Free energy of solid solutions and phase diagrams via quasiharmonic lattice dynamics

Abstract

We show how a configurational lattice dynamics technique, in which the free energy of a number of configurations is determined directly by means of a fully dynamic structural minimization, can be used to calculate thermodynamic properties of solid solutions and phase diagrams. No assumptions are made as to the nature of the solution and both configurational and vibrational entropy contributions are determined directly. Only a small number of configurations are required. We illustrate the method using MnO/MgO, for which our results support the recent experiments of Wood, Hackler, and Dobson ͓Contrib. Mineral. Petrol. 115, 438 ͑1994͔͒ who, in contrast to previous workers, suggest the formation of a complete solid solution at temperatures only above 1100 K.Solid solutions, alloys, and grossly nonstoichiometric compounds present considerable challenges to the theoretician, as does the calculation of absolute free energy. Ceramic solid solutions in particular are often strongly nonideal and approaches such as the cluster variation method ͑CVM͒, 1 widely used for metallic alloys, often perform poorly. In addition, despite the importance of accurate thermodynamic data for oxide solid solutions in such areas as ceramic fabrication and design and mineralogy, accurate thermodynamic data are rare.We have recently developed a highly efficient method for the fully dynamic structure optimization of large unit cells 2 that uses lattice statics and quasiharmonic lattice dynamics ͑QLD͒. We calculate the full set of free-energy first derivatives analytically, and a full minimization of the free energy with respect to all structural variables for large unit cells is possible. The accurate calculation of the free energy via QLD is quick and computationally efficient and does not resort to lengthy thermodynamic integration. Here we show this can be used for the free energies of solid solutions ͑in-cluding ⌬H mix and ⌬S mix ͒ and phase diagrams. The method is readily extended to elevated T and high P; no a priori assumptions are made regarding the configurational entropy contribution and the vibrational contribution is also evaluated straightforwardly.We illustrate our approach using MnO/MgO, for which not only are there several sets of experimental enthalpy data 3,4 but also conflicting reports of the phase diagram. As shown in Ref. 5, the experiments of Raghavan, Iyengar, and Abraham 6 suggest a consolute temperature, T c , as low as 600 K, whereas the results of Wood, Hackler, and Dobson 7 are consistent with a much larger T c ͑Ϸ1100 K͒ and a markedly asymmetric phase diagram. The data of Ref. 6 are indeed surprising given the CaO/MgO phase diagram, in which there is a large two-phase region, and the mismatch in ionic radii between Mn 2ϩ and Mg 2ϩ , which is substantial although smaller than between Ca 2ϩ and Mg 2ϩ . A further aim of this paper is to attempt to resolve this issue.In principle the solid solution can assume any state, i.e., each atom can be at any position. However, the only states of practical importance away from t...

Monte Carlo simulation of segregation in ceramic thin films: Comparison of the MgO/MnO {100} and {210} surfaces

“…No mean-field approximations are made. We have used this previously in studies of bulk MnO-MgO solid solutions [17] and the resulting phase diagram, calculated using exchange MC and the semigrand canonical ensemble, is in good agreement with experiment with a consolute temperature of E1150 K [18].…”

Configurational lattice dynamics: The phase diagram ofRh−Pd

“…To simulate disordered solid alloys we use the CLD as proposed in Ref. 12. In this method one generates a set of configurations k, in each of which the location of the Rh ͑or Pd͒ atoms on sites within the unit cell is chosen at random.…”

“…Non ideal effects are included in the second term. Details of this method, and its application to ionic solids, were already published 12 and not repeated here.…”

“…In this paper we propose the use of a very different method for calculating phase diagrams of alloys, namely configurational lattice dynamics ͑CLD͒. This method, used previously for ionic solid solutions, [12][13][14] is based on the generation of a large number of different atomic arrangements ͑configurations͒ on crystal lattice sites, followed by quasiharmonic lattice dynamic optimization of each one of them. The optimization gives for each configuration not only the relaxed atomic positions but also its free energy, and hence, its statistical probability.…”

Quasiharmonic free energy and derivatives for many-body interactions: The embedded atom method

“…The free energies of a number of configurations are determined directly by means of fully dynamic structural minimizations, and the thermodynamic properties of RhPd are then evaluated by suitable thermodynamic averaging. We shall see that lattice dynamics thus also provides an efficient route to the free energy of particular configurations of disordered solids, 16,17 and is particularly useful for assessing the importance of vibrational contributions to mixing properties.…”