Atomistic Modeling of Diffusion in the TiAl Compound

We combine atomistic modeling methods with kinetic Monte simulations to study selfdiffusion in the intermetallic compound L -TiAl. Atomic interactions in TiAl are modeled with a recently developed embedded-atom potential. The vacancy concentration in TiAl is obtained from a lattice gas model of non-interacting point defects. Molecular dynamics simulations are applied to determine vacancy migration mechanisms in the compound. A set of representative vacancy jumps is identified and their rate constants are computed using the harmonic transition state theory with the reaction path established by the nudged elastic band method. The rate constants are used as input to kinetic Monte Carlo simulations performed at several temperatures and alloy compositions. KMC simulations give us self-diffusion coefficients of Ti and Al, correlation factors and other diffusion characteristics. The results are in reasonable agreement with experimental data. We conclude that our methodology provides a viable approach to diffusion calculations in ordered intermetallic compounds. Introduction Titanium aluminide TiAl is a technologically important structural material due to its hightemperature applications in the aerospace industry. It crystallizes in the L structure containing alternating Ti and Al layers parallel a plane. Atomic diffusion has a significant impact on many properties of TiAl, including kinetics of phase transformations, creep resistance, oxidation resistance, etc. Ti self-diffusion, interdiffusion and diffusion of various impurities in TiAl have been studied over the recent years (see [1] for a review). It has been established by Ti radiotracer measurements that diffusion parallel to the -axis (i.e. normal to layers) is about a factor of ten slower than along layers [2, 3]. No reliable experimental data are available for Al self-diffusion due to the lack of a suitable radiotracer. Al diffusion in TiAl has only been estimated by inverting the Darken-Manning equation using interdiffusion and Ti self-diffusion data combined with experimental thermodynamic factors [1]. On the theoretical side, barriers of typical vacancy jumps in TiAl have been calculated with an embedded-atom method (EAM) potential and a preliminary evaluation of several plausible diffusion mechanisms has been made [1, 4]. Those barriers were also employed to explain the diffusion anisotropy in TiAl through vacancy jump correlations [2, 5]. Predictive capabilities of the previous calculations were limited for a number of reasons. The EAM potential used in [1, 4] was not reliable enough to produce more than rough estimates. Point-defect formation entropies were neglected and the frequency factors of all vacancy jumps were assumed to be identical. The diffusion mechanisms examined in [1, 4] were selected from geometric considerations or by analogy with other compounds. Perhaps more importantly, the relative contribution of different diffusion mechanisms was assessed from their estimated activation energies without calculating the absolute values of the relevant diffusion coefficients. The latter would require the incorporation of jump correlation effects, which Defect and Diffusion Forum Online: 2005-04-30 ISSN: 1662-9507, Vols. 237-240, pp 271-276 doi:10.4028/www.scientific.net/DDF.237-240.271 © 2005 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications Ltd, www.scientific.net. (Semanticscholar.org-13/03/20,21:57:18) was not done properly in the previous work. In this paper we revisit diffusion in TiAl with a more advanced approach based on more accurate atomistic simulations combined with the kinetic Monte Carlo (KMC) method. This approach allows us to analyze diffusion in TiAl in terms of absolute values of diffusion coefficients and compare them with experimental data. Methodology Point-defect concentrations. Atomic interactions in TiAl are described with an EAM potential fit to a large database of experimental and first-principles data [6]. This is believed to be the most accurate EAM potential currently available for this compound. In calculating the equilibrium vacancy concentration, we take into account that vacancies can occupy either of the two sublattices (V and V ) and are in dynamic equilibrium with antisites on both sublattices (Ti and Al ). We thus use a lattice-gas model in which the point defects are teated as a four-component gas in equilibrium with respect to all possible defect reactions [1]. The equilibrium defect concentrations are determined by numerically solving a set of three independent mass-action law relations augmented with a material conservation equation. The input data for the calculation are the free energy of the perfect lattice and free energies of the individual point defects V V Ti and Al . The energies and are obtained by molecular statics while the entropies and are calculated within the classical harmonic approximation [6]. This procedure allows us to find the total vacancy concentration as a function temperature and the alloy composition. The compositional dependence of turns out to be relative weak. The temperature dependence of does not follow the Arrhenius Law exactly but can be approximated by the Arrhenius relation in a chosen temperature interval to give the “effective” vacancy formation energy. It should be mentioned that is orders of magnitude smaller than both antisite concentrations. Vacancies prefer the Ti sublattice except in Al-rich compositions with at.% Ti. Diffusion mechanisms. To study diffusion mechanisms, molecular dynamics simulations (MD) are run on a perfect-lattice block containing a single vacancy. The computer code automatically saves a snapshot whenever large atomic displacements point to a possible vacancy jump. By examining multiple snapshots, which we combine into a movie, we are able to determine typical diffusion mechanisms of the vacancy. The MD simulations have been performed at temperatures , "! and K. The diffusion mechanisms observed do not depend on temperature and can be summarized as follows. A Ti vacancy prefers to move along Ti layers by exchanging with either Ti atoms or Al antisites. It occasionally jumps to an adjacent Al layer (thus leaving an Al antisite behind) but such jumps tend to be quickly reversed. In cases where they are not reversed, the vacancy typically makes one jump on the Al layer and returns to its latest position in the Ti layer, thus eliminating the Al antisite. This correlated jump sequence, identified in [1] as the three-jump cycle mechanism, results in an exchange of two Al atoms on the Al sublattice due to a Ti vacancy. An Al vacancy is observed to move along Al layers with occasional jumps to a neighboring Ti layer. Three-jump cycles of Al vacancies, although geometrically possible [1], have not been observed in the MD simulations. No six-jump cycles of either Ti or Al vacancies have been found. In contrast to B2-NiAl where vacancies can make collective jumps involving two atoms [7], only single-atom exchanges with a vacancy are observed in TiAl. As the vacancy moves, it first produces antisites, some of them are later eliminated, and the crystal approaches point-defect equilibrium. The presence of scattered antisites at later stages of the simulation does not give rise to any new diffusion mechanisms. Overall, the picture of diffusion can be described as correlated vacancy motion both along and between layers, with interlayer jumps resulting in the production or elimination of antisites. Jump correlations are significant but not strong enough for a domination of cycles. KMC catalog. To prepare catalog-based KMC simulations, we have identified a set of the most typical vacancy jumps. They include both intraand inter-sublattice jumps of Ti and Al vacancies, in each case for exchanges with either a regular atom or an antisite. Furthermore, we have taken into 272 Diffusion in Materials DIMAT2004