A comprehensive diffusion mobility database comprising 23 elements for magnesium alloys

Abstract

Reliable experimental diffusion coefficients of 10 key alloying elements in Mg obtained by the present authors together with experimental data in the literature enabled us to perform a systematic test of the reliability of diffusion coefficients obtained from DFT calculations. The computed activation energy values were found to be quite accurate (mostly within 0.2 eV) but the computed pre-factors were less reliable. Such insights allowed us to develop a practical and yet robust strategy to perform diffusion mobility assessments by adopting the computed activation energy while fitting only the pre-factor when available experimental data are limited to a narrow temperature range. The overall good agreement between the DFT data and experimental data also gave us the confidence to employ the computed data for those that were still missing or inaccessible from experimental measurements. A systematic assessment of both the measured and computed diffusion data in hcp Mg was performed using the above holistic approach to yield the most comprehensive open Mg mobility database to date, comprising 23 elements (Mg, Ag, Al, Be, Ca, Cd, Ce, Cu, Fe, Ga, Gd, In, La, Li, Mn, Nd, Ni, Pu, Sb, Sn, U, Y, Zn). This more reliable mobility database will contribute to future development of advanced Mg alloys. The holistic approach developed in this study will be very beneficial to the future establishment of reliable mobility databases for other alloy systems as well.

Introduction

Lightweight magnesium alloys have significant potential to reduce vehicle weight and thus increase fuel efficiency in the transportation sector to reduce greenhouse gas emissions. Mg alloys have already found applications in vehicles, but at a much lower volume in comparison with well-developed aluminum alloys. There are substantial efforts worldwide in developing Mg alloys with improved properties aiming at much enhanced applications in vehicles [1], [2], [3], [4], [5].

The Integrated Computational Materials Engineering (ICME) approach has now been widely employed to accelerate the development of new materials and improved processes [6]. The CALPHAD (CALculation of PHAse Diagram) approach is one of the powerful ICME tools for materials design and innovation [7]. Commercial software packages such as Thermo-Calc, Pandat and FactSage have thus been developed and widely used worldwide. Both thermodynamic databases and kinetic (mobility) databases are essential foundations for the use of the CALPHAD approach for alloy design and process optimization. Diffusion coefficients (atomic mobilities) are essential for understanding and simulating processes such as solidification and precipitation as well as properties such as creep resistance. The thermodynamic database for Mg-based systems is relatively well developed [8]; however, only one preliminary Mg mobility database based on limited experimental data published in 2014 [9] is available up to now in the open literature. On the other hand, the commercial MOBMG1 Mg-alloy mobility database [10] from Thermo-Calc Software was released in 2015 and has not been updated since then.

Systematic experimental diffusion measurements of 10 key alloying elements in Mg have been performed in the past few years [11], [12], [13], [14], [15] (including Nd diffusion data to be reported in this study) by the present authors, providing reliable experimental diffusion coefficients as new inputs to the establishment of a more reliable Mg mobility database. In the last few years, the density-functional theory (DFT)-based first-principles calculations [16], [17], [18] have also contributed additional computed dilute diffusion coefficients in Mg. A systematic analysis of all the experimental and computed diffusion data in Mg is thus performed to assess the mobility of 22 alloying elements in Mg in order to establish the most comprehensive mobility database to date. This mobility database will contribute to future development of advanced Mg alloys using the ICME approaches. In addition, our approach to integrate experimental and computational data is an example for future establishment of diffusion mobility databases for other alloy systems.

Self-diffusion coefficients, impurity (dilute) diffusion coefficients, tracer diffusion coefficients and interdiffusion coefficients are usually the types of diffusion data reported in the literature. Various experimental techniques have been applied to measure the diffusion coefficients in Mg. Tracer experiments were employed to determine the impurity/tracer diffusion coefficients as well as self-diffusion coefficients of several elements; unfortunately, only a few research groups nowadays have maintained the capabilities to perform such tedious and costly experiments. Moreover, the tracer method is not applicable to some elements such as Al and Ca since no suitable radioactive isotopes are readily available. Secondary ion mass spectrometry (SIMS) [19] was used to obtain Mg self-diffusion coefficient and Al impurity diffusion coefficient in Mg. However, the sputter induced roughening adds an error source and uncertainty of data obtained using this technique.

Diffusion couples [20] and more efficient diffusion multiples [21] are extensively used to obtain diffusion coefficients, especially the composition-dependent interdiffusion coefficients. Such experiments were also employed to extrapolate the impurity diffusion coefficients at dilute solute compositions. Diffusion couples and multiples are usually made using pure elements; and the diffusion anneal temperatures must be limited below the eutectic points to avoid liquid formation which would ruin the samples. However, the eutectic points are quite low for many Mg-based binary systems (e.g., 437°C for Mg-Al, 341°C for Mg-Zn, 203°C for Mg-Sn) and the annealing time would be very long to allow sufficient diffusion at low temperatures for reliable measurements. Moreover, pure elements such as Ca, Li and rare earth elements are not suitable to make such pure to pure diffusion samples due to their high chemical reactivity or/and tendency to oxidize. Mg-rich single-phase binary alloys (e.g. Mg-Al and Mg-Zn) can be used to assemble diffusion couples against pure Mg to avoid the low eutectics, but such alloys are difficult to make for some elements with very limited solubility in Mg, such as Ce and Ca. A novel and elegant liquid-solid diffusion couple (LSDC) approach [11] was developed recently and it intentionally takes advantage of the liquid formation to study the diffusion of 10 elements in Mg at elevated temperatures above the eutectic temperatures.

Altogether, experimental measurements of diffusion coefficients of a total 23 elements in Mg have been performed using the aforementioned approaches in past decades and the studies up to date are summarized in Table S1 ([11], [12], [13], [14], [15], [19], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48]) and Table S2 ([11], [12], [13], [14], [15], [22], [23], [32], [35], [36], [37], [43], [49]) in the Supplementary Information, including our own measurements of 10 elements. It is noted that the impurity diffusion of Al and Zn in Mg and the self-diffusion of Mg were studied much more extensively than the other elements. The interdiffusion coefficients in hcp Mg were measured for only 10 elements but the composition range is very limited for Ca, Ce, Mn and Nd (< 0.5 at.% in Mg). Furthermore, only two ternary diffusion couple studies are available for the hcp Mg-Al-Zn and Mg-Al-Sn systems [22,23]. A tracer study was conducted to extract the Zn tracer diffusion coefficients in hcp Mg-Al alloys [24]. All these experimental results were employed to the establishment of the new mobility database.

Since experimental measurements of diffusion coefficients in Mg are relatively challenging and only a number of measurements have been performed, especially before our systematic measurements of 10 alloying elements in recent years, first-principles calculations based on DFT have been employed to compute the dilute (impurity) diffusion coefficients of various elements in Mg to address the shortage of diffusion data for Mg alloys. Ganeshan et al. first computed the self-diffusion coefficient of Mg [50] and impurity diffusion coefficients of 4 solutes (Al, Ca, Sn, and Zn) in Mg [51] using the 8-frequency model. The agreement between their computed data and experimental results is understandably not the best since their study was the initial attempt of its kind, Fig. 1(a). Zhou et al. subsequently computed the diffusion coefficients of 48 elements in Mg using the 8-frequency model [16] with an improved exchange-correlation functional, PBEsol in DFT. Later Zhou reported additional diffusion data of 15 rare earth elements in his PhD dissertation [52]. Zhou's calculation achieved much better agreement with the experimental results, Fig. 1(b). At the same time, Wu et al. also predicted the diffusion coefficients of 50 elements in Mg using the 8-frequency model and further applied a correction based on the difference between the experimental and computed values of Mg self-diffusion coefficients [17]. Their computed results show impressive overall agreement with the experimental results, Fig. 1(c). Agarwal and Trinkle [53] revealed that the 8-frequency model is incomplete in its approximations of the vacancy transition states and the correlation factors since it does not include the drag ratios of the solutes. They computed the diffusion coefficients of 62 elements in Mg by employing a Green function approach as well as the 8- and 13-frequency models [18]. The calculation results from Agarwal and Trinkle are quite good, but still show appreciable discrepancies with experimental values for several elements, Fig. 1(d), albeit they did not do an artificial correction as Wu et al. have done.

The four first-principles calculations are compared to the available experimental diffusion data in Fig. 1 in which the dashed diagonal lines represent a perfect agreement between the computed and experimental data. Lines parallel to the dashed diagonal lines represent a perfect agreement between the computed and experimentally measured activation energy (this is not intuitive but true). One can clearly see from Fig. 1 that the first-principles calculations predict the diffusion activation energy values well (as represented by the slope of the lines) except for a few elements such as Fe, Ni and Cu. Such a comprehensive comparison allows us to leverage the computed activation energy values in mobility assessments when experimental measurements are limited or missing.

The distance between each line and the dashed diagonal line in Fig. 1 represents the deviation of the pre-factor (D₀) between the computed and experimentally measured diffusion coefficients. One can clearly see from Fig. 1 that the diffusion pre-factors predicted from first-principles calculations are not as reliable as the activation energy values. For this reason, experimentally measured diffusion coefficients are employed to fit the pre-factor only when the experimental data are available only over a narrow temperature range which would not allow accurate evaluation of the activation energy. Our experimental measurements of 10 elements in Mg enabled higher confidence in the experimental values for such comprehensive comparisons with the computed values for Mg alloys.

The available information of thermodynamic data, experimental diffusion data and computational diffusion data in the literature for 33 elements, including 23 elements that will be assessed in this study, is summarized in Table 1 and marked with different colors. These elements are classified into 7 groups based on the availability information. There are 29 elements in Groups I-V that are included in the commercial thermodynamic database TCMG5 from the Thermo-Calc Software company [54]. The experimental diffusion coefficients in hcp Mg are reported for 23 elements in Groups I-IV and Groups VI-VII. Binary interdiffusion coefficients are available for 10 elements only of Groups I-III. Ternary interdiffusion coefficients were reported only for the Mg-Al-Zn and Mg-Al-Sn systems. The experimental self-diffusion coefficients of 7 hcp elements (Mg, Zn, Y, Zr, Er, Be, Cd) were reported and used for the current mobility assessment. First-principles calculations on the dilute diffusion coefficients in hcp Mg covers the elements in Group I-VI. The self-diffusion coefficients in the hcp structure were computed for the elements in Groups I-VI except for Sb and Si. An Mg mobility database containing 23 elements is thus established based on the integration of available experimental diffusion data and supplementary computational data. It also includes additional 10 elements of Group V that are solely based on computed data from DFT. It should be noted that this Mg mobility database covers all 16 elements in the preliminary Mg database [9] and 22 of 24 elements (except K and Na which are not usual alloying elements in Mg alloys) in MOBMG1 Mg-alloy mobility database [10] from Thermo-Calc Software. In addition, our database is developed based on a holistic integration of comprehensive experimental and computational data, including those reported after the preliminary database was published and MOBMG1 was released.