From bond valence maps to energy landscapes for mobile ions in ion-conducting solids Stefan Adams ⁎ Department of Materials Science and Engineering, National University of Singapore, Singapore 117576 GZG, Abteilung Kristallographie, Universität Göttingen, Germany Received 3 July 2005; received in revised form 18 February 2006; accepted 31 March 2006 Abstract Applications of the bond valence method for the analysis of ion transport pathways in crystalline cation ion conductors with various mobile cations are reviewed and an extension of the approach to anion conductors is discussed. In both cases the discussion highlights structures, where special care is required in the interpretation of the pathway model. The extension of the bond valence approach enhances the application range of the method for the identification of the ion transport mechanisms to materials, where both cations and anions have to be considered as potentially mobile species, as demonstrated for the presumed trivalent cation conductor Sc 2 (WO 4 ) 3 . © 2006 Elsevier B.V. All rights reserved. Keywords: Bond valence analysis; Ion transport pathways; Anion conductors; Trivalent cation conductors; Anion conductors; Lithium lanthanum titanate 1. Introduction Ion transport in solids requires a network containing both occupied and vacant sites for the mobile ions. Only a subset of the ions can be highly mobile, while immobile ions maintain the stability of the crystal structure. In most cases (exceptions range from systems with a paddlewheel mechanism to proton conductors) the situation may be simplified assuming that immobile ions define an essentially static energy landscape for the mobile ions. In this scenario bond valence (BV) calculations (cf., e.g. [1]) can be effectively used to model ion transport pathways and mechanisms, especially when the assessment of non-equilibrium site energies is enhanced by a systematic adjustment of BV parameters to the bond softness [2,3]. Applications to amorphous ion conductors have recently been reviewed elsewhere [4–7]. In this work, transport pathway models for crystalline ion conductors with various mobile cations or anions are analyzed to identify the underlying general principles. Discussion focuses on materials, where a proper interpretation of BV maps or the comparison with experimental structure or property data requires special care. 2. Computational approach “Accessible” sites for mobile ions A in a local structure model are identified using empirical relationships between the bond length R and a so-called bond valence s A-X s A−X ¼ exp½ðR 0 −RÞ=b ð1Þ as sites where the mismatch of the bond valence sum V(A) jDV ðAÞj ¼      X x s A−X −V id ðAÞ      þ X x p A−X ð2Þ over the s A-X from all adjacent counterions X approaches its oxidation state V id (A). R 0 (bond length corresponding to unit valence, not to a single bond) and b are refined empirically from reference sets of well-ordered crystal structures (cf. [1–3]). To enhance the plausibility of BV “energy landscapes”, the BV mismatch is complemented by minimum distance constraints and the penalty function p A-X in Eq. (2), which discriminates against sites that achieve a matching V(A) by strongly Solid State Ionics 177 (2006) 1625 – 1630 www.elsevier.com/locate/ssi ⁎ Department of Materials Science and Engineering, National University of Singapore, Singapore 117576. E-mail address: mseasn@nus.edu.sg. 0167-2738/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2006.03.054