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 

When w = 0 the p phase can coexist stably with a and y. In that limit + T = 2ff = 2a = o, by the obvious symmetiy of the model. Let the common value of these four quantities be called r. Then by (3.13), with m (for purposes of this illustration) taken to be another constant parameter. 

The composition profile of the a-y interface when w 0 is found from (3.9) with (8.11). If we choose the direction of increasing distance z to be that of increasing x, and define z == 0 to be the point where x=0, we find  

The second expression for f comes from (8.12), in which a is the tension of the ay interface in the limit w = 0. We see in (8.13) or (8.14) that 4 sets the scale of distance in the interface. The dimensionless parameter e is an alternative to w as a measure of how far the state of the system is from one in which the phase would be stable in bulk. 

The smaller e is. that is, the closer a and y are to being in equilibrium with bulk fi, the longer and more nearly horizontal is the plateau in the composition profile. At the midpoint, the gradient of the composition is found (most simply, directly from (3.9), (8.11), and (8.15)) to be 

---

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. 

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. 

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. 

---