Many functional materials, in particular those that are used as electrodes in batteries or fuel cells, typically exhibit mixed conductivity stemming from ionic and electronic carriers. Transport in such systems is well understood, as long as only two carriers are involved which are directly coupled by the local electroneutrality condition. But when a third carrier comes into play, the situation becomes perceptibly more complex. A striking example is the behavior of Fe-doped SrTiO3 single crystals after an increase of water partial pressure pH2O  shown in Fig. 1.
Fig. 1: In-situ optical absorption images of a Fe-doped SrTiO3 single crystal after increase of water vapor pressure from 4 to 20 mbar in O2 at 650°C (the largest faces of the crystal are sealed by glass, restricting exchange with the gas phase to the left and right edges which are coated with porous Pt). The higher the concentration of Fe4+, the darker the image. The symbols show the corresponding change of normalized electrical conductance. Reproduced from  with permission, copyright Wiley-VCH (2007).[less]
Fig. 1: In-situ optical absorption images of a Fe-doped SrTiO3 single crystal after increase of water vapor pressure from 4 to 20 mbar in O2 at 650°C (the largest faces of the crystal are sealed by glass, restricting exchange with the gas phase to the left and right edges which are coated with porous Pt). The higher the concentration of Fe4+, the darker the image. The symbols show the corresponding change of normalized electrical conductance. Reproduced from  with permission, copyright Wiley-VCH (2007).
H2O + O2- → 2 OH-. (1)
While such reactions can easily take place on the surface, oxygen vacancies are required for this reaction to occur in the bulk of a material. Hereby the "OH–" part of the water occupies the vacancy while the "H+" part combines with a regular O2– to form a second OH-. This reaction does not only enable water absorption but also leads to proton conductivity (phonon-assisted proton hopping). Mechanistically this is written as
(denotes the oxygen vacancy, is a protonic defect, i.e. a hydroxide ion on an oxide ion site). Naively, one would expect that the modified carrier concentrations propagate into the sample by simultaneous motion of H+ and O2–, which mechanistically means ambipolar diffusion of and which are coupled through local electroneutrality. However, the changes of optical absorption (measuring the Fe4+ concentration) as well as conductivity (monitoring the concentration of electron holes) unambiguously show that two processes with different time scale take place: (i) fast hydrogen uptake, leading to a reduction of the sample and decreasing the hole conductivity below the final value. (ii) slow oxygen uptake, bringing the sample (almost) back to its initial oxidation state. Obviously, the overall water uptake by the acid-base reaction is decoupled into two redox reactions, leading to a non-monotonic change of the hole concentration. This is due to a sufficiently high concentration of a third mobile carrier (holes), which is able to charge compensate the fast uptake of protons (having a higher mobility than ) as well as the later filling od the oxygen vacancies.
On the component level this can be written as
H2O → 2 H (fast) + O (sluggish) (3)
While this is the most striking example, various other "anomalies" have been found in literature.
To quantitatively analyze transport in a three carrier system, we start out from the known flux equations relating the defect flux to the respective conductivity and gradient in the electrochemical potential of the carrier (assuming that the surface reaction is very fast):
The vacancy and proton flux can also be expressed with gradients in the oxygen and water chemical potential , as driving force (transference number ti = σi/σtot):
In a system comprising only protons and oxygen vacancies as carriers, these equations can be used to analytically calculate water uptake by coupled oxygen vacancy and proton diffusion. For a three carrier material the situation is more complex because, e.g., does not only depend on the oxygen concentration, but in a nontrivial way also on the water concentration in the material [C1]. To resolve this issue, the component potentials , are expressed in terms of defect concentrations (assuming ideally dilute behavior)
where Di denote the defect diffusivities. Unlike the chemical potentials of the components, the chemical potentials of the ions and hence of the charged defects are – under dilute conditions – unambiguous functions of the respective concentrations. The quite intricate behavior of three carrier systems arises from the fact that charge neutrality must be fulfilled only for the ensemble of all three defects (in two carrier systems it relates the two defect concentrations by a simple prefactor). This means that the flux of one defect is not only determined by its own gradient, but depends also on the independent gradient of one more defect as indicated in the equations above. The detailed solution is given in Ref. . In Fig. 2 we only want to show their use in terms of numerical calculations.
Fig. 2(a) demonstrates that the proton concentration changes (middle panel) penetrate into the material much quicker than the profile (top panel), and that transiently larger hole concentration changes occur although for complete relaxation the hole concentration almost comes back to its initial value. The time evolution of the integral defect concentrations in Fig. 2(b) also reflects this behavior. It shows that in the initial, fast process the proton concentration increase is charge-compensated by a corresponding decrease of holes (= reduction of the material). The slower annihilation of oxygen vacancies is to smaller degree compensated by further proton uptake, and to larger degree by an increase of the hole concentration (= sample oxidation).
Since the flux of one defect is not only driven by its own gradient, the other defect gradients can lead to situations where one defect moves opposite to its gradient. This apparent "uphill diffusion" is visible in Fig. 2(b) for holes during the first stage when the hole concentration continued to decrease although it is already lower than the final equilibrium value (note, however, that in terms of the defect's electrochemical potential the diffusion is always downhill). The simulation also allows us to rationalize the experimentally observed "moving boundary phenomenon" .
It naturally follows from the flux equations that the effective proton diffusivity is lower in the outer region, and higher in the inner region separated by the "moving boundary": in the outer region the contributions from and to the proton flux partially cancel owing to opposite signs of and , while in the inner region is zero. The "moving boundary" is located where vanishes (corresponding to the minimum in the hole profile, cf. the dotted line for the red profile in Fig. 2(a)), and slowly moves inward with increasing time.
Having analyzed the space and time evolution of the defect concentrations in a case of decoupled diffusion, let us now explore the conditions for such a decoupling. To obtain the equilibrium defect concentrations, the hydration reaction as well as the oxidation reaction
must be considered. The combination of reactions (2) and (8) describes proton uptake at expense of holes, i.e. a hydrogenation reaction:
For given mass action constants KO, KW of reactions (2), (8) and partial pressures all defect concentrations can be calculated. Results are shown in Fig. 3(a), which represents a "map of materials": each point in the base plane corresponds to a set of KO, KW values, and thus represents a material with a specific propensity for water and oxygen uptake. For example, materials located in the left part of the map are oxide ion and/or proton conducting electrolytes because formation of electronic carriers (holes) by reaction (8) is quite unfavorable. From the defect model one can furthermore calculate which defect mainly compensates the proton uptake. When for electrolyte-type materials the grey surface is higher than the pink surface (the concentration change is higher than that of the holes), proton incorporation happens at expense of by the hydration reaction. On the other hand, for materials with high redox activity close to the front right edge of the map it occurs predominantly at expense of holes by hydrogenation.
Fig. 3: (a) "Materials map": a set of mass action constants KO, KW in the base plane specifies the redox- and hydration activity of a material. Grey surface = oxygen vacancy concentration change, pink = hole concentration change upon pH2O increase. (b) Initial defect fluxes after a pH2O increase at constant pO2; grey surface = oxygen vacancy flux, cyan mesh = proton flux, pink surface = hole flux. The solid line in the base plane marks the border between water and hydrogen uptake as derived for equilibrium conditions, cf. (a); the dashed line is the demarcation between initial predominant water uptake (left hind) and hydrogen uptake (front right), obtained in the kinetic picture from the intersection of the grey and pink surfaces. In the zone in between, decoupled stoichiometry relaxation kinetics is observed. Reproduced from  with permission, copyright Wiley-VCH (2015).[less]
Fig. 3: (a) "Materials map": a set of mass action constants KO, KW in the base plane specifies the redox- and hydration activity of a material. Grey surface = oxygen vacancy concentration change, pink = hole concentration change upon pH2O increase. (b) Initial defect fluxes after a pH2O increase at constant pO2; grey surface = oxygen vacancy flux, cyan mesh = proton flux, pink surface = hole flux. The solid line in the base plane marks the border between water and hydrogen uptake as derived for equilibrium conditions, cf. (a); the dashed line is the demarcation between initial predominant water uptake (left hind) and hydrogen uptake (front right), obtained in the kinetic picture from the intersection of the grey and pink surfaces. In the zone in between, decoupled stoichiometry relaxation kinetics is observed. Reproduced from  with permission, copyright Wiley-VCH (2015).
While Fig. 3(a) gives the equilibrium picture, Fig. 3(b) show the defect fluxes at short time after the pH2O increase, calculated from eq. (5). Again one can recognize regions with predominant hydration (proton flux is twice the flux; left hind region) and hydrogenation (proton flux matches the h• flux; front right), but the demarcation line is different than in the equilibrium picture of Fig. 3(a). As a consequence we find a zone in the "materials map" where initially proton uptake occurs at expense of holes, but since the overall reaction is mainly water uptake a slower oxygen incorporation step must follow. This is exactly the decoupled regime as shown in Fig. 2. Numerical modelling shows that for the appearance of decoupled kinetics not only a perceptible hole transference number is required, but also a certain hole concentration (which is the higher the more protons the material contains). The present ansatz allows one to quantitatively determine the onset of this peculiar decoupled stoichiometry relaxation kinetics.
Finally it is important to note that in a material comprising protons, oxygen vacancies, and an electronic carrier, decoupled transport phenomena are not only expected after water partial pressure changes, but under certain conditions also after oxygen partial pressure changes. The derived flux equations together with the corresponding numerical simulations supply the tools to naturally explain the apparent uphill diffusion and "moving boundary phenomena" in a three carrier system. Such complex transport behavior may be encountered in numerous mixed conducting functional materials, and may become important e.g. with respect to ageing processes in fuel cells or permeation membranes. From a general viewpoint it is worth emphasizing that it is finally the complexity of the mechanistis situation and not the complexita of the individual flux equations or the magnitude of the driving force that leads to the observed striking non-monotonicities.