Phase Boundaries in Solids

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

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

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. 

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 ... 

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 ... 

In this case, we have given both the starting conditions and those of the intermediate stage of solid state reaction. It should be clear that A reacts with B, and vice versa. Thus, a phase boundary is formed at the interface of the bulk of each particle, i.e.- between A and AB, and between B and AB. The phase boundary, AB, then grows outward as shown above. Once the phase boundary is established, then each reacting specie must diffuse through the phase AB to reach its opposite phase boundary in order to react. That is- A must difiuse through AB to the phase boundary... 

We call each solid line in this graph a phase boundary. If the values of p and T lie on a phase boundary, then equilibrium between two phases is guaranteed. There are three common phase boundaries liquid-solid, liquid-gas and solid-gas. The line separating the regions labelled solid and liquid , for example, represents values of pressure and temperature at which these two phases coexist - a line sometimes called the melting-point phase boundary . 

Similar types of electric double layer may also be formed at the phase boundary between a solid electrolyte and an aqueous electrolyte solution [7]. They are formed because one electrically-charged component of the solid electrolyte is more readily dissolved, for example the fluoride ion in solid LaFs, leading to excess charge in the solid phase, which, as a result of movement of the holes formed, diffuses into the soUd electrolyte. Another possible way a double layer may be formed is by adsorption of electrically-charged components from solution on the phase boundary, or by reactions of such components with some component of the solid electrolyte. For LaFa this could be the reaction of hydroxyl ions with the trivalent lanthanum ion. Characteristically, for the phase boundary between two immiscible electrolyte solutions, where neither solution contains an amphiphilic ion, the electric double layer consists of two diffuse electric double layers, with no compact double layer at the actual phase boundary, in contrast to the metal electrode/ electrolyte solution boundary [4,8, 35] (see fig. 2.1). Then, for the potential... 

Another solid state reaction problem to be mentioned here is the stability of boundaries and boundary conditions. Except for the case of homogeneous reactions in infinite systems, the course of a reaction will also be determined by the state of the boundaries (surfaces, solid-solid interfaces, and other phase boundaries). In reacting systems, these boundaries are normally moving in space and their geometrical form is often morphologically unstable. This instability (which determines the boundary conditions of the kinetic differential equations) adds appreciably to the complexity of many solid state processes and will be discussed later in a chapter of its own. 

Following the introduction of basic kinetic concepts, some common kinetic situations will be discussed. These will be referred to repeatedly in later chapters and include 1) diffusion, particularly chemical diffusion in different solids (metals, semiconductors, mixed conductors, ionic crystals), 2) electrical conduction in solids (giving special attention to inhomogeneous systems), 3) matter transport across phase boundaries, in particular in electrochemical systems (solid electrode/solicl electrolyte), and 4) relaxation of structure elements. 

Studies on the degradation of polychlorinated biphenyls (PCBs) have indicated that dissolved PCBs are much more readily degraded by hydroxyl radical than are PCBs sorbed to sand [37]. PCBs sorbed to glass were also difficult to degrade [38]. These studies suggest that hydroxyl radical is unlikely to cross a phase boundary in order to react with the solid or sorbed pollutant. 

Contact angle measurement is probably the most common method for solid surface tension measurement in condensed state. Young [71] described the equilibrium at three-phase boundary in terms of the vectorial sum, as shown in Fig. 3, resulting in the following equation of equilibrium forces balance... 

Diffusion resistances can occur for Li in the electrode, but also for the salt in the electrolyte (if anion conductivity in the electrolyte is significant). Further effects are due to depletion of carriers at a phase boundary. In such cases, time dependencies of the electrical properties occur (in addition to Rs, effective capacitances Cs also appear). The same is true for impeded nucleation processes. Since any potential step of the electrochemical potential can be connected with current-dependent effective resistances and capacitances, the kinetic description is typically very specific and complex. As the storage processes in Li-based batteries are solid-state processes, the... 

Usually, the working electrode (W) is a porous metallic electrode in PEVD. Thus, reactant (B) in the vapor phase can reach the surface of the solid electrolyte for initial electrochemical reaction at a three-phase boundary of solid electrolyte (E), working electrode (W) and sink vapor phase (S) as shown in Eigure 3 (location II). All reactants for the sink side electrochemical reaction (1) or (2) are only available there. Subsequent reaction and deposition of the product (D) requires both electrons and ions to travel through product (D) to the surface to react with vapor phase reactant(s) electrochemically at location III in Eigure 3. 

As schematically shown in Figure 7a, initial PEVD reaction and product nucleation occurs at the three-phase boundary of solid electrolyte (E), working electrode (W) and the sink vapor phase (S) which contains vapor phase reactant (B). Only here are all reactants available for the half-cell electrochemical reaction at the sink side of a PEVD system. Although the ionic and electronic species can sometimes surface diffuse at elevated temperature to other sites to react with (B) in the vapor phase, the supply of the reactants continuously along the diffusion route is less feasible and the nuclei are too small to be stabilized under normal PEVD conditions. Only along the three phase boundary line are all the reactants available for further growth to stabilize the nuclei. Consequently, initial deposition in a PEVD process is restricted to certain areas on a substrate where all reactants for the sink electrochemical reaction are available. 

Dynamic contact angles are the angles which can be measured if the three-phase boundary (liquid/solid/vapor) is in actual motion. A Wilhelmy plate is used in dynamic contact angle measurements, and this method is also called the tensiometric contact angle method. It has been extensively applied to solid-liquid contact angle determinations in recent years. In practice, a solid substrate is cut as a thin rectangular plate, otherwise a solid material is... 

This chapter deals with the diffusional transfer of mass to and across a phase boundary. In particular, gas-liquid, gas-solid, and liquid-liquid phase combinations have been considered. Process applications include absorption, stripping, distillation, extraction, adsorption, and the diffusional aspects of chemical reactions on a solid surface. For steady-state transfer operations, the rates of mass transfer can be correlated by variations of Pick s first law, which states that the rate is directly proportional to the concentration driving force and the extent of interfacial area, and inversely proportional to the distance of movement of the mass to the interface. 

Unlike the Laplace equation, which can only be applied to fluid surfaces, the Kelvin equation is valid for any phase boundary, also solid-gas, solid-liquid, and even solid-solid. As for the Laplace equation, Eq. (10.9) can also be modified to accommodate nonspherical curved surfaces then 2/(l/i + l/i 2) should be used instead of r. The values for the solubility should give the thermodynamic activity of the substance involved, for the equation to be generally valid for ideal or ideally dilute solutions, concentrations can be used. Most gases in water show ideal behavior. It should further be noted that Eq. (10.9) applies for one component if a particle contains several components, it should be applied to each of them separately. Finally, there are some conditions that may interfere with the... 

Figure 3. A conceptual generalized physical/chemical phase diagram of solids at high pressures and temperatures, illustrating the melting maximum and phase boundaries in both solid and fluids.

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