Rate of nucleation

Volmer and Weber derived an equation for the nucleation rate.  

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An interesting point is that AH itself varies with r [10].] As is the case when P is varied, the rate of nucleation increases so strongly with the degree of supercooling that a fairly sharp critical value exists for T. Analogous equations can be written for the supercooling of a melt, where the heat of fusion AH/ replaces AH . 

Assuming that for water AGd is 7 kcal/mol, calculate the rate of nucleation for ice nuclei for several temperatures and locate the temperature of maximum rate. Discuss in terms of this result why glassy water might be difficult to obtain. 

By carefully controlling the precipitation reaction we can significantly increase a precipitate s average particle size. Precipitation consists of two distinct events nu-cleation, or the initial formation of smaller stable particles of precipitate, and the subsequent growth of these particles. Larger particles form when the rate of particle growth exceeds the rate of nucleation. 

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 As additional precipitant is created, nucleation is eventually superseded by particle growth.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. ... 

In spite of these obstacles, crystallization does occur and the rate at which it develops can be measured. The following derivation will illustrate how the rates of nucleation and growth combine to give the net rate of crystallization. The theory we shall develop assumes a specific picture of the crystallization process. The assumptions of the model and some comments on their applicability follow ... 

Figure 10.2 The time-temperature-nucleation curve showing the balance between the rate of nucleation and the critical radius which produces a maximum rate...

The suppliers of nylon 46 have laid particular emphasis on the fact that this polymer, with its highly symmetrical chain structure, leads to both a high level of crystallinity and a high rate of nucleation. In turn the high nucleation rate leads to a fine crystalline structure which in this case is claimed to lead to a higher impact strength (dry as moulded) than with nylons 6 and 66. 

Rates of nucleation and crystal growth may be given, respectively, as... 

Rates of nucleation, B , and crystal growth, G, are respectively evaluated as the functions of supersaturation. The level of supersaturation, AC, is determined... 

The uniformity of film thickness is dependent upon temperature and pressure. The nucleation rate rises with pressure, such that at pressures above atmospheric the high rate of nucleation can lead to comparatively uniform oxide films, while increase in temperature reduces the density of oxide nuclei, and results in non-uniformity. Subsequently, lateral growth of nuclei over the surface is faster than the rate of thickening until uniform coverage is attained, when the consolidated film grows as a continuous layer ... 

In fundamental terms, the transformation temperature affects both the driving force for the decomposition of austenite and the diffusion rate of carbon. In effect, therefore, the transformation temperature alters both the rate of nucleation and the rate of growth. This in turn manifests itself in... 

Phalloidin and phallacidin are cyclic peptides from the mushroom Amanita phalloides that stabilize F-actin. Phalloidin binds to residues 114-118 of an actin protomere and blocks nucleotide exchange without interfering with nucleotide hydrolysis. It enhances the rate of nucleation as well as that of elongation. It slowly penetrates the cell membrane and is used for immunocytochemical localization of F-actin. 

Growth theories of surfaces have received considerable attention over the last sixty years as summarized by Laudise et al. [53] and Jackson [54]. The well-known model of the crystal surface incorporating adatoms, ledges and kinks was first introduced by Kossel [55] and Stranski [56]. Becker and Doring [57] calculated the rates of nucleation of new layers of atoms, and Papapetrou [58] investigated dendritic crystallization. 

The growth rate of a nucleation-controlled face can be expressed, within certain limits, in terms of i and g without their explicit evaluation. The first limit we shall consider is that of large g so that a nucleus, having formed, spreads rapidly to cover the substrate before another nucleation event occurs, see Fig. 3.5 a. If the substrate has a width L, then the rate of nucleation is iL and the growth rate is given by ... 

The account of the formal derivation of kinetic expressions for the reactions of solids given in Sect. 3 first discusses those types of behaviour which usually generate three-dimensional nuclei. Such product particles may often be directly observed. Quantitative measurements of rates of nucleation and growth may even be possible, thus providing valuable supplementary evidence for the analysis of kinetic data. Thereafter, attention is directed to expressions based on the existence of diffuse nuclei or involving diffusion control such nuclei are not susceptible to quantitative... 

In solid state reactions, the rate of nucleation may be given by either of the expressions dN/dt = const, or dN/dt = t° const. For both expressions, the probability (pdf) is proportional to the total volume of the spherical layers at the instant t at the peripheries of nuclei which originated at time r. The radii of the spheres at the inner and outer boundaries of these layers are... 

Gamer and Hailes [462] postulated a chain branching reaction in the decomposition of mercury fulminate, since the values of n( 10—20) were larger than could be considered consistent with power law equation [eqn. (2)] obedience. If the rate of nucleation is constant (0 = 1 for the generation of a new nuclei at a large number of sites, N0) and there is a constant rate of branching of existing nuclei (ftB), the nucleation law is... 

Gamer and Jennings [431] studied nucleation during the dehydration of potassium and ammonium chromium alums. Detailed kinetic measurements were made for the relatively enhanced rate of nucleation which followed admission of water vapour to the solid after a period of vacuum nucleation. This catalytic effect of water vapour is ascribed to its participation in the reorganization of the lattice which had collapsed during previous treatment in vacuum. 

From microscopic measurements of the rates of nucleation and of growth of particles of barium metal product, Wischin [201] observed that the number of nuclei present increased as the third power (—2.5—3.5) of time and that the isothermal rate of radial growth of visible nuclei was constant. During the early stages of reaction, the acceleratory region of the a—time plot obeyed the power law [eqn. (2)] with 6 temperature coefficients of these processes were used by Wischin [201]... 

Although some progress has been made in determining the geometry of interface advance through interpretation of observed f(a)—time relationships for individual salts, the reasons for differences between related substances have not always been established. Nickel carboxylates, for which the most extensive sequence of comparative rate studies has been made [40,88,375,502,1106,1107,1109], show a wide variety of kinetic characteristics, but the controlling factors have not yet been satisfactorily determined. Separate measurements of the rates of nucleation and of growth are not usually practicable. 

The rate of polymerization with styrene-type monomers is directly proportional to the number of particles formed. In batch reactors most of the particles are nucleated early in the reaction and the number formed depends on the emulsifier available to stabilize these small particles. In a CSTR operating at steady-state the rate of nucleation of new particles depends on the concentration of free emulsifier, i.e. the emulsifier not adsorbed on other surfaces. Since the average particle size in a CSTR is larger than the average size at the end of the batch nucleation period, fewer particles are formed in a CSTR than if the same recipe were used in a batch reactor. Since rate is proportional to the number of particles for styrene-type monomers, the rate per unit volume in a CSTR will be less than the interval-two rate in a batch reactor. In fact, the maximum CSTR rate will be about 60 to 70 percent the batch rate for such monomers. Monomers for which the rate is not as strongly dependent on the number of particles will display less of a difference between batch and continuous reactors. Also, continuous reactors with a particle seed in the feed may be capable of higher rates. 

The first is diffusion capture. This theory was originally proposed by Fitch and Tsai (13) for the aqueous polymerization of methyl methacrylate. According to this theory, any oligomer which diffuses to an existing particle before it has attained the critical size for nucleation is irreversibly captured. The rate of nucleation is equal to the rate of initiation minus the rate of capture. The rate of capture is proportional to both the surface area and the number of particles. 

The nature of the deposit and the rate of nucleation at the very beginning of the deposition are affected, among other factors, by the nature of the substrate. A specific case is that of epitaxy where the structure of the substrate essentially controls the structure of the deposit.Plb lP ] Epitaxy can be defined as the growth of a crystalline film on a crystalline substrate, with the substrate acting as a seed crystal. When both substrate and deposit are of the same material (for instance silicon on silicon) or when their crystalline structures (lattice parameters) are identical or close, the phenomena is known as homoepitaxy. When the lattice parameters are different, it is heteroepitaxy. Epitaxial growth cannot occur if these stmctural differences are too great. 

However, we need to point out that the rate of nucleation, I, is defined as ... 

In Table 4-2, we show both phase-boundaiy controlled and diffusion-controlled nucleation reactions as a function of both constant and zero rate of nucleation. 

Ap.2.1.- Rate of nucleation(I) = the number of nuclei formed per unit time... 

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