The Nucleation Problem

The problem we wish to solve is as follows A vapor is initially carefully equilibrated at some pressure pi and temperature TV, where pi and T are such that no formation of droplets is visible. Then the pressure and temperature are altered (either by adiabatic expansion or isothermal compression) so that the new pressure p exceeds the equilibrium vapor pressure of the condensed phase Po at the new temperature T. One now asks How many droplets of some macroscopic visible size / v will be produced in a sample volume V during a time t This question can be formulated mathematically. 

(t) be the concentration of clusters of size i at time t. Let c, (t) be the average rate at which a cluster of size i captures a monomer to grow to size i +1 and let 6, (f) be the average rate at which a monomer escapes from a cluster of size i causing it to shrink to size i — 1. Then the time-dependent behavior of the system is described by a set of coupled differential equations  

Equation (1) can be transformed into a more useful form of determining the nucleation rate. The net rate of formation of clusters of size i +1 from clusters of size i is 

Further on we will solve Eq. (4) to obtain the steady state nucleation rate, but first a number of observations must be made. 

Equations (1) and (4) are complicated and we do not expect to be able to solve them quite generally except numerically on a computer. They can, however, be simplified to make them tractable. 

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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. 

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. 

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... 

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. 

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. 

To describe the formation of a nanosheU, we do not need to change the model equations - we change only the initial conditions. Namely, let us consider a sphere of pure B of radius tba surrounded by a shell of pure A. To avoid solving the nucleation problem, let us assume that the initial pure sphere A already contains a small void in the center. Of course, for a very big initial core, this assumption seems unreasonable since the first voids should nucleate in the vicinity of the initial contact between A and B. Yet, for nanoparticles, it is natural that the initial nanovoids coalesce very fast into a single central void. Thus, in our model, the initial B-profile is... 

Bursae, R., Sever, R., Hunek, B., 2009. A practical method for resolving the nucleation problem in lyophilization. Bioproc. Int. 7(9) 66-72. 

An increase in the time required to form a visible precipitate under conditions of low RSS is a consequence of both a slow rate of nucleation and a steady decrease in RSS as the precipitate forms. One solution to the latter problem is to chemically generate the precipitant in solution as the product of a slow chemical reaction. This maintains the RSS at an effectively constant level. The precipitate initially forms under conditions of low RSS, leading to the nucleation of a limited number of particles. As additional precipitant is created, nucleation is eventually superseded by particle growth. This process is called homogeneous precipitation. ... 

The nucleation rate is, in fact, critically dependent on temperature, as Fig. 8.3 shows. To see why, let us look at the heterogeneous nucleation of b.c.c. crystals at grain boundaries. We have already looked at grain boundary nucleation in Problems 7.2 and 7.3. Problem 7.2 showed that the critical radius for grain boundary nucleation is given by... 

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. 

Development of the theory of nucleation is the long-standing problem in the statistical physics. The kinetic approach to this problem was proposed by Zeldovich" . For the nucleation rate. 7. i.e. for the number of critical and supercritical embryos being formed in the unit volume per unit time at early stages of nucleation, he obtained the following expression... 

The importance of twinned crystals in demonstrating that nucleation is the relevant growth mechanism has been realized since 1949 [64, 99]6. They were first investigated extensively in polymer crystals by Blundell and Keller [82] and they have recently received increased attention as a means of establishing, or otherwise, the nucleation postulate for lamellar growth [90, 91, 95,100-102]. The diversity of opinion in the literature shows that it is very difficult to draw definite conclusions from the experimental evidence, and the calculations are often founded upon implicit assumptions which may or may not be justified. We therefore restrict our discussion to an introduction to the problem, the complicating features which make any a priori assumptions difficult, and the remaining information which may be fairly confidently deduced. 

The purpose of this review is to solve these two unresolved problems by confirming the nucleation during the induction period of nucleation and the important role of the topological nature with experimental facts regarding the molecular weight (M)- or number density of the entanglement (independence of nucleation and growth rates. 

The former problem is a general problem not only for polymers but also for any other materials (atomic or low molecular weight systems). Although nucleation is a well-known concept, it has never been confirmed by direct observation due to the low number density of the nuclei to be detected with present experimental techniques, such as small angle X-ray scattering (SAXS). Therefore, one of the most important unresolved problems for basic science is to obtain direct evidence to solve the nucleation mechanism of any material. 

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