How to Measure and Calculate the Transport Number in Solid Electrolytes

A practical guide to ionic and electronic transport numbers in ceramic electrolytes, covering EMF concentration cells, Hebb–Wagner polarization, conductivity separation, calculations, and experimental validation.

SOLID-STATE IONICS · EXPERIMENTAL METHODS 27

How to Measure and Calculate the Transport Number in Solid Electrolytes

The transport number (also called the transference number) quantifies the fraction of the total electrical current carried by a specified charge carrier. In a nominally ionic ceramic electrolyte, it is not enough to demonstrate a high total conductivity: the material must also carry most of that current through the intended ionic species, with minimal electronic leakage.

This distinction is essential for oxide-ion, protonic, and other solid electrolytes used in fuel cells, electrolysers, hydrogen pumps, and gas-separation membranes. A material can exhibit a high conductivity while still having an unacceptable electronic contribution. Conversely, a mixed ionic–electronic conductor (MIEC) may be desirable as an electrode but unsuitable as a dense electrolyte.

This guide explains how to define, measure, and calculate transport numbers in solid electrolytes. The main emphasis is on ceramic oxides. Methods developed for liquid or polymer electrolytes are discussed separately because their assumptions and equations are not automatically transferable to dense oxide ceramics.

1. Definition of the transport number

For a charge carrier species (i), the transport number is the fraction of the total current density carried by that species:

\[ t_i=\frac{j_i}{j_{\mathrm{total}}} \]

For a material in which the relevant partial conductivities add under the same local conditions,

\[ t_i=\frac{\sigma_i}{\sigma_{\mathrm{total}}} \]

and

\[ \sigma_{\mathrm{total}}= \sum_i\sigma_i \]

For a simple oxide-ion conductor with oxide ions and electronic carriers,

\[ t_{\mathrm{O}^{2-}}+ t_e+t_h=1 \]

where (t_e) and (t_h) represent electron and hole transport numbers. If electronic conduction is represented by a single combined contribution,

\[ t_{\mathrm{ion}}=1-t_{\mathrm{e}} \]

This last expression is valid only when the current is partitioned into the stated ionic and electronic contributions and all relevant carriers are included.

A transport number is dimensionless and, under the usual steady-state decomposition, lies between zero and one. It is not a conductivity: a material can have a transport number close to unity but a very low ionic conductivity.

Transport number versus conductivity

Consider two materials:

• Material A: ({}=10^{{-2} \mathrm{S,cm}{-1}}), with (t{}=0.99).
• Material B: ({}=10^{{-1} \mathrm{S,cm}{-1}}), with (t{}=0.60).

Material B has the higher total conductivity, but 40% of its current is electronic under the stated conditions. Whether it is the better electrolyte depends on the application, geometry, operating conditions, and acceptable Faradaic loss.

2. Specify which ion is being measured

Before selecting an experiment, identify the mobile species and the operating conditions.

Electrolyte class	Intended mobile carrier	Typical competing contribution
Oxide-ion conductor	(^{2-})	Electrons or holes
Proton conductor	(^{+}) (protonic defects)	Electrons, holes, and sometimes oxide ions
Lithium-ion ceramic	(^{+})	Electronic leakage and, in some systems, other mobile ions
Hydride-ion conductor	(^{-})	Electronic conduction and other ionic carriers

The transport number is condition-dependent. Temperature, oxygen partial pressure, water-vapour pressure, hydrogen partial pressure, and applied electrochemical potential can change defect concentrations and partial conductivities. Therefore, a value should always be reported together with the temperature, atmosphere, gas composition, and measurement method.

For a protonic oxide, for example, reporting a single proton transport number without specifying (p_{}), (p_{}), and temperature may be insufficient to reproduce the result.

3. Which experimental method should be used?

No single method is suitable for every solid electrolyte. The most useful options for ceramic materials are:

1. EMF (concentration-cell) method: measures the voltage developed by a chemical-potential gradient across a membrane with reversible electrodes. It is especially useful for ionic transport numbers when the ionic contribution is sufficiently large.
2. Hebb–Wagner polarization: uses an ion-reversible electrode and an ion-blocking electrode. After ionic polarization reaches steady state, the residual current can be used to quantify electronic conduction.
3. Conductivity as a function of oxygen or hydrogen chemical potential: separates ionic and electronic contributions using defect-chemistry models. It is useful for mapping transport over a broad atmosphere range but depends on model assumptions.
4. Faradaic-efficiency or permeation measurements: compare the measured gas production or permeation rate with the charge passed. These methods provide an application-relevant check on ionic current efficiency.
5. Hittorf/Tubandt-type chemical analysis: determines transport from composition changes after passing a known charge. It can be valuable where reliable chemical analysis of the polarized solid is possible, but is experimentally demanding.

For a dense oxide electrolyte, EMF and Hebb–Wagner measurements are often the most direct complementary pair: one probes the voltage response to a chemical-potential gradient, while the other isolates the steady electronic leakage under an appropriate blocking-electrode configuration.

4. Method 1 — EMF (concentration-cell) method

4.1 Principle

A dense electrolyte membrane separates two gas atmospheres with different chemical potentials. Reversible electrodes on both sides exchange the relevant species with the gas phase. If the electrolyte conducts only the target ion, the open-circuit voltage approaches the Nernst voltage.

For a mixed conductor, electronic carriers provide an internal leakage path and the measured open-circuit voltage is generally smaller than the ideal ionic Nernst voltage. Under the assumptions of the selected concentration-cell model, the ratio of measured EMF to theoretical Nernst voltage provides an estimate of the ionic transport number.

For a single ionic species with charge number \(z\), the Wagner relation for a mixed conductor is

\[ E_{\\mathrm{OCV}}= \\frac{1}{zF} \\int_{\\mu_1}^{\\mu_2}t_i(\\mu)\\,d\\mu \]

where \(\mu\) is the chemical potential of the component exchanged at the electrodes. If the transport number is effectively constant over the imposed chemical-potential interval, this reduces to \(E_{\mathrm{OCV}}\approx t_i E_{\mathrm N}\). The ideal reversible-electrode Nernst voltage is

\[ E_{\\mathrm N}=\\frac{\\Delta\\mu_i}{zF} \]

where \(\Delta\mu_i\) is the corresponding chemical-potential difference per mole, \(F\) is the Faraday constant, and the sign depends on the chosen cell polarity and direction convention.

For an oxygen concentration cell with reversible oxygen electrodes,

\[ E_{\mathrm N}= \frac{RT}{4F} \ln\left( \frac{p_{\mathrm{O_2},2}} {p_{\mathrm{O_2},1}} \right) \]

where (R) is the gas constant and (T) is absolute temperature. The electrode labels and voltage polarity must be kept consistent when comparing measured and theoretical values.

In the simplest idealized interpretation,

\[ t_{\mathrm{O}^{2-}}\approx \left|\frac{E_{\mathrm{OCV}}}{E_{\mathrm N}}\right| \]

This is an approximation, not a universal correction formula. Electrode polarization, interfacial resistance, gas leakage, mixed electrode kinetics, and non-equilibrium gas composition can all reduce or distort the measured EMF. Use the correction procedure appropriate to the cell model and report the raw OCV and the impedance data as well as the calculated transport number.

4.2 Cell configuration

A typical high-temperature ceramic EMF cell contains:

• a dense, gas-tight electrolyte disc;
• a reversible porous electrode on each face, often Pt or Au for suitable test conditions;
• two gas compartments with independently controlled atmospheres;
• gas-tight seals that prevent mixing between the compartments;
• a furnace and thermocouple;
• gas-flow controllers and, where appropriate, oxygen or dew-point sensors;
• a high-input-impedance voltmeter or potentiostat;
• an impedance analyser for measuring the cell resistance.

The electrolyte should be dense enough to prevent gas crossover through open porosity. A relative density of at least 95% is a common starting target for these protocols, but gas-tightness must be verified experimentally rather than inferred from density alone.

4.3 Practical measurement procedure

1. Prepare a dense disc with flat, parallel faces and measure its dimensions and relative density.
2. Apply compatible porous electrodes to both faces and fire them using a procedure that does not react with or damage the electrolyte.
3. Mount the disc so that the two gas chambers are hermetically separated.
4. Leak-test the fixture and seals before introducing reactive gases.
5. Heat to the selected temperature and allow the sample and fixture to equilibrate.
6. Flow the same gas mixture to both sides first. The OCV should approach zero after correcting for instrument offsets and thermoelectric background voltages.
7. Keep one side at a fixed reference gas composition and vary the relevant partial pressure on the other side.
8. Measure the stabilized OCV at each gas composition.
9. Measure EIS under the same temperature and gas conditions to determine the ohmic and total cell resistances and assess electrode polarization.
10. Repeat the measurements at several gas ratios and temperatures, including a return sequence to check for hysteresis.

Do not assume that a fixed waiting time guarantees equilibrium. Record the voltage as a function of time and use a stabilization criterion appropriate to the instrument and material.

4.4 Worked EMF calculation

Suppose an oxygen concentration cell is measured at (T=800 ^C) (1073 K), with

(p_{,2}/p_{,1}=10^4).

Using

\[ E_{\mathrm N}= \frac{RT}{4F}\ln(10^4) \]

with (R=8.314 ) and (F=96485 ),

\[ E_{\mathrm N}\approx0.213\ \mathrm V \]

If the stabilized, background-corrected OCV is 0.192 V, the simple ratio gives

\[ t_{\mathrm{O}^{2-}}\approx \frac{0.192}{0.213}=0.901 \]

The approximate ionic transport number is therefore 0.90 under the stated conditions. This result should not be reported as a fully corrected value unless electrode-polarization and interfacial effects have been evaluated. The remaining 0.10 can be assigned to electronic transport only if oxide ions and electronic carriers are the only relevant charge carriers.

4.5 Protonic EMF measurements

For proton-conducting oxides, the gas chemical potentials are controlled using combinations of hydrogen, water vapour, oxygen, and inert gas. The reversible electrode reaction and the gas equilibria must be specified before calculating the theoretical Nernst voltage.

A protonic concentration cell is not necessarily described by simply substituting (^{+}) into the oxygen-electrode equation. In materials where protonic and oxide-ion conduction coexist, the measured EMF can contain contributions from more than one ionic species. The correct cell equation must account for the imposed hydrogen and water-vapour chemical-potential gradients and the relevant transport numbers.

For a mixed protonic–oxide-ion conductor, do not interpret one OCV ratio as the proton transport number unless the experimental configuration and thermodynamic derivation isolate that contribution.

4.6 Strengths and limitations

Strengths

• Directly probes the electrochemical response to a controlled chemical-potential gradient.
• Can distinguish predominantly ionic from mixed ionic–electronic behaviour.
• Can be applied over a range of temperatures and gas compositions.
• Particularly relevant to oxide-ion and proton-conducting ceramic membranes.

Limitations

• Requires gas-tight sealing and well-defined gas compositions.
• Electrode polarization and interfacial resistance can cause systematic errors.
• The method becomes less reliable when electronic conductivity is comparable to or greater than ionic conductivity.
• Multiple mobile ionic species require a more complete EMF model.
• Gas crossover can mimic a reduced transport number.

5. Method 2 — Hebb–Wagner polarization

5.1 Principle

The Hebb–Wagner method separates electronic current from ionic current by using electrodes with different reversibility toward the mobile ion.

One electrode is reversible to the mobile ionic species, while the other blocks its transfer across the interface. Under an applied DC voltage, ions initially contribute to the current but accumulate or become depleted at the blocking interface. Once the ionic concentration polarization reaches steady state, the remaining current is predominantly electronic, provided the electrode selectivity and cell stability assumptions are satisfied.

For a simple cell with a constant electronic conductivity and a uniform electric field approximation,

\[ \sigma_e\approx\frac{I_{\mathrm{ss}}L}{A\Delta V} \]

where (I_{}) is the steady-state current, (L) is the electrolyte thickness, (A) is the electrode area, and (V) is the effective voltage drop across the specimen.

The electronic transport number is then estimated from

\[ t_e=\frac{\sigma_e}{\sigma_{\mathrm{total}}} \]

and, for a two-carrier ionic/electronic conductor,

\[ t_{\mathrm{ion}}=1-t_e \]

The constant-conductivity expression is a simplified geometry-based estimate. In a mixed conductor where electronic conductivity varies with oxygen chemical potential or local potential, the current–voltage response must be interpreted using the appropriate Wagner integral or a validated transport model. Do not assume that a single steady-state current at one voltage always gives an intrinsic, voltage-independent (_e).

5.2 Experimental configuration

A typical oxide-ion Hebb–Wagner configuration uses:

• one electrode reversible to oxygen exchange;
• one electrode that blocks oxide-ion transfer but allows electronic contact;
• a dense electrolyte specimen;
• a controlled atmosphere and temperature;
• a DC source or potentiostat;
• current monitoring over time;
• EIS or an independent total-conductivity measurement.

The choice of electrode materials is critical. The nominally blocking electrode must not become ion-reversible through a side reaction, diffusion, or an unintended catalytic process. The reversible electrode must maintain the required electrochemical equilibrium.

5.3 Practical measurement procedure

1. Prepare a dense, chemically stable specimen and measure its dimensions.
2. Apply the reversible and blocking electrodes to the intended faces.
3. Verify that the contacts are mechanically stable and that the electrode configuration does not short-circuit the specimen.
4. Equilibrate at the target temperature and atmosphere.
5. Measure the initial impedance and/or total conductivity.
6. Apply a small, controlled DC bias and record current as a function of time.
7. Continue until a reproducible steady-state current is reached. A plateau should be established by a stated current-drift criterion, not by an arbitrary fixed duration.
8. Repeat at several voltages and, where relevant, both polarities.
9. Check for nonlinearity, hysteresis, electrode degradation, and irreversible changes.
10. Calculate the electronic conductivity using the appropriate geometry and transport model, then calculate the transport number from partial and total conductivities.

5.4 Worked calculation

Consider a dense electrolyte with:

• thickness (L=1.0 =1.0^{-3} m);
• electrode area (A=0.50 =5.0^{{-5} \mathrm{m}2});
• effective specimen voltage (V=0.10 V);
• steady-state current (I_{}=5.0 A).

Using the simplified expression,

\[ \sigma_e\approx \frac{(5.0\times10^{-6})(1.0\times10^{-3})} {(5.0\times10^{-5})(0.10)} \]

\[ \sigma_e\approx1.0\times10^{-3}\ \mathrm{S\,m^{-1}} \]

If the total conductivity measured under the same conditions is

\[ \sigma_{\mathrm{total}}=0.10\ \mathrm{S\,m^{-1}} \]

then

\[ t_e=\frac{0.001}{0.10}=0.010 \]

and, for a two-carrier system,

\[ t_{\mathrm{ion}}=1-0.010=0.990 \]

The estimated ionic transport number is 0.99. This numerical example assumes that the measured steady-state current is electronic, that the effective voltage across the specimen is known, and that the simple uniform-conductivity approximation is valid.

5.5 Strengths and limitations

Strengths

• Directly targets electronic leakage in an otherwise ion-conducting electrolyte.
• Useful when the ionic conductivity is much greater than the electronic conductivity.
• Can be performed under controlled temperature and gas composition.
• Provides an independent complement to EMF measurements.

Limitations

• Requires a genuinely ion-blocking electrode and a reversible counter-electrode.
• Long polarization times may be needed.
• Electrode reactions, interfacial resistance, and chemical degradation can distort the steady current.
• The apparent electronic conductivity may depend on applied voltage and oxygen chemical potential.
• A simple (I_{}L/(AV)) calculation is not sufficient for every mixed conductor.

6. Method 3 — Conductivity versus oxygen partial pressure

Total conductivity can be measured as a function of oxygen partial pressure, temperature, and, where relevant, water-vapour or hydrogen partial pressure. Defect chemistry relates the concentrations of ionic and electronic defects to the gas chemical potential.

For an oxide-ion conductor, the total conductivity can be written as

\[ \sigma_{\mathrm{total}}= \sigma_{\mathrm{O}^{2-}}+ \sigma_e+\sigma_h \]

In a regime where oxide-ion conductivity is approximately independent of oxygen partial pressure, while electronic conductivity follows a known power law, the measured conductivity curve can be fitted to separate the contributions.

For example, under a particular defect-chemistry regime,

\[ \sigma_{\mathrm{total}}= \sigma_{\mathrm{ion}}+ K_e p_{\mathrm{O_2}}^{-1/4} + K_h p_{\mathrm{O_2}}^{1/4} \]

The exponents are not universal. They depend on the dominant defects, charge-neutrality condition, dopant chemistry, and thermodynamic regime. They must be derived from the defect model rather than assumed for every oxide.

Once the partial conductivities are obtained,

\[ t_{\mathrm{ion}}= \frac{\sigma_{\mathrm{ion}}} {\sigma_{\mathrm{total}}} \]

This approach is valuable for mapping a material’s transport behaviour across operating conditions. Its reliability depends on the validity of the defect-chemical model and on the assumption that the specimen equilibrates with the imposed atmosphere.

7. Method 4 — Faradaic efficiency and permeation measurements

For an electrolyte used in an electrochemical device, an independent check is to compare the measured gas production or permeation rate with the theoretical rate predicted by Faraday’s law.

For hydrogen production,

\[ \dot n_{\mathrm{H_2,theory}}= \frac{I}{2F} \]

For oxygen production,

\[ \dot n_{\mathrm{O_2,theory}}= \frac{I}{4F} \]

The Faradaic efficiency is

\[ \mathrm{FE}= \frac{\dot n_{\mathrm{measured}}} {\dot n_{\mathrm{theory}}} \times100\% \]

A low Faradaic efficiency can indicate electronic leakage, gas crossover, parasitic electrode reactions, or measurement losses. It should not automatically be equated to (t_{}), because the gas balance also depends on electrode kinetics, sealing, gas flow, and side reactions.

For a simple device where the target ionic current is the only desired Faradaic process and all products are quantitatively collected, Faradaic efficiency can provide a useful application-level estimate of ionic current utilization.

8. Transport number from partial conductivities: a complete example

Suppose a ceramic oxide has, at one specified temperature and atmosphere,

\[ \sigma_{\mathrm{ion}}=0.085\ \mathrm{S\,cm^{-1}} \]

and

\[ \sigma_{\mathrm{electronic}}=0.015\ \mathrm{S\,cm^{-1}} \]

The total conductivity is

\[ \sigma_{\mathrm{total}}=0.085+0.015 =0.100\ \mathrm{S\,cm^{-1}} \]

The ionic transport number is

\[ t_{\mathrm{ion}}= \frac{0.085}{0.100}=0.85 \]

The electronic transport number is

\[ t_{\mathrm{electronic}}= \frac{0.015}{0.100}=0.15 \]

The material therefore carries 85% of the current ionically and 15% electronically under the stated conditions.

If the application requires an ionic transport number above 0.99, this composition would not meet that target even though its total conductivity may appear attractive.

9. How to measure total conductivity correctly

Transport numbers calculated from partial conductivities are only as reliable as the total-conductivity measurement.

For a bar measured by a four-probe DC method,

\[ \sigma_{\mathrm{total}}= \frac{IL}{VA} \]

where (I) is the applied current, (V) is the measured voltage between the inner voltage probes, (L) is the distance between those probes, and (A) is the cross-sectional area.

For a pellet measured by impedance spectroscopy,

\[ \sigma_{\mathrm{total}}= \frac{L}{R_{\mathrm{bulk}}A} \]

Use the resistance appropriate to the quantity being reported. If grain-boundary resistance is included in the intended total through-sample response, use the corresponding total resistance rather than only the high-frequency bulk resistance. State clearly whether the reported conductivity is bulk, grain-boundary, or total effective conductivity.

Measurements of total and partial conductivity must be performed at matching temperature, atmosphere, hydration state, and chemical potential. Combining values measured under different conditions produces a transport number without a well-defined physical meaning.

10. What about the Bruce–Vincent method?

The Bruce–Vincent DC polarization method is widely used for estimating cation transference numbers in certain polymer electrolytes and lithium-metal cells. It combines the initial and steady-state current with impedance measurements before and after polarization.

A commonly used expression is

\[ t_{+}= \frac{I_{\mathrm{ss}}(\Delta V-I_0R_0)} {I_0(\Delta V-I_{\mathrm{ss}}R_{\mathrm{ss}})} \]

where (I_0) and (I_{}) are the initial and steady-state currents, and (R_0) and (R_{}) are the interfacial resistances estimated before and after polarization under the method’s assumptions.

This expression is not a general transport-number equation for oxide ceramics. Its validity depends on the electrolyte model, electrode reversibility, concentration polarization, interfacial stability, and the assumptions used in the original analysis. Do not apply it to a ceramic oxide simply because a DC polarization curve has an initial and steady-state current.

11. Common errors and how to avoid them

• Using total conductivity as ionic conductivity. Total conductivity includes every mobile charge carrier.
• Assuming (t_{}=1) because the material is called an electrolyte. Electronic leakage must be measured or bounded.
• Using the EMF ratio without checking electrode polarization. Report OCV, theoretical Nernst voltage, EIS data, and any correction model.
• Ignoring gas leakage. A leaking seal can reduce the measured voltage and mimic mixed conduction.
• Using the wrong Nernst equation. The equation must match the reversible electrode reaction and imposed chemical-potential gradient.
• Ignoring multiple ionic species. Protonic and oxide-ion contributions may coexist in hydrated oxides.
• Treating every steady-state DC current as electronic. This requires a validated ion-blocking interface and stable specimen.
• Using an arbitrary polarization voltage. Large bias can change defect concentrations, trigger nonlinear transport, or damage electrodes.
• Combining conductivity and transport-number data from different conditions. Both must refer to the same temperature and chemical state.
• Confusing Faradaic efficiency with a transport number. Gas losses and parasitic reactions also affect measured efficiency.
• Applying polymer-electrolyte equations to ceramic oxides without justification. The electrode reactions and transport models are different.

12. Recommended experimental strategy for ceramic oxides

For a new oxide-ion or proton-conducting ceramic, a defensible workflow is:

1. Confirm phase purity, density, microstructure, and chemical stability over the intended temperature and atmosphere range.
2. Measure total conductivity as a function of temperature and the relevant gas partial pressures.
3. Select an EMF cell configuration that isolates the target ionic species as closely as possible.
4. Verify gas-tightness and electrode reversibility; measure the zero-gradient background voltage.
5. Collect OCV and EIS data under identical conditions and evaluate interfacial polarization.
6. Where electronic leakage is expected to be small but important, perform a complementary Hebb–Wagner measurement with validated electrode selectivity.
7. Compare the resulting partial conductivities and transport numbers with defect-chemistry expectations.
8. For device-relevant materials, validate the result using gas permeation or Faradaic-efficiency measurements.
9. Repeat measurements and report uncertainty, sample-to-sample variation, and the operating conditions.

For predominantly ionic electrolytes, the EMF method is often the most straightforward starting point. For materials with measurable electronic leakage, combining EMF with Hebb–Wagner polarization gives a stronger assessment than relying on either method alone.

13. Reporting checklist

A transport-number result should include:

• exact composition and phase;
• target carrier and definition of the reported transport number;
• temperature and equilibration procedure;
• oxygen, hydrogen, and water-vapour partial pressures as relevant;
• sample dimensions, density, and evidence of gas-tightness;
• electrode composition, geometry, and firing procedure;
• cell configuration and polarity convention;
• raw OCV or current–time data;
• EIS spectra and the resistance values used;
• equations and assumptions used for the Nernst voltage or polarization model;
• total and partial conductivities under matching conditions;
• replicate measurements and uncertainty;
• evidence that the sample and electrodes remained chemically stable.

Further reading

• J. S. Hardy, A. P. Kulkarni, J. W. Stevenson and O. A. Marina, “Conductivity and Transference Number Determination Protocols for Solid Oxide Cell Materials,” Frontiers in Energy Research 10 (2022), 867783. https://doi.org/10.3389/fenrg.2022.867783.
• T. Norby, “EMF Method Determination of Conductivity Contributions from Protons and Other Foreign Ions in Oxides,” Solid State Ionics 28–30 (1988), 1586–1591. https://doi.org/10.1016/0167-2738(88)90424-9.
• I. Riess, “Four Point Hebb-Wagner Polarization Method for Determining the Electronic Conductivity in Mixed Ionic-Electronic Conductors,” Solid State Ionics 51 (1992), 219–229. https://doi.org/10.1016/0167-2738(92)90204-3.
• M. Liu and H. Hu, “Effect of Interfacial Resistance on Determination of Transport Properties of Mixed-Conducting Electrolytes,” Journal of The Electrochemical Society 143 (1996), L109–L112. https://doi.org/10.1149/1.1836892.
• “Review: Measurement of partial electrical conductivities and transport numbers of mixed ionic-electronic conducting oxides,” Journal of Power Sources (2022). https://www.sciencedirect.com/science/article/pii/S0378775322002208.
• P. G. Bruce, J. Evans and C. A. Vincent, “Conductivity and transference number measurements on polymer electrolytes,” Solid State Ionics 28–30 (1988), 918–922. https://doi.org/10.1016/0167-2738(88)90304-9.

Related articles

• Electrochemical impedance spectroscopy: a practical guide
• Impedance data analysis: equivalent circuits and DRT
• Mixed ionic–electronic conductors
• Proton-conducting oxides
• Ionic transport in solid oxides
• Oxygen nonstoichiometry measurement
• Electrochemical hydrogen pumping and separation
• Fuel cell and electrolyzer testing

A transport number is not a universal material constant. It is a condition-dependent fraction of current carried by a specified charge carrier. Always report the carrier, temperature, atmosphere, measurement method, and assumptions used in the calculation.