Boundary conditions with phase transformation

The reaction scheme at and near the phase boundary during the phase transformation is depicted in Figure 10-14. The width of the defect relaxation zone around the moving boundary is AifR, it designates the region in which the relaxation processes take place. The boundary moves with velocity ub(f) and establishes the boundary conditions for diffusion in the adjacent phases a and p. The conservation of mass couples the various processes. This is shown schematically in Figure 10-14b where the thermodynamic conditions illustrated in Figure 10-12 are also taken into account. The transport equations (Fick s second laws) have to be solved in both the a and p... 

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... 

If the external heat and mass transfer resistances are negligible, eqs 7 and 8 can be simplified by replacing the unknown surface values ci)S and Ts with the known conditions in the bulk fluid phase, and then transform to the following simple boundary conditions at the external pellet surface ... 

Solutes are carried through the soil zone, where they are subjected to a range of phase exchange (e g. sorption) and transformation processes. Along with the boundary conditions at the soil surface, these transport, phase exchange and transformation processes will determine the ultimate concentrations and fluxes of substances in the vadose zone and groundwater. 

It is known that an exact description of transfer processes in the aerosol particles-gas phase system with chemical or phase transformations on the particle surface for arbitrary particle sizes (and correspondingly for arbitrary Knudsen numbers) can be found only by solving the Boltzmann kinetic equation. However, the mathematical difficulties associated with the solution of the given equation lead to the necessity of obtaining rather simple expressions for mass and energy fluxes either on the basis of an approximate solution of the Boltzmann equation or with the use of simpler models. In particular, it is known that the use of the diffusion equation with appropriate boundary conditions on the particle surface leads to the equation that gives correct limiting cases with respect to the Knudsen number [2]. 

In summary, we have obtained a microscopic picture of the pressure-induced structural transformation of the polymorphs of silica at room temperature. In most cases the high-pressure phases highly retain structural order since the transformation occurs without diffusion process, although the tendency might be enhanced by the restricted size of the MD cell for the periodic boundary condition. It is conceivable that compression of a macroscopic sample produces small domains of ordered phases with random crystalline directions. The present result also suggest that the application of high pressure at room temperature may provide routes to new polymorphs, which cannot be obtained under equilibrium conditions. 

The above discussions indicate that not only the composition path and its position relative to the phase separation tine, but also the speed of the composition change along the path, govern the structure of the membrane formed after the gelation process. In order to have some idea on this speed, the complete solution of the differential equation, Equation 3.63, with initial conditions. Equations 3.66 and 3.67, and the boundary condition, Equation 3.68, is necessary. Assuming a special case of a constant diffusivity, the pseudobinary equations are simplifled considerably, and a complete analytical solution is possible. For O = Do, Equation 3.63 can be transformed by the substitution of... 

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