Nuclei, growth rate

Many mathematical models have bees advanced relating nucleation and nuclei growth rates to the overall kinetics of phase transformation, such as Johnson and Mehl [427], Avrami [428], Yerofyeyev [429], Kolmogorov [430] as well as Jacobs-Tompkins [431] or Mampel [432] and were agreeably suinmarized elsewhere [1,3,413,144, 421,422,423,426,43 I ]. 

There is increasing interest in nucleus growth mechanisms and many mathematical models have been advanced relating nucleation and nuclei growth rates to the kinetics of solid-state reactions (22-28 general form of the kinetic equations for nuclei growth models is... 

Rate constants for the reaction between rutile and strontium carbonate calculated using both the contracting cylinder equation and the nucleus growth rate equation with m = f yielded nearly horizontal plots of rate... 

Fig. 8. Arrhenius analysis of nucleus growth rate constants (m = ) for reaction between rutile and strontium carbonate, x Ti02 + SrCOs O TiOi.995 + SrCOa TiOi.983 + SrCOs TiOi.97s + SrCOs a TiOi.970 + SrCOa.

An intrinsic surface is built up between both phases in coexistence at a first-order phase transition. For the hard sphere crystal-melt interface [51] density, pressure and stress profiles were calculated, showing that the transition from crystal to fluid occurs over a narrow range of only two to three crystal layers. Crystal growth rate constants of a Lennard-Jones (100) surface [52] were calculated from the fluctuations of interfaces. There is evidence for bcc ordering at the surface of a critical fee nucleus [53]. 

A pre-factor 1/r contains a time scale r or a frequency which for instance corresponds to the hard phonon or to an atomic frequency. The growth rate of the crystal is proportional to this rate (23). As will be shown later, the nucleus once formed expands in a time scale shorter than the one necessary for nucleation. If the process consists of a series of sequential subprocesses, the global velocity is governed by the slowest one. Therefore, this nucleation process determines the growth rate of a faceted surface. 

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 ... 

Extended chain crystal (ECC) Folded chain crystal (FCC) Growth Growth rate Induction period Melt relaxation Molecular weight Nucleation Nucleation rate Nucleus Optical microscope (OM) Polyethylene Polymer Power law ... 

They were evaluated from our analysis of the primary nucleation and lateral growth rates and that of the l dependence to the melting temperature Tm using the Gibbs-Thomson equation. Insertion of the parameters given by Eq. 20 into Eq. 6 shows that the shape of a nucleus is a long thin rectangular parallelepiped with the ratio of... 

From the experimental fact that the lamellar thickening growth rate (17) of ECSC of PE is independent of l [8], we can regard ve on the end surface of the nucleus as constant and independent of l. For simplicity, we assume that the shape of ECSC is rectangular parallelepiped (Fig. 28). ve is given by the ratio of the volume of the crystal to the surface area of the crystal as shown below,... 

The incorporation of discreet nucleation events into models for the current density has been reviewed by Scharifker et al. [111]. The current density is found by integrating the current over a large number of nucleation sites whose distribution and growth rates depend on the electrochemical potential field and the substrate properties. The process is non-local because the presence of one nucleus affects the controlling field and influences production or growth of other nuclei. It is deterministic because microscopic variables such as the density of nuclei and their rate of formation are incorporated as parameters rather than stochastic variables. Various approaches have been taken to determine the macroscopic current density to overlapping diffusion fields of distributed nuclei under potentiostatic control. 

For each polymer molecule, the first step is to place its first stem at the growth surface, whose lateral dimension is taken as Lp. This step is assumed to be associated with a nucleation. After this step, the secondary nucleus spreads out laterally with the rate g. The thickness of the stem is a along the lateral direction and b along the growth direction with growth rate G. 

Suppose that nucleation is a multistep process characterized by rate constants k0, ku kp, and that once a nucleus containing p atoms of product is formed it becomes an active growth nucleus. The rate of formation of active growth nuclei is therefore... 

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