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A canonical stability–elasticity relationship verified for one million face-centred-cubic structures


Any thermodynamically stable or metastable phase corresponds to a local minimum of a potentially very complicated energy landscape. But however complex the crystal might be, this energy landscape is of parabolic shape near its minima. Roughly speaking, the depth of this energy well with respect to some reference level determines the thermodynamic stability of the system, and the steepness of the parabola near its minimum determines the system’s elastic properties. Although changing alloying elements and their concentrations in a given material to enhance certain properties dates back to the Bronze Age1,2, the systematic search for desirable properties in metastable atomic configurations at a fixed stoichiometry is a very recent tool in materials design3. Here we demonstrate, using first-principles studies of four binary alloy systems, that the elastic properties of face-centred-cubic intermetallic compounds obey certain rules. We reach two conclusions based on calculations on a huge subset of the face-centred-cubic configuration space. First, the stiffness and the heat of formation are negatively correlated with a nearly constant Spearman correlation4 for all concentrations. Second, the averaged stiffness of metastable configurations at a fixed concentration decays linearly with their distance to the ground-state line (the phase diagram of an alloy at zero Kelvin). We hope that our methods will help to simplify the quest for new materials with optimal properties from the vast configuration space available.

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Figure 1: Ground-state diagrams of binary Ni-W.
Figure 2: Spearman’s coefficients and for the correlation between stiffness and enthalpy excess β.
Figure 3: as a function of the enthalpy excess β before averaging over all structures at constant β.
Figure 4: Averaged stiffnesses using harmonic (Voigt-type) averaging for four different systems at five different compositions.


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Funding by the DFG (Deutsche Forschungsgemeinschaft) grant Mu1648/5 is acknowledged. We also thank the RRZ-Hamburg super-computing site for a generous amount of computational time and E. Kahnert and her team for support and advice related to the computing facilities. Additional computing resources from the German super-computing alliance HLRN are acknowledged.

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Authors and Affiliations



S.B.M. performed the density functional theory calculations for the NiW, NiTa and CuAl alloys, and M.H. for the NiAl alloy. S.B.M. calculated the cluster expansion Hamiltonians, performed the data post processing and wrote the paper. S.M. formulated the original problem and supervised the investigation. All authors participated in the manuscript preparation during all stages of the process.

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Correspondence to Stefan Müller.

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The authors declare no competing financial interests.

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Maisel, S., Höfler, M. & Müller, S. A canonical stability–elasticity relationship verified for one million face-centred-cubic structures. Nature 491, 740–743 (2012).

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