Common-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. 

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...

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. 

The local equilibrium surface alloy configuration and structure may be found by minimization of the surface free energy, or if several different phases may exist, by finding a convex hull of the lowest free energies of different phases at different alloy compositions (at T=0), or more generally by a common-tangent construction which is completely analogous to the usual treatment of the bulk systems. The procedure is illustrated in Fig. 4. 

Fig. 4. Sketch of a common-tangent construction for the surface energy of an alloy, c is the coverage of the deposited material.

Now suppose that intermetallic compounds are suppressed in Fig. 2.16. Consider the metastable equilibrium between an amorphous phase and the terminal solutions of a-Zr (bcc Zr) and Ni. Under these conditions, the common tangent construction yields a metastable equilibrium state which consists of a single-phase a-solution for xZr > x4, a two-phase region of liquid (glass)/a-... 

Between compositions M and N, the system s lowest energy state is given by the common tangent construction. In other words, the system s lowest free energy occurs when two phases (a solid phase and a liquid phase) coexist. 

---