Predicting grain boundary energies of complex alloys from ab initio calculations
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Predicting grain boundary energies of complex alloys from ab initio calculations
Changle Li a , Song Lu a, ∗ , Levente Vitos a,b,c a Applied Materials Physics, Department of Materials Science and Engineering, KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden b Department of Physics and Astronomy, Division of Materials Theory, Uppsala University, Box 516, SE-75120 Uppsala, Sweden c Research Institute for Solid State Physics and Optics, Wigner Research Center for Physics, P.O. Box 49, H-1525 Budapest, Hungary
Abstract
Investigating the grain boundary energies of pure fcc metals and their surface energies obtained from ab initio model-ing, we introduce a robust method to estimate the grain boundary energies for complex multicomponent alloys. Theinput parameter is the surface energy of the alloy, which can easily be accessed by modern ab initio calculations basedon density functional theory. The method is demonstrated in the case of paramagnetic Fe-Cr-Ni alloys for whichreliable grain boundary data is available.
Keywords:
Grain boundary energy, Surface energy,
Ab initio , fcc metalsGrain boundary (GB) plays a critical role in the microstructural evolution of polycrystalline materials [1, 2]. Thestructures and energetics of GBs are closely related to various physical and mechanical properties, e.g., alloyingsegregation, precipitation, coarsening, or crack. Particularly, tailoring the properties of GBs by, e.g., controlling theelement segregation / depletion, has been an important strategy for improving mechanical properties of alloys, and itis often referred to as the ’grain boundary engineering’ [3, 4]. The GB geometry is described by the five degrees offreedom (DOFs). For a complete description, three degrees are assigned to the vector relating the misorientation oftwo adjoining grains and the other two describe the GB inclination plane [5]. As a key feature, the GB energy (GBE,or γ GB ) varies significantly with both misorientation and inclination [6, 7], which makes accurate determination of theGBEs a great challenge in both experimental measurements and computational simulations.Experimentally, the mean GBE can be estimated from the geometries of the surface triple junctions or internalGB triple junctions in zero-creep experiments [8]. Often, it is the relative GBE or the GBE anisotropy that aredetermined [9, 10]. For example, measuring the groove angle at the equilibrium junction composed of grain boundaryand surface, the GBE to surface energy ratio can be determined [11–13]. Assuming that the surface energy is known,the GBE can be determined. Similar procedure has been applied for the evaluation of the low- and high-angle GBEs, aswell as the twin boundary energies [14–17]. Unfortunately, the surface energy itself is a quantity di ffi cut to determine ∗ Corresponding author
Email address: [email protected] (Song Lu)
Preprint submitted to Elsevier February 4, 2021 ccurately by experiments, especially at low temperatures. Most of the available room temperature experimentalsurface energy were obtained indirectly by extrapolating the surface tension measured in the liquid phase [18–20].The GBE can also be determined using di ff usion data. Borisov [21] proposed a semi-empirical relationship betweenthe increase of the self-di ff usion in the GB relative to the bulk and the absolute GBE. This method was examined byPelleg [22] and a satisfactory agreement between directly measured GBEs and the calculated ones was reported for afew close-packed metals. The same method has been applied to alloys (e.g., Au-Ta alloys [23, 24] and Ni alloys [25]).Despite the existing experimental methods for GBE determination, it is cumbersome to perform these measurementsconsidering the large number of GB types and their composition, temperature, and magnetic state dependences.Alternatively, GBs can be investigated by atomistic simulations using empirical potentials or density functionaltheory (DFT) calculations. Potentials using embedded atomic method (EAM) applied for the GB studies in puremetals [26–31] led to improved understanding of the atomic structures and their energetics. For example, Holm etal. [32, 33] calculated the GBEs for a large number of GBs in pure face-centered cubic (fcc) metals and found that theGBEs in di ff erent materials are strongly correlated. Olmsted et al. [33] showed that the GBEs are more influenced bythe grain boundary plane than the misorientation angle. Studies based on DFT are usually more accurate and have abetter predictive power as compared to methods based on empirical potentials. On the other hand, DFT calculationsare often limited to pure metals or simple alloys and to low-index coincidence site lattice types of GBs due to theextensive computational burden [34–36].In the present work, we adopt DFT calculations to study the correlation between the GBEs of various fcc metalsand between the GBEs and the surface energy. We show that the GBEs in a pair of fcc metals are strongly correlatedvia a material dependent factor ( δ ). This parameter can be estimated from the ratio of the low-index surface energies.Considering that DFT methods for studying the surfaces of pure metals [37, 38] and alloys [39] have readily beenestablished, the present development puts forward a robust method for predicting the GBEs of complex alloys usingtheir surface energies.Ten types of the [1¯10] tilt GBs in ten fcc metals (X = Al, Cu, Au, Ag, Ni, Pd, Pt, Co, Rh, and Ir) were calculatedby the Vienna ab initio
Simulation Package (VASP) [40] using the projector augmented wave (PAW) method [41].The atomic structures of these GBs are presented in Table S1 and Fig. S1 in the Supplementary Material (SM). Forthe exchange-correlation funtional we adopted the generalized gradient approximation parameterized by the Perdew,Burke, and Ernzerholf (PBE) [42]. The k -point meshes were carefully tested to ensure the convergence of the GBEswithin ∼ / m . Cuto ff energies were set to 500 eV for all metals. Full geometry relaxation was performed and theconvergence criteria for electronic energy and force calculations were 10 − eV and 0.02 eV / Å, respectively.In Fig. 1(a), we show the calculated γ GB for Cu as a function of the [1¯10] tilt angle θ , in comparison with the2revious DFT values [34, 43–49] as well as the EAM results [7]. The available experimental GBEs at high temperatureare also included. The DFT results agree well with each other, showing the typical shape of the γ -surface for the [1¯10]tilt GBs with two energy minima located at Σ θ = . Σ θ = . γ -surfaces show the same shape [7, 50] although the absolute values are somewhat smaller thanthe DFT results. We associate the deviations between the DFT and experimental values to thermal e ff ects, which wereneglected in the DFT calculations, and to the fact that the experimental values are relative results [50]. In Fig. 1(b),we compare the present GBEs for all fcc metals with the available DFT results [34, 43–49, 51–66] (numerical valuesare listed in Table S2 in SM). Overall, the present GBEs have an excellent agreement with the former DFT values,with R = .
97 and a standard error of ∼ / m . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) G B E G B ( J / m ) Tilt angle ( )
Present work EAM, Bulatov et al., [7] Zheng et al., [34] Expt. (1273 K), Miura et al., [50] Tanaka et al., [43] Olsson et al., [44] Jiang et al., [45] McKenna et al., [46] Widom et al., [47] Shibutani et al., [48] Ishibashi et al., [49] ( ) (a) R e f s . G B ( J / m ) Pres.GB (J/m ) R =0.970 (b) Fig. 1. (Color online) Theoretical and experimental GBEs for ten selected fcc metals. (a) Comparison between the present and previous γ GB values for Cu as a function of the [1¯10] tilt angle θ . The previous DFT (solid symbols) [34, 43–49], EAM (squares) [7] and the high temperatureexperimental (squares with plus) [50] data are indicated. On the top of the figure, we show the ten GBs for which the present ab initio calculationswere performed. (b) Comparison between the present ( γ Pres . GB ) and previous ( γ Refs . GB ) [34, 43–49, 51–66] DFT results obtained for the ten selectedmetals (shown in the legend) and ten tilt angles (not shown). Numerical values are listed in Table S2 in SM. The R value obtained for the linear fitconsidering all selected metals shows a high degree of correlation. Starting from the present DFT results for the GBEs, we examine the correlation between the GBEs in di ff erentmetals. Here, we choose Cu as the reference and compare the GBEs of the same GB structure in di ff erent materials.We notice however that choosing another metals as reference leads practically to the same conclusions. Results forfour elements, Al, Ni, Pd, and Pt, taken as examples of sp , 3 d , 4 d , and 5 d metals, respectively, are shown in Fig. 2.In order to strengthen our point, some previous DFT results for tilt and twist GBs [34] are also included in the figure.We emphasize that the following observations and discussions apply to all metals considered here and the detailed3esults can be found in SM (Fig. S2 and Table S2). First, we confirm that there is a clear correlation between theGBEs in di ff erent metals, as demonstrated by the previous EAM results [67]. All the boundary energies in a specificmetal locate approximately on a straight line passing through the origin (dashed line in Fig. 2), indicating that a singlematerial dependent factor ( δ ) may be used to correlate the GBEs in a pair of metals, i.e., γ AGB (DOF) ≈ δγ CuGB (DOF)(A stands for an fcc metal). The slopes ( δ A / CuGB(fit) ) of the linear fitting for all 9 metals are listed in Table 1. The nearlyperfect scaling relation between the GBEs in di ff erent metals highlights the critical roles played by the GB structuresin deciding the GBEs. The anisotropy of GBEs is, to the first order approximation, decided by the boundary structure,i.e., the five DOFs. In other words, despite of the existence of the di ff erence in local atomic structure configuration(or even in magnetic environment), the GBEs in di ff erent materials may be described by an universal functional of thefive geometric parameters in di ff erent materials. This observation follows closely the concept developed by Bulatov etal. [7]. AlGB , Pres.
AlGB , Ref. [34]
NiGB , Pres.
NiGB , Ref. [34]
PdGB , Pres.
PdGB , Ref. [34]
PtGB , Pres.
PtGB , Ref. [34]
AlS , Ref. [37]
NiS , Ref. [37]
PdS , Ref. [37]
PtS , Ref. [37] G B i n X ( J / m ) GB in Cu (J/m ) Fig. 2. (Color online) Pairwise comparison of the calculated γ GB (solid symbols) and the (111) surface energies (open symbols with plus) for Al,Ni, Pd, and Pt with Cu obtained at 0 K. The dashed lines are the linear fits to the GBEs. Previous DFT GBEs (open symbols) for tilt and twist GBsfrom Ref. [34] are included in the linear fitting. The DFT surface energies are taken from Ref. [37]. In literature, the mean GBEs for general GBs were proposed to scale with physical parameters like shear modulus( a c ) or Voight average shear modulus ( a µ ), cohesive energy ( E / a ), stacking fault energy (SFE, γ SF ) or theircombinations [32, 34, 67]. The shear moduli and cohesive energy were rescaled by the lattice parameter ( a ) togive the same units as the GBE. The rationale behind the relation between the GBE and shear modulus µ , e.g., γ GB ≈ ka µ ( k , a coe ffi cient) is from the Read-Shockley type of dislocation model [75]. There, the GBs with low4 able 1. Surface energies for the (100) and (111) surface facets ( γ S(100) and γ S(111) , respectively) and their ratios relative to that of Cu for theselected fcc metals. δ A / CuGB(fit) is the gradient of the linear fitting of the DFT GBEs in Fig. 2 and Fig. S2. γ GB ( δ A / CuS(100) ) and γ GB ( δ A / CuS(111) ) are thepredicted GBEs using the (100) and (111) surface energies, respectively. γ expt . GB are the experimental GBEs (references indicated). All the DFTcalculations correspond to the static state (0 K), whereas the experimental GBEs for the general GBs are obtained by linear extrapolation to 0 K(see Fig. S3 in SM). DFT Predicted Expt. γ S(100) γ S(111) δ A / CuS(100) δ A / CuS(111) δ A / CuGB(fit) ( R ) γ GB ( δ A / CuS(100) ) γ GB ( δ A / CuS(111) ) γ expt . GB (J / m ) (J / m ) (J / m ) (J / m ) (J / m )Cu 1.44 a a - - - - - 0.78 c Al 0.92 a a d Au 0.86 a a e Ag 0.84 a a f Ni 2.22 a a g Pd 1.51 a a h Pt 1.85 a a i Rh 2.35 a a a a b b a Ref. [37], b Ref. [38], c Ref. [24], d Refs. [16, 17, 68], e Ref. [24], f Refs. [16, 68], g Refs. [14, 69, 70]. h Refs. [68, 70–72]. i Refs. [73, 74].misorientation angles are considered to be composed of arrays of dislocations whose energies are proportional to theshear modulus [32]. Therefore, the material dependent scaling factor δ that connects the GBEs in two materials isthought to be related to the ratio of a c or a µ (denoted as δ A / B a c and δ A / B a µ in the following). Holm et al. [32] showedthat the ratios of both a c and a µ are very close to the actual slope of the linear fit of the EAM GBEs for metals withlow SFEs; while for metals with high SFEs like Al, the ratio of a c acts as a better scaling factor than that of a µ .However, Ratanaphan et al. [67] reported that the ratio of the cohesive energies ( δ A / B E / a ) is much better indicator thanthe ratios of a c or a µ in bcc metals. It is argued that the GBEs scale with the cohesive energy based on the brokenbond model of GBE [76]. But in fcc metals, EAM results [29, 32] indicate that the broken bond model does not givesatisfactory prediction of GBEs, i.e., GBEs do not scale with E / a , nor one can use the ratio of E / a to correlate theGBEs in di ff erent fcc metals. DFT calculations also confirm that despite a general positive correlation between theGBE and the cohesive energy may exist, the overall correlation is weak [34]. As for the SFE, it is strongly related tothe coherent twin boundary energy, γ SF ≈ γ tw , but correlates weakly with the general GBEs [32]. The above resultsindicate that the ratios of a c , a µ , E / a and SFE for a pair of materials are not likely to give a good prediction of δ that can be unambiguously used to correlate the GBEs in the randomly chosen fcc materials. In Fig. 3, we compare5 A / Cu a c and δ A / Cu a µ with δ A / CuGB(fit) in the studied metals with available DFT, EAM and experimental data (Table S3 in SM).Indeed, it shows that their agreement is strongly material dependent. For example, in Rh and Ir, δ A / Cu a c and δ A / Cu a µ areabout two times larger than δ A / CuGB(fit) . a c , Present a , Present a c , EAM a , EAM a c , Expt. a , Expt. (100)S , DFT (111)S , DFT A / C u a c , A / C u a , a nd A / C u S A/CuGB(fit) Ir Fig. 3. (Color online) Comparison of δ A / BGB(fit) with δ A / B a c , δ A / B a µ , and δ A / BS . The EAM and experimental shear moduli are from Refs. [32, 77]. Thesurface energies are from Refs. [37, 38]. Numerical values are listed in Table S5. In high-index GBs, the geometry near the boundary is less close packed and many bulk-like bonds are missingwhich resembles locally a surface-like packing. Because of that, we consider the surface energy of close packedsurface facets as an alternative indicator of the GBEs. Indeed, our analysis indicates that the ratio of the low-indexsurface energies ( δ A / BS ) gives a highly accurate prediction of δ . In Fig. 2, the (111) surface energies of pure metalslocate approximatively on the same lines as the GBEs. For Al, Au, Ag, Ni, Pd, Pt, Co, and Rh, the ratios of the (111)surface energies ( δ A / CuS(111) ) are very close the actually slopes of the GBEs ( δ A / CuGB(fit) ) with mean deviation of ∼ ∼ δ A / BGB(fit) ) and the ratio ofthe surface energies ( δ A / CuS(111) and δ A / CuS(100) , respectively). Indeed, a good agreement is reached for both (111) and (001)surface energies. The above result suggests an e ffi cient approach for predicting the GBEs, especially the general GB6or which the DOFs are not properly defined. We illustrate the approach by taking Cu as the reference system withmeasured general GBE of γ expt . CuGB . Then the general GBE of an fcc metal A can be predicted as γ AGB ≈ δ A / CuS × γ expt . CuGB where δ A / CuS is the ratio of the surface energies of metal A and Cu. This ratio changes very weakly with temperatureup to the room temperature (see SM) and thus the formula is expected to apply at both low and room temperature.Following this approach, we predicted the 0 K GBEs of all metals considered here. Results are listed in Table 1. Itturns out that the proposed surface energy-based scheme gives highly reliable predictions. (b)
AgAu Al PtPd CoRhIr DFT Expt. A / C u G B A/CuS(100) Ni (a) A/CuS(111)
IrRh CoNiPtPdAlAuAg
Fig. 4. (Color online) Comparison of the ratio of the GBEs with the ratios of the low-index surface energies, (a) (100) surface and (b) (111) surface,for fcc metals. The ratios of the experimental GBEs (0 K) for the general GBs are also plotted.
Most importantly, the same approach can be applied to complex alloys for predicting the general GBEs. Here weconsider the paramagnetic stainless steel 304 (Fe . Cr . Ni . , atomic concentration) as an example, for which the(111) surface energy was calculated to be 2.83 J / m at 0 K [39]. Taking Cu as the reference system again, the ratio ofthe surface energy δ Fe . Cr . Ni . / CuS(111) is 2.18 and using the room-temperature general GBE of Cu (0.74 J / m , Ref. [24]),for the general GBE of the 304 stainless steel we predict 1.61 J / m . Our value agrees well with the experimental oneof 1.67 J / m reported at room temperature by Murr et al. [78], especially when considering the errors associated withthe experimental values, such as the linear temperature dependence and the e ff ects from minor alloying elements. Infact, with the concentration dependent surface energy calculated, we can also provide a parameterized function ofpredicting the GBE with respect to the chemical variations. Pitkänen et al. [39] provided the regression function for7he (111) surface energy with respect to the composition in Fe − c − n Cr c Ni n stainless steel, viz. γ S(111) = . c − . n + .
588 (J / m ) , (1)where c and n are the atomic fractions of Cr and Ni contents, respectively. The above formula was established forcompositions 0 . c .
32 and 0 . n .
32. Now using the present approach based on surface energies, forthe general GBE of Fe − c − n Cr c Ni n alloys we obtain γ GB = . c − . n + .
471 (J / m ) . (2)The above linear expression in terms of weight percent (wt.%) becomes γ GB = . x − . y + .
474 (J / m ) , (3)where x and y are the weight percent of Cr and Ni contents, respectively. The variables are within the limits (11 x
30, 4 y
34, wt.%).In summary, we explored the correlation between the GBEs in fcc metals with ab initio calculations. Our resultsdemonstrated that the GBEs in fcc metals are strongly correlated, with a primary origin coming from the GB structure.A material dependent parameter δ is expected to scale the GBEs of the same GB structure in a pair of fcc metals.Here, we found that the ratio of the low-index surfaces can give a satisfactory estimation of δ . Using ab initio surfaceenergies and the reference data in Cu, we successfully predicted the general GBEs in other pure fcc metals and in acomplex solid solution alloy. The present work introduces a feasible method for the prediction of the GBEs using abinitio calculations. We envision that with more ab initio studies for the GBs with structures varying in the space of thefive DOFs in a reference metal, using the surface energy-based scaling parameters as proposed in the present work,the GBEs and anisotropy in complex alloys can be readily predicted. Acknowledgments
The present work is performed under the project "SuperFraMat" financed by the Swedish Steel Producers’ Asso-ciation (Jernkontoret) and the Swedish Innovation Agency (Vinnova). The Swedish Research Council, the SwedishFoundation for Strategic Research, the Swedish Energy Agency, the Hungarian Scientific Research Fund, the ChinaScholarship Council, and the Carl Tryggers Foundation are also acknowledged for financial support. The compu-tations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the8ational Supercomputer Centre (NSC) in Linköping partially funded by the Swedish Research Council through grantagreement no. 2018-05973.
References [1] A. P. Sutton, R. W. Ballufi, Interfaces in crystalline materials, Clarendon Press, Oxford, 1995.[2] G. S. Rohrer, E. A. Holm, A. D. Rollett, S. M. Foiles, J. Li, D. L. Olmsted, Comparing calculated and measured grain boundary energies innickel, Acta Mater. 58 (15) (2010) 5063–5069. doi:10.1016/j.actamat.2010.05.042 .[3] V. Randle, Twinning-related grain boundary engineering, Acta Mater. 52 (14) (2004) 4067–4081. doi:10.1016/j.actamat.2004.05.031 .[4] V. Randle, G. Owen, Mechanisms of grain boundary engineering, Acta Mater. 54 (7) (2006) 1777–1783. doi:10.1016/j.actamat.2005.11.046 .[5] A. R. Krause, P. R. Cantwell, C. J. Marvel, C. Compson, J. M. Rickman, M. P. Harmer, Review of grain boundary complexion engineering:Know your boundaries, J. Am. Ceram. Soc. 102 (2) (2019) 778–800. doi:10.1111/jace.16045 .[6] G. S. Rohrer, Grain boundary energy anisotropy: a review, J. Mater. Sci. 46 (18) (2011) 5881–5895. doi:10.1007/s10853-011-5677-3 .[7] V. V. Bulatov, B. W. Reed, M. Kumar, Grain boundary energy function for fcc metals, Acta Mater. 65 (2014) 161–175. doi:10.1016/j.actamat.2013.10.057 .[8] C. Herring, The physics of powder metallurgy, McGraw Hill, New York, 1951.[9] J. Kudrman, J. ˇCadek, Relative grain boundary free energy and surface free energy of some metals and alloys, Czech. J. Phys. B 19 (11)(1969) 1337–1342. doi:10.1007/BF01690833 .[10] C. S. Smith, Grains, phases, and interfaces: An introduction of microstructure, Trans. Metall. Soc. AIME 175 (1948) 15–51.[11] N. A. Gjostein, F. N. Rhines, Absolute interfacial energies of [001] tilt and twist grain boundaries in copper, Acta Metall. 7 (5) (1959)319–330. doi:10.1016/0001-6160(59)90198-1 .[12] G. C. Hasson, C. Goux, Interfacial energies of tilt boundaries in aluminium. experimental and theoretical determination, Scr. Metall. 5 (10)(1971) 889–894. doi:10.1016/0036-9748(71)90064-0 .[13] H. Mykura, An interferometric study of grain boundary grooves in tin, Acta Metall. 3 (5) (1955) 436–441. doi:10.1016/0001-6160(55)90131-0 .[14] L. E. Murr, R. J. Horylev, W. N. Lin, Interfacial energy and structure in fcc metals and alloys, Philos. Mag. 22 (177) (1970) 515–542. doi:10.1080/14786437008225841 .[15] T. A. Roth, The surface and grain boundary energies of iron, cobalt and nickel, Mater. Sci. Eng. 18 (2) (1975) 183–192. doi:10.1016/0025-5416(75)90168-8 .[16] G. F. Bolling, On the average large-angle grain boundary, Acta Metall. 16 (9) (1968) 1147–1157. doi:10.1016/0001-6160(68)90049-7 .[17] L. E. Murr, Twin boundary energetics in pure aluminium, Acta Metall. 21 (6) (1973) 791–797. doi:10.1016/0001-6160(73)90043-6 .[18] W. R. Tyson, W. A. Miller, Surface free energies of solid metals: Estimation from liquid surface tension measurements, Surf. Sci. 62 (1)(1977) 267–276. doi:10.1016/0039-6028(77)90442-3 .[19] F. R. De Boer, W. C. M. Mattens, R. Boom, A. R. Miedema, A. K. Niessen, Cohesion in metals, North-Holland, Amsterdam, 1988.[20] L. Vitos, A. Ruban, H. L. Skriver, J. Kollar, The surface energy of metals, Surf. Sci. 411 (1-2) (1998) 186–202. doi:10.1016/S0039-6028(98)00363-X .
21] V. T. Borisov, V. M. Golikov, G. V. Scherbedinskiy, Relation between di ff usion coe ffi cients and grain boundary energy, Phys. Met. Metall. 17(1964) 881–885.[22] J. Pelleg, On the relation between di ff usion coe ffi cients and grain boundary energy, Philos. Mag. 14 (129) (1966) 595–601. doi:10.1080/14786436608211954 .[23] D. Gupta, Grain-boundary energies and their interaction with ta solute from self-di ff usion in au and au-1· 2 at.% ta alloy, Philos. Mag. 33 (1)(1976) 189–197. doi:10.1080/14786437608221103 .[24] D. Gupta, Influence of solute segregation on grain-boundary energy and self-di ff usion, Metall. Trans. A 8 (9) (1977) 1431–1438. doi:10.1007/BF02642856 .[25] D. Prokoshkina, V. A. Esin, G. Wilde, S. V. Divinski, Grain boundary width, energy and self-di ff usion in nickel: e ff ect of material purity,Acta Mater. 61 (14) (2013) 5188–5197. doi:10.1016/j.actamat.2013.05.010 .[26] J. D. Rittner, D. N. Seidman, < > symmetric tilt grain-boundary structures in fcc metals with low stacking-fault energies, Phys. Rev. B54 (10) (1996) 6999. doi:10.1103/PhysRevB.54.6999 .[27] D. Wolf, Structure-energy correlation for grain boundaries in fcc metals—i. boundaries on the (111) and (100) planes, Acta Metall. 37 (7)(1989) 1983–1993. doi:10.1016/0001-6160(89)90082-5 .[28] D. Wolf, S. Phillpot, Role of the densest lattice planes in the stability of crystalline interfaces: A computer simulation study, Mater. Sci. Eng.A 107 (1989) 3–14. doi:10.1016/0921-5093(89)90370-5 .[29] D. Udler, D. N. Seidman, Grain boundary and surface energies of fcc metals, Phys. Rev. B 54 (16) (1996) R11133. doi:10.1103/PhysRevB.54.R11133 .[30] M. Shiga, M. Yamaguchi, H. Kaburaki, Structure and energetics of clean and hydrogenated ni surfaces and symmetrical tilt grain boundariesusing the embedded-atom method, Phys. Rev. B 68 (24) (2003) 245402. doi:10.1103/PhysRevB.68.245402 .[31] M. A. Tschopp, D. L. McDowell, Asymmetric tilt grain boundary structure and energy in copper and aluminium, Philos. Mag. 87 (25) (2007)3871–3892. doi:10.1080/14786430701455321 .[32] E. A. Holm, D. L. Olmsted, S. M. Foiles, Comparing grain boundary energies in face-centered cubic metals: Al, au, cu and ni, Scr. Mater.63 (9) (2010) 905–908. doi:10.1016/j.scriptamat.2010.06.040 .[33] D. L. Olmsted, S. M. Foiles, E. A. Holm, Survey of computed grain boundary properties in face-centered cubic metals: I. grain boundaryenergy, Acta Mater. 57 (13) (2009) 3694–3703. doi:10.1016/j.actamat.2009.04.007 .[34] H. Zheng, X. G. Li, R. Tran, C. Chen, M. Horton, D. Winston, K. A. Persson, S. P. Ong, Grain boundary properties of elemental metals, ActaMater. 186 (2020) 40–49. doi:10.1016/j.actamat.2019.12.030 .[35] D. Scheiber, R. Pippan, P. Puschnig, L. Romaner, Ab initio calculations of grain boundaries in bcc metals, Model. Simul. Mat. Sci. Eng.24 (3) (2016) 035013. doi:10.1088/0965-0393/24/3/035013 .[36] J. L. Wang, G. K. H. Madsen, R. Drautz, Grain boundaries in bcc-fe: a density-functional theory and tight-binding study, Model. Simul. Mat.Sci. Eng. 26 (2) (2018) 025008. doi:10.1088/1361-651X/aa9f81 .[37] J. Y. Lee, M. P. J. Punkkinen, S. Schönecker, Z. Nabi, K. Kádas, V. Zólyomi, Y. M. Koo, Q. M. Hu, R. Ahuja, B. Johansson, et al., The surfaceenergy and stress of metals, Surf. Sci. 674 (2018) 51–68. doi:10.1016/j.susc.2018.03.008 .[38] J. C. W. Swart, P. van Helden, E. van Steen, Surface energy estimation of catalytically relevant fcc transition metals using dft calculations onnanorods, J. Phys. Chem. C 111 (13) (2007) 4998–5005. doi:10.1021/jp0684980 .[39] H. Pitkänen, M. Alatalo, A. Puisto, M. Ropo, K. Kokko, L. Vitos, Ab initio study of the surface properties of austenitic stainless steel alloys,Surf. Sci. 609 (2013) 190–194. doi:10.1016/j.susc.2012.12.007 .[40] W. Kohn, L. J. Sham, Self-consistent equations including exchange and correlation e ff ects, Phys. Rev. 140 (4A) (1965) A1133. oi:10.1103/PhysRev.140.A1133 .[41] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50 (24) (1994) 17953. doi:10.1103/PhysRevB.50.17953 .[42] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (18) (1996) 3865. doi:10.1103/PhysRevLett.77.3865 .[43] T. Nishiyama, A. Seko, I. Tanaka, Application of machine learning potentials to predict grain boundary properties in fcc elemental metals,arXiv preprint arXiv:2007.15944 (2020).[44] H. Hallberg, P. A. T. Olsson, Investigation of microstructure evolution during self-annealing in thin cu films by combining mesoscale levelset and ab initio modeling, J. Mech. Phys. Solids 90 (2016) 160–178. doi:10.1016/j.jmps.2016.02.026 .[45] J. Xu, J. B. Liu, S. N. Li, B. X. Liu, Y. Jiang, Self-healing properties of nanocrystalline materials: a first-principles analysis of the role ofgrain boundaries, Phys. Chem. Chem. Phys. 18 (27) (2016) 17930–17940. doi:10.1039/C6CP02505F .[46] J. J. Bean, K. P. McKenna, Origin of di ff erences in the excess volume of copper and nickel grain boundaries, Acta Mater. 110 (2016) 246–257. doi:10.1016/j.actamat.2016.02.040 .[47] Q. Gao, M. Widom, First-principles study of bismuth films at transition-metal grain boundaries, Phys. Rev. B 90 (14) (2014) 144102. doi:10.1103/PhysRevB.90.144102 .[48] T. Tsuru, Y. Kaji, D. Matsunaka, Y. Shibutani, Incipient plasticity of twin and stable / unstable grain boundaries during nanoindentation incopper, Phys. Rev. B 82 (2) (2010) 024101. doi:10.1103/PhysRevB.82.024101 .[49] R. Z. Wang, M. Kohyama, S. Tanaka, T. Tamura, S. Ishibashi, First-principles study of the stability and interfacial bonding of tilt and twistgrain boundaries in al and cu, Mater. Trans. 50 (1) (2009) 11–18. doi:10.2320/matertrans.MD200820 .[50] H. Miura, M. Kato, T. Mori, Temperature dependence of the energy of cu [110] symmetrical tilt grain boundaries, J. Mater. Sci. Lett. 13 (1)(1994) 46–48. doi:10.1007/BF02352916 .[51] T. Uesugi, K. Higashi, First-principles calculation of grain boundary energy and grain boundary excess free volume in aluminum: role ofgrain boundary elastic energy, J. Mater. Sci. 46 (12) (2011) 4199–4205. doi:10.1007/s10853-011-5305-2 .[52] R. Mahjoub, K. J. Laws, N. Stanford, M. Ferry, General trends between solute segregation tendency and grain boundary character inaluminum-an ab inito study, Acta Mater. 158 (2018) 257–268. doi:10.1016/j.actamat.2018.07.069 .[53] M. Yamaguchi, K. I. Ebihara, M. Itakura, T. Tsuru, K. Matsuda, H. Toda, First-principles calculation of multiple hydrogen segregation alongaluminum grain boundaries, Comput. Mater. Sci. 156 (2019) 368–375. doi:10.1016/j.commatsci.2018.10.015 .[54] Y. Inoue, T. Uesugi, Y. Takigawa, K. Higashi, First-principles studies on grain boundary energies of [110] tilt grain boundaries in aluminum,Mater. Sci. Forum 561 (2007) 1837–1840. .[55] T. Tsuru, Y. Shibutani, Y. Kaji, Fundamental interaction process between pure edge dislocation and energetically stable grain boundary, Phys.Rev. B 79 (1) (2009) 012104. doi:10.1103/PhysRevB.79.012104 .[56] F. H. Cao, Y. Jiang, T. Hu, D. F. Yin, Correlation of grain boundary extra free volume with vacancy and solute segregation at grain boundaries:a case study for al, Philos. Mag. 98 (6) (2018) 464–483. doi:10.1080/14786435.2017.1408968 .[57] X. Y. Pang, N. Ahmed, R. Janisch, A. Hartmaier, The mechanical shear behavior of al single crystals and grain boundaries, J. Appl. Phys.112 (2) (2012) 023503. doi:10.1063/1.4736525 .[58] R. Janisch, N. Ahmed, A. Hartmaier, Ab initio tensile tests of al bulk crystals and grain boundaries: Universality of mechanical behavior,Phys. Rev. B 81 (18) (2010) 184108. doi:10.1103/PhysRevB.81.184108 .[59] A. F. Wright, S. R. Atlas, Density-functional calculations for grain boundaries in aluminum, Phys. Rev. B 50 (20) (1994) 15248. doi:10.1103/PhysRevB.50.15248 .[60] D. I. Thomson, V. Heine, M. W. Finnis, N. Marazi, Ab initio computational study of ga in an al grain boundary, Philos. Mag. Lett. 76 (4) doi:10.1080/095008397179039 .[61] D. I. Thomson, V. Heine, M. C. Payne, N. Marzari, M. W. Finnis, Insight into gallium behavior in aluminum grain boundaries from calculationon σ =
11 (113) boundary, Acta Mater. 48 (14) (2000) 3623–3632. doi:10.1016/S1359-6454(00)00175-0 .[62] J. Chen, A. M. Dongare, Role of grain boundary character on oxygen and hydrogen segregation-induced embrittlement in polycrystalline ni,J. Mater. Sci. 52 (1) (2017) 30–45. doi:10.1007/s10853-016-0389-3 .[63] Z. L. Pan, V. Borovikov, M. I. Mendelev, F. Sansoz, Development of a semi-empirical potential for simulation of ni solute segregation intograin boundaries in ag, Model. Simul. Mat. Sci. Eng. 26 (7) (2018) 075004. doi:10.1088/1361-651X/aadea3 .[64] D. J. Siegel, J. C. Hamilton, Computational study of carbon segregation and di ff usion within a nickel grain boundary, Acta Mater. 53 (1)(2005) 87–96. doi:10.1016/j.actamat.2004.09.006 .[65] O. M. Løvvik, D. D. Zhao, Y. J. Li, R. Bredesen, T. Peters, Grain boundary segregation in pd-cu-ag alloys for high permeability hydrogenseparation membranes, Membranes 8 (3) (2018) 81. doi:10.3390/membranes8030081 .[66] C. J. O’Brien, C. M. Barr, P. M. Price, K. Hattar, S. M. Foiles, Grain boundary phase transformations in ptau and relevance to thermalstabilization of bulk nanocrystalline metals, J. Mater. Sci. 53 (4) (2018) 2911–2927. doi:10.1007/s10853-017-1706-1 .[67] S. Ratanaphan, D. L. Olmsted, V. V. Bulatov, E. A. Holm, A. D. Rollett, G. S. Rohrer, Grain boundary energies in body-centered cubic metals,Acta Mater. 88 (2015) 346–354. doi:10.1016/j.actamat.2015.01.069 .[68] S. I. Prokofjev, Estimation of surface tension of grain boundaries in allotropes of elements, J. Mater. Sci. 54 (23) (2019) 14554–14560. doi:10.1007/s10853-019-03904-y .[69] S. V. Divinski, G. Reglitz, G. Wilde, Grain boundary self-di ff usion in polycrystalline nickel of di ff erent purity levels, Acta Mater. 58 (2)(2010) 386–395. doi:10.1016/j.actamat.2009.09.015 .[70] A. R. Miedema, Surface energies of solid metals, Z. Metallk. 69 (5) (1978) 287–292.[71] R. Birringer, M. Ho ff mann, P. Zimmer, Interface stress in polycrystalline materials: The case of nanocrystalline pd, Phys. Rev. Lett. 88 (20)(2002) 206104. doi:10.1103/PhysRevLett.88.206104 .[72] R. Birringer, M. Ho ff mann, P. Zimmer, Interface stress in nanocrystalline materials, Z. Metallk. 94 (10) (2003) 1052–1061. doi:10.3139/146.031052 .[73] M. McLean, E. D. Hondros, A study of grain-boundary grooving at the platinum / alumina interface, J. Mater. Sci. 6 (1) (1971) 19–24. doi:10.1007/BF00550286 .[74] E. D. Hondros, Metallic interfaces, Sci. Prog. 68 (269) (1982) 35–63.[75] W. T. Read, W. Shockley, Dislocation models of crystal grain boundaries, Phys. Rev. 78 (3) (1950) 275. doi:10.1103/PhysRev.78.275 .[76] D. Wolf, A broken-bond model for grain boundaries in face-centered cubic metals, J. Appl. Phys. 68 (7) (1990) 3221–3236. doi:10.1063/1.346373 .[77] G. Simons, H. Wang, Single crystal elastic constants and calculated aggregate properties, J. Grand. Res. Center. 34 (1) (1977) 269.[78] L. E. Murr, G. I. Wong, R. J. Horylev, Measurement of interfacial free energies and associated temperature coe ffi cients in 304 stainless steel,Acta Metall. 21 (5) (1973) 595–604. doi:10.1016/0001-6160(73)90068-0 . r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Supplementary Material forPredicting grain boundary energies of complex alloys from ab initio calculations
Changle Li a , Song Lu a, ∗ , Levente Vitos a,b,c a Applied Materials Physics, Department of Materials Science and Engineering, KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden b Department of Physics and Astronomy, Division of Materials Theory, Uppsala University, Box 516, SE-75120 Uppsala, Sweden c Research Institute for Solid State Physics and Optics, Wigner Research Center for Physics, P.O. Box 49, H-1525 Budapest, Hungary
Contents1 GB structures 22 The calculated GBEs 43 Material properties 5 ff erent metals 75 Temperature dependence of the experimental GBEs 96 Estimation of δ A / CuS at room temperature. 10 ∗ Corresponding author
Email address: [email protected] (Song Lu)
Preprint submitted to Elsevier February 4, 2021 . GB structures
In the present work, we focus on symmetric tilt GBs with a [1¯10] tilt axis. A set of 10 di ff erent GBs are listed inTable S1. All GBs are initiated from the tilt plane with certain misorientation angles. The schematics of the selected10 GB structures are presented in Fig. S1. Table S1.
The properties of the [1¯10] tilt GBs studied in present study.
Index Angle GB-plane Number of atoms Σ ◦ (1 1 1) 24 Σ ◦ (1 1 2) 46 Σ ◦ (1 1 4) 68 Σ ◦ (2 2 1) 34 Σ
11 50.48 ◦ (1 1 3) 22 Σ
11 129.52 ◦ (3 3 2) 82 Σ
17 86.63 ◦ (2 2 3) 62 Σ
17 93.37 ◦ (3 3 4) 62 Σ
19 26.53 ◦ (1 1 6) 68 Σ
19 153.35 ◦ (3 3 1) 362 ig. S1. Schematics of the atomic GB structures before relaxation. . The calculated GBEs The calculated GBEs for all studied metals are tabulated in Table S2. For comparison, the available DFT resultsin literature are also presented [1–23].
Table S2.
The calculated GBEs (in unit of J / m ) in comparison with available DFT results in literature. Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ . Material properties Table S3 shows the calculated lattice parameters, bulk modulus, elastic constants, and Voight average shear mod-ulus for the studied fcc metals in the present work. Available theoretical (DFT [25] and EAM [26]) and experimen-tal [27] data are also included for comparison. In general, the present results show a good agreement with othertheoretical and experimental data.
Table S3.
Comparison of lattice parameters ( a ), bulk modulus ( B ), elastic constants ( c , c , c ), and Voight average shear modulus ( µ ) forfcc pure metals between the present and the previous works. All theoretical and experimental data correspond to the static conditions (0 K), exceptthose experimental results at the room temperature marked by ∗ . Al Au Ag Cu Ni Pd Pt Rh Ir Co Functional Ref. a (Å) 4.04 4.16 4.15 3.64 3.52 3.94 3.97 3.82 3.87 3.52 PBE-GGA Present4.05 4.17 4.16 3.64 3.52 3.96 3.99 3.84 3.88 3.52 PBE-GGA [25]4.03 4.08 3.62 3.52 EAM [26]4.05 4.08 4.09 3.62 3.53 3.89 3.92 3.80 3.84 Expt. [27] B (GPa) 78.2 138.0 90.1 140.4 195.6 167.4 248.2 257.5 350.0 208.4 PBE-GGA Present74.3 137.6 91.3 137.5 195.6 163.7 243.4 253.4 342.8 210.2 PBE-GGA [25]80.9 167.0 138.3 180.4 EAM [26]79.4 180.3 108.7 142.0 187.6 195.4 288.4 267.0 354.7 Expt. [27] c (GPa) 108.9 154.7 108.2 177.0 274.2 199.3 309.5 411.6 586.2 291.4 PBE-GGA Present101.0 159.1 115.9 174.8 275.5 198.0 296.4 405.3 580.8 296.6 PBE-GGA [25]118.1 185.8 169.9 240.5 EAM [26]114.3 201.6 131.4 176.2 261.2 234.1 358.0 413.0 ∗ ∗ Expt. [27] c (GPa) 62.8 129.7 81.1 122.1 156.3 151.4 217.5 180.5 231.9 166.9 PBE-GGA Present61.0 136.7 85.1 122.8 160.1 155.7 225.6 185.5 232.0 171.9 PBE-GGA [25]62.3 157.1 122.6 150.3 EAM [26]61.9 169.7 97.3 124.9 150.8 176.1 253.6 194.0 ∗ ∗ Expt. [27] c (GPa) 32.6 23.3 43.0 75.5 131.2 57.7 58.4 185.1 258.9 142.6 PBE-GGA Present25.4 27.6 42.1 76.3 126.3 69.7 50.7 176.5 249.8 144.0 PBE-GGA [25]36.7 38.9 76.2 119.2 EAM [26]31.6 45.4 51.1 81.8 131.7 71.2 77.4 184.0 ∗ ∗ Expt. [27] µ (GPa) 28.8 19.0 31.2 56.3 102.3 44.2 53.3 157.3 226.2 110.5 PBE-GGA Present23.2 21.1 31.4 56.2 98.9 50.3 44.6 149.9 219.6 111.3 PBE-GGA [25]33.0 29.1 55.2 90.0 EAM [26]29.4 33.6 37.5 59.3 101.1 54.3 67.3 154.2 ∗ ∗ Expt. [27]
For surface energy, extensively theoretical works have been employed for fcc metals [28–35]. In Table S4, wecollect several works using di ff erent exchange-correlation functionals. For using the same exchange-correlation func-tional, it can be seen that the calculated surface energies show a good agreement with each other. To be consistent, weadopt the surface energies from the work with results for most fcc metals [28] and the other one for Co is taken fromRef. [32]. The corresponding values are in bold. 5 able S4. Surface energies of fcc metals. Those adopted in the present work are in bold.
Surface Al Au Ag Cu Ni Pd Pt Rh Ir Co Functional Ref.(100)
PBE-GGA [28]0.95 0.86 0.81 1.48 1.79 1.88 2.77 PBE-GGA [29]2.18 1.51 2.32 2.80 2.45 PBE-GGA [30]2.21 1.84 PBE-GGA [31]0.90 1.46 2.34
PBE-GGA [32]1.15 1.39 1.16 1.99 2.43 2.35 3.04 LDA-GGA [29]1.33 2.02 2.95 LDA-GGA [32]1.08 1.13 1.04 1.76 2.15 2.21 2.97 PBEsol-GGA [29]1.04 1.19 1.14 1.81 2.61 1.84 2.23 2.84 3.24 PBEsol-GGA [33]1.36 1.27 2.15 2.15 2.47 3.49 PBE-GGA [34](111)
PBE-GGA [28]0.77 0.75 0.78 1.33 1.36 1.56 2.09 PBE-GGA [29]1.91 1.33 2.00 2.28 2.07 PBE-GGA [30]0.71 0.71 0.74 1.29 1.93 1.36 1.51 2.02 2.31 PBE-GGA [35]0.76 1.92 1.33 1.48 2.42 PBE-GGA [31]0.70 1.32 2.02
PBE-GGA [32]0.99 1.24 1.13 1.81 1.88 1.98 2.67 LDA-GGA [29]1.11 1.85 2.46 LDA-GGA [32]0.99 1.10 1.00 1.59 1.63 1.85 2.40 PBEsol-GGA [29]0.93 1.01 1.07 1.67 2.28 1.59 1.81 2.44 2.56 PBEsol-GGA [33]1.14 1.15 1.94 1.90 2.00 2.78 PBE-GGA [34]6 . The correlation between GBEs in di ff erent metals Fig. S2 shows the pairwise comparison of the calculated GBEs and the (111) surface energies for the remainingelements, Ag, Au, Co, Rh, and Ir with Cu. Previous DFT GBEs (solid symbols) for tilt and twist GBs from Ref. [1]are included in the linear fitting. DFT surface energies are taken from Refs. [28, 32].
AgGB , Pres.
AgGB , Ref. [1]
AuGB , Pres.
AuGB , Ref. [1]
AgS , Ref. [28]
AuS , Ref. [28] G B i n X ( J / m ) GB in Cu (J/m ) (a) CoGB , Pres.
RhGB , Pres.
RhGB , Ref. [1]
IrGB , Pres.
IrGB , Ref. [1]
CoS , Ref. [32]
RhS , Ref. [28]
IrS , Ref. [28] G B i n X ( J / m ) GB in Cu (J/m ) (b) Fig. S2. (Color online) Pairwise comparison of the calculated γ GB (solid symbols) and the (111) surface energies (open symbols with plus) for Ag,Au, Co, Rh, and Ir with Cu obtained at 0 K. For clearance, the results are shown in two subfigures. The dashed lines are the linear fit of the GBEs.Previous DFT GBEs (open symbols) for tilt and twist GBs from Ref. [1] are included in the linear fitting. DFT surface energies are taken fromRefs. [28, 32]. able S5. The pairwise comparison of the ratios of a c , a µ , surface energies ( γ S ), and δ A / CuGB(fit) in fcc metals. The EAM, experimental, and VASPresults are from Refs. [26], [27], and [1, 28, 32], respectively. A δ A / Cu a c δ A / Cu a µ δ A / CuS(100) δ A / CuS(111) δ A / CuGB(fit) R δ A / CuGB(fit) R (Pres.) (EAM) (Expt.) (Pres.) (EAM) (Expt.) (VASP) (VASP) (VASP) (VASP) (Pres.) (Pres.)Au 0.353 0.576 0.626 0.386 0.595 0.639 0.597 0.546 0.501 0.995 0.486 0.989Ag 0.650 0.706 0.633 0.714 0.583 0.585 0.607 0.997 0.568 0.997Al 0.480 0.537 0.432 0.569 0.667 0.555 0.639 0.631 0.540 0.991 0.559 0.980Pd 0.829 0.935 0.852 0.984 1.049 1.023 1.014 0.999 0.964 0.998Pt 0.845 1.025 1.037 1.228 1.285 1.146 0.990 0.991 1.006 0.987Co 1.828 1.899 1.708 1.554 1.448 0.990Ni 1.685 1.523 1.570 1.762 1.588 1.661 1.542 1.477 1.441 0.994 1.384 0.995Rh 2.581 2.361 2.941 2.728 1.632 1.546 1.667 0.995 1.714 0.994Ir 3.655 3.320 4.282 3.954 1.972 1.585 2.036 0.988 2.119 0.994 . Temperature dependence of the experimental GBEs Experimentally, the mean GBE for high angle GBs can be obtained at elevated temperatures. As shown in Fig. S3,the mean (experimental) GBEs for di ff erent metals are collected [36–46]. Gupta et al. [42] measured the GBEs of Auand Cu at di ff erent temperatures and obtained the corresponding temperature coe ffi cients through the linear fitting. Toextrapolate the experimental GBEs to 0 K or room temperature, linear fitting based on the measured GBEs at elevatedtemperatures is used. The corresponding linear function and temperature coe ffi cient for pure metals are shown inFig. S3. expt. AlGB (T) = 0.436 0.000136T expt. CuGB (T) = 0.776 0.000124T expt. PdGB (T) = 0.801 0.000171T expt. PtGB (T) = 0.836 0.000186T Ni Cu Pt Pd Ag Au Al e xp t . G B ( J / m ) T (K) expt. NiGB (T) = 1.113 0.000200T expt. AuGB (T) = 0.397 0.000017T expt. AgGB (T) = 0.447 0.000049T
Fig. S3. (Color online) Temperature dependence of the mean (experimental) GBEs ( γ expt . GB ) in fcc metals. The symbols represent the measured γ expt . GB of the high-angle grain boundaries in Al [36–38], Ni [39–41], Cu [42], Pd [38, 41, 43, 44], Ag [37, 38], Pt [45, 46], and Au [42]. The solid linesrepresent the linear fit results. . Estimation of δ A / CuS at room temperature.
Following the work by Tyson et al. [47], we can estimate the change of δ A / CuS as we go from 0 K to room temper-ature. Tyson et al. computed the surface energy at low temperature using the surface tension data in the liquid phaseand the linear temperature dependence γ S (0) − γ S ( T m ) ≈ RT m / A , connecting the surface energy at melting temperature T m and at 0 K. Here R is the gas constant and A is the surface area per mole of surface atoms. Writing this expressionfor temperature T and taking the di ff erence, we arrive at γ S ( T ) − γ S (0) ≈ − α · T / T m , where α = RT m / A . Thus thetemperature dependence of δ A / CuS ( T ) in leading order in T can be written as δ A / CuS ( T ) ≈ δ A / CuS (0) + γ CuS (0) α Cu T Cu m · γ AS (0) γ CuS (0) − α A T A m · T , (1)where the γ CuS (0) and γ AS (0) are the 0 K surface energies for Cu and metal A, respectively. Using the α and γ S (0)values for pure metals by Tyson et al. [47], we obtain the change of δ A / CuS when going from 0 K to room temperature.The results for the present metals are listed in Table S6. The average change of ∆ δ A / CuS is ∼ Table S6.
Calculated δ A / CuS values for the selected fcc metals at 0 K and 298 K. A δ A / CuS (0) δ A / CuS (298) ∆ δ A / CuS %Al 0.640 0.636 0.004 0.6Au 0.843 0.842 0.001 0.1Ag 0.697 0.695 0.002 0.3Ni 1.331 1.324 0.007 0.5Pd 1.125 1.118 0.007 0.6Pt 1.398 1.383 0.015 1.1Co 1.410 1.400 0.010 0.7Rh 1.494 1.478 0.016 1.1Ir 1.712 1.689 0.023 1.310 eferences [1] H. Zheng, X. G. Li, R. Tran, C. Chen, M. Horton, D. Winston, K. A. Persson, and S. P. Ong, Acta Mater. , 40 (2020).[2] T. Uesugi and K. Higashi, J. Mater. Sci. , 4199 (2011).[3] R. Mahjoub, K. J. Laws, N. Stanford, and M. Ferry, Acta Mater. , 257 (2018).[4] T. Nishiyama, A. Seko, and I. Tanaka, arXiv preprint arXiv:2007.15944 (2020).[5] M. Yamaguchi, K. I. Ebihara, M. Itakura, T. Tsuru, K. Matsuda, and H. Toda, Comput. Mater. Sci. , 368 (2019).[6] Y. Inoue, T. Uesugi, Y. Takigawa, and K. Higashi, Mater. Sci. Forum , 1837 (2007).[7] T. Tsuru, Y. Shibutani, and Y. Kaji, Phys. Rev. B , 012104 (2009).[8] F. H. Cao, Y. Jiang, T. Hu, and D. F. Yin, Philos. Mag. , 464 (2018).[9] X. Y. Pang, N. Ahmed, R. Janisch, and A. Hartmaier, J. Appl. Phys. , 023503 (2012).[10] R. Janisch, N. Ahmed, and A. Hartmaier, Phys. Rev. B , 184108 (2010).[11] A. F. Wright and S. R. Atlas, Phys. Rev. B , 15248 (1994).[12] D. I. Thomson, V. Heine, M. W. Finnis, and N. Marazi, Philos. Mag. Lett. , 281 (1997).[13] D. I. Thomson, V. Heine, M. C. Payne, N. Marzari, and M. W. Finnis, Acta Mater. , 3623 (2000).[14] H. Hallberg and P. A. T. Olsson, J. Mech. Phys. Solids , 160 (2016).[15] J. J. Bean and K. P. McKenna, Acta Mater. , 246 (2016).[16] Q. Gao and M. Widom, Phys. Rev. B , 144102 (2014).[17] T. Tsuru, Y. Kaji, D. Matsunaka, and Y. Shibutani, Phys. Rev. B , 024101 (2010).[18] R. Z. Wang, M. Kohyama, S. Tanaka, T. Tamura, and S. Ishibashi, Mater. Trans. , 11 (2009).[19] J. Chen and A. M. Dongare, J. Mater. Sci. , 30 (2017).[20] Z. L. Pan, V. Borovikov, M. I. Mendelev, and F. Sansoz, Model. Simul. Mat. Sci. Eng. , 075004 (2018).[21] D. J. Siegel and J. C. Hamilton, Acta Mater. , 87 (2005).[22] O. M. Løvvik, D. D. Zhao, Y. J. Li, R. Bredesen, and T. Peters, Membranes , 81 (2018).[23] C. J. O’Brien, C. M. Barr, P. M. Price, K. Hattar, and S. M. Foiles, J. Mater. Sci. , 2911 (2018).[24] J. Xu, J. B. Liu, S. N. Li, B. X. Liu, and Y. Jiang, Phys. Chem. Chem. Phys. , 17930 (2016).[25] S. L. Shang, A. Saengdeejing, Z. G. Mei, D. E. Kim, H. Zhang, S. Ganeshan, Y. Wang, and Z. K. Liu, Comput. Mater. Sci. , 813 (2010).[26] E. A. Holm, D. L. Olmsted, and S. M. Foiles, Scr. Mater. , 905 (2010).[27] G. Simons and H. Wang, J. Grand. Res. Center. , 269 (1977).[28] J. Y. Lee, M. P. J. Punkkinen, S. Schönecker, Z. Nabi, K. Kádas, V. Zólyomi, Y. M. Koo, Q. M. Hu, R. Ahuja, B. Johansson, et al. ,Surf. Sci. , 51 (2018).[29] A. Patra, J. E. Bates, J. W. Sun, and J. P. Perdew, Proc. Natl. Acad. Sci. U. S. A. , E9188 (2017).[30] H. Lin, J. X. Liu, H. J. Fan, and W. X. Li, J. Phys. Chem. C , 11005 (2020).[31] R. Tran, Z. H. Xu, B. Radhakrishnan, D. Winston, W. Sun, K. A. Persson, and S. P. Ong, Sci. Data , 1 (2016).[32] J. C. W. Swart, P. van Helden, and E. van Steen, J. Phys. Chem. C , 4998 (2007).[33] S. H. Yoo, J. H. Lee, Y. K. Jung, and A. Soon, Phys. Rev. B , 035434 (2016).[34] J. Wang and S. Q. Wang, Surf. Sci. , 216 (2014).[35] X. Z. Wu, R. Wang, S. F. Wang, and Q. Y. Wei, Appl. Surf. Sci. , 6345 (2010).[36] L. E. Murr, Acta Metall. , 791 (1973).[37] G. F. Bolling, Acta Metall. , 1147 (1968).[38] S. I. Prokofjev, J. Mater. Sci. , 14554 (2019).[39] S. V. Divinski, G. Reglitz, and G. Wilde, Acta Mater. , 386 (2010).[40] L. E. Murr, R. J. Horylev, and W. N. Lin, Philos. Mag. , 515 (1970).[41] A. R. Miedema, Z. Metallk. , 287 (1978).[42] D. Gupta, Metall. Trans. A , 1431 (1977).[43] R. Birringer, M. Ho ff mann, and P. Zimmer, Phys. Rev. Lett. , 206104 (2002).[44] R. Birringer, M. Ho ff mann, and P. Zimmer, Z. Metallk. , 1052 (2003).[45] M. McLean and E. D. Hondros, J. Mater. Sci. , 19 (1971).[46] E. D. Hondros, Sci. Prog. , 35 (1982).[47] W. R. Tyson and W. A. Miller, Surf. Sci. , 267 (1977)., 267 (1977).