Nucleation, degree

Figure 10.7 Nucleation rate versus temperature for both homogeneous and heterogeneous nucleation. Degree of supercooling (AT) for each is also shown.

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 . 

In principle, nucleation should occur for any supersaturation given enough time. The critical supersaturation ratio is often defined in terms of the condition needed to observe nucleation on a convenient time scale. As illustrated in Table IX-1, the nucleation rate changes so rapidly with degree of supersaturation that, fortunately, even a few powers of 10 error in the preexponential term make little difference. There has been some controversy surrounding the preexponential term and some detailed analyses are available [33-35]. 

Polypropylene molecules repeatedly fold upon themselves to form lamellae, the sizes of which ate a function of the crystallisa tion conditions. Higher degrees of order are obtained upon formation of crystalline aggregates, or spheruHtes. The presence of a central crystallisation nucleus from which the lamellae radiate is clearly evident in these stmctures. Observations using cross-polarized light illustrates the characteristic Maltese cross model (Fig. 2b). The optical and mechanical properties ate a function of the size and number of spheruHtes and can be modified by nucleating agents. Crystallinity can also be inferred from thermal analysis (28) and density measurements (29). 

Sol-Gel Techniques. Sol-gel powders (2,13,15,17) are produced as a suspension or sol of coUoidal particles or polymer molecules mixed with a Hquid that polymerizes to form a gel (see Colloids SoL-GELtechnology). Typically, formation of a sol is foUowed by hydrolysis, polymerization, nucleation, and growth. Drying, low temperature calciaation, and light milling are subsequently required to produce a powder. Sol-gel synthesis yields fine, reactive, pseudo-crystalline powders that can be siatered at temperatures hundreds of degrees below conventionally prepared, crystalline powders. 

When a process is continuous, nucleation frequently occurs in the presence of a seeded solution by the combined effec ts of mechanical stimulus and nucleation caused by supersaturation (heterogeneous nucleation). If such a system is completely and uniformly mixed (i.e., the product stream represents the typical magma circulated within the system) and if the system is operating at steady state, the particle-size distribution has definite hmits which can be predic ted mathematically with a high degree of accuracy, as will be shown later in this section. 

Normally, when a pond of water freezes over, or when a metal casting starts to solidify, nucleation occurs at a temperature only a few degrees below T, . Flow do we explain... 

At higher temperatures, other degrees of freedom than the radius R must also be considered in the fluctuation. However, this becomes critical only near the critical point where the system goes through a phase transition of second order. The nucleation arrangement described here is for heterogeneous or two-dimensional nucleation on a flat surface. In the bulk, there is also the formation of a three-dimensional nucleation, but its rate is smaller ... 

This closeness of 0 to zero explains the existence of a gas-oversaturated solution area in the polymer melt, when P < Pg, but the entire volume of gas remains in the solution. The degree of oversaturation, particularly upon free foaming (not in flow) can be 2- to 3-fold. In real polymer compositions, there are always solid admixtures, which have poor wetting areas. This reduces the degree of oversaturation at the interface melt-molding tool. Moreover, bubble nuclei can result from fragmentation of gas bubbles in the polymer [16]. Another factor that promotes the formation of bubble nuclei is the presence of localized hot points in the polymer melt they act as nuc-leation centres. Hot points appear either after a chemical reaction in the melt polymer [17], or in overheated areas on the surface of metal equipment [18]. Density of nucleation can be improved via introduction of various agents that reduce tension of the polymer [19]. 

These various distributions reflect the variations in reactivity of the reacting sites. Johnson and Kotz [444] discuss in detail the Weibull and other distributions which find application when conditions of strict randomness of the exponential distribution are not satisfied. From an empirical point of view, the power transformation is a practical and convenient method of introducing a degree of flexibility into a model. Gittus [445] has discussed some situations in which the Weibull distribution may be expected to find application, including nucleation and growth processes in alloy transformations. 

We define a nucleation overpotential rjN EN E0 (Fig. 36) required to make the N0 oxidation nuclei appear. The nucleation overpotential is related to the degree of closure (compaction) of the polymeric entanglement ( ), expressed as the fraction of interchain free volume destroyed after polarization at a given potential Ec, compared with the amount of free volume present at Es. 

Thermodynamic and mechanical equilibrium on a curved vapor-liquid interface requires a certain degree of superheat in order to maintain a given curvature. Characteristics of homogeneous and heterogeneous nucleation can be estimated in the frame of classical theory of kinetics of nucleation (Volmer and Weber 1926 Earkas 1927 Becker and Doring 1935 Zel dovich 1943). The vapor temperature in the bubble Ts.b can be computed from equations (Bankoff and Flaute 1957 Cole 1974 Blander and Katz 1975 Li and Cheng 2004) for homogeneous nucleation in superheated liquids... 

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