Category Ceramic and Glass Materials

Phase Transitions and the Processing of Zirconia

Upon heating, the monoclinic phase in zirconia starts transforming to the tetragonal phase at 1,461 K, peaks at 1,471 K, and finishes at 1,480 K. On cooling, the transfor­mation from the tetragonal to the monoclinic phase starts at 1,326 K, peaks at 1,322 K, and finishes at 1,294 K, exhibiting a hysteresis behavior that is well known for this material [61-65]. This transformation can also be affected by irradiation with heavy ions, such as 300 MeV Ge [66] and 340 keV Xe [67].

The tetragonal phase transforms to the cubic fluorite structure at 2584 ± 15 K [68]. This transformation temperature has been found to be dependent on the atmosphere
in which the transformation is taking place [6]. In a reducing atmosphere, the trans­formation takes place at approximately 2,323 K, and in a neutral atmosphere, it takes place at approximately 2,563 K, which is in proximity to the highly accurate value found by Navrotsky et al. [68]. Continued heating of the material results in melting at a temperature of 2,963 K [63]. The phase stability as a function of pressure for this material in its pure form is shown in Fig. 20 [7].

The practical use of pure zirconia is restricted by the monoclinic to tetragonal transformation, as this transformation causes cracking and sometimes complete disin­tegration of the specimen. Depending on the orientation of the particular grain that is undergoing the transformation, there is a maximum strain in the lattice of ~4% [29], which is quite significant and promotes failure of the specimen when undergoing heating and cooling cycles.

This transformation has many characteristics of martensitic transformations in metals, with definite orientation relationships between the two structures. The orienta­tion relationships conform to the following [69-71]:

Подпись: (9)(100)J(110)bct and [01°]mll[001]bct ,

and by ^innmg (l00)m|1 (Ш)bct and [omjJ1 [00l]bct

Phase Transitions and the Processing of Zirconia

where m and t represent the monoclinic and tetragonal phases, and bct refers to the body-centered tetragonal structure. Possible variants of these twin relationships for small tetragonal particles are shown in Fig. 21. In this figure, the hashed areas represent the transformed monoclinic phase and the unhashed areas represent the

5 10 15 20 25 30 35 40

Phase Transitions and the Processing of Zirconia

Fig. 21 The four possible arrangements of twin-related variants together with the range of strain values predicted for the directions indicated (adapted from Kelly [76])

untransformed tetragonal phase. As the transformation progresses, the entire particle eventually forms the stable monoclinic phase for this material. The transformation progresses in two stages. The first stage involves a displacive transformation with small shifts of the atoms and the second stage involves a martensitic transformation in which both structures remain almost unchanged [72]. It is this latter transformation that has been studied the most thoroughly [73-75].

To avoid this destructive transformation, stabilization of the tetragonal and cubic structures of zirconia can be done at room temperature by the addition of trivalent dopant ions such as Y3+ and Ce3+, divalent dopant ions such as Ca2+, or tetravalent dopant ions. Doping of zirconia has enormous consequences not only for the mechani­cal properties of this material, but also for the electronic properties. In particular, Y3+ has a large solubility range in zirconia and can be used to stabilize both the tetragonal and cubic phases. To maintain charge neutrality, one oxygen vacancy must be created for each pair of dopant cations that are added to the structure. This results in large increases in ionic conductivity. Stabilization of the tetragonal and cubic structures requires differing amounts of dopants. The tetragonal phase is stabilized at lower dopant concentrations. The cubic phase is stabilized at higher dopant concentrations, as shown in the room temperature region of the ZrO2-Y2O3 phase diagram in Fig. 22

[77] . At higher Y2O3 doping, the material exhibits an ordered Zr3Y4O12 phase at 40 mol% Y2O3, a eutectoid at a temperature < 400°C at a composition between 20 and 30 mol% Y2O3, a eutectic at 83 ± 1 mol% Y2O3, and a peritectic at 76 ± 1 mol% Y2O3

[78] . Other zirconia phase diagrams have been developed by Stubican and Ray for ZrO2-CaO [79], Grain for ZrO2-MgO [80], Cohen and Schaner for ZrO2-UO2 [81], Mumpton and Roy for ZrO2-ThO2 [82], Barker et al [83] for ZrO2-Sc2O3, and Duwez and Odell for ZrO2-CeO2 [84], among others.

As mentioned briefly in Sect. 4, another way of stabilizing the tetragonal structure at room temperature is the formation of nanocrystalline powders or nanograined sintered specimens. To obtain powders of dense PSZ compacts at room temperature, the material has to contain crystals or grains below a certain critical size, which

Подпись: Fig. 22 Zirconia-rich end of the yttria-zirconia phase equilibrium diagram [77] (reprinted with permission)

increases as the dopant concentration increases. The critical size is 22.6 (also found to be -18 nm by Chraska et al. [85] and 15.3 nm by Garvie [86]), 41.7, 67, and 93.8 nm for yttria doping concentrations of 0, 0.5, 1.0, and 1.5 mol% [87]. The values decrease with increasing dopant concentration, consistent with the fact that yttria is a tetragonal – phase stabilizer. Changes in the transformation temperature with dopant concentration and crystallite size are shown in Fig. 23 [87], where it can be seen that the transforma­tion temperature decreases with decreasing crystallite size and increasing dopant con­centration. The dotted lines represent theoretical curves calculated according to:

10Ah rf

Подпись: transformation Phase Transitions and the Processing of Zirconia Подпись: (10)

DH v0l + J suf

where AH. is the volumetric heat of transformation, Ah f is the surface enthalpy dif­ference, dcritical is the critical crystallite size to stabilize the tetragonal phase at room temperature, AS. is the volumetric entropy of transformation, and As, is the surface

vol surf

entropy difference. The solid curves are from the standard ZrO2-Y2O3 phase diagram (Fig. 22). The solid circles represent experimental data on samples that happened to have crystallite sizes close to those for which the theoretical curves were calculated.

Phase Transitions and the Processing of Zirconia Подпись: AH T AH ~ (for powders) 6AYTb 6Ay Подпись: (11)

The stabilization of the tetragonal phase at room temperature due to a decrease in the crystallite size has been attributed to a surface energy difference and roughly obeys the relationships [88]:

Phase Transitions and the Processing of Zirconia

where dcritical is the critical crystallite/grain size, AH – is the enthalpy of the tetragonal – to-monoclinic phase transformation in a sample with infinite crystallite/grain size, T is the temperature of transformation, Ag is the difference in surface energy in pow­der crystallites, AS is the difference in interfacial energy in sintered pellets, Tb is the transformation temperature for an infinitely large-grained sample, and AUse is the strain energy involved in the transformation. From these equations, it can be seen that the same material in the solid form has a lower transformation temperature than in the powder form. This difference is due to the strain energy, AUse, involved in the trans­formation, which is present only in the pellets since there is a requirement for geomet­ric compatibility that is not present in the powders.

Diffusion Coefficients

Diffusion Coefficients Подпись: :9.73 ± 1.4 x 10-3 exp Подпись: 56.0 ± 2.4 kcal/mol 1 RT J (5)

Diffusion in zirconia is closely linked to ionic conductivity. Consequently, some diffusion data has already been presented in Sect. 5. This section will include additional results par­ticularly for monoclinic zirconia. Oxygen self-diffusion at a pressure of 300 Torr, as determined by testing zirconia spheres of diameters between 75 and 105 (dm, behaves as shown in Fig. 17 [57], where D is the diffusion coefficient, t is time, and a is the sphere radius. At a pressure of 700 Torr, the behavior changes to that shown in Fig. 18 [58]. In this case D* is the self-diffusion coefficient and the rest of the terms are as defined before, with a = 100-150 (dm. Both of these experiments were per­formed in an oxygen atmosphere of 18O-16O. The self-diffusion coefficients calculated from the diffusion data obey Arrhenius expressions as illustrated in Fig. 19 [57, 58]. The linear fits describing the diffusion coefficient at 300 and 700 Torr, are given by:

Diffusion Coefficients

According to Ikuma et al. [59], surface diffusion and lattice diffusion should be sepa­rated and result in the following diffusion coefficients:

Fig. 19 Arrhenius plot of oxygen self-diffusion in mono­clinic zirconia (adapted from Madeyski and Smeltzer [57] and Keneshea and Douglass [58])

Diffusion CoefficientsПодпись:Подпись: 87.0 kJ/mol RT (8)

These two expressions are not that very different. Hence, the macroscopic diffusion behavior of monoclinic zirconia can be approximated by lattice diffusion, while sur­face diffusion can be ignored.

Diffusion in pure tetragonal and cubic zirconia is experimentally challenging because it requires the higher temperatures at which the two phases are stable. However, simulations at temperatures between 1,273 and 2,673 K have been performed on cubic zirconia, showing noticeable, but not large, oxygen ion diffusion along the grain boundaries and a significant energy barrier to movement from the grain bounda­ries into the bulk, although at higher temperatures diffusion is obviously enhanced. However, even at higher temperatures, diffusion along the grain boundary is not as favorable as that across the grain boundary [60].

Electronic Properties

Cubic zirconia doped with oxides such as Y2O3 or CaO is the material of choice for many high temperature applications because of its extremely high ionic conductivity at intermediate and high temperatures. A review on the properties of these specialized rare-earth stabilized zirconia materials has been prepared by Comins et al. [50].

The oxygen pressure dependence of the conductivity in tetragonal zirconia can be seen in Fig. 13 [51]. This material is a mixed electronic and ionic conductor with a large ionic contribution except at very high temperatures or very low oxygen partial pressures. The electronic component of the conductivity arises from doubly-charged oxygen vacancies at lower oxygen pressures and a temperature of 1,400°C. Other contributions to conductivity are difficult to determine. The movement of oxygen vacancies can take place along two directions for the tetragonal structure: within the x-y plane along the [110] direction or perpendicular to this plane along the [001] direction. In both directions, the O-O distances are very similar (0.2640 nm within the (x, y) plane and 0.2644 nm in the direction perpendicular to that plane) [25]. From these numbers, it would appear that there is no preferential direction for diffusion.

Electronic Properties

Fig. 13 Conductivity isotherms for tetragonal zirconia as a function of oxygen pressure [51] (reprinted with permission)

However, the diffusion process is controlled by the Zr-Zr distance and not by the O-O distance, since the vacancy must move between two such ions to diffuse. Along the two relevant directions, these distances are 0.3655 nm for the [110] direction and 0.3645 nm for the [001] direction. The diffusion barriers for movement of a neutral vacancy along [110] and [001] are 1.35 and 1.43 eV, respectively. This is expected from the fact that there is a smaller gap between zirconium ions along the [001] direc­tion. Hence, diffusion along this direction proves to be more difficult. The diffusion barriers for movement of a doubly-charged vacancy along the two relevant directions are 0.22 and 0.61 eV, respectively. Again, movement along the [001] direction proves to be more difficult. This can be visualized in Fig. 14 [6].

Monoclinic zirconia is both an electron and ion conductor depending on the temperature and oxygen pressure (Fig. 15) [52-54]. At low pressures, it exhibits n – type behavior in which the charge carriers are double-charged oxygen vacancies, while at higher pressures it exhibits p-type behavior in which the charge carriers are singly-ionized oxygen interstitials. The transition from n-type to p-type is established by the change in sign of the conductivity curve. Assuming the -1/6 and 1/5 depend­ences in the two regions are good fits to the data, the total conductivity at 1,000°C can be represented by:

s 1,ooo°C = 8.5 X10-5 pOi 1/5 +1.1 x 10-9 pOi ~116 + 3.2 x 10-6. (4)

In addition, Vest et al. [53] determined the hole mobility at 1,000°C to be

m1 000°C = 1.4 x 10-6 cm2 • V-1 • s-1.

If the pressure is kept constant and the temperature is increased, the conductivity also increases (see Fig. 4 of Kumar et al. [52]). At lower temperatures (< 600°C), con­ductivity is predominantly ionic, and at higher temperatures (> 700°C), it is predomi­nately electronic. Between 600 and 700°C, both ionic and electronic conductivities are seen in this material. Values of the activation energies required for each type of

Electronic Properties

Fig. 14 Simplified representation of possible diffusion mechanism for oxygen atoms in tetragonal zirconia. (a) Tetragonal cell with two octahedral empty sites, marked with black squares 1 and 2, (b) Position of oxygen 1 during its motion past the zirconium 1-3 face, (c) possible off-centered position for oxygen 1 inside octahedral site 1 (adapted from [6])

Electronic Properties

Fig. 15 Oxygen pressure dependence of total conductivity for monoclinic zirconia at 990°C (adapted from Kumar et al. [52] and Vest et al. [53])

conductivity are still a matter of controversy because of the complexity of the conduc­tion processes. Earlier values include numbers such as 3.56 eV for n-type conductivity and 0.86 eV for p-type conductivity [55].

The conductivity of two high-pressure phases of zirconia is shown in Fig. 16 [56]. The discontinuities in the conductivity occur approximately at 1,000°C for the sample at 16.5 GPa and 1,050°C for the sample at 18.0 GPa. At the higher temperatures, the conductivity corresponds to a so-called “cubic” high-pressure and high-temperature phase of zirconia, although its exact nature was not determined by Ohtaka et al. [56].

Подпись: Fig. 16 Electrical conductivity of pure cubic zirconia at (a) 16.5 GPa and (b) 18 GPa [56] (reprinted with permission)

At the lower temperatures, the conductivity corresponds to the orthorhombic-II phase. From the Arrhenius plots in the figure, approximate activation energies for conduction can be obtained. For the “cubic” phase, the activation energies are 8.80 and 0.60 eV at pressures of 16.5 and 18.0 GPa, respectively, while for the orthorhombic-II phase they are 0.72 and 0.40 eV for the two pressures studied.


Using the strain rate data shown in Fig. 12, the activation energy for creep in mono­clinic zirconia has been found to be Qc ~ 330-360 kJ mol-1 [48, 49]. The measured stress exponent, n, from equation:




was found to be 1.7 by Roddy et al. [48] and 2.3-2.5 by Yoshida et al. [49]. In this equa­tion, A is a constant, G is the shear modulus, b is the Burger’s vector, d is the grain size,


Fig. 12 Stress dependence of monoclinic zirconia during creep deformation (adapted from Roddy et al. [48] and Yoshida et al. [49])

s is the applied stress, p is the grain size exponent, n is the stress exponent, and D is the diffusion coefficient. The value by Roddy et al. is intermediate between diffusional creep (n = 1) and superplastic deformation (n = 2), but closer to superplastic deformation, whereas the values by Yoshida et al. are higher than both, but close to the stress exponent for superplastic deformation. The grain size exponent, p, was found to be 2.8 by Roddy et al., which is closer to p = 3 for Coble creep (lattice diffusion) than p = 2 for superplastic or Nabarro-Herring creep (grain boundary diffusion). However, Yoshida et al. found values between 2.4 and 2.5. From the exponents found in both studies, it is likely that creep deformation in monoclinic zirconia is due to superplastic deformation.


The toughness of pure monoclinic zirconia is difficult to obtain because of problems encountered during sintering of these types of specimens. Generally, if a full density is desired for mechanical properties evaluation, the material needs to be heated to a temperature that is above the tetragonal-to-monoclinic transformation temperature (i. e., 1,471 K). This results in severe cracking upon cooling. However, there have been a few studies that have shown that nanocrystalline monoclinic zirconia can be sintered to full density at 1,273 K. In this case, microcracking during cooling can be avoided [35]. Unfortunately, these specimens have not been tested for toughness.

Experiments have been attempted with porous specimens of monoclinic zirconia and the fracture toughness has been extrapolated. A value of 2.06 ± 0.04 MPa m1/2 was found for a specimen of 92.2 ± 0.4 % relative density, from which a fracture toughness
of 2.6 MPa m1/2 was extrapolated for a specimen of full density [36]. Slightly higher numbers of 3.7 ± 0.3 MPa • m1/2 were found for specimens with > 95% density [32]. Evidently, the fracture toughness of this phase of zirconia is quite low. The toughness of cubic zirconia is also low, reported as 2.8 MPa m1/2 by Chiang et al. [37] and 1.8 ± 0.2 MPa • m1/2 by Cutler et al. [32].

The addition of alloying elements such as Y3+, Ce3+, and Mg2+ can result in stabili­zation of tetragonal zirconia, which results in an increase in the fracture toughness of the material via a process of transformation toughening. The addition of increasing amounts of the stabilizing elements results in the stabilization of the cubic phase, which does not have transformation-toughening behavior. Toughening requires the presence of the metastable tetragonal phase.

As can be seen in Fig. 10 [29], the fracture toughness in polycrystalline tetragonal zirconia (TZP) and partially-stabilized zirconia (PSZ) appears to reach a maximum. This indicates a transition from flaw-size control of strength to transformation-limited strength. Ranges of fracture toughness values for zirconia composites are given by Richerson [38].

The stability of the tetragonal structure can be controlled by three factors: the grain size [39, 40], the constraint from a surrounding matrix [41, 42], and the amount of dopant additions. Commonly, very small tetragonal particles are added as a reinforc­ing phase to a matrix of another material, which is usually brittle (i. e., pure cubic or monoclinic zirconia, alumina [43], Si3N4 [44], and others [45]) as shown in Fig. 11a. This results in a higher overall toughness for the composite. For example, Gupta et al. [46] has shown that the addition of small tetragonal particles to a matrix of monoclinic zirconia results in an increment of the toughness to values between 6.07 and 9.07 MPa • m1/2, in contrast to the low numbers observed for pure monoclinic zirconia. A review on the transformation toughnening of several zirconia composites has been prepared by Bocanegra-Bernal and Diaz De La Torre [42].

This toughening mechanism is associated with the increase in volume upon transformation to the monoclinic phase. Since the monoclinic phase occupies a larger volume compared with the tetragonal phase, it forces closure of any propagating cracks, greatly diminishing the catastrophic failure of the material due to fracture [47]. In addition, the transformation from tetragonal to monoclinic results in energy absorp­tion that blunts the crack.

The transformation is induced by an applied stress on the material. Initially, a ceramic composite may contain a crack that begins to propagate upon application of

ToughnessFig. 10 Strength vs. fracture toughness for a selection of ZrO2-toughened engineering ceramics [29] (reprinted with permission)

a stress (Fig. 11a). If the composite contains metastable tetragonal particles, the large stresses at the tip of the crack can force the tetragonal-to-monoclinic transformation of these particles increasing the volume of material in the region of the crack and forc­ing crack closure (Fig. 11b). The positive change in volume during the transformation is small but significant. If the positive change in volume is large, it can result in frag­mentation of the material. On the other hand, a negative change in volume will not result in strains that promote crack closure. Hence, zirconia is quite unique in that the monoclinic and tetragonal structures are very close in density such that exaggerated volume increases are avoided during transformation. As the amount of dopant is increased, the stability of the tetragonal phase is higher and the transformation becomes more sluggish.

Indeed, Bravo-Leon et al. [31] and Sakuma et al. [34] have found that the tough­ness is higher for samples with yttria concentrations lower than the typical 3 mol% used for this material. Fracture toughness values of 16-17 MPa m1/2 were reached by Bravo-Leon et al. for a 1 mol% yttria specimen with a grain size of 90 nm and a 1.5 mol% yttria specimen with a grain size of 110 nm. This can be attributed to the lower stability of the tetragonal phase with low dopant concentrations, which easily transforms to the monoclinic phase upon application of the stress.


The hardness for monoclinic zirconia is approximately 9.2 GPa [31] for samples with a density > 98% and 4.1-5.2 GPa [32] for samples with a density > 95% of theoretical, whereas hardness values for amorphous zirconia vary between 5 and 25 GPa [33]. The hardness increases slightly to values approaching 11 GPa for yttria-stabilized zirconia of 1.5 mol% yttria, which is stabilized in the tetragonal form [31]. Addition of larger amounts of yttria dopant results in hardness values approaching 15 GPa [34].

Elastic Properties

The measured elastic stiffness and compliance moduli for monoclinic zirconia have been summarized by Chan et al. [30]. The Young’s and shear moduli of this same

Table 4 Polycrystalline Young’s and shear moduli for monoclinic zirconia in GPa (adapted from Chan et al. [30]








































structure are given in Table 4 and were calculated using the Voigt, Reuss, and Hill approximations. The Voigt and Reuss approximations usually give the upper and lower bounds of these parameters. The maximum errors in these numbers are about 10% for most values, but can increase to greater than 20% for some of the transverse directions in the crystal.

For the monoclinic and tetragonal structures, the bulk modulus hovers around 150— 200 GPa. Cubic zirconia has higher bulk modulus somewhere around 171-288 GPa. The high-pressure phases have values around 224-273 GPa and 254-444 GPa, for the orthorhombic-I and orthorhombic-II phases, respectively.

Mechanical Properties

Measurements of the mechanical properties of pure tetragonal and cubic zirconia are exceedingly difficult because of the higher temperatures required for such measure­ments. Hence, only monoclinic zirconia has been thoroughly studied in pure form. The mechanical properties of tetragonal and cubic zirconia have been determined for many stabilized zirconias and, because of the importance of these materials in engineering applications, several reviews have been written [27-29].

Vacancy Defects

Oxygen vacancies in cubic zirconia result in a calculated displacement pattern as shown in Fig. 9 [2]. In this figure, the vacancy is depicted as a small cube, the oxygen

Vacancy Defects

Fig. 8 Charge density in the plane through ZrA, OA, and O., and a schematic diagram of a neutral oxygen interstitial (О,) near a triple-bonded oxygen (OA) in zirconia. Charge density is in 0.1 eV A-1 and all distances are in A [24] (reprinted with permission)

Vacancy Defects

Fig. 9 Displacement pattern of atoms around an isolated vacancy in a 95 atom supercell [2] (reprinted with permission)

atoms occupy the sites at the corners of the cubes, and metal cations occupy half of the sites at the center of the cubes. The six oxygen neighbors nearest to the vacancy move along <100> by 0.024 nm, while the zirconium atoms move outward along <111> by 0.018 nm. The oxygen atoms nearest to the zirconium, but not nearest to the vacancy, follow the displacement of the cation and move outwards along <111> by 0.004 nm. The oxygen atoms in the outermost right corner of the figure move inwards by 0.004 nm along <111>.

Tetragonal zirconia contains anion vacancies and may be written as ZrO2_x, with x varying from 0.001 at 1,925°C to 0.052 at 2,410°C [6]. To accommodate these vacan­cies, surrounding ions move toward the vacancy to reduce its size. The two zirconium ions move by an approximate amount of 0.008 nm and the oxygen ions by 0.013 nm. The energy gain due to the relaxation of these ions is 0.22 eV. These values are, of course, different depending on the charge of the vacancy [25].

For the case of a singly-charged vacancy, the structural distortion results in the movement of surrounding oxygen ions by an approximate amount of 0.022 nm toward the vacancy and the zirconium ions by 0.009 nm away from the vacancy. These values are modified for the case of a doubly-charged vacancy to 0.033 nm for the oxygen ions and 0.022nm for the zirconium ions. Obviously, the higher the positive charge of the vacancy, the greater the distortion towards or away from it. The energy gains due to the formation of singly – and doubly-charged vacancies are 1.0 and 3.3 eV, respectively.

Oxygen vacancies in monoclinic zirconia can occur in both the triple-planar and tetragonal geometries. When the vacancy is neutral, these vacancies have formation energies of 8.88 eV and 8.90 eV, respectively. Once the vacancy is singly charged posi­tively (i. e., V+) and in a tetrahedral position, the atomic relaxation energy is 0.47 eV. Creation of a doubly-charged positive vacancy (i. e., V2+) in a tetrahedral position causes further displacement of the four surrounding zirconium ions away from the vacancy by an additional 0.01 nm. This leads to a further decrease in energy of 0.74 eV. Creation of a singly-charged negative vacancy (i. e., V-) in the same tetrahe­dral position causes minimal displacement of the surrounding zirconium ions (by less than 0.002nm) and an energy decrease that is less than 0.1 eV, which clearly points to the fact that the additional electron is only weakly localized in the vicinity of the vacancy and, hence, has little influence on the surrounding ions. The lattice relaxation and formation energies in the case of a neutral zirconium vacancy are about 1.4 and 24.2 eV, respectively. The oxygen ions surrounding this type of vacancy are displaced outwards from their equilibrium positions by about 0.01-0.02 nm.

At higher temperatures (i. e., 1,000°C) and excess partial pressure of oxygen (i. e., 10-6to 1 atm.), monoclinic zirconia contains completely ionized zirconium vacancies [26]. At 1,000°C, zirconia is stoichiometric at a pressure of 10-16 atm. At this point, the concentration of oxygen vacancies is equal to twice the concentration of zirconium vacancies. As the partial pressure of oxygen increases, the stoichiometry changes such that for ZrO^ with the d value defined by:

d = 6 x 10-3 Vx0, (2)

where po2 is the oxygen partial pressure in atm.

Interstitial Defects

Interstitial defects in monoclinic zirconia have been modeled in detail by Foster et al. [24]. Using plane wave density functional theory, the tetragonal bonding and triple­planar bonding geometries of lattice oxygen ions were determined. In addition, it was determined that interstitial defects can form stable defect pairs with either type of lattice oxygen ions (i. e., tetragonal or triply bonded). The analysis looked at defect pairs formed by interstitial oxygen ions with three possible charge states: 0, -1, and -2, bonded to triple-planar lattice oxygen ions. An analysis of oxygen vacancies both in the triple-planar and tetragonal geometries was also undertaken.

A neutral oxygen interstitial forming a defect pair with a triple-bonded oxygen is illustrated in Fig. 8 [24]. Using the oxygen atomic energy as a reference, a single neutral oxygen can be incorporated in the lattice as an interstitial with an energy gain of -1.6 eV, if next to a triple-bonded lattice oxygen, and -0.8 eV, if next to a tetragonally bonded lattice oxygen. Figure 8 illustrates the fully relaxed charge density and positions of ions, showing that the interstitial and lattice oxygen form a strong covalent bond. The labels A and B associated with the lattice ions represent two different crystal planes within the structure. The lattice oxygen (OA), forming the defect pair with the interstitial oxygen, relaxes by up to 0.05 nm to accommodate the interstitial, distorting the triply-bonded oxygen with respect to the three zirconium ions bonded to it. The O-3Zr group has a slight pyramidal shape with its apex pointing away from the interstitial. The rest of the crystal remains more or less undisturbed, with the nearest zirconium (ZrA) only relaxing by about 0.005 nm. The case of a singly-charged oxygen interstitial forming a defect pair with a triply-bonded oxygen results in weakening of the covalent bond between the defect pair significantly. The extreme is the case of a doubly-charged oxygen interstitial in which the interstitial forms elongated bonds with the zirconium ions and occupies a new triple site, which is bond­ing with the ZrA, ZrB, and a new zirconium ion.