Double tangent construction

Figure 7.15 Illustrations of the Gibbs double-tangent construction for finding coexistence regions of G]iq(x), Gso1(jc) in the case of (a) a single crossing or (b) double crossing.

Fig. 6.31 Results from SCFI calculations for diblock/homopolymer blends (Matsen 1995b). (a) The dimensionless Helmholtz free energy Fu() as a function of homopolymer volume fraction at y X = 12, / = 0.45 and /3 = The dashed line shows the double tangent construction used to locate the binodal points denoted with dots. The dotted line is the free energy of non-interacting bilayers, (b) Phase diagram obtained by repeating this construction over a range of %N. The dots are the binodal points obtained in (a), and the diamond indicates a critical point below which two-phase coexistence does not occur. The disordered homopolymer phase is labelled dis, and the lamellar phase lam.

The function tp p, T) will typically have the shape illustrated in figure 2.13 below the critical point of the fluid it will have two minima and a double tangent construction defines the two coexisting phases, one liquid and one gas. 

Figure 2.13. The excess free energy density rp p, T) as a fimction of the density p for a fluid below its critical point. The double tangent construction defines the equilibrium densities of gas and liquid phases.

II. This is the equivalent, in the ii, p-plane, of the equal-areas construction in the p, u-plane shown in Fig. 1.8, and is the once differentiated version of the double-tangent construction in Fig. 3.1. 

Fig. 6. Schematic representation of the surface excess free energy Aa as a function of density. The double tangent construction defines the densities of the equilibrium coexisting phases, pgas and piiquid- The slope of the double tangent construction is equal to the chemical potential m. Ao for a particular density is defined as the difference between the free energy density curve and the double tangent line.

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