Complex Crystallization Conditions General Models

For 3D crystallization from instantaneous nuclei the conical zone in space and time is reduced to a 3D zone. 

Both the approaches just described are equivalent. In both cases the number of spherulites nucleated per unit volume around A increases with time  

Determination of the average number of spherulites in a portion of material requires integrating over spatial coordinates and dividing by an appropriate volume. 

In fact, both Equation (7.39) and Equation (7.40) are based on the assumption of radial growth, which is not always true, for instance in a temperature gradient, as will be described in Section 7.6.2. 

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The following describes results of three, relatively simple chemical reactions involving hydrocarbons on model single crystal metal catalysts that illustrate this general approach, namely, acetylene cyclotrimerization and the hydrogenation of acetylene and ethylene, all catalyzed by palladium. The selected reactions fulfdl the above conditions since they occur in ultrahigh vacuum, while the measured catalytic reaction kinetics on single crystal surfaces mimic those on reahstic supported catalysts. While these are all chemically relatively simple reactions, their apparent simplicity belies rather complex surface chemistry. 

Perhaps the most general mathematical treatment of the surface mass loading effect on bulk shear wave resonators has been presented by Kanazawa (13). In this work, a wave equation was developed for acoustic wave propagation within the deposited layer, assuming the material had both elastic and viscous properties. Boundary conditions between crystal and deposited mass were established by assuming shear forces and particle displacements were equal for both materials at the interface plane. This approach results in a fairly complex mathematical model, but simplified relationships were derived for purely elastic and purely viscous behaviour. 

A common approach to study heterogeneous catalyst materials is by means of surface science techniques, in particular spectroscopy [4], These techniques allow one to characterize and investigate surfaces and interfaces (routinely) and improved their understanding significantly [5, 6]. However, the techniques of surface science are mainly restricted to UH V pressure conditions, thus are in general only applicable under ideal conditions far from real catalyst environments. Further, for application of i.e. spectroscopy, usually model systems with reduced complexity and trying to mimic real catalysts are used in order to understand catalytic phenomena [7]. In the early days of surface science these systems were predominantly single crystal surfaces [8], that evolved into supported particles [9, 10], which still often lack the complexity of real catalyst materials. 

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