Quantifiable models for surface protonic conductivity in porous oxides – case of monoclinic ZrO 2 †

The surface protonic conductivity of porous monoclinic ZrO 2 sintered at temperatures in the range 700–1100 1 C yielding relative densities of around 60% and grain sizes of approximately 160 nm has been studied using impedance spectroscopy as a function of temperature well below the sintering temperature in wet atmospheres ( p H 2 O = 0.03 bar). The sum of two high-frequency impedance responses is argued to represent surface conductance according to a new model of impedance over curved surfaces. A simple brick layer model is applied to compare the measured macroscopic conductivities with predicted surface conductances. The well-faceted samples sintered at the highest temperatures exhibited activation enthalpies up to 58 kJ mol (cid:2) 1 of surface protonic conduction in wet atmospheres at temperatures above 300 1 C. We attribute this to the mobility of dissociated protons over surface oxide ions, and the high preexponential is in good agreement with a model comprising relatively strong dissociative chemisorption. With decreasing sintering temperature, the particles appear more rounded, with less developed facets, and we obtain activation enthalpies of surface protonic conduction in the chemisorbed layer down to around 30 kJ mol (cid:2) 1 , with correspondingly smaller preexponentials and an observed p 1 = 2H 2 O dependency. Supported by the thermogravimetry of adsorption, we attribute this to weaker and more molecular chemisorption on the more randomly terminated less faceted surfaces, providing water layers with fewer dissociated charge carrying protons, but also smaller activation enthalpies of mobility. Below 200 1 C, all samples exhibit a strongly inverse temperature dependency characteristic of conduction in the 1st physisorbed layer with increasing coverage. The preexponentials correspond well to the models of physisorption, with dissociation to and proton migration between physisorbed water molecules. The enthalpies fit well to physisorption and with enthalpies of dissociation and proton mobility close to those of liquid water. We have by this introduced models for proton conduction in chemisorbed and physisorbed water on ZrO 2 , applicable to other oxides as well, and shown that preexponentials are quantitatively assessable in the order-of-magnitude level to discriminate models via a simple brick layer model based topographical analysis of the ceramic microstructure.


Introduction
Adsorption of water on the outer and inner surfaces of dense and porous ceramics is well known and studied, which enables humidity sensors (see e.g. ref. 1 and references therein) and gives rise to protonic conduction that in combination with various airborne contamination decreases the performance of electrical insulators. 2 Surface protonic conduction in water adsorbed in porous nano-grained ceramics has more recently received renewed interest as potential electrolytes for fuel cells, [3][4][5] while it is also realised that it plays significant roles in kinetics of catalysts, photocatalysts, and electrocatalysts. 6,7 Surface protonics is particularly well studied for pure and doped dioxides of tetravalent cations. Much of our present knowledge on the adsorption of water and surface protonics in porous nano-grained ceramic oxides stems from studies of undoped monoclinic and cubic ZrO 2 , [8][9][10][11] and cubic Y-stabilised ZrO 2 (YSZ). [12][13][14][15][16][17][18] Miyoshi et al. 19,20 showed that the room temperature (RT) protonic conductivity of YSZ increased by 3 orders of magnitude when the grain size was reduced from 100 to 13 nm, while the concentration of Y 2 O 3 did not have a significant effect on the surface protonic conductivity.
They attributed the proton conductivity to water adsorbed at grain boundaries and surfaces. CeO 2 is of particular importance in catalysis; the water adsorption has been well characterised, 21,22 and surface protonic conduction has been measured, 23,24 but it exhibits a complex surface chemistry with slow equilibria and considerable hysteresis. [25][26][27] TiO 2 has also been well studied [28][29][30] lately because protonic migration in adsorbed water appears to be central for the use of TiO 2 photocatalysts. 31,32 Recently, Kang et al. 33 investigated the role of surface orientation on water adsorption and surface protonic conduction in anatase TiO 2 materials synthesised under conditions that favour different surface facets. They found by in situ FT-IR and conductivity measurements that simple {100} and {001} surfaces with accessible cations and anions favour dissociative chemisorption and the conductivity suggested that the subsequent physisorbed water layer was rigidly bonded and solid (ice-like). In comparison, the {101} surface favours molecular (associative) chemisorption, with a more loosely bonded subsequent physisorbed layer. The two types of surfaces have characteristic differences in surface protonic conductivityespecially its temperature dependency.
Gregori et al. 23 proposed a brick layer model that qualitatively estimates the surface protonic conductivity of porous oxides based on the pore size, volume fraction of the open porosity, and thickness and bulk conductivity of the adsorbed water layer. They further assumed that dissociation into charge carriers follows autoprotolysis of water. Stub et al. 15,16 suggested that proton transport can occur over grain surfaces (intra) and over grain boundary intersects (inter). They treated surface protonic conduction quantitatively based on their own measurements, standard models for adsorption, and transport terms for YSZ developed by Raz et al. 12 These recent studies and the growing understanding of the phenomena involved have led us to propose a quantitative approach to parameterise models of adsorption, dissociation, and migration, which can also be applied to other porous materials. This in turn leads to prediction and interpretation of the macroscopic surface protonic conductivity of the porous sample, employing both enthalpies and, unlike non-quantitative approaches, the preexponentials. To do this, we define and employ a defect chemical notation for surface species, merge adsorption theory and defect equilibrium thermodynamics, apply normal theory of proton diffusion and migration, and link the surface conductance through the macroscopic sample conductivity by a simple topographical analysis -a brick layer model (BLM) -of the microstructure.
The surfaces of monoclinic ZrO 2 show a preferred termination along the {111} plane after sintering at high temperatures. 34 The literature on adsorption is limited to a review and study of CO adsorption 35 in addition to the aforementioned studies on the adsorption of water and stability of hydroxylated ZrO 2 surfaces. 8,11 Surface protonic conductivity has not been reported for porous monoclinic ZrO 2 and only scarcely from the perspective of volume conduction in undoped ZrO 2 with a relatively high density (480%) and a coarse microstructure. 36 We herein report the surface protonic conductivity in adsorbed water layers in nano-grained porous monoclinic ZrO 2 , with focus on the effect of sintering temperature and hence small differences in the microstructure and degree of faceting of the surfaces. Samples are characterised by X-ray diffraction (XRD), scanning electron microscopy (SEM), thermogravimetry (TG) and electrochemical impedance spectroscopy (EIS). Results are interpreted according to the model for activation enthalpies and preexponentials of protonic conduction in chemisorbed and physisorbed water layers, supported by the observed isothermal p H 2 O dependence of conductivity, and the effect of faceting is considered. A discussion on the appearance of two high-frequency time responses in the impedance spectra is provided, which our samples have in common with most porous materials with surface protonic conduction.

Experimental
In order to prepare ZrO 2 ceramic pellets, ZrO 2 powder (99.99% metal basis except Hf, CAS no. 1314-23-4, Aladdin Industrial Corporation, China) and a binder (a mixture of polyvinyl, glycerol, ethanol and DI water) were mixed in the weight ratio of 7 : 1 and pressed into disks with a diameter of 12.7 mm and a thickness of approx. 2.0 mm under a pressure of 760 MPa. They were sintered for 24 h at 600, 700, 800, 900, 1000, or 1100 1C, with a heating and cooling ramp rate of 5 1C min À1 . Samples sintered at 600 1C were of insufficient mechanical strength for conductivity measurements, and only samples sintered at 700-1100 1C were characterised further and were denoted as ST700-ST1100 accordingly, see Table 1. Table 1 Sample notation with sintering and microstructural parameters. Factor c is the ratio between the macroscopic conductivity of the porous sample and the grain surface conductance according to the brick layer model (BLM) that we will come back to later in eqn (3) with a percolation power x = 1. Also, volumetric, gravimetric, and molar specific surface areas are estimated to first approximation based on the BLM (see ESI  Powder XRD patterns were collected using a Bruker D8 Advance Diffractometer (Laguna Hills, CA, USA) with Cu Ka radiation (l = 1.5418 Å). Microstructures were observed by SEM using a Hitachi SU 8010 Boerne (Hitachi, Tokyo, Japan).
Thermogravimetric data were acquired using a Netzsch 449 F1 Jupiter s thermal analyser (GmbH, Germany) on ST700 and ST1000, representing samples sintered at low and high temperatures, respectively. Samples were first degassed during heating to 650 1C (ST700) or 900 1C (ST1000) at 3 K min À1 using bottle-dry 99.999% N 2 as both the carrier and protective gas, and further held at the given temperature for 2 h. (In our experience, even very dry gases typically end up with 30 ppm H 2 O in high-temperature apparatus. 37 ) Then, the uptake of water was measured by flowing wet (p H 2 O = 0.026 bar) N 2 over the samples during stepwise cooling to 26 1C. The water uptake was obtained by subtracting the background measurements in dry N 2 gas carried out under otherwise identical conditions.
For electrical measurements, silver paste (SOFCMAN, China) was painted on both sides of the pellets, and heat-treated at 800 1C in an ambient atmosphere for 2 h, except for the sample sintered at 700 1C for which the heat-treatment was made at 600 1C. The pellets were mounted in a ProboStatt (NORECS, Norway) sample holder with a 4-wire 2-electrode configuration. AC impedance spectra were collected in the range 1 MHz-1 Hz at 0.7 V rms applied voltage with a CHI604E electrochemical workstation (Shanghai Chenhua Instruments Ltd, Shanghai, China) or at 3 V rms using a Solartron SI 1260 frequency response analyser (Solartron Analytical, Farnborough, UK). For both, the ProboStatt base unit chassis was connected to instrument ground to reduce noise and eliminate parasitic parallel conduction, and it was controlled that both instruments yielded equivalent spectra. The atmosphere was flowing Ar (99.999%) either bottle-dry or wetted to p H 2 O E 0.03 bar by bubbling through de-ionized water at room temperature or using a HumiStat gas-mixer and humidifier (NORECS, Norway). The impedance spectra were collected after keeping the sample at each temperature for at least 1 h during cooling and analysed with ZView software (Scribner Associates, Inc. NC, USA). The electrical conductivity of each porous sample is calculated from its resistance, thickness, and electrode area.

Microstructure characterisation
The geometrical dimensions after sintering were close to those of green bodies, i.e., little sintering takes place. From the dimensions and weight, the relative densities were calculated to vary only between 60% and 70% for the samples sintered at 700 and 1100 1C, respectively, see Table 1. Fig. 1 shows the XRD results for the precursor powder and powder ground off the sintered samples. They all correspond to the monoclinic polymorph of ZrO 2 . Analyses of the main peaks shows that the full width at half maximum (FWHM) values are close, at about 0.1801 for the samples annealed at 700-1100 1C, see Table 1, indicating from the Debye-Scherrer formula average crystallite sizes around 50 nm. 38 Fig . 2 shows the SEM images of the precursor powder and sintered samples, revealing the openly porous microstructure in agreement with the densities from geometry and weight. From the statistical analyses of the images, we estimate average grain sizes of 150 nm for the precursor powder and 150-190 nm for the ST700-ST1100 samples. Hence, both sintering and grain growth are modest at the sintering temperatures applied here.
Closer inspection of the grains suggests no differences in the aspect ratio and modest differences in connectivity (necks). However, while the mostly spherical shape of the precursor particles has been retained in the samples sintered at low temperatures (below 900 1C), the particles of samples sintered at higher temperatures show more faceted surfaces, apparently a distribution over normal {100}, {110}, and {111} surfaces (in cubic framework) as the regular aspect of the crystallite is maintained. Fig. 3 shows the TG result in terms of the amount of water (chemisorbed and physisorbed) adsorbed on ZrO 2 ST700 and ST1000, with area specific coverage calculated based on the estimates of specific surface areas listed in Table 1. The uptake of chemisorbed water saturates towards 200 1C at around 4-5 H 2 O molecules per nm 2 , which is considered a monolayer. 8 We note that the coverage is not yet complete at 400 1C for the ST700 sample, suggesting relatively weak and hence molecular chemisorption, while the ST1000 sample achieves coverage at somewhat higher temperatures, indicating stronger chemisorption on its more developed (facetted) surfaces, as observed also for CeO 2 . 39 Physisorption sets in below 200 1C, and a first physisorbed layer is complete around 80 1C. Qualitatively, the water uptake modelled as the sum of a completed chemisorbed layer and physisorbed layers yields adsorption enthalpies similar to that of condensation to liquid water, 12 as expected. The total water uptake corresponds well to one reported in the literature, when its surface area of 80 m 2 g À1 , a factor of 4 higher than in our samples is taken into account. 40

Electrochemical impedance spectroscopy (EIS).
Representative impedance spectra are displayed in ESI 2 and Fig. S1 (ESI †). All samples exhibited two overlapping semicircles in the high frequency domain, with capacitances in the lower and higher end of the 10 À11 F range, irrespective of sintering and measurement temperature or atmosphere (dry or wet). The literature does not provide a consistent rationalisation of this, but in our present understanding, it reflects surface conduction over the curved surface of the porous microstructure, namely concave regions with small capacitance of the gas phase over the gap and convex regions with higher capacitance due to the dielectric solid phase. We provide a more detailed discussion of the origin and capacitances of the two responses in ESI 3 (ESI †), but here take it that the sum of the two resistances represents the total resistance of the surface transport.
The two responses are present also in dry atmospheres at the highest measurement temperatures. This suggests that also the native, non-protonic conduction is dominated by surface transport, which is reasonable in these fine-grained porous samples.
At the highest measurement temperatures, there is an additional response at the lowest frequencies, with characteristic capacitances of the order of magnitude of 10 À8 F, both in wet and dry atmospheres. This may be a grain boundary impedance as reported by Stub et al. 16 for YSZ, but it may also be an electrode response. The contact area between the adsorbed water layer and the metal electrode is small compared with a normal solid electrolyte, hence the normal capacitance of 10 À6 F for samples of this size attributed to the double layer in parallel with charge transfer may be reduced by orders of magnitude and be what we observe.
3.3.2 Surface protonic conductivities. An Arrhenius plot of the electrical conductivity of all samples under wet and dry Ar (separated) is provided in ESI 4 and Fig. S2 (ESI †). The conductivities in wet atmosphere generally behave like typical surface protonic isobars with considerable conductivities at high temperatures, a minimum at around 200 1C, and conductivities increasing with decreasing temperature towards room temperature. In bottle-dry atmospheres, the measured conductivities were similar for all samples, with activation energies of around 100 kJ mol À1 at the highest temperatures, typical of oxide vacancy mobility in ZrO 2 materials. 36,41 We attribute this to native conductivity of the sample, apparently dominated by surface transport based on the above-mentioned interpretation of the impedance spectra in dry atmospheres. Fig. 4 shows the surface protonic conductivities obtained in wet atmospheres after subtraction of native conductivity, plotted as log(sT) vs. 1/T as we shall use that for extracting enthalpies and preexponentials, listed in Table 2.
The surface protonic conductivities in wet atmospheres above 200 1C are typical of conduction in the chemisorbed  The enthalpies in this region are approaching 60 kJ mol À1 for ST1000 and ST1100, which we for now note is comparable to the enthalpy of mobility of protons in the lattices of MO 2 oxides, e.g. 58 kJ mol À1 for polycrystalline rutile TiO 2 42 , and 43 kJ mol À1 as an estimate for 50% La-doped CeO 2 . 43 With lower sintering temperatures, the enthalpies decrease, to around 30 kJ mol À1 for ST700, and conductivities are higher suggesting that the enthalpy of migration is lower and/or the exothermic adsorption is at play, i.e., the layer is not complete and still filling up with decreasing temperature for these samples, in agreement with the TG results, and to be discussed further later. Below 150 1C, we have a strong increase in conductivity, as a first molecular physisorbed water layer starts to build on top of the chemisorbed layer, providing easier transfer of protons. The activation enthalpies are now found to be in the range of À40 to À20 kJ mol À1 . This is qualitatively dominated by the exothermic enthalpy of the condensation of water vapour, and we shall quantify it when we discuss models for transport in this layer later.
As the physisorbed layer builds, it expectedly gets more liquid-like and the physisorption enthalpy approaches that of pure liquid water. This gives rise to an additional strong increase in the surface protonic conductivity in many porous ceramics, 15,28 including some facets of TiO 2 33 as we approach RT and relative humidity (RH) surpasses 60%. For our monoclinic ZrO 2 samples, however, there is no such secondary increase in conductivity. This is similar to the {101} facet of TiO 2 33 and our on-going work on undoped CeO 2 39 where it even increases less steeply above RHs of 60%. As our data do not extensively explore the region at RH 4 60%, we refrain from analysing that region further and it is not part of dependency, same as that of undoped nanoscopic CeO 2 sintered at 550 1C and measured at 100 1C. 39 We will discuss models that may rationalise it below.

Models and quantitative interpretation
In the following, we provide a deeper theoretical analysis of the preexponentials and activation enthalpies of surface conduction in porous oxides, with focus on MO 2 type oxides and ZrO 2 and the actual samples and data presented here. For this, we will go through some definitions and clarifications of conductivity contributions, considerations of geometrical factors for surface conduction in porous materials, estimation of charge mobility of protonic species, nomenclature for surface species, and thermodynamics of adsorption and dissociation, leading up to expressions for surface conduction and sample macroscopic conductivity. A list of symbols and subscripts is provided in the ESI † (ESI 1 and Table S1).

Conductivity and geometry
The total macroscopic conductivity of a porous material s M can be taken to have bulk and surface contributions. Under our experimental conditions, we take it that the bulk conductivity of ZrO 2 is due to native oxide ion transport, and while it appears that this is negligible, the surface conductivity also has a contribution from oxide ions in addition to protonic conduction. The latter may be taken to arise from migration in chemisorbed, first rigid physisorbed, and second liquid-like physisorbed water layers: We will treat the three water layers as one, dominated by the outermost one, in the sense that the layers adsorb one by one, to a first approximation, and that the conductivity of the outermost layer for the purpose of modelling totally outcompetes that of any layers underneath. The conductivities in dry atmospheres at high temperatures exhibit activation energies of around 100 kJ mol À1 , attributable to inherent conduction by oxide ions in gross agreement with literature data for bulk conductivity of monoclinic ZrO 2 . 36,41 If it is a bulk conduction process, the measured conductivity is empirically proportional to the square of the relative density, making factors such as 0.65 2 = 0.42, but this does not change the comparison significantly. As mentioned earlier, impedance spectra suggest that the inherent conduction is also a surface process due to the presence of the overlapping high-frequency semicircles with capacitances of the order of 10 À11 F (see ESI 2 and Fig. S1(e, f), ESI †). Dual time constants have also been reported for YSZ, suggested to reflect intra-and inter-grain surface protonic resistances. 16 It is anyway not of our interest here, and we subtract it where it is significant at high temperatures and can neglect it at lower temperatures, so that Fig. 4 represents s M = s M,s,H +.
The macroscopic protonic surface conductivity s M,s,H + of the sample material must be related to the adsorbed water surface layer volume conductivity s s,H + along an appropriate geometric model. For this, we need the protonic surface layer conductance, G s,H +. It has unit S (Siemens) and is the in-plane conductance of a square of a surface layer with thickness t. It is independent of the width w and the length l of the square, since they are equal and cancel. The surface protonic conductance G s,H + with average volume concentration c H + and charge mobility u H + in the water layer is then related to the protonic surface layer volume conductivity s s,H + by where we have also replaced the protonic surface layer volume concentration c H + (in mol cm À3 ) with surface concentration g H + (in mol cm À2 ) and layer thickness t in c H + = g H +/t. As stated above, we here treat the surface layer as one, and we have deliberately used subscript s to distinguish conductance and conductivity of the surface layer from other geometries, while we have omitted the superscript s for concentration and mobility terms to avoid build-up of too many subscripts later on. From a simple brick layer model (BLM, see ESI 5, ESI †), we may make an order-of-magnitude approximation to the macroscopic conductivity contribution s M,s of a porous material from the surface conductance G s , the relative density r r , and assumed equal grain and pore size d g : The exponent x adjusts for percolation. For materials of regular and equal shapes of grains and pores, and densities around 50%, it will be x = 1-2 and we here use x = 1. The effect is still less than half an order of magnitude for densities around 50%. The factor c = s M,s /G s in eqn (3) calculated on this basis is listed for our samples in Table 1. In this work, we hence calculate the predicted macroscopic sample surface protonic conductivity from s M,s,H + = cG s,H +.

Charge mobility of protonic species
We now consider the diffusivity and charge mobility of protons, which we will need to couple thermodynamics and concentrations with conductance and conductivity. Since the proton is a defect on oxide surfaces, its random diffusivity is in the classic approximation we will use here a simple activated function, multiplied by the chance X O that the oxide ion it jumps to is available to accept the proton: X O is the fractional occupancy of any oxide-ion containing species, O 2À , OH À , or H 2 O, that can accept a proton, depending on the mechanism. The charge mobility is via the Nernst-Einstein relation then: Generally, the preexponential of diffusivity is given by D 0 ¼ where A is a geometrical factor of the order of unity, s is the jump distance, o 0 is the vibrational attempt frequency, and DS m is the entropy part of the energy barrier. The proton always jumps from its host oxide ion to a nearest one, so we may take the effective jump distance to be 2.8 E 3 Å. The vibrational attempt frequency is in practice that of the oxide ion host, of the order of 10 12 s À1 (not the 10 13 s À1 of the proton itself). With this, we get D H + 0 E 1 Â 10 À3 cm 2 s À1 as an order of magnitude estimate, irrespective of the mechanism or host system, and in turn,

Nomenclature for surface species
Now that we start our thermodynamic analysis with the aim to combine it with charge mobility to get conductivity, we choose to use a Kröger-Vink compatible notation for surface chemistry in which superscripts Â , 0 , and denote, respectively, neutral, negative, and positive effective charges compared to the charge of the clean, ''perfect'' surface. For compatibility with other binary oxides, we furthermore choose to denote cations by the general symbol M, so that here M = Zr and surface cation and oxide ion sites are hence denoted as M s and O s .

The chemisorbed layer
We start by attempting to predict the level and behaviour of surface protonic conductivity in the chemisorbed layer, which dominates typically in wet atmospheres at high temperatures down to around 200 1C. The samples behave differently in terms of activation enthalpy and preexponentials, and we will derive two models that can rationalise the two. One is based on weak, molecular chemisorption, and it divides further into two sub-models with different dissociation and migration behaviours, of which one will be shown to be applicable to sample sintered at the lowest temperatures. The other model is based on strong, dissociative chemisorption, which appears to apply to the samples sintered at higher temperatures. The models are schematically depicted in Fig. 6. 4.4.1 Weak, molecular chemisorption. A previous in situ FTIR study shows that a high surface concentration of hydroxyl groups is missing from water chemisorption on monoclinic ZrO 2 between RT and 1173 K over a pressure range of 10 À5 -24 mbar, in contrast to highly hydroxylated YSZ and Y 2 O 3 . 18 Room temperature IR measurements on high surface area ZrO 2 after annealing at 873 K reveal a single broad peak at 3690 cm À1 , attributed to the O-H stretch mode of H 2 O molecules, in addition to a characteristic molecular water band at B1630 cm À1 due to H-O-H bending. 40 Ab initio molecular dynamics calculations suggest that while only dissociative adsorption is observed on tetragonal ZrO 2 {110} surfaces, 44 both undissociated and dissociated water species coexist on the two most stable surfaces of monoclinic ZrO 2 , i.e. {101} and {111}. 11 These experimental and computational studies lead us to propose that H 2 O adsorbs on a surface M 4+ cation but does not easily dissociate protons H + onto surface oxide ions O 2À or to other water molecules. The lack of dissociation, i.e., an endothermic dissociation enthalpy, leaves the overall chemisorption weak (only modestly exothermic adsorption enthalpy, dominated by unfavourable entropy) and makes an only partly Fig. 6 Schematic illustration of the five models derived for adsorption, dissociation and protonic migration in this work. ''Surface'' shows the bare MO 2 surface. ''chm'' shows weak molecular chemisorption and dissociation (for reasons of space limitations shown as half covered and half dissociated) with resulting protonic migration (arrows) in the molecular dissociated layer. ''chm-s'' is similar, but protons are dissociated to and migrate with the help of surface oxide ions. ''chd'' shows strong dissociative chemisorption with proton migration on surface oxide ions. ''chm-ph1'' shows partial coverage of the 1 st physisorbed layer with protons dissociated from a molecular chemisorbed layer underneath. ''ph1'' shows partial coverage of the 1 st physisorbed layer with weak dissociation over a strongly adsorbed and dissociated chemisorbed layer underneath. covered and largely molecular layer (''chm'' and ''chm-s'' in Fig. 6). The molecular chemisorption is in our nomenclature written as follows: with equilibrium coefficient Here, X represents the fractional occupancy (which ideally corresponds to activity and also to surface site coverage), p is the partial pressure, and g is the surface concentration e.g. in mol cm À2 . The concentration of adsorbed molecular water according to this model follows simple adsorption isotherm treatments. We are interested in partial dissociation, either protonating other adsorbed molecules (''chm'') or surface oxide ions (''chm-s'') and thereby providing protonic conductivity. We shall derive expressions for both, and compare with the experimental data.

Dissociation within molecular water
As suggested by e.g. Raz et al., 12 and the equilibrium constant can then be written as follows: If this dominates, we have the following simple electroneutrality: Moreover, cation surface site balance requires g M Â Ms þ g M Ms OH Â 2 þ g M Ms OH 0 þ g M Ms OH 3 ¼ g Ms , but under the assumption of weak molecular adsorption (low coverage and limited dissociation), we have g M Â Ms % g Ms ) g M Ms OH Â 2 ) g M Ms OH 0 þ g M Ms OH 3 , and we can insert and simplify to get In estimating the surface conductance based on this dissociation model, we assume that jumps of protons from dissociated protons H 3 O + to H 2 O and jumps from H 2 O to dissociated OH À contribute equally, and from the electroneutrality the surface protonic conductance in the chemisorbed layer according to eqn (2) can be expressed as the product of charge, surface concentration, and charge mobility: The charge mobility of the charged defects contains the chance that a proton in an H 3 O + group finds an adjacent water molecule to jump to, or that a water molecule is there to offer a proton to jump to OH À . Hence, we get which, by combination with eqn (7) and (11), yields We rewrite this as where G 0 s;H þ chm 0 is the standard preexponential of surface protonic conductance at p H 2 O = 1 bar. We emphasize that G 0 s;H þ chm 0 is hence the most fundamental parameter specific for a surface and a particular adsorption and transport model, in this case weak molecular adsorption with dissociation and transport in the molecular layer. To derive such a fundamental parameter is only meaningful once a model is chosen, so that the p H 2 O dependency is clear. Now, we attempt to estimate G 0 s;H þ chm 0 , G s;H þ chm 0 , and eventually s M;s;H þ chm 0 . The standard entropy change DS 0 a chm plays a major role and represents the biggest uncertainty. Condensation of water vapour to liquid water has a standard entropy change of À109 J mol À1 K À1 at 100 1C. 45 It decreases strongly with temperature, e.g., to À44 J mol À1 K À1 at 300 1C which is relevant for our chemisorption temperature range. On the other hand, the entropy of condensation to ice is larger than that to water (À144 vs. À118 J mol À1 K À1 at RT) and the entropy of molecular adsorption to hydroxylated a-Fe 2 O 3 , i.e., physisorption, has been reported to be À138 J mol À1 K À1 . 46 We will hence in this work for the most part use À109 J mol À1 K À1 from the entropy of condensation to water at 100 1C 45 as our value for the standard entropy of molecular adsorption DS 0 a chm , both here for chemisorption and later for physisorption. This implies uncertainties of several orders of magnitude in our calculations for conductances, but helps to emphasise differences between different models.
We next assume DS 0 d chm ¼ 0 J mol À1 K À1 for dissociation (since it does not involve gas species or solvation of ions). Using furthermore an adsorption site surface concentration g M s E 5 nm À2 = 8 Â 10 À10 mol cm À2 , and u H + 0 E 10 cm 2 K V À1 s À1 as derived earlier, we get G 0 s;H þ chm 0 % 7 Â 10 À15 S K for p H 2 O = 1 bar and G s;H þ chm 0 % 6 Â 10 À18 S K at p H 2 O = 0.03 bar. By applying s M;s;H þ chm 0 ¼ cG s;H þ chm 0 from eqn (3) and values of c from Table 1, we get predicted preexponentials for macroscopic conductivity of around s M;s;H þ chm 0 % 4 Â 10 À13 S K cm À1 for our samples at p H 2 O = 0.03 bar. This low value, dominated by the variable and uncertain but grossly negative entropy of adsorption, is several orders of magnitude below the experimental ones for the surface protonic conductivity in the chemisorbed layer at higher temperatures ( Table 2) and the ''chm'' model can be ruled out.

Dissociation to surface oxide ions
In the other possibility of dissociation, the molecule dissociates into an adsorbed hydroxide ion and a surface hydroxide ion, i.e., it protonates a surface oxide ion (''chm-s'' in Fig. 6), as suggested by AIMD simulations on cubic ZrO 2 {110} surfaces 9 : The same analysis as above now yields We may further assume that the proton on surface oxide in the OH Os defect may migrate by jumping to other surface oxide ions, since M M s OH 0 it left behind from the dissociation is rare to find. The surface protonic conductance will then be Since we have little adsorption and little dissociation, surface oxide ions are generally available, X O Â Os % 1, so that we by inserting eqn (19) get and hence Considering that the concentration of surface oxide adsorption sites in ZrO 2 is twice that of cation sites, i.e., g Os = 2g Ms , and other parameters are the same as before, we predict now G 0 s;H þ chms 0 % 2 Â 10 À6 S K, G s;H þ chms 0 % 3 Â 10 À7 S K, and s M;s;H þ chms 0 % 2 Â 10 À2 S K cm À1 . Sample ST700 has a preexponential of macroscopic conductivity of around 5 Â 10 À3 S K cm À1 (Table 2), i.e., within an order of magnitude of prediction, suggesting that weak molecular adsorption and dissociation to protons transported along surface oxide ions may apply. In this case, the experimental enthalpy should be interpreted accord- The standard enthalpy of molecular chemisorption has been determined calorimetrically as DH 0 a chm ¼ À76 AE 5 kJ mol À1 for YSZ relatively independent of the Y content. 13 If we adopt a dissociation enthalpy of 22.1 kJ mol À1 from ref. 47 used by Raz et al. 12 for YSZ, we get an enthalpy for surface mobility of protons of DH m;H þ chms ¼ 54 kJ mol À1 .
The model proposes a proportionality to p 1=2 H 2 O of surface protonic conduction for weakly adsorbed and dissociated chemisorbed water according to eqn (21). This is in line with the experimental results obtained for the ST700 sample at 400 1C (Fig. 5).
All in all, partial coverage of molecularly chemisorbed water, partially dissociated into mobile protonic species, forms a credible model for the surface protonic conductivity at temperatures above 200 1C for samples sintered at the lowest temperatures. Involvement of surface hydroxide as a dissociation product and mobile species fits much better with the observed preexponentials than transport between adsorbed water species. We suggest that this may apply also to other oxides where a similar behaviour is observed, and is a result of surfaces that are relatively amorphous or rounded (vs. faceted) or have terminations where dissociation is unfavourable. While the transfer of a proton to a surface oxide ion is not very favourable, the transfer between oxide ions may be correspondingly easier.
4.4.2 Strong, dissociative chemisorption. Next, we assume as before that H 2 O adsorbs on a surface M 4+ cation (here Zr 4+ ) and dissociates one proton H + onto a surface oxide ion O 2À . However, now this dissociation step is favourable (exothermic) and strengthens the overall chemisorption (making it more exothermic) and makes a fully covered and dissociated layer at relatively high temperatures (low RH). Dissociative chemisorption (''chd'' in Fig. 6) is then a combination of eqn (6) and (17): with equilibrium coefficient being the product of eqn (7) and (9): While we earlier used data for weak molecular chemisorption on ZrO 2 , there exist data also for strong dissociative adsorption. From first principles calculations, the enthalpy of dissociative chemisorption on the {001} surface of monoclinic ZrO 2 has been reported to be À165 kJ mol À1 for half-monolayer coverage, and À109 kJ mol À1 for full-monolayer coverage 48 . Radha et al. 8 measured the amount of chemisorbed water on monoclinic ZrO 2 and found that full coverage corresponds to 3.8-5 H 2 O per nm 2 . From high-temperature oxide melt solution calorimetry on samples annealed at 600 and 800 1C, they found integral enthalpies of dissociative chemisorption at half monolayer coverage (2.2 H 2 O per nm 2 ) and full monolayer coverage of À142 kJ mol À1 and À119.5 AE 1.7 kJ mol À1 , respectively. With this and aforementioned estimates of the entropy, eqn (25) and the electroneutrality and site balances yield the coverage of cation adsorption sites at p H 2 O = 0.03 bar varying from above 90% at the lower end of the temperature range considered for strong dissociative chemisorption here to 50% at higher temperatures. For the interpretation of the exponential changes in conductivity with temperature, these variations are minor (of the order of a factor of 2) in the range where we assign conductivity to transport in the chemisorbed layer. The Arrhenius plots of the surface protonic conductivity for the samples sintered at higher temperatures are indeed linear in the range above 300 1C (Fig. 4). The electroneutrality and assumption of full coverage leads directly to g M Ms OH 0 ¼ g OH Os % g Ms (26) indicating that the concentration of dissociated protons is independent of temperature as well as RH and p H 2 O , and that the temperature dependence is attributable solely to the mobility of dissociated protons. We assume that the dominating transport mechanism for dissociated protons on ZrO 2 surfaces within the chemisorbed water layer is their jump between their surface oxide host sites, and that these have an average occupancy of X O Â Os ¼ 0:5 since half of them are occupied with dissociated protons. The surface protonic conductance then becomes With the same assumptions as before, this yields G s;H þ chd 0 ¼ G 0 s;H þ chd 0 % 4 Â 10 À4 S K, and s M;s;H þ chd 0 % 20 S K cm À1 , to be compared with the corresponding experimental preexponentials (Table 2) for the chemisorbed layer of 0.1-0.4 S K cm À1 for the samples sintered at the highest temperatures. The two orders of magnitude difference may be ascribed to the many approximations along the way, but may also mean that the assumptions of the model comprising fully covered and dissociated chemisorbed water are not fully met. However, we may assign the activation enthalpies of up to 60 kJ mol À1 to the diffusion barrier for protons migrating between oxide ions on the faceted surfaces of monoclinic ZrO 2 covered with dissociatively chemisorbed water. The aforementioned comparison to the enthalpy of mobility of protons of 58 kJ mol À1 for polycrystalline rutile TiO 2 42 (29) with equilibrium coefficient which we may write as follows: If we assume that u H + 0 is still of the order of magnitude of 10 cm 2 K V À1 s À1 , and that the standard entropy of physisorption is À109 J mol À1 K À1 , and that of dissociation is negligible, we obtain G 0 s;H þ chm -ph1 0 % 2:3 Â 10 À12 S K, G s;H þ chm -ph1 0 % 1:2 Â 10 À14 S K, and s M;s;H þ chm -ph1 0 % 7 Â 10 À10 S K cm À1 . If we assign a negative standard entropy to dissociation, like a small part of what it is in liquid water, the preexponentials decrease further. This is in rough agreement with the experimentally observed preexponentials of conductivity in the 1 st physisorbed layer for the samples sintered at the lowest temperatures (B10 À9 SK cm À1 , see Table 2). The corresponding apparent activation enthalpies of around À30 kJ mol À1 contain in this case 3/2 times the standard adsorption enthalpy estimated at À45 kJ mol À1 as explained earlier, plus half the dissociation enthalpy, which as a first estimate now may be taken as that in water at 100 1C of +42 kJ mol À1 , leaving the activation enthalpy of proton mobility at around +17 kJ mol À1 , credible in being just somewhat higher than that in liquid water of 12 kJ mol À1 .
Furthermore, ST700 revealed an approximate p The surface protonic conductance from this process is then where the preexponential of surface protonic conductance is Under the same assumptions as above, we obtain G 0 s;H þ ph1 0 % 6 Â 10 À15 S K, G s;H þ ph1 0 % 6 Â 10 À18 S K, and s M;s;H þ ph1 0 % 3:4Â 10 À13 S K cm À1 , possibly somewhat lower if the entropy of dissociation is negative like in liquid water. This corresponds well with the experimentally observed preexponentials for the samples sintered at the highest temperatures, see Table 2. The corresponding apparent activation enthalpies approaching À50 kJ mol À1 are to compare with twice the estimated standard adsorption enthalpy of À45 kJ mol À1 plus half the dissociation enthalpy, which as a first estimate may be taken as that in liquid water at 100 1C of +42 kJ mol À1 , leaving the activation enthalpy of proton mobility at 19 kJ mol À1 , again a credible assessment. The difference to the preceding model is the expected effect of half the adsorption enthalpy.
There is a p 2 H 2 O -dependency of this latter model vs. a p 3=2 H 2 O -dependency of the preceding one. Sun et al. 39 report p 2 H 2 O -dependency for surface conduction in the 1 st physisorbed layer on undoped more facetted CeO 2 , supporting our assumption that ZrO 2 samples sintered at high temperatures take place by this last model, i.e., by dissociation and migration in the physisorbed layer. For the sake of completion, we mention here again that as RH surpasses 60%, i.e., below 50 1C at p H 2 O = 0.03 bar, a 2 nd liquid-like physisorbed layer is expected with strong increase in conductivity as temperature decreases further, as observed for, e.g., porous YSZ 15 and TiO 2 , 33 but this was not evidenced in this study of monoclinic ZrO 2 .

Summarising discussion
The temperature dependency of the conductivity can be measured and modelled at constant RH instead of constant p H 2 O .
With the exception of the modest effect of c BET , a constant RH yields a constant coverage, as one may see in the mathematics derived here. All models, except strong dissociative chemisorption, then come out with an activation enthalpy given by half the dissociation enthalpy plus the migration enthalpy: 15 did this for YSZ in the region of physisorption, and for constant RHs in the range of 20-60% obtained enthalpies of 43-34 kJ mol À1 . Colomer 50 studied the proton conductivity of nanoporous anatase TiO 2 thin films as a function of temperature (from 25 to 80 1C) at different RH values, and found the activation enthalpies to be 36 kJ mol À1 at RH of 33%, and 33 kJ mol À1 at RH of 58%. These results are in agreement with our assessment for ZrO 2 of 38-40 kJ mol À1 measured at constant p H 2 O after subtraction of the estimated enthalpies of adsorption according to the two models applied for surface protonic conduction in the 1 st physisorbed layer.
It appears that the surface protonic conductivity in the porous samples of ZrO 2 sintered at low and high temperatures can be rationalised with simple models of adsorption, dissociation, and transport. But why do well-sintered and presumably wellfaceted surfaces show a stronger more dissociative chemisorption, while the less developed surfaces show weaker, molecular chemisorption? During annealing at high temperatures, the system lowers its excess surface and grain boundary energies by sintering and grain growth, but the type of surface and hence area-specific surface energy and adsorption behaviour may still be the same. In faceting, on the other hand, random surfaces from particle growth during synthesis, or from milling, transform into more stable, usually low-index surfaces. However, the lower-energy facets should adsorb water less strongly, so this explanation seems not to hold. But if the low-sintered samples have a large amount of amorphous surfaces, not yet faceted at all, they are believed from general adsorption theory to have low surface energies by the freedom to arrange terminations variably along the surface, with less gain in adsorbing gases like water. This would explain our Table 3 Derived p H 2 O dependencies and predicted preexponentials for the five models of protonic surface transport in the chemisorbed layer and 1 st physisorbed layer, for porous MO 2 oxides in general, and in the case of conditions and microstructures employed in this study, compared with the experimental values of the same parameters for ST700 and ST1100. The preexponentials are rounded off to one significant digit, but the realistic predictive power is much less accurate, rather within 1-2 orders of magnitude due to the many assumptions mainly in adsorption thermodynamics and transport parameters observations. But why do amorphous surfaces crystallise and facet if they already have low energies? This would be because the subsurface lattice decreases its energy in crystallisation, while the surface itself may increase in energy.
If our interpretations hold, the samples sintered at low temperatures, exhibiting weak molecular chemisorption, have physisorbed water that dissociates protons from the chemisorbed water underneath (''chm-ph1'' in Fig. 6). In comparison, the samples sintered at high temperatures have dissociation from the physisorbed water itself (''ph1'' in Fig. 6), because the strongly dissociated chemisorbed water on the high temperature sintered samples has no more protons to offer; they have all gone to the oxide surface, where they are immobile due to their high activation energy and the now low temperature. Hence, the conduction in the physisorbed layer can now only arise from dissociation within the physisorbed layer itself. It appears intuitive that the steep increase in conductivity with increasing coverage of this layer then must imply that mobility has less activation enthalpy here than on the oxide surface, and indeed the derived enthalpies of mobility go from those typical of solid-state proton mobility towards those typical of liquid water. We note that both still reflect Grotthuss proton jumps.

Conclusions
Our impedance spectrometric study of the surface protonic conductivity vs. temperature in wet atmospheres of porous undoped monoclinic ZrO 2 sintered at five different temperatures has allowed us to forward models for interpretation and parameterisation of surface protonic conduction that may be applied to porous oxides in general. The dual time constants in the high frequency part of the impedance spectra with capacitances in the geometric (bulk) range are omnipresent and attributable to different capacitances over concave and convex surface paths of porous microstructures. A simple brick layer model (BLM) connects measured preexponentials of macroscopic surface protonic conductivity s M,s,H + 0 to the predicted values from credible models of preexponentials G s,H + 0 of surface protonic conductance. Along with p H 2 O dependencies and credible enthalpies, this enables discrimination between models.
The surface protonic conductivity in wet atmospheres above 200 1C is attributed to the chemisorbed water monolayer. Samples sintered at low temperatures have activation enthalpies as low as 30 kJ mol À1 , along with an observed p 1=2 H 2 O dependency, and preexponentials expected for water weakly chemisorbed to surface cations and weakly dissociated into protons on and migrating between surface oxide ions. (Dissociation to and migration between adsorbed water molecules have a predicted preexponential orders of magnitude away from experimental values, and can be ruled out.) Samples sintered at higher temperatures display higher activation energies and preexponentials, expected for strong, near complete dissociative chemisorption, driven by a more exothermic dissociation to surface oxide ions. The activation enthalpy of conductivity of up to 58 kJ mol À1 can then be interpreted as that of mobility of protons on surface oxide ions. The smaller enthalpy for the samples sintered at low temperatures can be understood as a combination of incomplete exothermic molecular adsorption, unfavourable endothermic dissociation, and easier mobility of protons on surface oxide ions than in the case of the surface with strong dissociative chemisorption. In other words, the well-faceted surfaces are relaxed by protons that bond strongly and are hard to move, while the random surfaces have less affinity for protons, allowing those that are there to move more easily.
The surface protonic conductivity in wet atmospheres below around 150 1C and down to around 50 1C can be attributed to the first physisorbed water monolayer. The apparent negative enthalpies are readily attributed to a lower activation energy of mobility now overtaken by the negative enthalpy of physisorption contributing strongly because more than one water molecule is now involved to dissociate and migrate a proton. The samples sintered at low temperatures have preexponentials and p 3=2 H 2 O dependency suggesting that the dissociated protons come from the chemisorbed layer, while for the samples sintered at high temperatures, they seem to come from the physisorbed layer itself (autoprotolysis), supported by the difference in enthalpies. The measured enthalpies fit to the models if physisorption is assumed to have an enthalpy of À45 kJ mol À1 , dissociation has +42 kJ mol À1 like in water, and mobility has around 17 kJ mol À1 , close to that in liquid water. To our understanding, the difference between samples of ZrO 2 sintered at low and high temperatures must be related to crystallisation and faceting from a more amorphous state of the surface.
All in all, surface protonic conductivity data can be credibly modelled with simple thermodynamics and transport. It is noteworthy how all thermodynamics of adsorption and dissociation and proton mobility terms change completely from the chemisorbed to the physisorbed layers, giving them their widely different temperature dependencies. In the chemisorbed layer, we have favourable adsorption, weakly or strongly favourable dissociation, and difficult migration. In the physisorbed layer, adsorption is weak and dissociation is hard, but migration is easy. We point out how preexponentials and isothermal p H 2 O dependencies of conductivity can discriminate between models.

Author contributions
JG and XS fabricated the samples and performed characterisation and electrical measurements. XS and JG interpreted and plotted the results and fitted them to the models. XS performed and interpreted TG. DH and TN conceived and supervised the experimental study. XS and TN developed the models and wrote the first manuscript draft. All authors contributed to revision and checking of the final manuscript.

Conflicts of interest
There are no conflicts to declare.