Phonon-assisted diffusion in bcc phase of titanium and zirconium from first principles

Sara Kadkhodaei and Ali Davariashtiyani

Phys. Rev. Materials 4, 043802 – Published 13 April 2020

Abstract

Diffusion is the underlying mechanism for many complicated materials phenomena, and understanding it is basic to the discovery of novel materials with desired physical and mechanical properties. Certain groups of solid phases, such as the bcc phase of IIIB and IVB metals and their alloys, which are only stable when they reach high enough temperatures and experience anharmonic vibration entropic effects, exhibit “anomalously fast diffusion.” However, the underlying reason for the observed extraordinary fast diffusion is poorly understood and due to the existence of harmonic vibration instabilities in these phases the standard models fail to predict their diffusivity. Here, we indicate that the anharmonic phonon-phonon coupling effects can accurately describe the anomalously large macroscopic diffusion coefficients in the bcc phase of IVB metals and therefore yield understanding on the underlying mechanism for diffusion in these phases. We utilize temperature-dependent phonon analysis by combining ab initio molecular dynamics with lattice dynamics calculations to provide an approach to use the transition state theory beyond the harmonic approximation. We validate the diffusivity predictions for the bcc phase of titanium and zirconium with available experimental measurements, while we show that predictions based on harmonic transition state theory severely underestimate diffusivity in these phases.

Received 13 October 2019
Revised 11 January 2020
Accepted 17 March 2020

DOI:https://doi.org/10.1103/PhysRevMaterials.4.043802

Physics Subject Headings (PhySH)

Anharmonic lattice dynamics Anomalous diffusion Phonons Transport phenomena

First-principles calculations

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Sara Kadkhodaei ^* and Ali Davariashtiyani

Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA

^*Author to whom correspondence should be addressed: sarakad@uic.edu

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Issue

Vol. 4, Iss. 4 — April 2020

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Images

Figure 1
DFT harmonic phonon dispersion and density of states (DOS) for a $6 \times 6 \times 6$ supercell of bcc titanium including 215 atoms and a monovacancy (a) at the bcc site and (b) migrated halfway to the nearest neighbor bcc site along the [111] direction. Phonon instabilities are depicted as negative frequencies in the dispersion curves. The atomic configuration for the initial and intermediate states of vacancy diffusion are illustrated in (c) and (d), respectively. A conventional supercell is presented for visualization simplicity.
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Figure 2
Phonon dispersion for a bulk defect-free $6 \times 6 \times 6$ supercell of bcc titanium including 216 atoms at (a) 0 K, (b) 1300 K, and (c) 1600 K obtained using the SCAILD method. (d) The longitudinal phonon branch along the $[ξ ξ ξ]$ reduced wave vector (between H and P high-symmetry points) from 1300 K to 1600 K. The star marks in (d) represent the negative dip on the phonon branch. (e) The trigonal distortion of (111) atomic planes along the [111] direction associates with the eigenvector of $ξ = \frac{2}{3}$ phonon mode along the $Γ$ -P-H branch. A complete collapse of atomic planes 1 and 3 results in the hexagonal $ω$ phase.
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Figure 3
The anharmonic phonon dispersion and density of states (DOS) for (a) a $6 \times 6 \times 6$ supercell of bcc titanium including 216 atoms and (b) bcc Ti with a monovacancy including 215 atoms at 1600 K. (c) The temperature-dependent DOS for bulk bcc Ti and (d) the bcc phase with a monovacancy at different temperatures.
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Figure 4
(a) The vacancy formation enthalpy in bcc Ti as a function of temperature. The circles are the statistical average formation enthalpy obtained from AIMD simulation using the PBE exchange-correlation functional. The crosses are the formation enthalpy values including the explicit intrinsic surface correction term. (b) The formation entropy and (c) equilibrium vacancy concentration versus temperature for monovacancy in bcc Ti. For bulk bcc we used a $6 \times 6 \times 6$ supercell of bcc titanium including 216 atoms and for the defected system we used bcc Ti with a monovacancy including 215 atoms.
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Figure 5
Complete DFT-based prediction of diffusion coefficient for (a) bcc Ti and (b) bcc Zr obtained according to anharmonic phonon analysis (circled curves), validated with experimental measurements (solid curve) extracted from Ref. [40]. Diffusion coefficient for bcc Cr (dashed curve) is also extracted from Ref. [40] for comparison. Calculations based on standard harmonic phonon analysis (crosses) underestimate the diffusion coefficient by several orders of magnitude. $T_{m}$ is equal to 1966 K and 2153 K for Ti and Zr, respectively. All the parameters for calculating the diffusion coefficient are presented in the tables. The internal surface correction terms for vacancy formation and migration enthalpies are presented in parentheses. $ν^{*}$ for each temperature is also represented by star marks in Fig. 2 for bcc Ti and Fig. S6 for bcc Zr.
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