the nucleation problem. This trick is particularly useful for solving very complex nucleation problems on a computer. While we have previously used this trick, which is more powerful than that of McDonald, to solve the problem of void nucleation in nuclear reactors (discussed in Section 6), it has never been fully explained in the literature. We will do so now. The exposition in this section, combined with the further explanation of using this trick in Section 5, will enable the reader to solve almost any complex nucleation problem without having to find a closed-form expression for the nucleation rate as was done in the previous section using McDonald s trick. 

When the equations are written in this form, the total number of clusters is conserved. More correctly, the right-hand side of the first line of Eq. (16) should be gk ng+2e2 2 2ci/ii +Le,ni — as in Eq. (1) and the right-hand side of the last line should be Cg-iHg-i-gk g. If the equations were written this way, molecules would be conserved rather than clusters. As we will obtain the steady state nucleation rate, where all of the dn /dt are zero, conserving clusters is equivalent to conserving molecules. We write the nucleation equations in the form of Eq. (16) as this is more convenient mathematically to solve, particularly for very complex nucleation problems. 

In the previous sections we have obtained an expression for the steady state homogeneous gas phase nucleation rate. Two approaches were examined. The first, that of McDonald, yields an anal)dic result in a straightforward way. The second approach, which is less direct, is more suited to numerical solutions to complex nucleation problems. It yields the same result for homogeneous nucleation as obtained with McDonald s approach. We will now examine a few alternative forms of Eq. (13) for the homogeneous gas phase nucleation rate. These forms are often referred to in the literature and it is useful to be familiar with them, even though they will not be discussed in this paper. 

The simplest nucleation problem of practical interest is the condensation of water vapor on ions. This occurs in the upper atmosphere and is also of importance in air pollution. Experimentally it is found that condensation of water vapor on ions will nucleate at much lower supersaturation. 

The nucleation problems discussed so far have been those where we want to make quantitative predictions of nucleation rates. There are other very complex problems for which even a qualitative understanding can be useful. In this final section we will discuss briefly one such problem, which will also illustrate the power of the general procedure of Section 1.3 for calculating steady state nucleation rates from complicated rate equations. 

There was one problem with this program. Various experiments suggested that the voids in the neutron (reactor) irradiated materials contained helium. Helium is formed rapidly during neutron irradiation as a decay product. Harkness et had shown that the swelling is a nucleation-controlled process. As the accelerator ion bombardment experiments, intended to mimic the reactor neutron bombardments experiments, do not produce helium within the sample, it is important to ask what effect helium has on the void nucleation problems. 

The microscopic approach to nucleation problems has apparently not yet been carried out. There have been a number of mesoscopic developments for homogeneous nucleation [2.19,36-38]. The mesoscopic approach is successful in giving information on fluctuations, which are, of course, central to the process of nucleation. In this, the mesoscopic approach improves on the macroscopic approach. However, the transition probability is not known from "first principles" and, therefore, must retain some phenomenological elements. 

Both the nucleation of supercritical anti-solvent bubbles in a polymer+organic solvent-rich phase in the supercritical anti-solvent process (SAS) (or, equivalently, precipitation with a compressed antisolvent PCA) (e.g., [76]) and the nucleation of bubbles of a dissolved supercritical fluid from a saturated and nozzle-expanded solution containing a solute to be precipitated, in the formation of particles from gas-saturated solutions (PGSS) [77] are bubble nucleation problems, to which the above ideas apply. In the latter case, the nucleation of bubbles occurs simultaneously with that of solid particles within the bulk supersaturated solution. 

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