A thermodynamic interpretation of the size-ratio limits for laves phase formation
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ation is closely related to the type of bonding. The enthalpy of formation of an intermetallic compound is usually negative because the free energy change must be negative and the entropy-temperature product is small. For this reason, the magnitude of the enthalpy of formation of a compound can be used as a measure of its stability. The enthalpies of formation of Laves phases are often experimentally determined using calorimetry (including direct-reaction calorimetry, solution calorimetry, etc.).[18,19,20] Electromotive force (emf) measurements and vapor pressure measurements are also used for estimating DH values of Laves phases; however, DH values obtained using these methods are generally less reliable than those determined by calorimetry. Several approximation methods on the basis of empirical or semiempirical models have been proposed to estimate the enthalpies of formation of Laves phases, including Pasturel’s and Miedema’s methods.[19,21] The atomic size factor in controlling the occurrence and stability of Laves phases is well established, i.e., the ideal atomic size ratio for Laves phase formation is 1.225, with a range of 1.05 to 1.67 typically observed.[3] The atomic size effect is clearly demonstrated in Figure 1, where the numbers of binary Laves phases are plotted as a function of atomic size ratio, RA/RB, where RA and RB are the metallic atom radii of A and B atoms, respectively. The atomic radii for a coordination number of 12[22] are used in this article, consistent with the geometric analyses of other studies on the stability and occurrence of Laves phases. The number of Laves phases at different RA/RB is calculated using the list of binary Laves phases in Reference 23. In general, the number of compounds is increased as RA/RB is relatively close to the ideal ratio, and all the Laves phases exist in the RA/RB range of 1.05 to 1.67. This range for Laves phase formation was previously noted by Berry and Raynor,[24] Dwight,[25] and Nevitt.[26] Outside this range, no Laves phases are formed. The present survey of the currently available data for binary Laves phases, as plotted in Figure 1, does not change the atomic size-ratio limits (1.05 to 1.67) for the Laves phase formation. The fact that Laves phases can only be stabilized within an RA/RB range of 1.05 to 1.67 is obviously related to the geometric packing condition in the Laves phases. However, a systematic analysis of thermodynamic information, especially enthalpies of formation of Laves phases, may offer a new approach to understand this size-ratio effect in controlling Laves phase formation. A complete literature survey has been undertaken to obtain all the data on the enthalpies of formation of Laves phases.[27] The effect of atomic size ratio, RA/RB, on the enthalpy of formation of binary Laves phases, with the C15, C14, and C36 structures, is shown in Figure 2. Interestingly, the enthalpies of formation for binary Laves phases vary significantly, i.e., from 0 to about 2350 kJ/ mole (kJ/mole denotes kJ per mole of molecules here), indicating c
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