First‐Principles Calculations of the Thermal Expansion Path of the Co₇Mo₆ μ‐Phase

Abstract

The thermal expansion path is calculated for the Co7Mo6 compound, with the structure of μ‐phase, from T = 0 to 1500 K which is below its decomposition temperature. The method allowing to reduce the problem to a 1D case and considering the energy as depending on only one variable, the volume, is used. The thermodynamic properties of Co7Mo6 are studied using first‐principles‐based quasiharmonic Debye–Grüneisen approximation and by comparing the free energies calculated along different paths of thermal expansion. The electronic, vibrational, and magnetic energy contributions to the free energy are accounted. The thermal expansion of Co7Mo6 is not isotropic. The influence of electronic, magnetic, and vibrational entropy on the stability of Co7Mo6 is analyzed. The isobaric heat capacity, thermal expansion, elastic constants, bulk modulus, sound wave velocities, and Debye and Curie temperatures are reported. The calculated results are analyzed and are in agreement with the available experimental and theoretical data.

Thermal expansion anisotropy of Fe23Mo16 and Fe7Mo6 μ-phases predicted using first-principles calculations

“…The lattice parameters of the stable Fe 23 Mo 16 m-phase calculated at elevated temperatures correspond well with the experimental data. 12,13,16 In another theoretical work, 25 where a thermal expansion of a stable Co 7 Mo 6 m-phase was studied using the same approach, good correspondence with experimental data is also observed. Thus, the STEP approach may adequately describe the thermodynamics of the metastable m-phase.…”

“…Currently, a large number of materials are known to have NTE, a review of this phenomenon in functional materials is given in the study by Chen et al 44 The studies [24][25][26] using the STEP method provide the following results: (i) The shape of the energy surface DE(a, c) of a compound reflects possible directions of thermal expansions. (ii) The Debye temperatures calculated along different thermal pathways may give us an insight as which trajectory has more chances of being the most energetically favorable pathways of thermal expansion.…”

“…12,13,16 with those theoretically calculated in this work at T = 0 K, shows that these compounds should have anisotropic thermal expansion. Thus, the approach of searching a thermal expansion path (STEP) [24][25][26] used to predict the thermal expansions of TCP Laves Fe 2 Mo and m-Co 7 Mo 6 phases was employed. The scheme used in STEP is as follows.…”

“…First-principles calculations based on density functional theory (DFT) combined with the quasi-harmonic Debye-Gru ¨neisen (QDG) approximation 28 were used. A new simplified method of searching a thermal expansion path (STEP) 24,25 was employed in this work. This method allows simplifying calculations by reducing the problem from a two-dimensional task to a one-dimensional case.…”

“…It was shown in ref. 25 and 26 that accounting the magnetic and electronic subsystems is a necessary condition for the correct thermodynamic description of compounds containing elements with magnetic properties. In addition, the thermal expansion pathways of the μ-phase have not been studied.…”