Effect of carbon vacancies on thermodynamic properties of TiC–ZrC mixed carbides V.I. Razumovskiy a,b,n , A.V. Ruban b , J. Odqvist b , D. Dilner b , P.A. Korzhavyi b a Materials Center Leoben Forschung GmbH, Roseggerstraße 12, A-8700 Leoben, Austria b Department of Materials Science and Engineering, KTH Royal Institute of Technology, Brinellvägen 23, SE-100 44 Stockholm, Sweden article info Article history: Received 18 September 2013 Received in revised form 21 February 2014 Accepted 23 February 2014 Available online 7 March 2014 Keywords: Ab initio calculation Carbides Thermodynamics Metal and alloys Phase diagram abstract Thermodynamic properties of a TiZrC mixed carbide system are investigated by first-principles methods within density functional theory. Carbon vacancies are found to have a significant contribution to the thermodynamics of TiZrC mixed carbides. The temperature effect on the thermodynamic properties of the system is calculated taking into consideration the corresponding electronic and vibrational thermal excitations. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Titanium and zirconium carbides belong to a group of refractory materials characterized by extreme hardness, high wear resistance, and extraordinarily high melting point [1,2]. Phase separation of a mixed carbide (Ti,Zr)C into TiC-rich and ZrC-rich fractions (possibly via a spinodal mechanism) can be used in order to achieve even greater mechanical hardness of the material, see e.g., [3]. Investiga- tion of this phenomenon demands an accurate thermodynamic and kinetic description of the system that represents a quite special case due to the sub-stoichiometric nature of these carbides characterized by a vast number of carbon vacancies which can be as high as 50% [4]. So far very little is known about the effect of the sub- stoichiometry on the basic thermodynamic properties of the system, and therefore an investigation of this effect is of great interest from both physical and materials science points of view. Previous ab initio-based thermodynamic studies of these carbides [5–8] were focused on the stoichiometric compositions without carbon vacancies (V C ). For example, Ivashchenko et al. [5] performed an extensive thermodynamic study of the elastic properties and miscibility gap of stoichiometric pseudo-binary TiC–ZrC alloys. How- ever, the miscibility gap calculated on the basis of stoichiometric enthalpies of formation is overestimated by a factor of two compared to the one observed in the experiments and modeled by the Calphad method [7]. At the same time, Adjaoud et al. [8] performed ab initio based cluster expansion calculations of the TiC–ZrC binodal and found it to be much closer to the experimental data when the vibrational contribution to the free energy was taken into considera- tion. Nevertheless, their results still predict stronger phase separation than what is observed experimentally. This is an indication that the stoichiometric representation of the carbides is not sufficient for an accurate description of the thermodynamic properties of the system. The experimentally obtained carbides always show some deviation from stoichiometry [4] and its effect on thermodynamic properties of mixed carbides has not been well studied. In our previous works on TiC and ZrC [9–12], we found that V C have relatively small (or even negative in the case of TiC) formation energies, which is in good agreement with the experimentally observed off-stoichiometry, and a vast number of vacancies on the carbon sublattice, and have a great impact on the thermodynamics of point defects implicating that V C strongly affect thermodynamics of TiC–ZrC mixed carbide system as well. The main aim of this paper is to investigate the effect of structural V C on the basic thermodynamic properties of the mixed carbide system. 2. Methodology The results of the present study were obtained using ab initio calculations of the electronic structure and energetic properties of TiC–ZrC random alloys in the NaCl crystal structure. The calculations Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/calphad CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry http://dx.doi.org/10.1016/j.calphad.2014.02.005 0364-5916 & 2014 Elsevier Ltd. All rights reserved. n Corresponding author at: Materials Center Leoben Forschung GmbH, Roseggerstraße 12, A-8700 Leoben, Austria. E-mail address: razvsevol@yahoo.com (V.I. Razumovskiy). CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 87–91