Tangent construction

Transition pressures were determined by equating enthalpies of different polymorphs via the common tangent construction. We found that ratile transforms to columbite structure at 11.8 GPa with a small volume change of 3%. These results agree well with experimental... 

In Fig. 7.83 using the common tangent construction the equilibrium compositions of phase I-phase II at their boundary are found, from the points of contact, to be respectively Xg — A, and Xg = X2. 

Figure 4.2 Gibbs energy curves for the liquid and solid solution in the binary system Si-Ge at 1500 K. (a) A common tangent construction showing the compositions of the two phases in equilibrium, (b) Tangents at compositions that do not give two phases in equilibrium. Thermodynamic data are taken from reference [2],...

At an even lower temperature, T, a sample in equilibrium will consist of the crystalline phase h-Y2C>3(ss), MgO(ss) or a two-phase mixture of these (see Figure 4.8(f)). The compositions of the two phases in equilibrium are again given by the common tangent construction. 

Figure 11.17. Parallel tangent construction used to derive the maximum driving force (Gm) to form a from a 7 alloy of composition x. ...

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.

The time required and the difficulty of accurate tangent construction makes triangulation a method that cannot be recommended under any circumstance. 

C(n) is related to v(h) in the same way that the capillarity vector, , is related to 7(ri) and is constructed in the same way. The Wulff construction applied to v(n) produces the asymptotic growth shape. This and other relations between the Wulff construction and the common-tangent construction for phase equilibria are discussed by Cahn and Carter [16]. 

Figure 15.1 Effect of /3-phase particle size on the concentration, Xeq, of component B in the a phase in equilibrium with a /3-phase particle in a binary system at the temperature T. assuming that, 6 is pure B. (a) Schematic free-energy curves for a phase and three /3-phase particles of different radii, R > R2 > Rs. The free energies (per mole) of the particles increase with decreasing radius due to the contributions of the interfacial energy, which increase as the ratio of interfacial area to volume increases, (b) Corresponding phase diagram. The concentration of B in the a phase in equilibrium with the /0-phase particles, as determined by the common-tangent construction in (a), increases as R decreases, as shown in an exaggerated fashion for clarity, (c) Schematic concentration profiles in the a matrix between the three /3-phase particles.

The equilibrium order parameters X g and rjeq minimize AF subject to any system constraints. Supposing that the system s composition is fixed, the method of Lagrange multipliers leads to a common-tangent construction for AF with respect to XB—or equivalently, equality of chemical potentials of both A and B. Two compositions, Xjj and X +, will coexist at equilibrium for average compositions XB in the composition range Xe < XB < -X Bq+ if they satisfy... 

Solution. As the hemispherical tip radius becomes smaller (at constant temperature), the equilibrium concentration, cLS, will decrease. This is demonstrated in Fig. 22.7 by employing the common-tangent construction used in Fig. 15.1a. Furthermore, the... 

Figure 22.7 Common-tangent construction showing concentration of B, Xeq, X, and...

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.

Discuss in terms of chemical potentials why phase equilibrium requires the tangent constructions described in this section. 

Prove that the common tangent construction is equivalent to the equality of chemical potentials of the phases whose compositions are given by the points of tangency. 

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