Symmetric binary bulk mixtures

We begin the discussion with bulk mixtures, which shall serve as a reference for confined binary mixtures to be discussed below in Section 4.7.2. For a more comprehensive discussion of the phase behavior of general bulk 

For higher ab = 0-5, the phase diagram differs qualitatively from the previous one. This can be seen from Fig. 4.14(b) where a bifurcation appears (i.e., at a triple point) for ptr — —2.25 and Tjr 1.075 at which a gas phase coexists simultaneously with both a mixed and a demixed fluid phase. Consequently a critical point exists (peb — -2.25, Tcb — 1.15) at which the line of first-order transitions between mixed liquid and gas states ends. The line of first-order transitions involving mixed and demixed liquid states ends at a higher temperatinre and chemical potential of ptri — —2.00 and Ttri — 118, and the A-line is shifted toward lower temperatures as one can see from the plot in Fig. 4.14(b). This type of phase diagram comports with the one shown in Fig. 1(b) of Wilding et al. [87]. 

A further slight increase of eab to 0.56 does not cause the phase diagram to change qualitatively but quantitatively firom the previously discussed case. This can be seen in Fig. 4.14(c) where for sab = 0.56 the triple point is shifted to a lower temperature and chemical potential compared with ab = 0.50. Likewise, the line of first-order tran.sit.ions between gas and mixed liquid appears at lower chemical potential but is somew hat longer because the critical point is elevated to a higher Tcb — 118. The opposite is true for the coexistence between mixed and demixed liquid phases as one can see from Figs. 4.14(b) and 4.14(c). 

Eventually, as ab becomes sufficiently large, first-order transitions be- 

In the limit ab = 10, the symmetric binary mixture degenerates to a pure fluid. In this case Tcep — 0 and the A-linc becomes formally indistinguishable from the /r-axis (and therefore physically meaningless). The remaining coexistence line /Xxb = -3 = /icb (i e-, the phase diagram) involving gas (G) and liquid phases (L) becomes parallel with the T-axis and ends at the critical point where Tcb = as expected for the bulk lattice gas [16] [see, for example, Fig. 4.12(a)]. 

Eventually, as eab becomes sufficiently largo, first-order transitions bc- 

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Fig. 19. Monte Carlo result for the phase diagrams of an off-lattice bead rod model of a symmetric binary polymer mixture with N=20, in the plane of variables reduced temperature T =kBT/ AA and volume fraction of component A, denoted here as xx. Data are for bulk systems (full dots), and for confined films of thicknesses D=10.5a (squares) and 5a (triangles), respectively. Dashed curves represent fits to xx—xlc oc T-Tc fil, where the Ising model exponent [229,230] was chosen as Pj=l/3. From Kumar et al. [39]...

In a large part of what we have discussed above, we considered binary polymer mixtures. However, the situation is somewhat different, if instead of polymer blends, thin films of block copolymers are investigated. Due to the molecular connectivity of the different blocks, the inherent length scale is now determined by the size of the molecules. Early experiments focussed on the thin film morphology in symmetric diblock copolymers, where surface interactions tend to orient the block copolymer lamellae parallel to the boundary surfaces. In contrast to most bulk specimens, the planar interfaces lead to the formation... 

It may be established on the basis of the results described herein that the multimolecular character of adsorption from binary hquid mixtures is rapidly enhanced as the point of demixing is approached. Layer thickness itself, however, is less dependent on the polarity of the surface than on the properties of the bulk phase. Layer thickness has a maximum at a composition of X] 0.5 which is a direct consequence of the symmetric character of the lattice model, assuming molecules of identical size. The maximum of layer thickness will probably be shifted in the case of mixtures of components consisting of molecules of various sizes or with nonspherical shapes, or asymmetric in other respects. 

The implications for films cast from mixtures of enantiomers is that diagrams similar to those obtained for phase changes (i.e., melting point, etc.) versus composition for the bulk surfactant may be obtained if a film property is plotted as a function of composition. In the case of enantiomeric mixtures, these monolayer properties should be symmetric about the racemic mixture, and may help to determine whether the associations in the racemic film are homochiral, heterochiral, or ideal. Monolayers cast from non-enantiomeric chiral surfactant mixtures normally will not exhibit this feature. In addition, a systematic study of binary films cast from a mixture of chiral and achiral surfactants may help to determine the limits for chiral discrimination in monolayers doped with an achiral diluent. However, to our knowledge, there has never been any other systematic investigation of the thermodynamic, rheological and mixing properties of chiral monolayers than those reported below from this laboratory. 

Figure 5 summarizes these effects in a qualitative way for a binary symmetric mixture [13]. Near the critical temperature T, the coexistence curve in the bulk is described by a power law for the concentrations r,4 of A at the coexis-... 

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