Ay interface

Therefore, if A(a = 0, the cation flux changes its density at the AX/AY interface. This means that this interface (by application of a sufficiently strong electric field) acts either as an A sink or as an A source depending on the direction of the A flux. In the first case, metallic A will be precipitated at the AX/AY interface. Since AtA = A/e> the difference in electric current, A7e, will supply the necessary electrons for the (internal) reduction of the A cations. In the second case, the AX/AY interface operates as an A source and the lattice molecules AX or AY will be decomposed. Consequently, either X(Y) atoms or X2(Y2) molecules are formed and the corresponding reactions read... 

In any case, crystal lattices are destroyed by the field-driven decomposition. If the original AX/AY interface remains coherent, stresses develop which will consume some driving force. In other words, the AX/AY interface is then polarized. A determination of the amount (=jA/A-dt) of decomposed AX(AY) at the interface should give a very sensitive method to measure extremely small differences in the elec-... 

A similar process occurs if we electrolyze the phase sequence AX/AY, using A-metal electrodes. AX and AY are immiscible ionic crystals. This time we focus on the AX/AY interface. Since there is always a finite electronic partial conductivity and the very small transference numbers te (AX) and te (AY) are normally different, the AX side of the AX/AY interface serves either as an anode (oxidizing) or as a cathode (reducing). The difference (te(AY)-te(AX)) is proportional to the anodic (cathodic) current in AX. The cathodic interface is expected to obtain similar morphologies as have been described for the A-metal cathode in the previous paragraph. It is immobile as long as Dx,Dymorphological instability is therefore due to the A precipitates which cause the perturbations. 

We remarked in the previous section that macroscopically the locus of points in which three phases meet is one-dimensional and locally linear, even though Rt the molecular level it has a three-dimensional structure. In Fig. 8.1(a) are shown schematically three phases, a, and y, occupying the dihedral angiles between locally planar interfaces, which in turn meet in the three-phase line. We shall represent the dihedral angles by a, p, and y, thus naming them after the phases they contain. Those dihedral angles are the contact angles in whidi the a, Py, and ay interfaces meet. We have... 

In Fig. 8.1(c) we see in cross-section a drop of p phase resting on the otherwise planar ay interface. Hie three-phase line is then a drde, and is seen as sudi when the drop is viewed finnn above. On the cirde, any arc much shenter than its radius may be treated as linear, just as on any of the two-phase interfaces any area with linear dimensions much smaller than... 

At equilibrium, the net force on any element of the three-phase line vanishes. Resolving this force in directions that lie, respectively. In the ap, Py, and ay interfaces and are perpendicular to the three-phase line (hence, in the directions of the lines in Fig. 8.1(b)), we have... 

The equilibration of the phases a and y with the phase ensures that each of the chemical substances present in [3 comes to be present in a and y at the same thermodynamic activity as in even though, in cases of extreme insolubility, that may mean in very small amounts. But even those small amounts—which, spread through the bulk a and y phases, might be in undetectably low concentration—could, if sufficiently concentrated in the ay interface, affect profoundly the properties of that interface, and make it wholly different from one that was not equilibrated with B. 

Any trajectory is locally of minimum action, but when there are two, the action on one may be lower than that on the other. The one of lower action is the one that yields the stable, equilibrium structure of the ay interface x and y vary with each other throu the interface as on the trajectory of minimum action in the x, y-plane. The interface may for some time, and with some degree of stability, assume a structure corresponding to a local but not absolute minimum in the interfacial tension (action), but it would then be only metastable, and would ultimately, and spontaneously, undergp transformation to the structure of absolutely lowest tension. 

On the indirect path from a to y via, the particle in the dynamical analogy has vanishing velocity at the p peak, and so spends infinite time there. Thus, if that is the path of lower action, the equflibrium ay interface consists of a macroscopically thick layer of phase. Also, since the action on the indirect path is the sum of the actions on the a and y paths, we would then have [Pg.220]

We note that it is virtually a truism in this picture that the equilibrium 0- cannot exceed a +structure given by the direct path even when the tension associated with that structure exceeds or +(r but that, as remarked earlier, would be a metastable, not an equilibrium, condition. It is when the actions on the two paths in Fig. 8.6 are equal that we have a Cahn transition, which is the subject of 8.5. 

We turn now to a further consideration of the structure of the ay interface—particularly when 0 is not stable as a bulk phase, but when the two-phase states are in close proximity to three-phase states in which p is stable in bulk and spreads at the ay interface. 

In such a state, then, what is the nature of the ay interface We may determine that from the van der Waals theory, based on an appropriate modification of the potential W in Fig. 8.5. Tte simplest W(x) with the required properties is... 

As we saw, the natural scale of distance in the composition profile is f, related to the model s parameters by (8.1S). That basic distance is amplified by the logarithmic factor in (8.18). as, with decreasing e, the ay interface comes increasingly to resemble bulk 3 and its thickness becomes macroscopic. The reason wc may identify the interface thickness h with the distance between the two maxima in the composition gradient when e is small, is that, once past those maxima in its gradient, the composition goes exponentially rapidly—with the unamplified decay length (—to that of the bulk a and y phases. Wc see that from (8.14) as z - ac. 

The Cahn transition is the particular case in which the transition is between the wetting and non-wetting of an ay interface by p phase ( 8.3)—or by indpient phase if is not stable in bulk ( 8.4). It is thus the transition between two alternative structures of the ay interface one in which it consists of a macroscopic layer of bulk /3 (or a microscopic layer of incipient bulk ), and another in which it does not. 

The existence of such a transition—both when is and when it is not stable in bulk—was predicted by Cahn. He assumed the proximity of a critical point of Py (or aP) phase equilibrium, and then referred to the transition from non-wetting to wetting of the ay interface by a bulk or incipient p phase as critical-point wetting. We paraphrase here his argument that such a transition is to be expected in the neighbourhood of, say, the Py critical point. 

The important conclusion from this argument is that if a Cahn transition in the ay interface occurs near a y (or, equally well, an aP) critical point, the states in which p spreads at the ay interface are those which are nearer the critical point, while those in which does not spread are those which are further. That was brilliantly verified in the experiments of Moldover and Cahn. As a corollary of great practical importance, we note that sufficiently near to a or a critical point, the... 

In Fig. 8.11, which we have adapted from Cahn and from Teletzke er al., we show the temperature (T) vs composition (x) coexistence curve for the equilibrium of the phases B and y (two liquids, say), while these are also in equilibrium widi a third phase, a, which is not shown in the diagram (a vapour phase, say, or a solid boundary). The By critical point is at C. The points marked y and B tmd shown connected by a tieline are a general pair of equilibrium y and B phases. The tieline labelled P marks the Cahn transition in the three-phase (apy) region, and corresponds to P in Hg. 8.10. In the three-phase region above P, that is, dcmr to the critical point C, the ay interface is wetted by B, below P it is not. 

We saw that we may have a Cahn transition also in the two-phase (ay) region, where B is not stable in bulk. Ihe curve P C in Fig. 8.11 is the locus of these transition points. On the side of P C that is toward C, the ay interface consists of a layer of incipient B on the other side of the locus it does not. It is at the coexistence curve that B becomes stable in bulk. As any point of the coexistence curve between F and C is approached, the thickness of the B layer diverges, and does so proportionally to ln(l/E), where e is a measure of the distance from the coexistence curve. We saw that in 8.4 and it is also as found by Cahn. Note the asymmetry there is no locus corresponding to FC at the B side of the coexistence curve. That is because the high-tension interface... 

Tbe point C is the critical point of this interfiuaal phase transition, also predicted by Cahn. Along P C the two alternative structures of the ay interface are of equal tension. As C e approached, those two equally stable but dtetinct structures become gradually more alike, in die way we saw in our description of the critical points of general first-order interfa-dal phase transitions. At C they have become identical. 

The theory of Calm that led to Fig. 8.11 and to the foregoing interpretation was based on the van der Waals theory, but instead of describing the three-phase equilibrium as we do here, with a three-peak potential W, Cahn described it as a two-phase (fiy) equilibrium, with tbe third phase (a) introduced only via a boundary condition specifying the composition at the a surface. Minimizing the free energy then led either to only one possible value of that boundary composition, associated with the bulk composition x, or to two possible values of it, one associated with the bulk composition and the other with the bulk ccmiposition x The former implied that there was an intruding layer at the ay interface, the latter that the a, (3, and y phases met at a three-phase contact line with contact angle 3 > 0. 

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