A. Structural characterization of SnO₂ nanoparticles

Figure 1 shows XRD patterns for the as-synthesized samples annealed through different heat treatments (named P₁, P₂, and P₃), revealing SnO₂ tetragonal phase (JCPDF# 41-1445), absence of second phases, and broad peaks due to small crystallite sizes. The morphology of SnO₂ nanoparticles was observed by TEM and STEM as shown in Fig. 2. The images suggest relatively narrow size distribution and isotropic shapes. The clusters of nanoparticles suggest agglomeration, which is expected when using the polymeric precursor method, as described elsewhere. ^{Reference Rufner, van Benthem and Castro23} The microstructure implies a significant area of grain boundary existing in each sample after annealing.

FIG. 1.

X-ray diffraction pattern of nanocrystalline tin dioxide synthesized by polymeric precursor method at different conditions. P₁ is treated at 500 °C for 15 h, P₂ was further treated at 600 °C for 5 h, P₃ treated at 600 °C for 15 h, and P_S was sintered at 1300 °C for 30 s.

FIG. 2.

The SnO₂ nanoparticles annealed at different heat procedures: P₁ at 500 °C for 15 h, P₂ at 500 °C for 15 h then 600 °C for 5 h, P₃ at 600 °C for 15 h, and P_S at 1300 °C for 30 s. The morphology is observed from TEM and STEM images.

The grain sizes for each of the samples calculated from the surface area measured by BET, refinement of XRD peaks, and direct measurements from TEM/STEM images are listed in Table I. The grain sizes obtained from XRD patterns (GS_XRD) increase from P₁ to P₃ systematically as expected (increasing temperature/time), and in good agreement with the observation of the TEM images (GS_TEM). However, the grain sizes calculated from BET measurement (GS_BET) were in average 10 nm higher as compared with either GS_XRD or GS_TEM. This is due to the significant agglomerations of particles that have conflicted with the premises of Eq. (10) that particles have to be isolated.

TABLE I.

Grain sizes, interfacial areas, water content, and calorimetric data acquired for the different SnO₂ samples. ^a

^a All symbols in this table and Eqs. (1)–(7) have exactly equivalent definitions.

^b Grain sizes are evaluated by BET measurement, XRD peaks, and TEM images, respectively. GS_BET was back-calculated from A _S by Eq. (10).

^c A _I is calculated from GS_XRD by Eq. (10); A _S is determined by the BET measurement; A _GB is calculated by Eq. (9).

^d Coverage is defined as number of water molecules per unit area, which can be evaluated from the total number of water molecules (derived from x) divides by total surface area (derived from A _S) on a mole of SnO₂.

^e ΔH _ds is the experimental result directly obtained from the high-temperature oxide melt drop solution calorimeter; ΔH _ds = ΔH ₁.

^f ΔH _anhydrous is the heat content after corrected the contribution of adsorbed water; ΔH _anhydrous = ΔH ₁ + xΔH ₂ + ΔH ₃, where ΔH ₂ is given as $- 29.05\;{\rm{kJ}}\;\mathop {{\rm{mo}}{{\rm{l}}_{{{\rm{H}}_2}{\rm{O}}}}}\nolimits^{ - 1}$ ; ΔH ₃ was determined by the integral of heat of water adsorption up to corresponding θ upon the data shown in Fig. 3(b).

^g ΔH _exs = ΔH _ds,bulk − ΔH _anhydrous = ΔH ₄ – (ΔH ₁ + xΔH ₂ + ΔH ₃).

The sintered sample (P_S) showed consistent XRD pattern with as-synthesized samples as shown in Fig. 1. Short-term time sintering did not apparently change its composition or phase. Figure 2 shows its high-angle annular dark field (HAADF) scanning transmission electron microscope (STEM) image. It is apparent that more agglomerations and coarsening occur in the nanoparticles after sintering. The average grain size from a series of STEM images is 39.95 ± 12.78 nm, which is consistent with that obtained from XRD. The big uncertainty in the result from STEM is because of the broad size distribution and the difficulty in determining grain edges in the images due to the porosity. This uncertainty will not affect energetic measurements as detailed later in the text.

B. Heats of water adsorption

The energetics of water interactions with the fully oxidized (P₁) and sintered (P_S) SnO₂ samples were studied by water adsorption microcalorimetry. Figure 3(a) shows the adsorption isotherm as a function of relative pressure for both samples. The isotherm curves of both samples displayed three clear adsorption stages, consistently with adsorption mechanisms proposed by Asay and Kim. ^{Reference Asay and Kim18} The first stage (below 0.016 of relative pressure) exhibits a rapid increase of coverage with the increased pressure. This behavior indicates a strong chemical interaction between water molecules and anhydrous surfaces of samples, representing typically a dissociative behavior where water molecules split into hydroxyl groups. The second stage displays a smaller slope of the isotherm up to about 0.4 of relative pressure. The decrease of slope suggests that the influence of particles' surface on adsorbed water molecules becomes weaker as coverage increases. At this point, one expects a change from a chemisorption (dissociative) behavior to a physisorption (nondissociative or molecular) one. After that, the slope of isotherm raises again to the third stage up to higher pressure. At this pressure, the surface is expected to be in the liquid-water like state, as we are approaching the saturation vapor pressure, as supported by molecular dynamics. ^{Reference Castrillon, Giovambattista, Aksay and Debenedetti24}

FIG. 3.

(a) The adsorption isotherm of water and (b) the heat of adsorption plotted versus water coverage of fully oxidized and reduced samples. The black dash line marked in (b) points out the heat of adsorption of bulk water at −44 kJ mol⁻¹.

Looking closer into the absorption behavior one notes that the curves overlap for low coverage (low pressures), but P₁ shows a clear lower adsorption as pressure increases, suggesting that the onset points for the water adsorption transitioning behavior occur at lower pressures for P_S than P₁, therefore a different surface chemistry.

Figure 3(b) shows the heat of adsorption plotted versus water coverage for P_S and P₁. Both curves show high heat of adsorption at lower coverage, with a rapid drop with increasing coverage and leveling at −44 kJ mol⁻¹ for coverage around 7H₂O nm⁻². This behavior is consistent with the three stages of the isotherm; with heats of adsorption consistent with a dissociative behavior with proton transferring to oxygen bridges on the surface of SnO₂ as predicted by Bandura et al. at low coverage, ^{Reference Bandura, Sofo and Kubicki25} and a systematic decrease and leveling in a liquid water-like state. The highest measured heat was −218 kJ mol⁻¹ for P₁ and −215 kJ mol⁻¹ for P_S. This corresponds to the anhydrous surface state being exposed to 2 μmol g⁻¹ of water. At this point, the surface is high reactive as suggested by the measured heats, and suggests a fully dissociative behavior in both cases.

For comparison with theoretical calculations available in the literature, one needs to determine the heat of adsorption for a monolayer of water. Assuming a molecule of water typically occupies 25 × 10⁻²⁰ m², one may calculate the coverage corresponding to a monolayer, and after integration, the heats are −124 kJ mol⁻¹ for P₁ and −104 kJ mol⁻¹ for P_S. Theoretical calculations on heat of water adsorption for a monolayer coverage of a stoichiometric surface have been reported to be −134 to −170 kJ mol⁻¹ for (110) surface with dissociative water behavior, ^{Reference Batzill, Bergermayer, Tanaka and Diebold26} and −146 kJ mol⁻¹ for (101), also with dissociative behavior. For molecular adsorption, adsorption energy on (110) is reported from −112 kJ mol⁻¹ and −95.06 kJ mol⁻¹ for (101). ^{Reference Hahn, Tricoli, Santarossa, Vargas and Baiker27} The numbers are very consistent with the value measured from water adsorption calorimetry and suggest that the surface of the particle is a composition of planes and even at low coverage most likely showing a mix of dissociative and molecular adsorption behavior.

It is important to note that the results present here contrast with those recently reported by Ma et al. using similar experimental techniques. ^{Reference Ma, Castro, Zhou and Navrotsky16} Their work supports the adsorption energy as −152 kJ mol⁻¹ for the first dose (2 μmol g⁻¹), which is about 70 kJ mol⁻¹ smaller in modulus than the data presented here. By comparing the degassing procedures, one may note that Ma et al. have not considered a possible reduction upon degassing, as no reoxidizing step was used to assure stoichiometric samples. On the other hand, it is not straight forward to compare results of particles prepared from different techniques. That is, while we have used a polymeric precursor method to synthesize our samples, Ma et al. have used an oxidation procedure from metallic tin. The method of preparation is well known to affect exposed planes and surface conditions, such that a direct comparison cannot be made without appropriate in-depth microstructural analysis.

The less exothermic heat of adsorption for the P_S samples at the monolayer coverage suggests a different surface chemistry. The hypothesis is that at this sintering condition SnO₂ surface is most likely reduced. By comparing the diagram proposed by Bergermayer and Tanaka, ^{Reference Varela, Whittemore and Longo5,Reference Bergermayer and Tanaka9,Reference Batzill, Bergermayer, Tanaka and Diebold26} (101) is expected to be severed reduced and (110) only slightly as the sintering conditions lie on the top of the predicted critical temperature and pressure for reduction.

The adsorption energy for the reduced (101) surface has been reported as ∼ −35 kJ mol⁻¹ by theoretical calculations, ^{Reference Bandura, Sofo and Kubicki25} which is significantly smaller as compared to the stoichiometric surface. This was attributed to the fact that dissociative adsorption is less likely to occur. On the other hand, other authors have reported that for the (110) surface, reduced conditions will benefit dissociative behavior. ^{Reference Santarossa, Hahn and Baiker28} This is because upon exposure to water molecules, the reduced state is first oxidized. This scenario suggests that reduced states would have higher heats of adsorption as the energy for oxidation is exothermic, and contrasts with the data we have acquired. However, we believe that at the experimental conditions during water adsorption microcalorimetry, i.e., at room temperature, there will not be enough activation energy for the full oxidation process to take place. The measured −104 kJ mol⁻¹ value for the sintered samples would be therefore a composition of partially reduced surfaces and stoichiometric ones.

C. Surface energies from water adsorption

If the surface of the sintered sample shows signs of reduction, a different surface energy is expected. As proposed by Castro and Quach, ^{Reference Castro and Quach17} the water adsorption calorimetry method can be used to determine surface energy, but the precise coverage where the surface begins to behave like bulk water must be determined. To quantify this point, we looked for a coverage corresponding to where the heat of adsorption leveled at −44 kJ mol⁻¹ in Fig. 3(b). At this point, the second derivative of the adsorption isotherm curve is zero. ^{Reference Castro and Quach17,Reference Al-Abadleh and Grassian29} The corresponding coverage is 10.13 H₂O nm⁻² for P₁ and 5.68 H₂O nm⁻² for P_S; the lower coverage of P_S is in good agreement with the observation on isotherm curves that the transition region begins a small coverage on P_S. By using Eq. (8), where γ_S was set as 0.072 J m⁻², anhydrous surface energies for both stoichiometric and sintered (reduced) samples were listed in Table II. The surface energies of P₁ and P_S are 1.2 ± 0.02 J m⁻² and 0.94 ± 0.03 J m⁻², respectively, where the uncertainty of them is about 3%.

TABLE II.

Surface and grain boundary energies of SnO₂.

Other available data from the literature were also listed in Table II for reference. Ab-initio thermodynamics calculations have shown the surface energy of stoichiometric SnO₂(110), (100), and (101) planes as 1.04–1.21, 1.29, and 1.44–1.60 J m⁻², respectively. ^{Reference Batzill, Katsiev, Burst, Diebold, Chaka and Delley8,Reference Bergermayer and Tanaka9} Other assessments of those energies upon first-principle calculation provide 1.04, 1.14, and 1.33 J m⁻², respectively. ^{Reference Oviedo and Gillan30} The trends are consistent with a lower energy for (110) surface, which suggests that during experiments, SnO₂ particles are most likely composed by (110), as the thermodynamics will direct the probability of finding the planes as predicted by Wulff's figures. ^{Reference Chiang, Birnie and Kingery31} Therefore, one can infer that the average surface energy of SnO₂ particles is close to stoichiometric SnO₂(110) or higher.

Ma et al. have reported on the surface energy of SnO₂ nanoparticles using drop solution calorimetry. ^{Reference Ma, Castro, Zhou and Navrotsky16} The result (1.72 J m⁻²) is considerably higher than most assessments in the literature or our present work. We believe that the difference was caused by a miscalculation of the interfacial area. For instance, a sample in Ref. Reference Ma, Castro, Zhou and Navrotsky16 was reported that its grain size is 7.5 nm corresponding to surface area around 8682 m² mol⁻¹. However, if one calculates its interface area by Eq. (10), one can get the number 17,347 m² mol⁻¹, much higher than its surface area. This indicates strong agglomeration between particles, and significant grain boundary area existence (∼4332 m² mol⁻¹ estimated by Eq. (9)). The enthalpy contribution from the grain boundary to the excess energy measured during oxide melt solution calorimetry cannot be ignored, and possibly led to the observed overestimation on surface energy.

The surface energy of the sintered sample (P_S) calculated from the water adsorption using Eq. (8) is 20% lower than fully-oxidized/stoichiometric samples (Table II). This is consistent with the water adsorption energy, reinforcing the reduction of the surface under sintering conditions. The literature shows convincing experiments and calculations that the surface energy of the reduced surface is lower at high temperatures (highly negative oxygen chemical potential) and our results suggest that not only the surface was indeed reduced during sintering, but also the quenching procedure was successful in maintaining the reduced condition from the high temperature state. Note that during the measurement of surface energy using water dosage, there is an increase in the oxygen pressure for the water molecules, which suggest that a partial reoxidation can be taking place during the water adsorption experiment. As mentioned before though, because the experiment is carried out at room temperature, minimal reoxidation is expected. Moreover, even if oxidation occurs, the surface energy calculations are based on a thermochemical cycle, inferring that the surface energy calculations should not be affected.

D. Grain boundary energies from oxide melt drop solution calorimetry

Water adsorption microcalorimetry can satisfactorily assess the surface energy of oxide nanoparticles. However, it can give no information on grain boundary energies. Drop solution calorimetry can however assess both interfacial energies (grain boundary and surface) by a correlation of the heat of dissolution with the microstructure quantification. ^{Reference Castro and Wang15} To do this, a set of samples with different interfacial areas is needed. For this purpose, P₁ to P₃ were prepared as described in the experimental section. Results of microstructural characterization and heat of drop solution are listed in Table I. The surface and grain boundary energies calculated using the thermochemical cycle described in the experimental section are listed in Table II.

To determine interfacial energies, the enthalpy of anhydrous samples (ΔH _anhydrous) was assessed from the enthalpy of dissolution of SnO₂ nanoparticles (ΔH _ds) after proper water correction, and the excess enthalpy (ΔH _exs) obtained by subtracting ΔH _anhydrous from the dissolution enthalpy of bulk (coarsened) SnO₂, 69.69 ± 1.17 kJ mol⁻¹. A series of mathematical expressions correlating the microstructure (surface and grain boundary areas) of each sample and the excess enthalpy was built [Eq. (7)] and solved such that a unique solution for the surface and grain boundary energies was obtained by linear regression analysis. The surface energy is calculated as 1.03 ± 0.24 J m⁻² and the grain boundary energy 0.80 ± 0.47 J m⁻². The surface energy is consistent with the results from the water adsorption data, but with a larger error due to the small dispersion of interface areas. Similarly, due to the low grain boundary areas, the error for the grain boundary energy is more than 50%.

One of the reasons for the large errors in the calculations is that two unknowns are solved at the same time using the set of equations. To improve the results for the grain boundary energy, the reliable surface energy (smaller error) was imported from the water adsorption calorimetry experiments. A linear correlation between γ_GB and ΔH _exs is then achieved:

where the excess enthalpies are exclusively related to surface or grain boundary energy as marked with subscripts “S” or “GB”, respectively. Solving, the grain boundary energy is 0.71 ± 0.08 J m⁻², which lies within the uncertainty range of the result achieved without the imported surface energy, but with significant more accuracy. To the best of our knowledge, there is no available data on other literature for SnO₂ to compare this number with. Terwilliger and Chiang have reported on the grain boundary energy of TiO₂ doped with 1% SnO₂. ^{Reference Terwilliger and Chiang32} The data were acquired by grain growth study in the differential scanning calorimeter, and it was clear that there was a temperature influence on the grain boundary energy. For low temperature ranges (600–780 °C), it was reported to be 0.5–1 J m⁻². This temperature range is consistent with those used to prepare samples P₁, P₂, and P₃, and the energy measured here lies within their estimations.

Unfortunately, due to the special surface conditioning requirement for the sintered samples, it was not possible to perform reliable and meaningful drop solution experiment to acquire surface and grain boundary energies from the sintered samples (P_S).

E. Dihedral angles on thermodynamics of sintering

According to the assessments of interface energies from the two calorimetric methods, the surface energy and grain boundary energy of reoxidized (stoichiometric) SnO₂ is 1.20 ± 0.02 and 0.71 ± 0.08 J m⁻², respectively. Moreover, the surface energy of SnO₂ sintered at 1300 °C (reduced) is 0.94 ± 0.03 J m⁻². As hypothesized, the results suggest that upon sintering conditions (high temperatures), there is an actual change in the predicted driving force for sintering as compared to room temperature. Temperature has been recently reported to affect the driving force for sintering in doped systems by ion migration. ^{Reference Chang, Rufner, van Benthem and Castro33} In our case, we see that the surface redox state can also be affected by temperature and hence affect densification driving forces.

Conventional sintering theory suggests that sintering is driven by the surface energy and the local effect of curvatures in mass transport. Densification is therefore always thermodynamically favorable, but different mechanisms of mass transport would be responsible for nondensification behavior, as that described for SnO₂ (evaporation–condensation and surface diffusion). However, our results suggest that for SnO₂ there is a decrease in the driving force as well behind the observed behavior.

Though not typically considered in sintering models, the dihedral angle has been proven to play a significant role in the microstructure evolution during sintering. Lange et al. have demonstrated that during neck formation, particle arrays of the system may face an energy minimum, ^{Reference Lange34} and the smaller the angle is, the sooner this minimum is reached. Kellet et al. also demonstrated that the sintering stress increases with increase in dihedral angles at whichever contact angle, revealing higher sinterability for samples with larger dihedral angle. ^{Reference Kellett and Lange35} For SnO₂, using the data presented here, one can calculate the dihedral angle at low temperatures and under sintering conditions. If the grain boundary energy is assumed to be equal for both stoichiometric and sintered samples, based on Young's relation, ^{Reference Lange34} the dihedral angle for low temperature (oxidized) condition is 145.58°, and at the sintering condition it would be 135.6°. This implies that there is higher driving force for densification at lower temperatures than at higher temperatures due to the reduction of the surface in the latest.

The assumption that the grain boundary energy is equal for the low temperature and high temperature condition is, however, not justifiable. This is because the grain boundary is also expected to suffer effects from oxygen deficiency, as demonstrated by Terwilliger and Chiang for TiO₂ grain boundaries. ^{Reference Terwilliger and Chiang32} Unfortunately, we could not measure the grain boundary energy of the sintered sample due to experimental limitations (small grain boundary area even after only 30 s of sintering, and instability of surface chemistry condition that would not allow a reliable drop solution experiment). Therefore, to assess the energetic contribution from the grain boundaries, instead of the usage of the calorimetric measurements, the contact/dihedral angle was measured directly from HAADF STEM images. Only clear contact angles were counted, as exemplary shown in the insert image in Fig. 4. The criteria for angle selection were no tilt and no overlapping grains. All details of analysis were shown in our previous work. ^{Reference Chang, Rufner, van Benthem and Castro33} Note that it is particularly difficult to define if the contact angle has reached its equilibrium state (dihedral angle) or not. Ideally, the measurement of angles at different temperatures would allow a better definition if the angle is still changing or not (determining the dihedral angle). However, we faced a fundamental experimental issue: many reports have shown that SnO₂ starts to evaporate at ∼1300 °C. Therefore, we cannot prolong sintering time at such high temperature to avoid the contact angles decreasing due to mass loss. On the other hand, the chemical diffusion coefficient of oxygen in tin dioxide at 1300 °C is 2.58 × 10⁻⁶ cm²/s. ^{Reference Kamp, Merkle, Lauck and Maier36} Therefore, the theoretical diffusion length is 1.7 × 10⁵ nm after only 30 s. At this rate, significant mass flow is expected, reinforcing that based on the extremal principle the measured contact angle is probably very close to dihedral angle.

FIG. 4.

Cumulative counting of dihedral angle measure from STEM images for sintered sample (P_S). The insert image shows a clear dihedral angle with no tilt and overlapping grains. The average dihedral angle is about 100°.

After analysis, the average contact angle on the sintered sample (P_S) is 100 ± 18.76°; its cumulative frequency is shown in Fig. 4. Upon Young's relation, the back calculated grain boundary energy is 1.2 ± 0.24 J m⁻², significantly higher than reoxidized SnO₂ (0.71 ± 0.08 J m⁻²). This is in good agreement with the statement of Terwilliger and Chiang ^{Reference Chiang, Birnie and Kingery31,Reference Terwilliger and Chiang32} who have measured the grain boundary of TiO₂ doped with SnO₂ and showed that the grain boundary energy for a nanocrystalline dense sample systematically increases with temperature. Though SnO₂ is not so susceptible to reduction as TiO₂, still shows observable change in the grain boundary energy as expected. This suggests highly stable pores and explains the low sinterability of SnO₂ from a thermodynamic basis. Note that dihedral angles from the nonsintered samples could not be determined by this technique for comparison because the contact angles are not easily identifiable (obeying criteria of no tilt and no overlapping grains), making any attempt to measure them would be unreliable.

In fact, a simple practical experiment can prove the hypothesis of decreased driving force at high temperatures: comparatively sintering pellets at different atmospheres. Three sintering conditions were tested; one under air (like conventional sintering), another sintered under 42 mL/s pure oxygen flow, and the other sintered under pure argon flow at the same rate. All of them were heated up from 25 to 1300 °C at 5 °C/min, dwelled for 30 s, and then cooled down to 25 °C at the same rate. The densities were measured by Archimedes method. To obtain higher accuracy of density measurement, large pellets were used in this study (∼9.6 mm in diameter, ∼1.2 mm in thickness, ∼200 mg per pellet and 31% relative density for green density). The results are shown in Fig. 5(a). The one sintered with oxygen flow achieves higher density, at least 6% higher in relative density as compared with the sample sintered under air. In contrast, the pellet sintered under argon achieved density 14% lower than the one sintered under air.

FIG. 5.

(a) Relative density of SnO₂ samples sintered under three different atmospheric conditions. One was sintered in air and the others under 42 mL/s pure oxygen or argon flow. To obtain error bar, each sample was measured three times. Relative densities were defined as the ratio of experimental densities to theoretical densities. (b) Surface energy depends on temperature with varying atmospheric conditions. The solid line and dash line represent the prediction of stoichiometric and reduced SnO₂ surface energies, respectively. (c) The plot of normalized surface stresses versus contact angles. The arrow between two stars denotes the possible enhancement while applying extremely high oxygen pressure.

An important question at this point is how much of this change is actually caused by the dihedral angle change, and the answer can be found in calculated sintering stresses. As mentioned before, Lange et al. ^{Reference Lange34,Reference Kellett and Lange35} have shown that a particulate system reaches a meta-equilibrium condition when the angle between two particles meets the dihedral angle. Hence, the sintering stress increases as the ratio of dihedral angles to contact angles increases, meaning higher sinterability for samples with larger dihedral angle. Figure 6 shows STEM images of sintered SnO₂ showing isolated pores where apparently the contact angles are defining their stability, suggesting that indeed this mechanism could be playing a role in the densification behavior. An equation revealing the increased sinterability was deduced by Kellett and Lange, and for a linear array of spheres, it can be expressed as:

where σ _s is sintering stress; R is the sphere size normalized by its initial size; γ_s is surface energy; ϕ and ϕ_e represent contact and dihedral angle, respectively. ^{Reference Kellett and Lange35} Figure 5(c) shows the calculated sintering stress as a function of contact angle using this equation and showing the decrease in sinterability with increasing dihedral angle. We have plotted considering the two angles determined by our experiments for SnO₂, i.e., 145.58° and 100°, representing the stoichiometric and the sintering conditioned surfaces, respectively. A decrease in the sinterability is remarkable, such that for a given contact angle, for instance 40°, sintering stress decreases to half when comparing the two dihedral angles. The plot thus suggests that more stoichiometric samples (larger dihedral angles) will be more prone to densification, evidencing a strong effect of the atmosphere on the densification driving force.

FIG. 6.

STEM images of sintered tin dioxide pellets suggesting pore stability is a governing factor during densification.

The data in Fig. 5(a) (density for sintering under different atmospheres) are aligned with the predicted decrease in sinterability, showing higher densification for the oxygen rich atmospheres. From a practical point of view, it would be interesting to keep the stoichiometric condition at high temperatures to induce potentially higher densification. However, maintaining stoichiometry (large dihedral angles) is not elementary and exceedingly high oxygen pressures would be necessary. To quantify this, we have plotted the surface energy as a function of temperature as predicted by Eqs. (11) and (12) in Fig. 5(b). The plot shows that at low temperatures the surface energy is constant (stoichiometric value, 1.2 J m²) for any given atmosphere because the predicted surface energies for nonstoichiometric surfaces are larger. However, there are though critical temperatures, as indicated in the figure, at which the surface energy under reduced conditions is more stable. For instance, under air flow, above around 800 °C the net surface energy is expected to be smaller than the stoichiometric value. As the temperature increases further, the surface energy decreases more and finds the 0.9 J m⁻² value at 1300 °C, as we have measured. Under argon, one expects an even lower temperature at which the surface energy is decreased, but under oxygen flow this is shifted toward the stoichiometric value.

Oxygen flow is however not close to enough to allow high sinterability of SnO₂. Using the diagram in Fig. 5(b), a pressurized chamber with an oxygen pressure of 10³ atm would be theoretically required to allow fully stoichiometric surfaces at 1300 °C. The predictions of higher densification for the samples under oxygen atmosphere are also consistent with previous results on the sintering of SnO₂ compacts under argon versus oxygen atmospheres. Varela et al. ^{Reference Varela, Whittemore and Longo5} have shown a decrease in average pore size for the oxygen atmosphere sintering condition, suggesting a change in the pore stability thermodynamics, that is directly dependent on the dihedral angle. ^{Reference Kellett and Lange35}

However, the change in atmosphere and temperature will also affect kinetic aspects of densification. Leite et al. suggested that three mechanisms act during sintering of SnO₂, grain boundary/surface diffusion; lattice diffusion, and evaporation condensation. ^{Reference Leite, Cerri, Longo, Varela and Paskocima6} At temperatures above 1300 °C, Hoening and Searcy have shown that the decomposition of SnO₂ into tin and oxygen is very likely, evidencing an active evaporation/condensation mechanism at that condition. ^{Reference Hoenig and Searcy37} An increase in the oxygen pressure is then expected to decrease the kinetics of evaporation by shifting the reaction equilibrium, suppressing this nondensification mechanism and enabling slightly higher densities.

The kinetic argument about the shifted densification behavior can be even strengthened if considering that in general the surface energy of oxides is expected to decrease with increasing temperature. This would be true even for materials that show densification behaviors upon sintering. For instance, Tsoga and Nikolopoulos have proposed an equation describing the surface energy decrease of yttria-stabilized zirconia (YSZ) as a function of temperature, ^{Reference Tsoga and Nikolopoulos38} but it is well known that this system can be sintered up to high densities using conventional sintering techniques (pressureless under air). However, the difference between this case and SnO₂ is the expected size of the change. For YSZ, a small shift is observed (∼0.1 J m⁻²) for an interval of temperature of 300 °C. For SnO₂, an abrupt and significant drop on the surface energy (∼0.26 J m⁻²) is observed when the surface suffers partial reduction. Moreover, a change on the grain boundary energy is also expected due to local oxygen deficiencies, creating unfavorable dihedral angles, as suggested by the STEM images.

The very limited densities achieved in Fig. 5(a) suggest, however, that even if the sintering stresses are increased by a more favorable dihedral angle under high oxygen pressures (∼10³ atm), it is unlike that high densities will be achieved under natural sintering. This is because temperature will increase significantly the defect generation, which will consequently increase the rate of nondensification mechanisms, which are independent on the dihedral angle and an intrinsic feature of tin dioxide as previously reported. ^{Reference Leite, Cerri, Longo, Varela and Paskocima6} A more detailed study quantitatively addressing kinetics and thermodynamics aspects should be performed to clearly state the contribution of each to the nondensification behavior. We believe though that this paper presented the required tools for such studies by providing a quantitative thermodynamic description of SnO₂ microstructure during sintering.