Crystal axes system

In Section 4.4.2 some concepts were developed which allow us to quantitatively treat transport in ionic crystals. Quite different kinetic processes and rate laws exist for ionic crystals exposed to chemical potential gradients with different electrical boundary conditions. In a closed system (Fig. 4-3a), the coupled fluxes are determined by the species with the smaller transport coefficient (c,6,), and the crystal as a whole may suffer a shift. If the external electrical circuit is closed, inert (polarized) electrodes will only allow the electronic (minority) carriers to flow across AX, whereas ions are blocked. Further transport situations will be treated in due course. 

In Section 9.4.1, we introduced internal electrochemical reactions by considering heterophase AX/AY assemblages. We now discuss the more general case of internal electrochemical reactions which occur in inhomogeneous systems having various types of disorder. From the foregoing discussion, we expect internal reactions to occur in a crystal matrix whenever the condition V/jon = 0 is not met. The extreme is a transition from n- (or p-) type conduction to ionic conduction (which for brevity we shall call a (n-i) junction). 

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. 

Let us inspect more closely the inhomogeneous binary system in Figure 11-3 without external forces. At t = 0, the two crystals A and B (AX and BX) are brought in contact. As t oo, the crystals a and fi have equilibrated. This means that either the a/ft boundary has been shifted to its final position or one of the reactant crystals has been consumed (which only depends on the initial volume ratio VA(t = 0)/VB(t = 0)). 

With an open system to which electrodes are attached, we can study the stability of interface morphology in an external electric field. A particularly simple case is met if the crystals involved are chemically homogeneous. In this case, Vfij = 0, and the ions are essentially driven by the electric field. Also, this is easy to handle experimentally. The counterpart of our basic stability experiment (Fig. 11-7) in which the AO crystal was exposed to an oxygen chemical potential gradient is now the exposure of AX to an electric field from the attached electrodes. In order to define the thermodynamic state of AX, it is necessary to apply electrodes with a predetermined... 

The discussion up to this point has been concerned essentially with metal alloys in which the atoms are necessarily electrically neutral. In ionic systems, an electric diffusion potential builds up during the spinodal decomposition process. The local gradient of this potential provides an additional driving force, which acts upon the diffusing species and this has to be taken into account in the derivation of the equivalents of Eqns. (12.28) and (12.30). The formal treatment of this situation has not yet been carried out satisfactorily [A.V. Virkar, M. R. Plichta (1983)]. We can expect that the spinodal process is governed by the slower cation, for example, in a ternary AX-BX crystal. The electrical part of the driving force is generally nonlinear so that linearized kinetic equations cannot immediately be applied. 

The phase diagram of the AX-BY system has four planes, pc(AX), pc(BX), pc(AY), and pc(B Y), which are the projections of the crystallization planes of the components AX, AY, BX, and BY, respectively. There are five boundary lines, representing the simultaneous crystallization of two compounds. The dashed line marks the stabile system AX-BY, which divides the reciprocal system into two simple eutectic ones. 

If we have a mixture AX-AY-BY, the composition of which is in Figure 3.35 shown by the figurative point Xi, the crystallization path at its cooling is completely similar as in the case of a simple ternary eutectic system. The component BY begins to crystallize first, the composition of the melt moves towards point Mi, where also component AX starts to crystallize. At the ensuing cooling, both the components fall out from the melt simultaneously and the composition of the melt moves on the boundary line etj — et2 from point Mi up to the eutectic point etj, where also component AY starts to crystallize and where also the whole system will solidify. 

The binary compound BCX3 divides the AX-BX-CX2 and DX2-BX-CX2 ternary systems into four simple ternary subsystems. The crystallization paths of ternary mixtures end in one of the six ternary eutectic points E , where the ternary mixtures solidify. The crystallization path of any quaternary mixture follows the dotted boundary lines inside the concentration tetrahedron and ends in one of the two quaternary eutectic points Eq,. 

The presence of solid solutions has some important implications, as will be illustrated for a highly simplified example of two components, depicted in Figure 15.24. Assume that we have a liquid fat of composition a3 at temperature T. Cooling it to T2, crystals will form (provided that nucleation occurs). Now a liquid phase of composition a2 will result and a solid phase (crystals) a5. The molar ratio of solid to liquid will be given by the ratio (a3 — a2)/(a5 — a3). Assume now that the mixture is cooled further to T3. The crystals of composition a5 remain, and the liquid a2 will separate into a liquid and crystals of composition ax and a5, respectively. Note that the composition of the system differs from that resulting from cooling directly to T3. In other words, there is no equilibrium (i.e., within the polymorphic form, probably p ). There is, however, a surface equilibrium the crystals a4 will probably be formed at and around the existing crystals... 

A different sequence of operations is shown in another section of Figure 4.33b. Point w on curve FQ represents a solution saturated with salts AX and BY at the higher temperature. At the lower temperature, however, point w lies in the BY field of the diagram. If the correct amount of water is present in the system, pure BY crystallizes out on cooling, and the solution composition is given by point x located on line BY/w produced to meet curve PQ. A cyclic process can now be planned. 

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