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Ag migration pathways in crystalline and glassy solid electrolytes AgI–AgMxOy
Solid State Ionics 105 (1998) 67–74
Ag migration pathways in crystalline and glassy solid electrolytes AgI–AgM x O y a, b Stefan Adams *, Joachim Maier a
b
¨ Min.-Kristallograph. Institut, Univ. Gottingen , Germany ¨ Max-Planck-Institut f. Festkorperforschung , Stuttgart, Germany
Abstract Conduction pathways for silver ions in crystalline model structures of the AgI–AgM x O y electrolytes are derived from a bond valence sum model, which proved successful for a-AgI. Models for the silver transport in the related glassy phases are discussed in the light of our results. Keywords: Ion conductivity; Conduction pathways; Valence sum pseudopotentials; Silver ion conductors; Glassy electrolytes
1. Introduction Several years ago, the ionic conduction in AgI– AgM x O y glasses was mostly attributed to Ag 1 with a AgI-type co-ordination. Spectroscopic studies seemed to support this interpretation by the detection of different Ag 1 populations with distinguishable mobilities for local hopping (see e.g. references cited in reviews [1,2]). The pathway model presented in this paper, which is based on bond valence pseudopotential calculations, demonstrates that Ag 1 ions with mixed iodide–oxide co-ordinations crucially contribute to the ionic conduction in these systems. The investigations started with the glass system 4AgI–Ag 4 V2 O 7 , from which the compound Ag 8 I 4 V2 O 7 can be produced by congruent crystallization as well as by slow cooling of the melt of stoichiometric composition [3]. The close similarities
*Corresponding [email protected]
author.
Fax:
49
551
393825;
e-mail:
between the glassy and crystalline phase with respect to mass density, overall chemical composition, absolute conductivity, activation energy etc. [3,4] suggest that a determination of the conduction pathways in the crystalline phase should also be helpful for a discussion of the conduction mechanism in the glassy state.
2. Valence sum pseudopotential method Owing to the intrinsic disorder, single crystal structural data of Ag 1 ionic conductors are typically of moderate quality, so that a direct determination of Ag 1 pathways from a refinement of anharmonic contributions to the atomic displacement parameters (‘temperature factors’) remains restricted to simple structures such as a-AgI [5]. Pathways of lowest energy for the Ag 1 migration may, however, be estimated from the well-defined anion positions by means of simple valence sum pseudopotentials. The valence sum
0167-2738 / 98 / $19.00 1998 Elsevier Science B.V. All rights reserved. PII S0167-2738( 97 )00450-5
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
68
Os
V5
(1)
Ag2X
3. Ag 1 pathways in the crystalline state
X
for Ag is calculated for a grid of points throughout the asymmetric unit from empirical bond length bond valence equations. The parameter set of Eq. (2) by ¨ Radaev, Fink and Tromel [6] has been chosen among the various parameter sets in literature (see refs. cited in [4]), because it includes the influence of higher co-ordination shells:
F F
G G
˚ 2 RAg2O 1.89 A sAg2O 5 exp ]]]]] , ˚ 0.33 A ˚ 2 RAg2I 2.08 A sAg2I 5 exp ]]]]] . ˚ 0.53 A
(2)
˚ from the All anions up to a distance of 8 A respective site are included. The sum of radii of Ag and M in metals is assumed to be the shortest possible Ag–M distance. Pathways of lowest activation energy are assumed to be those connections between equilibrium positions, for which the maximum valence mismatch DV5uV2Vequilibrium u of the Ag bond valence is as small as possible. In the case of a-AgI this simple formalism closely reproduces the Ag density distribution results of neutron diffraction studies (Fig. 1, cf. Ref. [5]).
3.1. Ag8 I4V2 O7 From our recent single crystal structure determi] nation of Ag 8 I 4 V2 O 7 (space group P62m, a512.595 ˚ c59.119 A) ˚ [7], we derived Ag 1 migration A, pathways for this crystalline analogue of the 4AgI– 1 Ag 4 V2 O 7 glass [4]. The Ag sites clearly fall into two classes with distinct co-ordination shells: fully occupied sites with a nearly octahedral 4I 2 12O 22 co-ordination and 2O 22 13I 2 co-ordinated sites with occupation factors close to n52 / 3. The co-ordination of none of the Ag sites resembles the tetrahedral iodide co-ordination in a-AgI. Pathways of low valence mismatch (i.e. low activation energy) connect sites of the same type, but they permit only hops within closed loops that cannot account for the dc conductivity (cf. Fig. 2). Any reasonable long-range migration pathway includes both types of silver sites (see Fig. 3). Thus the Ag 1 conduction predominantly takes place in layers that extend perpendicular to the hexagonal axis via jumps between the fully occupied positions at z¯ 60.27 and the partially occupied positions at z50. Subsequently we extended the pseudopotential
Fig. 1. Network of Ag 1 conduction pathways in a-AgI (l.h.s.) and g-AgI (r.h.s.) as computed from the valence sum pseudopotentials. Equipotential contours are drawn only for the front half of the unit cell to reduce overlap.
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
69
Fig. 2. Sections of the Ag valence sum pseudopotential map of Ag 8 I 4 V2 O 7 (a) at z50 and (b) at z50.27.
Fig. 3. Detail from an Ag valence sum pseudopotential map of Ag 8 I 4 V2 O 7 for DV #0.45 (after Ref. [4]). Arrows mark the connections between the fully occupied sites (hatched contours) and the partially occupied sites (unhatched contours).
calculations to other crystalline compounds which may be understood as crystallized AgI2AgM x O y glasses. The ICSD crystal structure database reports two further crystal structures of related compounds: Ag 26 I 18 W4 O 16 and Ag 16 I 12 P2 O 7 . Isolated oxyacid 42 82 42 anions (V2 O 7 , W4 O 16 or P2 O 7 ) are present in all these compounds, so that a comparison of the results with the situation in glassy phases might be limited to ‘molecular glasses’, whereas no crystalline model structure is known for AgI2 AgM x O y glasses containing a polymeric network.
3.2. Ag26 I18 W4 O16 In the monoclinic room temperature phase of ˚ b5 Ag 26 I 18 W4 O 16 (space group C2, a516.76(3) A, ˚ ˚ 15.52(3) A, c511.81(2) A, b 5103.98) isolated 82 W4 O 16 ions reside in channels that are formed by 2 the framework of I ions. According to the structure refinement by Chan and Geller [8] the Ag 1 ions are distributed over 46 crystallographically different sites with a mean occupation factor of only 28.6% (Fig. 4).
70
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
Fig. 4. Crystal structure of Ag 26 I 18 W4 O 16 viewed along the c-axis 82 [8]. (W4 O 16 : polyhedra; Ag 1 : ellipsoids; I 2 network: sticks).
Moreover, their refinement results reflect a strong preference of the Ag 1 ions for mixed co2 ordination. The 90 polyhedra with pure I - co-ordination host only 22% of the 52 silver ions, while 78% of the Ag 1 reside in 92 polyhedra with mixed coordinations. In accordance with the prevailing theory at that time, the experimental results were taken as evidence for the immobilisation of Ag 1 by an oxide co-ordination. Still the authors argue that some of the Ag 1 with mixed coordinations will contribute to the conduction either as sources of mobile ions or as bypasses for complicated pathways through the iodide polyhedra. Under the assumption that only iodide co-ordinated Ag 1 should be mobile, the most direct Ag 1 pathway should exist along c*. This hypothesis was ruled out by a subsequent study of the anisotropic conductivity [9], since the conductivity along the a-direction is ca. 40% higher than along b and c* at ambient temperature. Due to a somewhat higher activation energy for the c*-direction, the conductivity along this direction reaches the conductivity along a at about 450 K. In agreement with the rather low overall anisotropy, valence sum calculations yield a three-dimensional network of pathways, for a valence mismatch of DV $0.19 (see Fig. 5b). A minor preference for
Fig. 5. Pathway models for the Ag 1 migration within the asymmetric unit of Ag 26 I 18 W4 O 16 : (a) linear connection of the refined Ag 1 positions (sticks are light within iodide polyhedra, dark within mixed polyhedra), (b) valence sum pseudopotential map (DV50.19).
the conduction along the a-direction might be derived from the somewhat larger cross area of these pathways.
3.3. Ag16 I12 P2 O7 The crystallographic description of Ag 16 I 12 P2 O 7 ˚ c57.43 A) ˚ [10], (space group P6 /mcc, a512.054 A, appears unsuitable for a straightforward estimation of migration pathways. Due to the disordered nature of the compound, structure determination yields only an average structure employing numerous sites with low occupation factors for both Ag 1 cations and pyrophosphate anions (cf. Fig. 6). The migration pathways rather depend on the actual local co-ordinations of the silver ions.
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
71
Fig. 6. Schematic representation of the crystal structure of Ag 16 I 12 P2 O 7 viewed along the hexagonal axis. (Ag 1 site network shown as grey rods; I 2 network as dark stick-and-ball; disordered P2 O 7 as light stick-and-ball).
Therefore, a molecular mechanics (MM) simulation based on the crystallographic information was applied to derive a chemically meaningful arrangement (especially for the pyrophosphate ions within the structure channels). MM calculations have been performed in a 23233 superstructure with the ‘quench molecular dynamics’ routine of the CERIUS 2 program suite using a slightly modified UNIVERSAL force field parameter set with constant
partial charges. The resulting local structure model (Fig. 7) reflects the expected features, e.g. the tilt angle of the pyrophosphate anions. The valence sum pseudopotential calculations yield a network of pathways, which exhibits close similarities to the models for the previously discussed compounds. Local closed loops of various diameters connect Ag 1 sites with a predominant iodide co-ordination (see Fig. 8). A rather low
Fig. 7. Anion framework within the local structure model of Ag 16 I 12 P2 O 7 from MM calculations (P2 O 7 : double tetrahedra; I 2 framework: stick-and-ball).
72
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
Fig. 8. Projections of the valence sum pseudopotential map of Ag 16 I 12 P2 O 7 for DV50.12 (l.h.s.) and the corresponding region of the structure model (r.h.s.). Anions are drawn as in Fig. 7, the network of Ag 1 sites is sketched as rods.
valence mismatch (DV50.12) is sufficient to link these loops and construct an infinite network of pathways, as necessary for long-range migration. These interconnecting pathways involve sites in the vicinity of the divanadate ions (Fig. 9). Direct hops between the loops avoiding sites with mixed coordinations require considerably higher activation energies. The unoccupied possible Ag 1 site with a pure iodide co-ordination at the center of the larger loops is not attached to the network.
4. Ag 1 pathways in the disordered state The preceding calculations were performed for models with a high degree of ordering within the anion substructure due to energy minimisations after the molecular dynamics runs. As an oversimplified approach to transfer the results of the study to the glassy state, we investigated also snapshots of the Ag 16 I 12 P2 O 7 structure during molecular dynamics runs such as the one shown in Fig. 10.
Fig. 9. Views of the 3-dim. network of Ag 1 pathways in crystalline Ag 16 I 12 P2 O 7 . Projections parallel to the hexagonal axis (r.h.s.) highlight links along the diphosphate ions between the loops of iodide co-ordinated Ag 1 sites.
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
Fig. 10. Snapshot of the local structure model for Ag 16 I 12 P2 O 7 during a molecular dynamics run (ions are drawn as in Fig. 8).
The corresponding valence sum pseudopotential map is given in Fig. 11. In this particular case pathways across the modeled region form already for a valence mismatch of DV50.10. A comparison between the pathway models for the ordered and disordered structures (Figs. 9 and 11) reveals that the essential features of the ordered model, such as local loops and links utilizing Ag 1 with mixed co-ordinations, are found as well in the more disordered state.
5. Concluding remarks The proposed technique produces vivid representa-
73
tions of possible Ag 1 migration pathways based on a very simple pseudopotential ansatz. For highly symmetrical crystalline compounds, where neutron diffraction data are available, the calculations closely mimic the experimental results [10]. The confidence in the method is raised by a clear relation between the minimum valence mismatch to create an infinite network and the experimental activation energy for the dc conductivity (Fig. 12). The activation energies of compounds with Ag 1 disorder seem to be roughly proportional to the logarithm of the minimum valence mismatch. For ordered compounds such as g-AgI the experimental value of DE is considerably higher than predicted from the ln(DV )2DE relation for the disordered compounds, suggesting that the low probability of successful jumps plays an important role in that case. Generally, the pathway model calculations suggest that mobile ions do not prefer large cages, but sites with the matching valence sum. Thereby Ag 1 ions often prefer mixed oxide–iodide coordinations. These sites are also generally involved in the most probable Ag 1 migration pathways, which are responsible for dc conductivity in these ion conducting systems. Additional contributions to the ac conductivity in the high frequency regime originate from closed loops of Ag 1 sites, among which a hopping requires a very low activation energy (cf. Fig. 13). Loops with different length scales exist even in the completely crystallized state.
Fig. 11. Views of the 3-dim. network of Ag 1 pathways in a snapshot of the local Ag 16 I 12 P2 O 7 model during the MD simulations (cf. Fig. 10).
74
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
Fig. 12. Correlation between the minimum valence mismatch for an infinite pathway network (from the calculations described in the text) and the activation energy for Ag 1 conduction. (Compounds with Ag 1 disorder: filled squares; g-AgI: hollow square).
An introduction of anion disorder into the model (as attempted in the preceding paragraph) complicates the graphs but leaves the basic features qualitatively unchanged. The Ag 1 transport follows specific energetically favourable pathways also in the disordered state. Still, this fundamental discrepancy with a free volume approach will hardly show up in experimentally observable quantities. None of our model structures contains an anion polymer network,
so that our conclusions might be limited to ‘molecular’ glasses. Recent RMC simulations support our findings suggesting that even in glasses with polymer anions (e.g. AgI–AgPO 3 ) ‘amorphous silver iodide’ (i.e. regions of iodide co-ordinated silver), the traditionally assumed prerequisite for high ionic conduction in glasses, is not found in relevant quantities [11]. Moreover, the calculations visualize that equilibrium sites in disordered systems should not be identified with lattice points. In many cases, where energy barriers between the loops of formal equilibrium sites become negligible (as in Figs. 2 and 13), the whole loop may, on certain time scales, be taken as the real ‘equilibrium site’. It appears noteworthy that the centers of these loops are generally not filled by an anion but the interactions of anions above and below the center exclude this region from the pathways.
Acknowledgements Stimulating discussions with A. Magistris and J. Kawamura are gratefully acknowledged.
References
Fig. 13. Detail from the valence map of Ag 26 I 18 W4 O 16 showing a closed loop of Ag 1 pathways attached to the infinite pathway network (the solid line indicates the direction of the a-axis).
[1] T. Minami, J. Non-Cryst. Solids 95–96 (1987) 107. [2] C.A. Angell, Solid State Ionics 18–19 (1986) 72. [3] St. Adams, K. Hariharan, J. Maier, Solid State Ionics 75 (1995) 193. [4] St. Adams, K. Hariharan, J. Maier, Solid State Ionics 86–88 (1996) 503. [5] S. Hoshino, T. Sakuma, Y. Fujii, Solid State Commun. 22 (1977) 763. ¨ [6] S.F. Radaev, L. Fink, M. Tromel, Z. Kristallogr. Suppl. 8 ¨ (1994) 628; Tromel, M., private communication. [7] St. Adams, Z. Kristallogr. 211 (1996) 770. [8] L.Y.Y. Chan, S. Geller, J. Solid State Chem. 21 (1977) 331. The corresponding ICSD entry contains an updated version from 1985. [9] S. Geller, G.F. Wilber, G.F. Ruse, J.R. Akridge, A. Turkovic, Phys. Rev. B 21 (1980) 2506. [10] J.D. Garrett, J.E. Greedan, R. Faggiani, S. Carbotte, I.D. Brown, J. Solid State Chem. 42 (1982) 183. ¨ [11] J. Swenson, L.R. McGreevy, L. Borjesson, J.D. Wicks, Solid State Ionics 105 (1998) 55, this issue.
Ag migration pathways in crystalline and glassy solid electrolytes AgI–AgM x O y a, b Stefan Adams *, Joachim Maier a
b
¨ Min.-Kristallograph. Institut, Univ. Gottingen , Germany ¨ Max-Planck-Institut f. Festkorperforschung , Stuttgart, Germany
Abstract Conduction pathways for silver ions in crystalline model structures of the AgI–AgM x O y electrolytes are derived from a bond valence sum model, which proved successful for a-AgI. Models for the silver transport in the related glassy phases are discussed in the light of our results. Keywords: Ion conductivity; Conduction pathways; Valence sum pseudopotentials; Silver ion conductors; Glassy electrolytes
1. Introduction Several years ago, the ionic conduction in AgI– AgM x O y glasses was mostly attributed to Ag 1 with a AgI-type co-ordination. Spectroscopic studies seemed to support this interpretation by the detection of different Ag 1 populations with distinguishable mobilities for local hopping (see e.g. references cited in reviews [1,2]). The pathway model presented in this paper, which is based on bond valence pseudopotential calculations, demonstrates that Ag 1 ions with mixed iodide–oxide co-ordinations crucially contribute to the ionic conduction in these systems. The investigations started with the glass system 4AgI–Ag 4 V2 O 7 , from which the compound Ag 8 I 4 V2 O 7 can be produced by congruent crystallization as well as by slow cooling of the melt of stoichiometric composition [3]. The close similarities
*Corresponding [email protected]
author.
Fax:
49
551
393825;
e-mail:
between the glassy and crystalline phase with respect to mass density, overall chemical composition, absolute conductivity, activation energy etc. [3,4] suggest that a determination of the conduction pathways in the crystalline phase should also be helpful for a discussion of the conduction mechanism in the glassy state.
2. Valence sum pseudopotential method Owing to the intrinsic disorder, single crystal structural data of Ag 1 ionic conductors are typically of moderate quality, so that a direct determination of Ag 1 pathways from a refinement of anharmonic contributions to the atomic displacement parameters (‘temperature factors’) remains restricted to simple structures such as a-AgI [5]. Pathways of lowest energy for the Ag 1 migration may, however, be estimated from the well-defined anion positions by means of simple valence sum pseudopotentials. The valence sum
0167-2738 / 98 / $19.00 1998 Elsevier Science B.V. All rights reserved. PII S0167-2738( 97 )00450-5
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
68
Os
V5
(1)
Ag2X
3. Ag 1 pathways in the crystalline state
X
for Ag is calculated for a grid of points throughout the asymmetric unit from empirical bond length bond valence equations. The parameter set of Eq. (2) by ¨ Radaev, Fink and Tromel [6] has been chosen among the various parameter sets in literature (see refs. cited in [4]), because it includes the influence of higher co-ordination shells:
F F
G G
˚ 2 RAg2O 1.89 A sAg2O 5 exp ]]]]] , ˚ 0.33 A ˚ 2 RAg2I 2.08 A sAg2I 5 exp ]]]]] . ˚ 0.53 A
(2)
˚ from the All anions up to a distance of 8 A respective site are included. The sum of radii of Ag and M in metals is assumed to be the shortest possible Ag–M distance. Pathways of lowest activation energy are assumed to be those connections between equilibrium positions, for which the maximum valence mismatch DV5uV2Vequilibrium u of the Ag bond valence is as small as possible. In the case of a-AgI this simple formalism closely reproduces the Ag density distribution results of neutron diffraction studies (Fig. 1, cf. Ref. [5]).
3.1. Ag8 I4V2 O7 From our recent single crystal structure determi] nation of Ag 8 I 4 V2 O 7 (space group P62m, a512.595 ˚ c59.119 A) ˚ [7], we derived Ag 1 migration A, pathways for this crystalline analogue of the 4AgI– 1 Ag 4 V2 O 7 glass [4]. The Ag sites clearly fall into two classes with distinct co-ordination shells: fully occupied sites with a nearly octahedral 4I 2 12O 22 co-ordination and 2O 22 13I 2 co-ordinated sites with occupation factors close to n52 / 3. The co-ordination of none of the Ag sites resembles the tetrahedral iodide co-ordination in a-AgI. Pathways of low valence mismatch (i.e. low activation energy) connect sites of the same type, but they permit only hops within closed loops that cannot account for the dc conductivity (cf. Fig. 2). Any reasonable long-range migration pathway includes both types of silver sites (see Fig. 3). Thus the Ag 1 conduction predominantly takes place in layers that extend perpendicular to the hexagonal axis via jumps between the fully occupied positions at z¯ 60.27 and the partially occupied positions at z50. Subsequently we extended the pseudopotential
Fig. 1. Network of Ag 1 conduction pathways in a-AgI (l.h.s.) and g-AgI (r.h.s.) as computed from the valence sum pseudopotentials. Equipotential contours are drawn only for the front half of the unit cell to reduce overlap.
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
69
Fig. 2. Sections of the Ag valence sum pseudopotential map of Ag 8 I 4 V2 O 7 (a) at z50 and (b) at z50.27.
Fig. 3. Detail from an Ag valence sum pseudopotential map of Ag 8 I 4 V2 O 7 for DV #0.45 (after Ref. [4]). Arrows mark the connections between the fully occupied sites (hatched contours) and the partially occupied sites (unhatched contours).
calculations to other crystalline compounds which may be understood as crystallized AgI2AgM x O y glasses. The ICSD crystal structure database reports two further crystal structures of related compounds: Ag 26 I 18 W4 O 16 and Ag 16 I 12 P2 O 7 . Isolated oxyacid 42 82 42 anions (V2 O 7 , W4 O 16 or P2 O 7 ) are present in all these compounds, so that a comparison of the results with the situation in glassy phases might be limited to ‘molecular glasses’, whereas no crystalline model structure is known for AgI2 AgM x O y glasses containing a polymeric network.
3.2. Ag26 I18 W4 O16 In the monoclinic room temperature phase of ˚ b5 Ag 26 I 18 W4 O 16 (space group C2, a516.76(3) A, ˚ ˚ 15.52(3) A, c511.81(2) A, b 5103.98) isolated 82 W4 O 16 ions reside in channels that are formed by 2 the framework of I ions. According to the structure refinement by Chan and Geller [8] the Ag 1 ions are distributed over 46 crystallographically different sites with a mean occupation factor of only 28.6% (Fig. 4).
70
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
Fig. 4. Crystal structure of Ag 26 I 18 W4 O 16 viewed along the c-axis 82 [8]. (W4 O 16 : polyhedra; Ag 1 : ellipsoids; I 2 network: sticks).
Moreover, their refinement results reflect a strong preference of the Ag 1 ions for mixed co2 ordination. The 90 polyhedra with pure I - co-ordination host only 22% of the 52 silver ions, while 78% of the Ag 1 reside in 92 polyhedra with mixed coordinations. In accordance with the prevailing theory at that time, the experimental results were taken as evidence for the immobilisation of Ag 1 by an oxide co-ordination. Still the authors argue that some of the Ag 1 with mixed coordinations will contribute to the conduction either as sources of mobile ions or as bypasses for complicated pathways through the iodide polyhedra. Under the assumption that only iodide co-ordinated Ag 1 should be mobile, the most direct Ag 1 pathway should exist along c*. This hypothesis was ruled out by a subsequent study of the anisotropic conductivity [9], since the conductivity along the a-direction is ca. 40% higher than along b and c* at ambient temperature. Due to a somewhat higher activation energy for the c*-direction, the conductivity along this direction reaches the conductivity along a at about 450 K. In agreement with the rather low overall anisotropy, valence sum calculations yield a three-dimensional network of pathways, for a valence mismatch of DV $0.19 (see Fig. 5b). A minor preference for
Fig. 5. Pathway models for the Ag 1 migration within the asymmetric unit of Ag 26 I 18 W4 O 16 : (a) linear connection of the refined Ag 1 positions (sticks are light within iodide polyhedra, dark within mixed polyhedra), (b) valence sum pseudopotential map (DV50.19).
the conduction along the a-direction might be derived from the somewhat larger cross area of these pathways.
3.3. Ag16 I12 P2 O7 The crystallographic description of Ag 16 I 12 P2 O 7 ˚ c57.43 A) ˚ [10], (space group P6 /mcc, a512.054 A, appears unsuitable for a straightforward estimation of migration pathways. Due to the disordered nature of the compound, structure determination yields only an average structure employing numerous sites with low occupation factors for both Ag 1 cations and pyrophosphate anions (cf. Fig. 6). The migration pathways rather depend on the actual local co-ordinations of the silver ions.
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
71
Fig. 6. Schematic representation of the crystal structure of Ag 16 I 12 P2 O 7 viewed along the hexagonal axis. (Ag 1 site network shown as grey rods; I 2 network as dark stick-and-ball; disordered P2 O 7 as light stick-and-ball).
Therefore, a molecular mechanics (MM) simulation based on the crystallographic information was applied to derive a chemically meaningful arrangement (especially for the pyrophosphate ions within the structure channels). MM calculations have been performed in a 23233 superstructure with the ‘quench molecular dynamics’ routine of the CERIUS 2 program suite using a slightly modified UNIVERSAL force field parameter set with constant
partial charges. The resulting local structure model (Fig. 7) reflects the expected features, e.g. the tilt angle of the pyrophosphate anions. The valence sum pseudopotential calculations yield a network of pathways, which exhibits close similarities to the models for the previously discussed compounds. Local closed loops of various diameters connect Ag 1 sites with a predominant iodide co-ordination (see Fig. 8). A rather low
Fig. 7. Anion framework within the local structure model of Ag 16 I 12 P2 O 7 from MM calculations (P2 O 7 : double tetrahedra; I 2 framework: stick-and-ball).
72
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
Fig. 8. Projections of the valence sum pseudopotential map of Ag 16 I 12 P2 O 7 for DV50.12 (l.h.s.) and the corresponding region of the structure model (r.h.s.). Anions are drawn as in Fig. 7, the network of Ag 1 sites is sketched as rods.
valence mismatch (DV50.12) is sufficient to link these loops and construct an infinite network of pathways, as necessary for long-range migration. These interconnecting pathways involve sites in the vicinity of the divanadate ions (Fig. 9). Direct hops between the loops avoiding sites with mixed coordinations require considerably higher activation energies. The unoccupied possible Ag 1 site with a pure iodide co-ordination at the center of the larger loops is not attached to the network.
4. Ag 1 pathways in the disordered state The preceding calculations were performed for models with a high degree of ordering within the anion substructure due to energy minimisations after the molecular dynamics runs. As an oversimplified approach to transfer the results of the study to the glassy state, we investigated also snapshots of the Ag 16 I 12 P2 O 7 structure during molecular dynamics runs such as the one shown in Fig. 10.
Fig. 9. Views of the 3-dim. network of Ag 1 pathways in crystalline Ag 16 I 12 P2 O 7 . Projections parallel to the hexagonal axis (r.h.s.) highlight links along the diphosphate ions between the loops of iodide co-ordinated Ag 1 sites.
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
Fig. 10. Snapshot of the local structure model for Ag 16 I 12 P2 O 7 during a molecular dynamics run (ions are drawn as in Fig. 8).
The corresponding valence sum pseudopotential map is given in Fig. 11. In this particular case pathways across the modeled region form already for a valence mismatch of DV50.10. A comparison between the pathway models for the ordered and disordered structures (Figs. 9 and 11) reveals that the essential features of the ordered model, such as local loops and links utilizing Ag 1 with mixed co-ordinations, are found as well in the more disordered state.
5. Concluding remarks The proposed technique produces vivid representa-
73
tions of possible Ag 1 migration pathways based on a very simple pseudopotential ansatz. For highly symmetrical crystalline compounds, where neutron diffraction data are available, the calculations closely mimic the experimental results [10]. The confidence in the method is raised by a clear relation between the minimum valence mismatch to create an infinite network and the experimental activation energy for the dc conductivity (Fig. 12). The activation energies of compounds with Ag 1 disorder seem to be roughly proportional to the logarithm of the minimum valence mismatch. For ordered compounds such as g-AgI the experimental value of DE is considerably higher than predicted from the ln(DV )2DE relation for the disordered compounds, suggesting that the low probability of successful jumps plays an important role in that case. Generally, the pathway model calculations suggest that mobile ions do not prefer large cages, but sites with the matching valence sum. Thereby Ag 1 ions often prefer mixed oxide–iodide coordinations. These sites are also generally involved in the most probable Ag 1 migration pathways, which are responsible for dc conductivity in these ion conducting systems. Additional contributions to the ac conductivity in the high frequency regime originate from closed loops of Ag 1 sites, among which a hopping requires a very low activation energy (cf. Fig. 13). Loops with different length scales exist even in the completely crystallized state.
Fig. 11. Views of the 3-dim. network of Ag 1 pathways in a snapshot of the local Ag 16 I 12 P2 O 7 model during the MD simulations (cf. Fig. 10).
74
S. Adams, J. Maier / Solid State Ionics 105 (1998) 67 – 74
Fig. 12. Correlation between the minimum valence mismatch for an infinite pathway network (from the calculations described in the text) and the activation energy for Ag 1 conduction. (Compounds with Ag 1 disorder: filled squares; g-AgI: hollow square).
An introduction of anion disorder into the model (as attempted in the preceding paragraph) complicates the graphs but leaves the basic features qualitatively unchanged. The Ag 1 transport follows specific energetically favourable pathways also in the disordered state. Still, this fundamental discrepancy with a free volume approach will hardly show up in experimentally observable quantities. None of our model structures contains an anion polymer network,
so that our conclusions might be limited to ‘molecular’ glasses. Recent RMC simulations support our findings suggesting that even in glasses with polymer anions (e.g. AgI–AgPO 3 ) ‘amorphous silver iodide’ (i.e. regions of iodide co-ordinated silver), the traditionally assumed prerequisite for high ionic conduction in glasses, is not found in relevant quantities [11]. Moreover, the calculations visualize that equilibrium sites in disordered systems should not be identified with lattice points. In many cases, where energy barriers between the loops of formal equilibrium sites become negligible (as in Figs. 2 and 13), the whole loop may, on certain time scales, be taken as the real ‘equilibrium site’. It appears noteworthy that the centers of these loops are generally not filled by an anion but the interactions of anions above and below the center exclude this region from the pathways.
Acknowledgements Stimulating discussions with A. Magistris and J. Kawamura are gratefully acknowledged.
References
Fig. 13. Detail from the valence map of Ag 26 I 18 W4 O 16 showing a closed loop of Ag 1 pathways attached to the infinite pathway network (the solid line indicates the direction of the a-axis).
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