Grain boundary kinetic equations

With the simplihed model for nearly spherical isolated pores on the grain boundary, the equations for the grain growth kinetics can be derived, if it is assumed that the grain growth is controlled by the pore mobility, i.e., pore control. If pore migration occurs through surface diffusion, there is... 

Although the flux equations for grain boundary and volume transport are of the same type, the creep kinetics are different because the boundary conditions of the transport differ for the two models (Fig. 14-3). Finally, we observe that creep in compound crystals requires the simultaneous motion of all components [R.L. Coble (1963)] so that the slow ones necessarily determine the creep rate. 

Equation 14.17 can be used to model grain-boundary grooving kinetics when vapor transport is the dominant mechanism. The normal velocity dh/dt is related... 

The stability of the Cu/Mo multilayered structures was analyzed in terms of the kinetics of thermal grooving of grain boundaries normal to the layer interface, as given by the following equation [36] ... 

The nucleation rate, growth rate, and transformation rate equations that we developed in the preceding sections are sufficient to provide a general, semiquantitative understanding of nucleation- and growth-based phase transformations. However, it is important to understand that the kinetic models developed in this introductory text are generally not sufficient to provide a microstructurally predictive description of phase transformation for a specific materials system. It is also important to understand that real phase transformation processes often do not reach completion or do not attain complete equilibrium. In fact, extended defects such as grain boundaries or pores should not exist in a true equilibrium solid, so nearly all materials exist in some sort of metastable condition. Many phase transformation processes produce microstructures that depart wildly from our equilibrium expectation. The limited atomic mobilities associated with solid-state diffusion can frequently cause (and preserve) such nonequilibrium structures. In this section, we will focus more deeply on solidification (a liquid-solid phase transformation) as a way to discuss some of these issues. In particular, we will examine a few kinetic concepts/models... 

Following the analysis by Coble (10), we shall outline the derivation of the kinetic equations for sintering by lattice diffusion and grain boundary diffusion. 

For the simplified model consisting of nearly spherical isolated pores on the grain boundary, we can derive equations for the grain growth kinetics. Consider the... 

However, if finely crystalline CaCOa is used, then the rate of decomposition is found to be proportional to the instantaneous area of the phase boundary CaC03/Ca0. From this it may be concluded that the decomposition kinetics are controlled by a phase boundary reaction which proceeds at a constant rate. In the case of fine-grained material, the nucleation of oxide is no longer a rate-determining kinetic step. Apparently there are always enough active sites for nucleation. The experimental rate constant varies inversely as the partial pressure of CO2. Since the decomposition process must cease when the CO2 pressure is equal to its equilibrium value, it follows that the rate equation for the phase boundary reaction must be of the form... 

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