Coupling between Ionic and Electronic Transport: Thermodynamic Perspectives

A thermodynamic framework for mixed ionic-electronic conduction in functional oxides, including electrochemical potentials, defect equilibria, transport coupling, and practical measurements.

1. Why ionic and electronic transport must be considered together

Many functional oxides conduct more than one type of charge carrier. Protons, oxide ions, and oxygen vacancies may contribute to ionic transport, while electrons and holes provide electronic transport. When two or more carrier populations are mobile, the material is described as a mixed ionic-electronic conductor (MIEC).

This situation is important in oxygen-permeation membranes, solid-oxide electrochemical cells, catalytic electrodes, protonic ceramics, and some oxide thermoelectrics. The measured conductivity is not necessarily the conductivity of a single carrier. Instead, it is the combined response of the mobile species under an electric field and, where relevant, chemical-potential gradients.

Thermodynamics provides the framework for determining the driving forces and equilibrium constraints. Transport theory describes how quickly species respond to those forces. Keeping these two levels distinct is essential: equilibrium defect concentrations do not by themselves determine mobility, and a high conductivity does not by itself establish which species carries the current.

2. Electrochemical potential: the driving force for charged species

For a species (i) with chemical potential (_i), charge number (z_i), and electric potential (), the electrochemical potential is

[ ]

when chemical potentials are expressed per mole and (F) is Faraday’s constant. For a single particle, the equivalent expression uses charge (q_i=z_ie) and energy units per particle.

The chemical contribution describes the tendency of a species to move down its chemical-potential gradient. The electrical contribution accounts for its interaction with the electric field.

For a charged species, the electrochemical-potential gradient is therefore

[ _i =_i+z_iF. ]

At equilibrium, the electrochemical potential of each independently mobile species is spatially uniform, subject to the relevant constraints and interfaces. A concentration gradient can therefore be balanced by an electric-potential gradient; zero net flux does not require every individual gradient to vanish.

3. Fluxes and transport coefficients

Near equilibrium, a common phenomenological description relates species fluxes to gradients of electrochemical potential:

[ _i=-j L{ij}(), ]

where (L_{ij}) are phenomenological transport coefficients. The exact form depends on the chosen fluxes and thermodynamic forces. Under isothermal conditions, the temperature factor can be absorbed into the coefficients, giving a simpler proportionality between flux and electrochemical-potential gradient.

The diagonal coefficients describe direct response, while off-diagonal coefficients represent coupled transport between different fluxes and forces. Onsager reciprocity constrains cross-coefficients close to equilibrium when the relevant microscopic reversibility conditions apply.

For practical oxide transport, simplified independent-carrier models are often useful. They should, however, be recognized as approximations that may neglect cross-coupling, concentration polarization, or interfacial limitations.

4. Ionic and electronic partial conductivities

If several carrier types respond independently to the same electric field in the linear-response regime, their partial conductivities add:

[ ]

where (_{}) is the total ionic contribution and (_e) is the electronic contribution. If both electrons and holes are present,

[ _e=_n+_p. ]

Likewise, if protons and oxide ions both contribute,

[ {}={^+}+_{{2-}}, ]

with the understanding that the symbols represent effective contributions from the mobile defects and their charge.

For a carrier with number density (n_i), charge magnitude (|q_i|), and mobility (u_i), a simple dilute-carrier expression is

[ _i=n_iq_i^2u_i. ]

This equation separates the effects of carrier population and mobility. Thermodynamics strongly influences the equilibrium population, while migration barriers, microstructure, and correlated motion influence mobility.

5. Defect chemistry couples ionic and electronic populations

In an oxide, the concentrations of vacancies, interstitials, protons, electrons, and holes are coupled by mass-action relations and charge neutrality.

For example, oxygen loss may be represented in Kröger–Vink notation as

[ ++2e’. ]

The equilibrium constant is related to the activities of the participating species. Under a simplified ideal dilute-defect approximation,

[ K_{}= a_{} a_{e’}^2 ()^{1/2}, ]

where the activity of the oxygen lattice site is incorporated into the standard-state convention. If defect activities are approximated by normalized concentrations, this expression can be converted into familiar concentration power laws.

The equilibrium concentrations must also satisfy charge neutrality. For a simple oxide containing doubly charged oxygen vacancies, electrons, and fixed acceptor dopants,

[ 2[V_{}^{}]+[h^{}] = [e’]+[A’] , ]

where the terms must be adapted to the actual defect species and sign convention of the material. This coupling means that changing oxygen partial pressure can alter both ionic and electronic carrier populations.

For a broader treatment of oxygen chemical potential and defect populations, see Oxygen Chemical Potential and pO₂-Dependent Defect Chemistry.

6. Protonic oxides: hydration links ionic and electronic defects

In a proton-conducting oxide, water incorporation can be written as

[ + + . ]

The equilibrium proton concentration depends on temperature, water-vapor pressure, oxygen-vacancy availability, and the dopant and defect chemistry. The hydration equilibrium therefore changes the population of an ionic carrier.

Electronic defects may also be present. Depending on composition and atmosphere, electrons or holes can compensate dopants or participate in redox equilibria. Consequently, proton concentration and electronic conductivity can respond differently to the same change in temperature or atmosphere.

A material with high proton concentration is not automatically a good proton electrolyte: electronic leakage can reduce the electrochemical efficiency of a cell. Hydration thermodynamics is discussed in Hydration Thermodynamics in Proton-Conducting Oxides.

7. Transport number: the fraction of current carried by each species

The transport number of carrier (i) is the fraction of total current attributed to that carrier under a specified set of conditions:

[ ]

For a material with ionic and electronic conduction,

[ t_{}= {_{}+e}, t_e= {{}+_e}. ]

Under this simple parallel-conduction description,

[ t_{}+t_e=1. ]

Transport numbers are not immutable material constants. They can vary with temperature, oxygen partial pressure, water-vapor pressure, composition, and the state of the sample. They may also depend on the measurement protocol if the sample is not at steady state or if electrode polarization is significant.

The experimental determination of transport numbers is covered in Transport Number: Measurement and Calculation.

8. Why ionic conductivity and ionic transport number are different

A high ionic conductivity does not necessarily mean that ionic transport dominates. Consider two illustrative materials:

• Material A: (_{}=0.01;) and (_e=0.0001;).
• Material B: (_{}=0.10;) and (_e=0.20;).

For A,

[ t_{}=. ]

For B,

[ t_{}=. ]

Material B has a higher absolute ionic conductivity, but only about one-third of its total conductivity is ionic in this simplified example. For an electrolyte, Material A may therefore be more selective for ionic transport, despite its lower ionic conductivity.

This distinction is particularly important when comparing electrolytes with electrodes or membrane materials, for which the desired balance of ionic and electronic conduction is different.

9. Ambipolar transport and internal electric fields

When oppositely charged species move together under a chemical gradient, local charge separation can generate an electric field. That field couples their fluxes and may constrain the net charge current.

A familiar example is ambipolar diffusion in an electronic semiconductor, where electron and hole motion become coupled by the internal field required to maintain approximate charge neutrality. Related electrochemical coupling can occur in mixed-conducting oxides containing mobile ionic and electronic defects.

The resulting chemical diffusion coefficient is generally not equal to a single tracer diffusion coefficient. It depends on the mobilities of the participating species and on thermodynamic factors describing how chemical potentials respond to concentration changes.

For a single component under restricted assumptions, chemical diffusion is often expressed schematically as

[ D_{}=D_{},, ]

where (D_{}) represents an appropriate mobility-related diffusivity and () is a thermodynamic factor. In multicomponent systems, the relation becomes matrix-valued and requires a coupled treatment. See Thermodynamic Factors in Ionic Transport.

10. Chemical diffusion in mixed conductors

Oxygen nonstoichiometry is a useful example of coupled transport in an oxide. A change in oxygen chemical potential at a surface can change the local oxygen-vacancy and electronic-defect concentrations. The resulting concentration gradients drive redistribution through the material.

The measured chemical diffusion coefficient reflects both the mobility of the relevant defects and the thermodynamic response of the defect population. It is therefore sensitive to composition and oxygen partial pressure.

In a mixed conductor, chemical diffusion can involve coupled ionic and electronic motion even when the experiment monitors only oxygen uptake or release. The electronic carriers provide the charge compensation needed for oxygen ions or vacancies to redistribute without sustaining a large macroscopic charge separation.

11. Open-circuit and short-circuit conditions

Boundary conditions strongly influence the measured response of a mixed conductor.

Open circuit means that the net external electrical current is zero. It does not necessarily mean that every individual carrier flux is zero. Ionic and electronic partial currents can, in principle, counterbalance each other while the total current vanishes.

Short circuit means that the electrical potential difference across the external terminals is constrained to zero. Internal electric fields and concentration gradients may still exist, and partial carrier fluxes can remain nonzero.

This distinction is central to interpreting oxygen permeation, concentration-cell, and Hebb–Wagner-type measurements. The measured voltage or flux depends on which species can cross each electrode or interface and on the boundary conditions imposed by the experimental configuration.

12. Electrochemical potential and the Nernst relation

For a reversible electrode reaction, equilibrium requires the electrochemical potentials of reactants and products to balance. This condition leads to the Nernst equation.

For an ideal oxygen concentration cell with reversible electrodes and a purely oxide-ion conducting electrolyte, the magnitude of the open-circuit voltage is

[ E= (), ]

where the pressure ratio is defined consistently with the electrode labeling. The factor four follows from the four electrons involved per oxygen molecule in the overall oxygen-electrode reaction.

If the electrolyte has appreciable electronic conductivity, the measured voltage can be lower than the ideal Nernst value. The reduction is associated with internal electronic leakage and the resulting mixed-conduction response. The exact relation depends on the electrode and transport model.

Thus, a deviation from the ideal Nernst voltage may indicate electronic leakage, but it should not be interpreted without checking electrode reversibility, gas sealing, temperature uniformity, and other experimental sources of error.

13. Materials design: electrolyte versus electrode

The desired ionic-electronic balance depends on the component’s function.

Component	Desired transport characteristics	Reason
Solid electrolyte	High ionic conductivity and low electronic conductivity	Suppress internal electronic leakage and maintain electrochemical potential difference
Fuel-cell or electrolyzer electrode	Adequate ionic and electronic conductivity	Extend reaction zones and provide both charge-transport pathways
Oxygen-permeation membrane	Significant ionic and electronic conductivity	Sustain coupled oxygen-ion and electron transport through the membrane
Mixed-conducting catalyst support	Application-dependent balance	Couple charge transfer with surface reaction and gas access

A material should therefore not be labeled universally “better” because its total conductivity is larger. The relevant performance metric depends on whether electronic conduction is beneficial, parasitic, or required.

14. Experimental methods for separating ionic and electronic contributions

No single measurement is sufficient for every mixed conductor. A defensible assessment combines complementary methods.

14.1 Impedance spectroscopy

Electrochemical impedance spectroscopy can resolve contributions associated with bulk, grain boundaries, and electrode processes when their characteristic responses are sufficiently separated. However, an impedance spectrum does not automatically identify a measured resistance as purely ionic or electronic.

Equivalent-circuit assignments should be supported by geometry, temperature dependence, electrode dependence, and independent measurements. See Practical EIS Data Analysis.

14.2 Hebb–Wagner polarization

A suitable ion-blocking/electron-reversible electrode configuration can be used to estimate electronic conductivity over a defined chemical-potential range. The interpretation requires well-characterized electrodes, controlled atmosphere, steady-state current, and a valid transport model.

14.3 Concentration cells and EMF measurements

The open-circuit voltage of a concentration cell can be compared with the Nernst prediction. A reduced voltage may provide evidence of mixed conduction, provided electrode kinetics and gas leakage are excluded.

14.4 Transference-number methods

Steady-state polarization and related electrochemical methods can estimate partial transport contributions. The chosen method must match the carrier species and electrode selectivity of the material.

14.5 Thermogravimetry and chemical diffusion

Thermogravimetric measurements under controlled oxygen partial pressure can quantify oxygen uptake or release. Relaxation measurements can then provide chemical diffusion and surface-exchange information when the geometry and kinetic model are appropriate.

15. A practical analysis workflow

For a new oxide suspected of mixed conduction, a useful sequence is:

1. Define the relevant mobile species. Identify possible ionic defects and electronic carriers from composition, atmosphere, and known defect chemistry.
2. Write the defect reactions. Establish mass-action relations and charge neutrality for the dominant regimes.
3. Map the thermodynamic conditions. Specify temperature, oxygen partial pressure, water-vapor pressure, and composition.
4. Measure total conductivity. Use a suitable four-probe or impedance method and verify the geometric factors.
5. Separate partial contributions. Select transport-number, polarization, or concentration-cell methods suited to the material and electrode configuration.
6. Check consistency. Compare the inferred partial conductivities with total conductivity and ensure the transport numbers satisfy the relevant sum rule.
7. Assess chemical diffusion and stability. Determine whether the material maintains its composition and phase during operation.
8. Connect transport to function. Evaluate whether electronic leakage is harmful or whether mixed conduction is essential for the intended device.

16. Common interpretation errors

• Equating total conductivity with ionic conductivity. Total conductivity may contain substantial electronic contributions.
• Assuming zero external current means zero partial flux. Counterflowing species can carry opposing partial currents under open-circuit conditions.
• Treating transport numbers as fixed constants. They can depend strongly on temperature and atmosphere.
• Inferring mobility from equilibrium concentration alone. Conductivity depends on both carrier concentration and mobility.
• Using a dilute-defect approximation outside its range. Defect interactions, association, and site saturation can invalidate simple power laws.
• Interpreting every sub-Nernst voltage as electronic leakage. Electrode polarization, gas leakage, and temperature gradients can also reduce measured EMF.
• Comparing materials without matching conditions. Partial conductivities measured at different temperatures or oxygen partial pressures are not directly comparable.

17. Key equations

Concept	Expression
Electrochemical potential	(_i=_i+z_iF)
Independent parallel contributions	(_{}=_i_i)
Partial conductivity, simple carrier model	(_i=n_iq_i^2u_i)
Transport number	(t_i=i/{})
Ionic transport number	(t_{}={}/({}+_e))
Ideal oxygen concentration-cell EMF magnitude	(E=(RT/4F)(p_{,2}/p_{,1}))
Restricted single-component chemical diffusion	(D_{}=D_{})

18. Conclusions

Ionic and electronic transport in functional oxides are coupled through electrochemical potentials, defect equilibria, charge neutrality, and internal electric fields. Thermodynamics determines the equilibrium populations and chemical driving forces; transport coefficients determine the response of those populations to the forces.

The distinction between total conductivity, partial conductivity, and transport number is fundamental. An electrolyte generally requires ionic selectivity, whereas an electrode or oxygen-permeation membrane may benefit from mixed conduction. Meaningful material assessment therefore requires atmosphere-dependent defect chemistry together with measurements that separate ionic and electronic contributions.

The practical objective is not simply to maximize total conductivity. It is to establish the carrier balance required by the intended function and to verify that this balance persists under the operating temperature, atmosphere, and electrochemical boundary conditions.

19. Further reading

• Thermodynamic Factors in Ionic Transport
• Defect Formation Energy and Equilibrium Concentrations in Oxides
• Oxygen Chemical Potential and pO₂-Dependent Defect Chemistry
• Hydration Thermodynamics in Proton-Conducting Oxides
• Transport Number: Measurement and Calculation
• Oxygen Surface Exchange and Chemical Diffusion