Nucleation stationary

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. 

In order to obtain the current consumed during the nucleated relaxation process under a constant potential, we assume that a stationary density of charge (<, ) will be stored in the polymer at the polarization potential E. The storage of these charges is controlled by both conformational relaxation (3r) and diffusion ( processes, so... 

Kast, W., 1964, Significance of Nucleating and Non-stationary Heat Transfer in the Heat Exchanger during Bubble Vaporization and Droplet Condensation, Chem. Eng. Tech. 36(9) 933-940. (2) Katto, Y., 1981, General Features of CHF of Forced Convection Boiling in Uniformly Heated Rectangular Channels, Ini. J. Heat Mass Transfer 24.14131419. (5)... 

Crystallization can be divided into three processes the primary nucleation process, the growth process, and the overgrowth process. The growth process is mainly controlled by the secondary nucleation mechanism. The steady (stationary) primary and secondary nucleation mechanisms of atomic or low molecular weight systems have been well studied since the 1930s by applying the classical nucleation theory (CNT) presented by Becker and Doring, Zeldovich, Frenkel and Turnbull and Fisher and so on [1-4]. 

The primary nucleation process is divided into two periods in CNT one is the so called induction period and the other is the steady (or stationary) nucleation period (Fig. 2) [16,17]. It has been proposed by CNT that small (nanometer scale) nuclei will be formed spontaneously by thermal fluctuation after quenching into the supercooled melt, some of the nuclei could grow into a critical nucleus , and some of the critical nuclei will finally survive into macroscopic crystals. The induction period is defined as the period where the nucleation rate (I) increases with time f, whereas the steady period is that where I nearly saturates to a constant rate (fst). It should be noted that I is a function of N and t,I = I(N, t). In Fig. 2, N and N mean the size of a nucleus and that of the critical nucleus, respectively. The size N is defined... 

Fig. 2 Illustration of the induction and the steady (stationary) periods during the nucleation process. Small clusters exist in the supercooled melt at t = 0. During the induction period (t < r,), isolated nuclei of size N, smaller than the critical nuclei (named nanonuclei or embryo), are formed. The nuclei grow larger and larger with increase of time and some of them attain a much larger size than the critical size, N ...

The molecular weight (M) dependence of the steady (stationary) primary nucleation rate (I) of polymers has been an important unresolved problem. The purpose of this section is to present a power law of molecular weight of I of PE, I oc M-H, where H is a constant which depends on materials and phases [20,33,34]. It will be shown that the self-diffusion process of chain molecules controls the Mn dependence of I, while the critical nucleation process does not. It will be concluded that a topological process, such as chain sliding diffusion and entanglement, assumes the most important role in nucleation mechanisms of polymers, as was predicted in the chain sliding diffusion theory of Hikosaka [14,15]. 

The experimental data for the rate of stationary two-dimensional nucleation in electrocrystallization of cadmium on the surface of the Cd(OOOl) crystal face in 2.5 M CdS04 aqueous solution at 45 °C were presented [234]. The overpotential dependence of the nucleus size was determined. 

The assumption of a stationary nucleation rate is justified by the fact that in situations not too far from equilibrium the relaxation time is small compared to those timescales which determine the hydrodynamical evolution of the shell. Yet this is not true for the growth process which always has to be treated as a time dependent problem (c.f. Gail and Sedlmayr, 1987c). 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

The kinetics of nucleation of one-component gas hydrates in aqueous solution have been analyzed by Kashchiev and Firoozabadi (2002b). Expressions were derived for the stationary rate of hydrate nucleation,./, for heterogeneous nucleation at the solution-gas interface or on solid substrates, and also for the special case of homogeneous nucleation. Kashchiev and Firoozabadi s work on the kinetics of hydrate nucleation provides a detailed examination of the mechanisms and kinetic expressions for hydrate nucleation, which are based on classical nucleation theory. Kashchiev and Firoozabadi s (2002b) work is only briefly summarized here, and for more details the reader is referred to the original references. 

Nucleation — Atomistic theory of nucleation — Figure 2. Supersat-uration dependence of the stationary nucleation rate Jo according to the atomistic theory of nucleus formation (a schematic representation)... 

Apart from the purely thermodynamic analysis, the description of the -> electro crystallization phenomena requires special consideration of the kinetics of nucleus formation [i-v]. Accounting for the discrete character of the clusters size alteration at small dimensions the atomistic nucleation theory shows that the super saturation dependence of the stationary nucleation rate /0 is a broken straight line (Figure 2) representing the intervals of Ap within which different clusters play the role of critical nuclei. Thus, [Ap, Apn is the supersaturation interval within which the nc -atomic cluster is the critical nucleus formed with a maximal thermodynamic work AG (nc). 

Activation of latent sites — The theory accounts for non-stationary effects due to the appearance and disappearance of active nucleation sites on the electrode surface as a result of chemical and/or electrochemical reactions parallel to the process of nucleus formation. Under such circumstances the time dependence of the number N(t) of nuclei is given by a second-order differential equation ... 

Zel dovich theory — The theory determines the time dependence of the nucleation rate 7(f) = d N (f )/df and of the number N(t) of nuclei and derives a theoretical expression for the induction time T needed to establish a stationary state in the supersaturated system. The -> Zel dovich approach [i] (see also [ii]) consists in expressing the time dependence of the number Z(n,t) of the n-atomic clusters in the supersaturated parent phase by means of a partial differential equation ... 

Fields of interest adsorption, catalysis, cavitation, nuclear and thermonuclear weapons, shock waves, nuclear physics, particle physics, astrophysics, physical cosmology, and general relativity. Andrei Sakharov named him a man of universal scientific interests and Stephen W. Hawking said to Zel dovich Before I met you here, I believed you to be a collective author , like Bourbaki. See also Zel dovich theory in -> nucleation, subentry -> non-stationary nucleation, and -> Roginskii-Zeldovich kinetics in adsorption kinetics. 

The problem of crystal growth from supercooled liquids has been formulated in terms of a similar model based on the interfacial tension of microcrystals in the solution. A number of experimental studies which have been made have given further support to the qualitative concepts of the model. The time lag in nucleation required for the distribution of nuclei to change from the equilibrium value at saturation to the stationary concentration at supersaturation has been discussed in some detail by Kantrowitz. ... 

The stationary layer thickness (i ) has to be much larger than the particle size. Usually, d is believed to he on the order of 10 fan, decreasing with increasing stirring intensity. For particles in the nucleation range (2-20 nm) this condition is very well fulfilled for seeded experiments with larger particles the assumption may be doubtful. 

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