Line tension approach

We will resort to the line tension approximation repeatedly since as was noted above, much may be learned about the key features of a given problem on the basis of such arguments, which are largely geometrical. Again, what is especially appealing about the line tension approach is the prospect for making analytic headway on fully three-dimensional problems. 

Following our approach in sec. 5.6, all contributions of nature (ii) are subsumed in the line tension, which has already been discussed. Hence, for the present section only item (i) remains. For the LG and LjL2 phase boundaries we have already treated the corresponding tensions in some detail in chapter 2. There is no need to repeat that. Interpretation of the two tensions involving solids is avoided because these are unmeasurable. The only issue left is how the various models and interpretations work out for a set of three phase boundaries. To that end, let us itemize the various interpretations of chapter 2. Where relevant, it is assumed that is the equilibrium value the SG interface may carry an adsorbate. 

The main criterion here is that the normal component of the pressure must remain constant as the surface line is approached, as required by the mechanics. The deficiency of tension in the transverse tensor should be analyzed. 

Eor the Plateau border we have z > / , 11 -> 0, o -> o = const., and both Equations 5.150 and 5.151 rednce to Eqnation 5.101 with AP = P. The macroscopic contact angle, a, is defined as the angle at which the extrapolated meniscns, obeying Equation 5.101, meets the extrapolated film snrface (see the dashed line in Fignre 5.16). The real surface, shown by solid line in Figure 5.16, differs from this extrapolated (idealized) profile, because of the interactions between the two film snrfaces, which is taken into acconnt in Eqnation 5.150, but not in Eqnation 5.101. To compensate for the difference between the real and idealized system, the line and transversal tensions are ascribed to the contact line in the macroscopic approach. In particnlar, the line tension makes up for the differences in snrface tension and running slope angle ... 

Suresh et al. have argued that the formation of dendritic structures during compression of tetradecanoic acid monolayers at the LE-LC transition is related to the vanishing of the line tension between the phases with the approach to a critical point. They believe that for tetradecanoic acid lies at about 31°C, which, if we apply the rule of thumb that there is a 10 K change in the phase diagram per CHj, would place an LE-LC critical point for PDA at 40°C. If the LC and LE phases do not have the same symmetry, e.g., if the LC phase is an ordered solid, then the coexistence region cannot... 

A further interaction between the dislocation and the solid solution atom is due to the different strength of the atomic bond between the dissolved atom and its neighbours, resulting in a locally changed elastic modulus in the vicinity of the solid solution atom. The line tension of the dislocation thus either increases or decreases when it approaches the atom, causing another obstacle effect known as modulus interaction. 

We start by supplying a thermodynamic definition of line tension [3]. Consider the three-phase system as shown in Fig. 13. Around the line of three-phase contact a cylinder is drawn with length L and radius R, and implicitly we assume that R and L approach infinity. The total free energy inside the cylinder comprises terms of the form pressure times volume, surface tension times surface area, and, finally, line tension times line length... 

Here, S, and Nf are linear excesses, k is the line tension in the detailed approach,... 

In Figure 11-2, line ER connects the composition of an extract phase with the composition of a raffinate phase with which it is in equilibrium. Such lines are known as tie lines. These tie lines move vertically as the compositions of the two phases approach each other, until only a single phase exists (as shown in Figure 11-3). The point on curve JRPEK where a single liquid phase is formed is called the plait point (point P). The interfacial tension approaches zero as the plait point is approached. 

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