Boundaries, in solids

THE ROLE OF PHASE BOUNDARIES IN SOLID STATE REACTIONS... 

The Role of Phase Boundaries in Solid State Reactions 132... 

Abeyaratne, R., and Knowles, J.K., 1991, Kinetic relations and the propagation ofphase boundaries in solids, hrc/i. Rat. Meek Anal. 114 119. 

In Chapter 3 we described the structure of interfaces and in the previous section we described their thermodynamic properties. In the following, we will discuss the kinetics of interfaces. However, kinetic effects due to interface energies (eg., Ostwald ripening) are treated in Chapter 12 on phase transformations, whereas Chapter 14 is devoted to the influence of elasticity on the kinetics. As such, we will concentrate here on the basic kinetics of interface reactions. Stationary, immobile phase boundaries in solids (e.g., A/B, A/AX, AX/AY, etc.) may be compared to two-phase heterogeneous systems of which one phase is a liquid. Their kinetics have been extensively studied in electrochemistry and we shall make use of the concepts developed in that subject. For electrodes in dynamic equilibrium, we know that charged atomic particles are continuously crossing the boundary in both directions. This transfer is thermally activated. At the stationary equilibrium boundary, the opposite fluxes of both electrons and ions are necessarily equal. Figure 10-7 shows this situation schematically for two different crystals bounded by the (b) interface. This was already presented in Section 4.5 and we continue that preliminary discussion now in more detail. 

K. Thornton, J. Agreen, P.W. Woorhees, Modeling the evolution of phase boundaries in solids at the meso- and nano-scales, Acta Mater., 51, 5675-5710 (2003). 

Ishibashi, Y. and Iwata. M. (1999) A theory of morphotropic phase boundary in solid-solution systems of perovskite-type oxide ferroelectrics. Jpn. J. Appl. Phys., 38 (2A), 800-804. 

Progress in modelling and analysis of the crack problem in solids as well as contact problems for elastic and elastoplastic plates and shells gives rise to new attempts in using modern approaches to boundary value problems. The novel viewpoint of traditional treatment to many such problems, like the crack theory, enlarges the range of questions which can be clarified by mathematical tools. 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

Fig. 10.4. Ball bearings can be used to simulate how atoms are packed together in solids. Our photograph shows a ball-bearing model set up to show what the grain boundaries look like in a polycrystalline material. The model also shows up another type of defect - the vacancy - which is caused by a missing atom.

Nucleation in solids is very similar to nucleation in liquids. Because solids usually contain high-energy defects (like dislocations, grain boundaries and surfaces) new phases usually nucleate heterogeneously homogeneous nucleation, which occurs in defect-free regions, is rare. Figure 7.5 summarises the various ways in which nucleation can take place in a typical polycrystalline solid and Problems 7.2 and 7.3 illustrate how nucleation theory can be applied to a solid-state situation. 

Fig. 7.5. Nucleation in solids. Heterogeneous nucleotion con take place at defects like dislocations, grain boundaries, interphase interfaces and free surfaces. Homogeneous nucleation, in defect-free regions, is rare.

In solid state reactions, the rate of nucleation may be given by either of the expressions dN/dt = const, or dN/dt = t° const. For both expressions, the probability (pdf) is proportional to the total volume of the spherical layers at the instant t at the peripheries of nuclei which originated at time r. The radii of the spheres at the inner and outer boundaries of these layers are... 

As shown schematically in Figure 1.4, ions arriving under the influence of the applied current or potential at the three-phase boundaries catalyst/solid electrolyte/gas form there adsorbed species (0(a), Na(a)) which have only three possibilities ... 

Several approaches have been proposed to measure the three phase boundary (tpb) length, Ntpb in solid state electrochemistry. The parameter Ntpb expresses the mol of metal electrode in contact both with the solid electrolyte and with the gas phase. More commonly one is interested in the tpb length normalized with respect to the surface area, A, of the electrolyte. This normalized tpb length, denoted by Ntpb,n equals Ntpt/A. 

In solid electrochemistry electrochemical (change transfer) reactions take place primarily at the three-phase-boundaries (tpb) metal-electrolyte-gas, e.g. ... 

We consider the porous metal catalyst film shown in Figure 11.12 which is interfaced with an O2" conductor. When a positive current, I, is applied between the catalyst and a counter electrode, oxide ions O2 are supplied from the solid electrolyte to the three phase boundaries (tpb) solid electrolyte-metal-gas at a rate I/2F. Some of these O2 will form 02 at the tpb and desorb ... 

The use of periodic boundary conditions also allows for an efficient evaluation of the ion-ion interaction. Ewald developed a method to compute the Coulomb energy associated with long range ion-ion interactions in solids. The Coulomb energy due to interactions between an ion at position R2 and an array of ions positioned at Rj+i is given by... 

Fig. 14.4 ASmGeSe4 compounds viewed parallel to the SmCeSe4 layers. Dashed lines indicate unit cell boundaries in each case. CeSe4 tetrahedra are drawn as tetrahedral solids, and the SmSe environments are drawn using a ball-stick model. In all...

The phase transition boundaries (phase envelope) of adamantane need to be investigated and constmcted. Predictable and diverse geometries are important features for molecular self-assembly and pharmacophore-based dmg design. Incorporation of higher diamondoids in solid-state systems and polymers should provide high-temperature stability, a property already found in polymers synthesized from lower diamondoids. 

---