Magnetic-induced tricritical points in alloys
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I.
INTRODUCTION
A S early as 1963, Meijering I predicted the appearance of different types of phase equilibria in binary alloys due to magnetic interaction. More recently Nishizawa, Hasebe, and Ko 2 carried out phase diagram calculations of several binary Fe-X systems including magnetic contribution to the thermodynamic properties of the alloy phases. They found phase separation at low temperatures as a result of the relative interplay of the chemical and magnetic terms of the free energy of the alloy phase. In all the cases, two spinodal curves were reported to exist at the tricritical point, and then extended to lower temperatures. In their review papers, Inden 3 and Miodownik 4 discussed the various possible phase equilibria in alloys due to magnetic interaction. Most recently Allen and Cahn 5 6' discussed the phase equilibria of a binary system in the vicinity of a tricritical point. Using a Landau-type expansion of the free energy in terms of temperature, composition, and order parameters, Allen and Cahn showed that a spinodal curve for the ordered phase exists from the tricritical point down to lower temperatures. But a spinodal curve for the disordered phase is absent in the vicinity of a tricritical point. The unstable region is bound by metastable extension of the second-order phase transition temperature and the spinodal curve for the ordered phase. Yet at lower temperatures, the spinodal curve for the disordered phase does appear as has been found for the fcc (Fe, Ni) phase due to magnetic interaction. 7 In the present study, we will discuss (1) the thermodynamic rationalization for the existence of a spinodal curve for the disordered phase and (2) the relative interplay of the chemical and magnetic contributions to the stabilities of various types of phase equilibria in binary alloys.
causes the formation of a miscibility gap. Figure 1 shows a schematic phase diagram displaying a tricritical point. At temperatures higher than the tricritical point (T2 shown in Figure 1), the phase undergoes a second-order phase transformation from a ferromagnetic state to a paramagnetic state.* At temperatures lower than the tricritical point, the *In the present study, we have assumed the magnetic transformation to be second order.
ferromagnetic phase undergoes a first-order phase transformation to a paramagnetic phase with different compositions. The spinodal curve for the ordered phase sp(o) exists at the tricritical point and to lower temperatures but the spinodal curve for the disordered phase sp(d) does not appear at the tricritical point. It exists only at temperatures lower than the tricritical point (e.g., at T -< T4 shown in Figure 1). We will discuss the appearance of these equilibria in terms of the free energy curve G, the first derivative of the free energy with composition curve, G' = dG/dx, and the second derivative of the free energy with composition curve, G" = d2G/dx 2 for the ordered and disordered phases. Figures 2(a) through (e) show schematically G, G', and G" as a function of composition for
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