Home Physical Sciences Predicting conductivities of alkali borophosphate glasses based on site energy distributions derived from network former unit concentrations
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Predicting conductivities of alkali borophosphate glasses based on site energy distributions derived from network former unit concentrations

  • Marco Bosi and Philipp Maass EMAIL logo
Published/Copyright: November 30, 2021
Zeitschrift für Physikalische Chemie
From the journal Zeitschrift für Physikalische Chemie Volume 236 Issue 6-8

Abstract

For ion transport in network glasses, it is a great challenge to predict conductivities specifically based on structural properties. To this end it is necessary to gain an understanding of the energy landscape where the thermally activated hopping motion of the ions takes place. For alkali borophosphate glasses, a statistical mechanical approach was suggested to predict essential characteristics of the distribution of energies at the residence sites of the mobile alkali ions. The corresponding distribution of site energies was derived from the chemical units forming the glassy network. A hopping model based on the site energy landscape allowed to model the change of conductivity activation energies with the borate to phosphate mixing ratio. Here we refine and extend this general approach to cope with minimal local activation barriers and to calculate dc-conductivities without the need of performing extensive Monte-Carlo simulations. This calculation relies on the mapping of the many-body ion dynamics onto a network of local conductances derived from the elementary jump rates of the mobile ions. Application of the theoretical modelling to three series of alkali borophosphate glasses with the compositions 0.33Li2O–0.67[xB2O3–(1 − x)P2O5], 0.35Na2O–0.65[xB2O3–(1 − x)P2O5] and 0.4Na2O–0.6[xB2O3–(1 − x)P2O5] shows good agreement with experimental data.

Keywords: activation energy; conductivity; glasses; hopping motion; ion transport; transport theory
Corresponding author: Philipp Maass, Fachbereich Physik, Universität Osnabrück, Barbarastraße 7, 49076 Osnabrück, Germany, E-mail:
Dedicated to Paul Heitjans on the occasion of his 75th birthday.

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: 428906592

Acknowledgement

We sincerely thank the members of the DFG Research Unit FOR 5065 for fruitful discussions.

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, Project No. 428906592).

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

1. Fergus, J. W. Sensing mechanism of non-equilibrium solid-electrolyte-based chemical sensors. J. Solid State Electrochem. 2011, 15, 971–984; https://doi.org/10.1007/s10008-010-1046-4.Search in Google Scholar

2. Yoo, S. J., Lim, J. W., Sung, Y.-E. Improved electrochromic devices with an inorganic solid electrolyte protective layer. Sol. Energy Mater. Sol. Cells 2006, 90, 477–484; https://doi.org/10.1016/j.solmat.2005.04.033.Search in Google Scholar

3. Tervonen, A., Honkanen, S. K., West, B. R. Ion-exchanged glass waveguide technology: a review. Opt. Eng. 2011, 50, 1–16.10.1117/1.3559213Search in Google Scholar

4. Samui, A., Sivaraman, P. Polymer electrolytes. In Woodhead Publishing Series in Electronic and Optical Materials; Sequeira, C., Santos, D., Eds. Woodhead Publishing: Sawston, UK, 2010; Chapter 11, pp. 431–470.Search in Google Scholar

5. Kim, J. G., Son, B., Mukherjee, S., Schuppert, N., Bates, A., Kwon, O., Choi, M. J., Chung, H. Y., Park, S. A review of lithium and nonlithium based solid state batteries. J. Power Sources 2015, 282, 299–322; https://doi.org/10.1016/j.jpowsour.2015.02.054.Search in Google Scholar

6. Koettgen, J., Grieshammer, S., Hein, P., Grope, B. O. H., Nakayama, M., Martin, M. Understanding the ionic conductivity maximum in doped ceria: trapping and blocking. Phys. Chem. Chem. Phys. 2018, 20, 14291–14321; https://doi.org/10.1039/c7cp08535d.Search in Google Scholar

7. Meyer, M., Jaenisch, V., Maass, P., Bunde, A. Mixed alkali effect in crystals of β- and β″-alumina structure. Phys. Rev. Lett. 1996, 76, 2338–2341; https://doi.org/10.1103/physrevlett.76.2338.Search in Google Scholar

8. Dyre, J. C., Maass, P., Roling, B., Sidebottom, D. L. Fundamental questions relating to ion conduction in disordered solids. Rep. Prog. Phys. 2009, 72, 046501; https://doi.org/10.1088/0034-4885/72/4/046501.Search in Google Scholar

9. Maass, P. Towards a theory for the mixed alkali effect in glasses. J. Non-Cryst. Solids 1999, 255, 35–46; https://doi.org/10.1016/s0022-3093(99)00422-6.Search in Google Scholar

10. Hunt, A. G. Mixed-alkali effect: some new results. J. Non-Cryst. Solids 1999, 255, 47–55; https://doi.org/10.1016/s0022-3093(99)00423-8.Search in Google Scholar

11. Maass, P., Meyer, M., Bunde, A., Dieterich, W. Microscopic explanation of the non-Arrhenius conductivity in glassy fast ionic conductors. Phys. Rev. Lett. 1996, 77, 1528–1531; https://doi.org/10.1103/physrevlett.77.1528.Search in Google Scholar PubMed

12. Baranovskii, S. D., Cordes, H. On the conduction mechanism in ionic glasses. J. Chem. Phys. 1999, 111, 7546–7557; https://doi.org/10.1063/1.480081.Search in Google Scholar

13. Dyre, J. C., Schrøder, T. B. Universality of ac conduction in disordered solids. Rev. Mod. Phys. 2000, 72, 873–892; https://doi.org/10.1103/revmodphys.72.873.Search in Google Scholar

14. Porto, M., Maass, P., Meyer, M., Bunde, A., Dieterich, W. Hopping transport in the presence of site-energy disorder: temperature and concentration scaling of conductivity spectra. Phys. Rev. B 2000, 61, 6057–6062; https://doi.org/10.1103/physrevb.61.6057.Search in Google Scholar

15. Dieterich, W., Maass, P. Non-Debye relaxations in disordered ionic solids. Chem. Phys. 2002, 284, 439–467; https://doi.org/10.1016/s0301-0104(02)00673-0.Search in Google Scholar

16. Habasaki, J., Hiwatari, Y. Molecular dynamics study of the mechanism of ion transport in lithium silicate glasses: characteristics of the potential energy surface and structures. Phys. Rev. B 2004, 69, 144207; https://doi.org/10.1103/physrevb.69.144207.Search in Google Scholar

17. Lammert, H., Banhatti, R. D., Heuer, A. The cationic energy landscape in alkali silicate glasses: properties and relevance. J. Chem. Phys. 2009, 131, 224708; https://doi.org/10.1063/1.3272273.Search in Google Scholar PubMed

18. Vogel, M. Identification of lithium sites in a model of LiPO3 glass: effects of the local structure and energy landscape on ionic jump dynamics. Phys. Rev. B 2004, 70, 094302; https://doi.org/10.1103/physrevb.70.139902.Search in Google Scholar

19. Adams, S., Swenson, J. Determining ionic conductivity from structural models of fast ionic conductors. Phys. Rev. Lett. 2000, 84, 4144–4147; https://doi.org/10.1103/physrevlett.84.4144.Search in Google Scholar PubMed

20. Adams, S., Swenson, J. Bond valence analysis of reverse Monte Carlo produced structural models; a way to understand ion conduction in glasses. J. Phys. Condens. Matter 2005, 17, S87–S101; https://doi.org/10.1088/0953-8984/17/5/010.Search in Google Scholar

21. Müller, C., Zienicke, E., Adams, S., Habasaki, J., Maass, P. Comparison of ion sites and diffusion paths in glasses obtained by molecular dynamics simulations and bond valence analysis. Phys. Rev. B 2007, 75, 014203.10.1103/PhysRevB.75.014203Search in Google Scholar

22. Karlsson, M., Schuch, M., Christensen, R., Maass, P., Martin, S. W., Imberti, S., Matic, A. Structural origin of the mixed glass former effect in sodium borophosphate glasses investigated with neutron diffraction and reverse Monte Carlo modeling. J. Phys. Chem. C 2015, 119, 27275–27284; https://doi.org/10.1021/acs.jpcc.5b09176.Search in Google Scholar

23. Schuch, M., Christensen, R., Trott, C., Maass, P., Martin, S. W. Investigation of the structures of sodium borophosphate glasses by reverse Monte Carlo modeling to examine the origins of the mixed glass former effect. J. Phys. Chem. C 2012, 116, 1503–1511; https://doi.org/10.1021/jp2085654.Search in Google Scholar

24. Adams, S., Swenson, J. Bond valence analysis of transport pathways in RMC models of fast ion conducting glasses. Phys. Chem. Chem. Phys. 2002, 4, 3179–3184; https://doi.org/10.1039/b111310k.Search in Google Scholar

25. Schuch, M., Trott, C., Maass, P. Network forming units in alkali borate and borophosphate glasses and the mixed glass former effect. RSC Adv. 2011, 1, 1370–1382; https://doi.org/10.1039/c1ra00583a.Search in Google Scholar

26. Silver, A. H., Bray, P. J. Nuclear magnetic resonance absorption in glass. I. Nuclear quadrupole effects in boron oxide, soda-boric oxide, and borosilicate glasses. J. Chem. Phys. 1958, 29, 984–990; https://doi.org/10.1063/1.1744697.Search in Google Scholar

27. Eckert, H. Spying with spins on messy materials: 60 years of glass structure elucidation by NMR spectroscopy. Int. J. Appl. Glass Sci. 2018, 9, 167–187; https://doi.org/10.1111/ijag.12333.Search in Google Scholar

28. Youngman, R. NMR spectroscopy in glass science: a review of the elements. Materials 2018, 11, 476; https://doi.org/10.3390/ma11040476.Search in Google Scholar PubMed PubMed Central

29. Bosi, M., Fischer, J., Maass, P. Network-forming units, energy landscapes, and conductivity activation energies in alkali borophosphate glasses: analytical approaches. J. Phys. Chem. C 2021, 125, 6260–6268; https://doi.org/10.1021/acs.jpcc.0c09960.Search in Google Scholar

30. Ambegaokar, V., Halperin, B. I., Langer, J. S. Hopping conductivity in disordered systems. Phys. Rev. B 1971, 4, 2612–2620; https://doi.org/10.1103/physrevb.4.2612.Search in Google Scholar

31. Dörfler, F., Bullo, F. Kron reduction of graphs with applications to electrical networks. IEEE Trans. Circuits Syst. I Regul. Pap. 2013, 60, 150–163.10.1109/TCSI.2012.2215780Search in Google Scholar

32. Zielniok, D., Cramer, C., Eckert, H. Structure property correlations in ion-conducting mixed-network former glasses: solid-state NMR studies of the system Na2O–B2O3–P2O5. Chem. Mater. 2007, 19, 3162–3170; https://doi.org/10.1021/cm0628092.Search in Google Scholar

33. Rinke, M. T., Eckert, H. The mixed network former effect in glasses: solid state NMR and XPS structural studies of the glass system (Na2O)x(BPO4)1−x. Phys. Chem. Chem. Phys. 2011, 13, 6552–6565; https://doi.org/10.1039/c0cp01590c.Search in Google Scholar PubMed

34. Larink, D., Eckert, H., Reichert, M., Martin, S. W. Mixed network former effect in ion-conducting alkali borophosphate glasses: structure/property correlations in the system [M2O]1/3[(B2O3)x (P2O5)1−x]2/3 (M = Li, K, Cs). J. Phys. Chem. C 2012, 116, 26162–26176; https://doi.org/10.1021/jp307085t.Search in Google Scholar

35. Peibst, R., Schott, S., Maass, P. Internal friction and vulnerability of mixed alkali glasses. Phys. Rev. Lett. 2005, 95, 115901; https://doi.org/10.1103/physrevlett.95.115901.Search in Google Scholar

36. Maass, P., Peibst, R. Ion diffusion and mechanical losses in mixed alkali glasses. J. Non-Cryst. Solids 2006, 352, 5178–5187; https://doi.org/10.1016/j.jnoncrysol.2005.12.061.Search in Google Scholar

37. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., Teller, E. Equation of state calculations by fast computing machines. J. Chem. Phys. 1953, 21, 1087–1092; https://doi.org/10.1063/1.1699114.Search in Google Scholar

38. Hoshen, J., Kopelman, R. Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm. Phys. Rev. B 1976, 14, 3438–3445; https://doi.org/10.1103/physrevb.14.3438.Search in Google Scholar

39. Storek, M., Böhmer, R., Martin, S. W., Larink, D., Eckert, H. NMR and conductivity studies of the mixed glass former effect in lithium borophosphate glasses. J. Chem. Phys. 2012, 137, 124507; https://doi.org/10.1063/1.4754664.Search in Google Scholar PubMed

40. Christensen, R., Olson, G., Martin, S. W. Ionic conductivity of mixed glass former 0.35Na2O + 0.65[xB2O3 + (1 − x)P2O5] glasses. J. Phys. Chem. B 2013, 117, 16577–16586; https://doi.org/10.1021/jp409497z.Search in Google Scholar PubMed

41. Pollak, M. Effect of carrier–carrier interactions on some transport properties in disordered semiconductors. Discuss. Faraday Soc. 1970, 50, 13–19; https://doi.org/10.1039/df9705000013.Search in Google Scholar

42. Efros, A. L., Shklovskii, B. I. Coulomb gap and low temperature conductivity of disordered systems. J. Phys. C Solid State Phys. 1975, 8, L49–L51; https://doi.org/10.1088/0022-3719/8/4/003.Search in Google Scholar

43. Möbius, A., Richter, M., Drittler, B. Coulomb gap in two- and three-dimensional systems: simulation results for large samples. Phys. Rev. B 1992, 45, 11568–11579.10.1103/PhysRevB.45.11568Search in Google Scholar

44. Müller, M., Pankov, S. Mean-field theory for the three-dimensional Coulomb glass. Phys. Rev. B 2007, 75, 144201.10.1103/PhysRevB.75.144201Search in Google Scholar

45. Heitjans, P., Indris, S., Wilkening, M. Solid-state diffusion and NMR. Diffus. Fund. 2005, 2, 45.Search in Google Scholar

46. Vinod Chandran, C., Heitjans, P. In Annual Reports on NMR Spectroscopy; Webb, G. A., Ed. Academic Press: Cambridge, Massachusetts, USA, Vol. 89, 2016; pp 1–102.10.1016/bs.arnmr.2016.03.001Search in Google Scholar

47. Rinn, B., Dieterich, W., Maass, P. Stochastic modelling of ion dynamics in complex systems: dipolar effects. Philos. Mag. B 1998, 77, 1283–1292; https://doi.org/10.1080/13642819808205021.Search in Google Scholar

48. Schäfer, M., Weitzel, K.-M. Site energy distribution of ions in the potential energy landscape of amorphous solids. Mater. Today Phys. 2018, 5, 12–19.10.1016/j.mtphys.2018.05.002Search in Google Scholar

49. Schäfer, M., Budina, D., Weitzel, K.-M. Site energy distribution of sodium ions in a sodium rubidium borate glass. Phys. Chem. Chem. Phys. 2019, 21, 26251–26261.10.1039/C9CP05194ESearch in Google Scholar

50. Weitzel, K.-M. Charge attachment-induced transport – toward new paradigms in solid state electrochemistry. Curr. Opin. Electrochem. 2021, 26, 100672; https://doi.org/10.1016/j.coelec.2020.100672.Search in Google Scholar
Received: 2021-10-15
Accepted: 2021-11-06
Published Online: 2021-11-30
Published in Print: 2022-06-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Preface
  3. Special issue on the occasion of the 75th birthday of Paul Heitjans
  4. Contribution to Special Issue dedicated to Paul Heitjans
  5. Unusual cation coordination in nanostructured mullites
  6. A novel high entropy spinel-type aluminate MAl2O4 (M = Zn, Mg, Cu, Co) and its lithiated oxyfluoride and oxychloride derivatives prepared by one-step mechanosynthesis
  7. Two new quaternary copper bismuth sulfide halides: CuBi2S3Cl and CuBi2S3Br as candidates for copper ion conductivity
  8. Sintering behavior and ionic conductivity of Li1.5Al0.5Ti1.5(PO4)3 synthesized with different precursors
  9. Status and progress of ion-implanted βNMR at TRIUMF
  10. How Li diffusion in spinel Li[Ni1/2Mn3/2]O4 is seen with μ ±SR
  11. Nuclear magnetic resonance (NMR) studies of sintering effects on the lithium ion dynamics in Li1.5Al0.5Ti1.5(PO4)3
  12. Anion reorientations and cation diffusion in a carbon-substituted sodium nido-borate Na-7,9-C2B9H12: 1H and 23Na NMR studies
  13. Site preferences and ion dynamics in lithium chalcohalide solid solutions with argyrodite structure: I. A multinuclear solid state NMR study of the system Li6PS5-xSexI and of Li6AsS5I
  14. Site preferences and ion dynamics in lithium chalcohalide solid solutions with argyrodite structure: II. Multinuclear solid state NMR of the systems Li6PS5−x Se x Cl and Li6PS5−x Se x Br
  15. Independent component analysis combined with Laplace inversion of spectrally resolved spin-alignment echo/T 1 3D 7Li NMR of superionic Li10GeP2S12
  16. How the cation size impacts on the relaxational and diffusional dynamics of supercooled butylammonium-based ionic liquids: DPEBA–TFSI versus BTMA–TFSI
  17. Solid-state NMR studies of non-ionic surfactants confined in mesoporous silica
  18. Inorganic-organic hybrid materials based on the intercalation of radical cations: 2-(4-N-methylpyridinium)-4,4,5,5-tetramethyl-4,5-dihydro-1H-imidazol-1-oxyl-3-N-oxide in fluoromica clay
  19. Lithium tracer diffusion in near stoichiometric LiNi0.5Mn1.5O4 cathode material for lithium-ion batteries
  20. On the CaF2-BaF2 interface
  21. The ionic conductivity of alkali aluminum germanium phosphate glasses – comparison of Plasma CAIT with two electrode DC measurements
  22. Thin-film chemical expansion of ceria based solid solutions: laser vibrometry study
  23. Predicting conductivities of alkali borophosphate glasses based on site energy distributions derived from network former unit concentrations
  24. Ionic transport in K2Ti6O13
  25. F anion transport in nanocrystalline SmF3 and in mechanosynthesized, vacancy-rich Sm1—x BaxF3—x
  26. An overview of thermotransport in fluorite-related ionic oxides
https://doi.org/10.1515/zpch-2021-3137
Keywords for this article
activation energy; conductivity; glasses; hopping motion; ion transport; transport theory

Articles in the same Issue

  1. Frontmatter
  2. Preface
  3. Special issue on the occasion of the 75th birthday of Paul Heitjans
  4. Contribution to Special Issue dedicated to Paul Heitjans
  5. Unusual cation coordination in nanostructured mullites
  6. A novel high entropy spinel-type aluminate MAl2O4 (M = Zn, Mg, Cu, Co) and its lithiated oxyfluoride and oxychloride derivatives prepared by one-step mechanosynthesis
  7. Two new quaternary copper bismuth sulfide halides: CuBi2S3Cl and CuBi2S3Br as candidates for copper ion conductivity
  8. Sintering behavior and ionic conductivity of Li1.5Al0.5Ti1.5(PO4)3 synthesized with different precursors
  9. Status and progress of ion-implanted βNMR at TRIUMF
  10. How Li diffusion in spinel Li[Ni1/2Mn3/2]O4 is seen with μ ±SR
  11. Nuclear magnetic resonance (NMR) studies of sintering effects on the lithium ion dynamics in Li1.5Al0.5Ti1.5(PO4)3
  12. Anion reorientations and cation diffusion in a carbon-substituted sodium nido-borate Na-7,9-C2B9H12: 1H and 23Na NMR studies
  13. Site preferences and ion dynamics in lithium chalcohalide solid solutions with argyrodite structure: I. A multinuclear solid state NMR study of the system Li6PS5-xSexI and of Li6AsS5I
  14. Site preferences and ion dynamics in lithium chalcohalide solid solutions with argyrodite structure: II. Multinuclear solid state NMR of the systems Li6PS5−x Se x Cl and Li6PS5−x Se x Br
  15. Independent component analysis combined with Laplace inversion of spectrally resolved spin-alignment echo/T 1 3D 7Li NMR of superionic Li10GeP2S12
  16. How the cation size impacts on the relaxational and diffusional dynamics of supercooled butylammonium-based ionic liquids: DPEBA–TFSI versus BTMA–TFSI
  17. Solid-state NMR studies of non-ionic surfactants confined in mesoporous silica
  18. Inorganic-organic hybrid materials based on the intercalation of radical cations: 2-(4-N-methylpyridinium)-4,4,5,5-tetramethyl-4,5-dihydro-1H-imidazol-1-oxyl-3-N-oxide in fluoromica clay
  19. Lithium tracer diffusion in near stoichiometric LiNi0.5Mn1.5O4 cathode material for lithium-ion batteries
  20. On the CaF2-BaF2 interface
  21. The ionic conductivity of alkali aluminum germanium phosphate glasses – comparison of Plasma CAIT with two electrode DC measurements
  22. Thin-film chemical expansion of ceria based solid solutions: laser vibrometry study
  23. Predicting conductivities of alkali borophosphate glasses based on site energy distributions derived from network former unit concentrations
  24. Ionic transport in K2Ti6O13
  25. F anion transport in nanocrystalline SmF3 and in mechanosynthesized, vacancy-rich Sm1—x BaxF3—x
  26. An overview of thermotransport in fluorite-related ionic oxides
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