1. Skjalg Erdal 1 Dept. of Chemistry, Centre for Materials Science and Nanotechnology, University of Oslo, FERMIO, Gaustadalleen 21, NO-0349 Oslo, Norway 1 skjalg.erdal@smn.uio.no Mixed conduction Outline Defects Derivation of flux equations Flux of a particular species Fluxes in a mixed proton, oxygen ion, electron conductor Fluxes in a mixed proton, electron conductor –Various defect situations Fluxes in a mixed proton, oxygen ion conductor Some materials of interest Potential issues

2. Defects Defects are crucial to the functional properties of materials  Provide paths for transport  Provide traps for other species  Alters the solubility of alien species  Act as donors/acceptors  Become dominating migrating species themselves

3. Defects Defects typically dealt with:  Oxygen vacancies: V o ¨  Oxygen interstitials:O i ´´  Hydroxide ion on oxygen site:OH o ˙  Metal vacancies:V M ´´  Metal interstitials:M i ¨  Addition of aliovalent element to structure:Ca La ´  Electronic defects:e´ and h˙

4. Equilibria K = Equilibrium constant ΔH 0 = Stand. Enthalpy Change ΔS 0 = Stand. Entropy Change p X = Partial pressure of gas X R = Ideal gas constant T = Absolute temperature

5. Transport of charged species Nernst-Einstein: j i = flux density B i = mech. mobility c i = concentration z i = elem. charge F = Faradays const. μ i = chem. potential φ = el. potential σ i = conductivity D i = diff. coefficient i i = current density

6. Transport of charged species Then, use the definition of total conductivity and transport number to find an expression for the electrical potential gradient in terms of transport numbers and chemical potential gradient of all charge carriers: and

7. Transport of charged species We now need to represent the chemical potential of charged species as the chemical potential of their neutral counterparts. Assume chemical equilibrium between neutral and charged species and electrons, the electrochemical red-ox reaction: E ~ neutral chemical entity, z ~ + or - Equilibrium expression is in terms of products and reactants: Substitute into expression for The chemical potential for the neutral species can be expressed via activities, or partial pressures., Voltage over the sample: assume no gradient in electron chemical potential  last term becomes zero.

8. Transport of charged species Calculate flux density of species i, in the company of other species Represents the flux at a particular point in the membrane. We need the steady state condition  constant flux everywhere in membrane

9. Transport of charged species Expression for SS flux. Still with chemical potentials of charged species  substitute in expressions for neutrals Corresponding neutral species All charged species

10. Fluxes in mixed H +, O 2-, el-conductors Possible contributions to proton transport:  Ambipolar with oxygen ions:  Ambipolar with electrons: - driven  If conduction of oxygen ions in an oxygen gradient, with charge compensating flow of protons: - dep.

11. Fluxes in mixed H +, O 2-, el-conductors Equilibrium between hydrogen, oxygen and water: and

12. Fluxes in mixed H +, O 2-, el-conductor Equilibrium between hydrogen, oxygen and water:

13. Ambipolar proton-electron conduction Assume: Need to know how the proton conductivity and the electron transport number vary with in order to integrate the expression.

14. Ambipolar proton-electron conduction  Examples of partial pressure dependencies with varying defect situations Assuming:Protons minority defects Electronic transport no. ~1 Assuming:Protons majority defects, compensated by electrons Electronic transport no. ~1

15.  Examples of partial pressure dependencies with varying defect situations Assuming:Protons majority defects, compensated by acceptor dopants Electronic transport no. ~1 Ambipolar proton-electron conduction Proton concentration and conductivity independent of p H2

16. Ambipolar proton-electron conduction If the transport number of protons ~1: If protons charge compensate acceptors, the electronic conductivities have dependencies:

17. Ambipolar proton-electron conduction  Examples of partial pressure dependencies with varying defect situations Assuming:Protons majority defects, compensated by acceptors Protonic transport no. ~1 Limiting n-type conductivity Assuming:Protons majority defects, compensated by acceptors Protonic transport no. ~1 Limiting p-type conductivity

18. Ambipolar proton-oxygen ion conduction If the material is a mixed proton-oxygen ion conductor with negligible electronic transport number: Water vapor pressure ~ driving force If material is acceptor doped, and protons or oxygen vacancies can be majority (compensating) defects: If v O ¨ compensating:If H˙ compensating:

19. TITANATES TUNGSTATES NIOBATES TANTALATES Total conductivity / S/cm 2 3 4 5 6 7 0.001 2 3 4 5 10 -20 10 -15 10 -10 10 -5 10 0 La 0.99 Ca 0.01 NbO 4 Oxygen pressure, p O 2 / atm 1200 800 1000 LaTaO 4 LaNbO 4 La 6 WO 12 La 2 Ti 2 O 7 Oxygen pressure, p O 2 / atm 10 -20 10 -15 10 -10 10 -5 10 0 La 0.99 Ca 0.01 TaO 4 Total conductivity / S/cm 2 3 4 5 6 7 0.001 2 3 4 5 1200 1000 800 Material examples R. Haugsrud,2007

20. Material examples (pH2) 1/2 -dep ln (pH2)-dep.

21. Material examples Partial conductivities modeled under reducing conditions La 6 WO 12 Protons dominate until ~ 800 °C All conductivities rise with rising T, until ~ 800 °C Total conductivity rise with rising T, increase in entire T-window We have a small T-region wheredominates

22. Potential Issues Neutral H diffusion could lead to ambipolar conductivity measurements not telling the whole tale about hydrogen transport Wet H 2 Wet Ar H2H2 H+H+ e - H 2 = 2H + +2e - H2H2 H 2 O= H 2 +1/2O 2 O 2- Water splitting and oxygen conduction giving hydrogen on wrong side  difficult to measure correct hydrogen flux Dry sweep? H 2 = 2H H H2H2

23. Potential Issues What if the electronic transport number is dependent on oxygen partial pressure gradients? How do we integrate the expression for the flux density in such a case? Integration by parts over a beer, anyone?

24. Sources Norby, T. and Haugsrud, R., 2007, Membrane Technology Vol. 2: Membranes for energy conversion, Weinheim: WILEY-VCH Kofstad, P. and Norby. T, 2006, Defects and transport in crystalline solids, University of Oslo Haugsrud, R. 2007, New High-Temperature Proton Conductors (HTPC)-Applications in Future Energy Technology, New Materials for Membranes, GKSS Serra, E., Bini, A.C., Cosoli, G. and Pilloni, L., 2005, Journal of the American Ceramic Society, 88, 15-18 Cheng, S., Gupta, V.K. and Lin, J.Y.S., 2005, Solid State Ionics, 176, 2653-2663 Hamakawa, S., Li, L., Li, A. and Iglesia, E., 2002, Solid State Ionics, 148, 71-83