Cahn transition

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 shall first review very briefly the thermodynamics of first-order phase transitions between ordinary bulk phases at equilibrium. We shall then be able to deseribe in similar terms the closely analogous phase transitions in interfaces. Among these is the Cahn transition, our present subject. 

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 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... 

So far, the only Cahn transitions that have been seen with certainty in fluid interfaces are in three-phase systems, - where they are visible as transitions between spreading and non-spreading. Hie phmiomenon may manifest itself as in Rg. 8.4, as a transition between a negative value of [Pg.231]

As a matter of historical interest. Buff and Saltsburg, in their paper on the theory of the three-phase contact line and Neumann s triangle (ref. 3), also reported, briefly and qualitatively, their observation that in the three-liquid system water-aniline-heptane, the middle, aniline-iich phase is transformed from a spread layer to a lens upon the addition of a detergent to the system. That was an early observation of a Cahn transition. 

Figure 6.10. Dilatotneuic record of a sample of a Ni-AI-Fe alloy in the neighbourhood of an order-disorder transition temperature (Cahn ei al. 1987).

Wood Hill (1991b) induced phase-separation in the clear glasses by heating them at temperatures above their transition temperatures. They found evidence for amorphous phase-separation (APS) prior to the formation of crystallites. Below the first exotherm, APS appeared to take place by spinodal decomposition so that the glass had an intercoimected structure (Cahn, 1961). At higher temperatures the microstructure consisted of distinct droplets in a matrix phase. 

Dilatometric methods. This can be a sensitive method and relies on the different phases taking part in the phase transformation having different coefficients of thermal expansion. The expansion/contraction of a sample is then measured by a dilatometer. Cahn et al. (1987) used dilatometry to examine the order-disorder transformation in a number of alloys in the Ni-Al-Fe system. Figure 4.9 shows an expansion vs temperature plot for a (Ni79.9Al2o.i)o.s7Feo.i3 alloy where a transition from an ordered LI2 compound (7 ) to a two-phase mixture of 7 and a Ni-rich f c.c. Al phase (7) occurs. The method was then used to determine the 7 /(7 + 7O phase boundary as a function of Fe content, at a constant Ni/Al ratio, and the results are shown in Fig. 4.10. The technique has been used on numerous other occasions,... 

Dynamical study of the phase transition of the gels in spinodal regimes was described. The evolution of intensity of light scattered from the gels indicated the applicability of Cahn s linearized theory to the phase transition. Our work offers a basis for the determination of diffusion coefficient of gels in their spinodal regimes. 

Deprotonation of OJ-alkylated acetic acid esters (e. g., the propionic acid ester of Figure 13.16) with LDA at-78 °C selectively yields the "if "-enolates. The quotation marks indicate that this application of the term is based on an extension of the /f/X-n omencIature here, the Cahn-Ingold-Prelog priority of the 0 Li substituent is considered to he higher than the priority of the OR group. The deprotonation of the ester shown in Figure 13.16 occurs via the strain-free transition state A. The alternative transition state B is destabilized by a 1,3-diaxial interaction. 

This wetting picture (Fig. 12 [6]) of surface-induced ordering in block copolymer melts has been considered recently by Milner and Morse [60]. They considered the transition from the state of weak surface-induced order (Fig. 12a) to the case of strong-surface induced order (Fig. 12b) and pointed out that typically a first-order transition may occur between these states, in analogy to the "prewetting transition" first proposed by Cahn [226] (Fig. 14a). This prewetting-type first-order transition may persist in a thin film (Fig. 14b), but it ends in a triple point where the surface excess ( ) is still finite, of course, since no diver-... 

Fig. 19. From the intersections in this figure, the diffusion coefficients were determined and are presented in Fig. 20. By extrapolating the D pp value to the horizontal axis, the spinodal temperature can be determined. Applying this procedure, we could determine a spinodal temperature of 34.2 °C which was sHghtly higher than the phase transition temperature of 34 °C. Thus, it was verified that the initial sta of the phase separation with the NIPA gel was expressed by Cahn s linearized theory.

As already stated in the discussion of [2.5.30] the two terms on the r.h.s. of [2.6.5] are dependent. Reduction of the square gradient contribution x(dx/dz) to y can only be achieved at the expense of Introducing more material to the transition zone at non-equilibrium composition, which would increase A/(x). Equilibrium is attained if the r.h.s. of [2.6.5] is a minimum. Cahn and Hillicird demonstrate that this equilibrium is attained if the two terms are identical... 

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