Point-Defect Segregation and Space-Charge Potentials at the Σ5(310)[001] Grain Boundary in Ceria

Abstract

The atomistic structure and point-defect thermodynamics of the model $\Sigma 5\left(310\right)\left[001\right]$ grain boundary in CeO₂ were explored with atomistic simulations. An interface with a double-diamond-shaped structural repeat unit was found to have the lowest energy. Segregation energies were calculated for oxygen vacancies, electron polarons, gadolinium and scandium acceptor cations, and tantalum donor cations. These energies deviate strongly from their bulk values over the same length scale, thus indicating a structural grain-boundary width of approximately 1.5 nm. However, an analysis revealed no unambiguous correlation between segregation energies and local structural descriptors, such as interatomic distance or coordination number. From the segregation energies, the grain-boundary space-charge potential in Gouy–Chapman and restricted-equilibrium regimes was calculated as a function of temperature for dilute solutions of (i) oxygen vacancies and acceptor cations and (ii) electron polarons and donor cations. For the latter, the space-charge potential is predicted to change from negative to positive in the restricted-equilibrium regime. For the former, the calculation of the space-charge potential from atomistic segregation energies is shown to require the inclusion of the segregation energies for acceptor cations. Nevertheless, the space-charge potential in the restricted-equilibrium regime can be described well with an empirical model employing a single effective oxygen-vacancy segregation energy.

1. Introduction

The best solid-state ionic conductors generally consist of a highly conductive bulk phase and less conductive grain boundaries. Depending on a variety of factors—the specific material, its composition, and its thermal and chemical history—the diminished grain-boundary conductivity may stem from resistive impurity phases [1,2,3], from decreased charge-carrier concentrations in space-charge zones adjacent to the interface [4,5,6,7,8,9,10], or from the intrinsic structural resistance of the interface [11,12]. Improving the overall rate of ion transport in such materials requires the origin of the diminished grain-boundary conductivity to be identified and understood in detail.

One prototypical system in this regard is the oxide-ion conductor substituted CeO₂, a material utilised as an electrolyte in Intermediate Temperature Solid Oxide Fuel Cells (IT-SOFC) [13,14,15,16]. Initial work indicated that the diminished grain-boundary conductivity came from a second phase at the grain boundaries [17,18,19,20]. Later work on substantially purer samples found that even in the absence of impurity phases, low grain-boundary conductivity remained [21,22,23], and this effect was attributed to the presence of space-charge zones in which the ionic charge carriers (oxygen vacancies) are depleted [7,23]. Treating this phenomenon in a thermodynamic framework based on point-defect segregation requires space-charge layer formation to be driven by the segregation of oxygen vacancies from the bulk to the grain-boundary core. Indeed, such a model has been shown to describe quantitatively not only isothermal resistances for dilute [4,24,25] and concentrated compositions [8,26], but also resistances for a dilute composition as a function of temperature and oxygen partial pressure [27]. In addition, the model accounts for grain boundaries being oxygen-deficient [28,29,30,31] (they contain an excess of oxygen vacancies) and for the accumulation of acceptor dopants in the space-charge zones [28,30,31] (aliovalent species react to the electric field set up by the charged vacancies’ redistribution from bulk to grain boundary). Furthermore, it also explains [32] why faster cation diffusion along grain boundaries, with an activation enthalpy close to that for bulk diffusion, is observed [33]. In all cases [4,8,24,25,26,27], quantitative modelling yielded an effective Gibbs segregation energy for oxygen vacancies to the grain boundaries of ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}=-\left(1\mathrm{to}2\right)\mathrm{eV}$. NB: the sign of the segregation energy is sometimes reversed in the literature (e.g., refs. [34,35]). We define ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{def}}={\Delta }_{\mathrm{f}}{g}_{\mathrm{def}}^{\mathrm{gb}}-{\Delta }_{\mathrm{f}}{g}_{\mathrm{def}}^{\mathrm{b}}$, i.e., as the difference in the Gibbs formation energy of a point defect between grain boundary and bulk, so that a negative segregation energy indicates a thermodynamic driving energy for defect segregation from the bulk to the grain boundary.

Atomistic simulation studies of selected grain boundaries in CeO₂ [31,34,36,37,38] and other fluorite-related oxides [31,35,39,40,41,42,43,44,45] found that the boundaries are characterised not by a single site of preferential formation, as typically assumed in the thermodynamic model, but by a number of symmetry-inequivalent sites, each with its own specific segregation energy. Some sites exhibit a negative segregation energy (${\Delta }_{\mathrm{seg}}{u}_{\mathrm{def}}<0$), meaning that oxygen-vacancy segregation to these sites is thermodynamically favourable; other sites exhibit positive segregation energies; i.e., oxygen vacancies will preferably redistribute from these sites to the bulk. In addition, the studies indicate favourable segregation energies for acceptor dopants (${\Delta }_{\mathrm{seg}}{u}_{\mathrm{a}}<0$). Space-charge potentials have not yet been calculated from such simulation data (${\Delta }_{\mathrm{seg}}{u}_{\mathrm{v}}<0$, ${\Delta }_{\mathrm{seg}}{u}_{\mathrm{a}}<0$), but it is clear that the segregation of acceptor cations cannot govern space-charge formation; otherwise, a negative space-charge potential (with accumulated oxygen vacancies in the space-charge zones) would result, and consequently, the experimentally observed hindrance of oxide-ion transport across boundaries would not be accounted for. The effect of combined ${\Delta }_{\mathrm{seg}}{u}_{\mathrm{v}}<0$ and ${\Delta }_{\mathrm{seg}}{u}_{\mathrm{a}}<0$ on the space-charge potential remains, therefore, to be elucidated.

The magnitude of the defect-segregation energies has been analysed in detail in previous studies [34,35,37,46,47] and different influences on the segregation energies have been discussed. In particular, the segregation energies of the cations have been analysed in terms of the mismatch between the ionic radius of the dopant and that of the host cation [34,35,46]. Besides this, bond lengths and coordination numbers in the boundary have also been claimed as important parameters [35,47]. However, no clear correlation between the segregation energies and these structural descriptors was found.

In addition to the question of the origin, there is the question of the grain-boundary width. This quantity, it is noted, can be defined in a number of ways [48]. The structural or crystallographic width refers to the finite distance over which the mismatch between two grains is accommodated [49]; the chemical width refers to the region over which the composition is different to the bulk composition [50,51,52,53]; the electrical width refers to distance obtained from impedance spectroscopy measurements [53,54]; and the diffusional width refers to the region along which diffusion may occur faster than in the bulk [27,55,56]. Here, we examine various methods to determine the structural width from atomistic simulations.

In this way, we come to the three main questions of this study: first, how the structural width of a grain boundary can be extracted from atomistic simulations; second, whether segregation energies can be related to local structural descriptors; and third, how a variety of segregation energies, for more than one point-defect species, affects the space-charge potential. To this end, we used Molecular Statics (MS) and Molecular Dynamics (MD) simulations to model the structure and the segregation thermodynamics of a grain boundary in CeO₂. Subsequently, we used Finite Element Method (FEM) calculations to obtain from the segregation energies the space-charge potential at the boundary as a function of temperature.

We chose the symmetric $\Sigma$5(310)[001] tilt grain-boundary as a model system because the boundary is highly symmetric and has a small repeat unit, thus making it accessible to atomistic simulations. A variety of point defects may be present in varying concentrations in CeO₂, depending on the type and concentration of dopant, the temperature, and the oxygen partial pressure. Previous studies [57,58,59] indicated that in acceptor-doped systems, the dominant defects are oxygen vacancies (and, under reducing conditions, also electron polarons); in donor-doped systems, the dominant defects are electron polarons, and oxygen interstitials [60,61,62]. To probe defects on anion and cation sublattices, we therefore determined ${\Delta }_{\mathrm{seg}}{u}_{\mathrm{def}}$ for oxygen vacancies (${\mathrm{v}}_{\mathrm{O}}^{••}$), Sc and Gd acceptors (${\mathrm{Sc}}_{\mathrm{Ce}}^{\prime }$ and ${\mathrm{Gd}}_{\mathrm{Ce}}^{\prime }$), electron polarons (${\mathrm{Ce}}_{\mathrm{Ce}}^{\prime }$), and Ta donors (${\mathrm{Ta}}_{\mathrm{Ce}}^{•}$). (Defects are given in Kröger–Vink notation [63,64,65].)

2. Computational Methods

2.1. Atomistic Simulations

Atomistic simulations were based on the Born model of ionic solids [66], which treats ions as classical particles interacting through long-range electrostatic interactions and short-range parameterised Buckingham pair potentials (see Table 1) [67,68,69]. This description has been successfully applied to simulating CeO₂ in diverse cases [67,70,71,72,73]. Both Molecular Statics (MS) and Molecular Dynamics (MD) simulations were performed with the Lammps code [74]. Energy minimisations were performed at constant pressure $p=0$; MD simulations were performed within the isothermal–isobaric ($NpT$) ensemble over simulation times of $3ns$.

Table 1. Parameters for the short-range Buckingham pair potentials.

Pair	$\mathit{A}/\mathrm{eV}$	$\mathit{\rho }/\AA$	$\mathit{C}/\left(\mathrm{eV}{\AA }^6\right)$	Ref.
${\mathrm{O}}^{2-}-{\mathrm{O}}^{2-}$	$22764.3$	$0.149$	$27.89$	[67]
${\mathrm{Ce}}^{4+}-{\mathrm{O}}^{2-}$	$1986.83$	$0.35107$	$20.4$	[67]
${\mathrm{Ce}}^{3+}-{\mathrm{O}}^{2-}$	$1731.61808$	$0.36372$	$14.43256$	[67]
${\mathrm{Gd}}^{3+}-{\mathrm{O}}^{2-}$	$1336.8$	$0.3551$	–	[68]
${\mathrm{Sc}}^{3+}-{\mathrm{O}}^{2-}$	$1299.4$	$0.3312$	–	[68]
${\mathrm{Ta}}^{5+}-{\mathrm{O}}^{2-}$	$1315.572$	$0.36905$	–	[69]

Table 1. Parameters for the short-range Buckingham pair potentials.

Pair	$\mathit{A}/\mathrm{eV}$	$\mathit{\rho }/\AA$	$\mathit{C}/\left(\mathrm{eV}{\AA }^6\right)$	Ref.
${\mathrm{O}}^{2-}-{\mathrm{O}}^{2-}$	$22764.3$	$0.149$	$27.89$	[67]
${\mathrm{Ce}}^{4+}-{\mathrm{O}}^{2-}$	$1986.83$	$0.35107$	$20.4$	[67]
${\mathrm{Ce}}^{3+}-{\mathrm{O}}^{2-}$	$1731.61808$	$0.36372$	$14.43256$	[67]
${\mathrm{Gd}}^{3+}-{\mathrm{O}}^{2-}$	$1336.8$	$0.3551$	–	[68]
${\mathrm{Sc}}^{3+}-{\mathrm{O}}^{2-}$	$1299.4$	$0.3312$	–	[68]
${\mathrm{Ta}}^{5+}-{\mathrm{O}}^{2-}$	$1315.572$	$0.36905$	–	[69]

Using the gosam software (https://gosam.hepforge.org/) [75], we generated an initial grain-boundary supercell with 23,040 ions containing two anti-parallel $\Sigma$5(310)[001] grain boundaries. This involved the rotation of two fluorite-structured ($Fm\overline{3}m$) ceria grains around the $\left[001\right]$ axis by $\theta ={36.9}^{\circ }$, so that their $\left(310\right)$ planes coincided. Subsequently, we performed three-dimensional rigid-body translations of the two grains, i.e., we shifted the grains relative to one another in steps of 1 Å, and performed at each step an energy optimisation, in this way scanning the so-called $\gamma$-surface. For each configuration, we then carried out a high-temperature anneal (at $T=2000$ K) in an MD simulation, followed by a final static energy optimisation. The grain-boundary structure with the lowest energy is shown in Figure 1. The corresponding supercell had dimensions of ca. $65.3 \AA \times 68.7 \AA \times 70.3 \AA$; the two anti-parallel boundaries are thus separated by a distance of $d_{\mathrm{gb}}-\mathrm{gb}=35.1$ Å. This grain-boundary simulation cell was employed in a previous communication [12].

We found scanning the $\gamma$-surface and a sufficiently high temperature in the MD run to be decisive for obtaining the low-energy configuration. We recommend, as other authors have [76], a three-step procedure involving first scanning the $\gamma$-surface; subsequently carrying out a high-temperature MD run for each rigid body translation over extended simulation times; and finally conducting a static energy minimisation. In particular, we emphasise the need to use sufficiently large cells, and to combine different techniques to obtain the minimum-energy configuration.

A single point defect was then introduced at various symmetry-inequivalent positions in the supercell (its charge being compensated by a uniform background charge), and all ion positions were subsequently relaxed (at constant volume) to zero net force with a steepest-descent algorithm. The defect formation energies (${\Delta }_{\mathrm{f}}{u}_{\mathrm{def}}$) were determined at all positions at zero Kelvin.

Figure 1. Simulation supercell with two anti-parallel $\Sigma 5\left(310\right)\left[001\right]$ grain boundaries, one at the centre of the cell and one half at each edge, viewed along the [001] axis (corresponding to the z direction of the simulation cell), visualised with the Molara package [77]. The light-yellow spheres denote cerium ions, and the red spheres, oxide ions. The closeups show the characteristic structural motif, which has a double-diamond shape, as indicated by the blue and green sketched diamonds; the ions occupy alternating positions both along x and y directions. In the closeup on the right, the alternating cerium positions are highlighted in light green and dark green; and the alternating oxygen positions are highlighted in light red and dark red.

Figure 1. Simulation supercell with two anti-parallel $\Sigma 5\left(310\right)\left[001\right]$ grain boundaries, one at the centre of the cell and one half at each edge, viewed along the [001] axis (corresponding to the z direction of the simulation cell), visualised with the Molara package [77]. The light-yellow spheres denote cerium ions, and the red spheres, oxide ions. The closeups show the characteristic structural motif, which has a double-diamond shape, as indicated by the blue and green sketched diamonds; the ions occupy alternating positions both along x and y directions. In the closeup on the right, the alternating cerium positions are highlighted in light green and dark green; and the alternating oxygen positions are highlighted in light red and dark red.

2.2. Structural Analysis

We take as structural descriptors the nearest-neighbour distances around the respective site ($d_{\min }^{\mathrm{Ce}-\mathrm{Ce}}$, $d_{\min }^{\mathrm{Ce}-\mathrm{O}}$, $d_{\min }^{\mathrm{O}-\mathrm{Ce}}$, $d_{\min }^{\mathrm{O}-\mathrm{O}}$), and a modified version of the Effective Coordination Number (ECoN) [78,79]. We introduced a modified ECoN in order to make it more resilient against the strong structural distortions in grain-boundary regions: in these regions, distortions may be severe, and the smallest interatomic distance will generally dominate the average coordination distance, $d_{\mathrm{avg}}$. That is, a particle might be identified as being outside of the coordination environment if other particles are even closer, even if the same distance would be well within the coordination environment in the bulk phase. Hence, before calculating coordination distances and numbers for the simulation cell as a whole, we calculated an average coordination distance $d_{\mathrm{avg}}$ for the particles in the bulk phase.

The average coordination distance for a given site, $d_{\mathrm{avg}}^s$, is obtained by iteratively solving as follows:

$$\begin{array}{c}\hfill d_{\mathrm{avg}}^s={\displaystyle \frac{{\displaystyle \sum _j{d}_{js}exp\left[1-{\left(\frac{d_{js}}{d_{\mathrm{avg}}^s}\right)}^6\right]}}{{\displaystyle \sum _{j}exp\left[1-{\left(\frac{d_{js}}{d_{\mathrm{avg}}^s}\right)}^6\right]}}}=:{\displaystyle \frac{{\sum }_j{d}_{js}{w}_{js}}{{\sum }_j{w}_{js}}},\end{array}$$

(1)

where $d_{js}$ denotes the distance between site j and site s, and $w_{js}$ is short for the exponential weight based on said distance. In the original approach by Hoppe [78,79], the ECoN is then obtained as $\mathrm{ECoN}\left(s\right)={\sum }_j{w}_{js}$. For the reasons laid out above, we take mean values of the bulk-phase distances from Equation (1) as a reference, instead of using the interatomic distances for each site individually. The use of fixed reference distances implies, however, that particles closer than the reference distance could be counted as multiple particles in the ECoN. To avoid this, we apply a cutoff value of 1 to the weights. With this, the modified weight terms $w_{js,\mathrm{mod}}$ read

$$\begin{array}{c}\hfill w_{js,\mathrm{mod}}=min\left\{1,exp\left[1-{\left({\displaystyle \frac{d_{js}}{d_{\mathrm{avg}}^{\mathrm{ref}}}}\right)}^6\right]\right\},\end{array}$$

(2)

where $d_{\mathrm{avg}}^{\mathrm{ref}}$ stands for the appropriate reference coordination distances ($d_{\mathrm{avg}}^{\mathrm{Ce}-\mathrm{Ce}}$, $d_{\mathrm{avg}}^{\mathrm{Ce}-\mathrm{O}}$, $d_{\mathrm{avg}}^{\mathrm{O}-\mathrm{Ce}}$, or $d_{\mathrm{avg}}^{\mathrm{O}-\mathrm{O}}$). From this, we calculate the modified effective coordination number:

$$\begin{array}{c}\hfill {\displaystyle {\mathrm{ECoN}}_{\mathrm{mod}}=\sum _j{w}_{js}^{*}.}\end{array}$$

(3)

For the sake of a manageable computational cost, the summations in Equations (1) and (3) were only carried out over the neighbouring particles within $7\AA$ of the respective site. The neighbouring particles within the cutoff radius were identified with k-d-tree search algorithms [80] from the SciPy package [81] for Python. A detailed comparison of the original ECoN approach with other definitions of a coordination number can be found in ref. [82].

2.3. Continuum Modelling of Space-Charge Layers

The calculation of the space-charge potential employs a one-dimensional continuum approach, and it is based on the electrochemical potential of a defect building unit in the bulk and in the grain-boundary [6] taking the form

$$\begin{array}{c}\hfill {\eta }_{\mathrm{def}}^{ }={\Delta }_{\mathrm{f}}{g}_{\mathrm{def}}+k_{\mathrm{B}}Tln\left({\displaystyle \frac{n_{\mathrm{def}}^{ }}{1-n_{\mathrm{def}}^{ }}}\right)+z_{\mathrm{def}}e\varphi ,\end{array}$$

(4)

with ${\Delta }_{\mathrm{f}}{g}_{\mathrm{def}}$ denoting the Gibbs formation energy and $z_{\mathrm{def}}$ the charge number of the point defect on the specific type of site (note that ${\Delta }_{\mathrm{f}}{g}_{\mathrm{def}}$ is also referred to as the standard chemical potential ${\mu }_{\mathrm{def}}^{⦵}$ in the literature). Defect–defect interactions are neglected. Defect concentrations are treated in terms of site fractions n ($n_{\mathrm{v}}$ for ${\mathrm{v}}_{\mathrm{O}}^{••}$; $n_{\mathrm{a}}$ for ${\mathrm{Acc}}_{\mathrm{Ce}}^{\prime }$; $n_{\mathrm{e}}$ for ${\mathrm{Ce}}_{\mathrm{Ce}}^{\prime }$; $n_{\mathrm{d}}$ for ${\mathrm{Ta}}_{\mathrm{Ce}}^{•}$).

In electrochemical equilibrium [${\eta }_{\mathrm{def}}\left(x\right)={\eta }_{\mathrm{def}}^{\mathrm{b}}$], the site fractions are thus directly related to the electric potential. In the case of acceptor-doped CeO₂, we ignore that CeO₂ will be reduced (i.e., will contain electron polarons) at very high temperatures, even in oxidising conditions; bulk electroneutrality is therefore approximated by $4n_{\mathrm{v}}=n_{\mathrm{a}}$. For the space-charge zones, we thus take into account the site fractions of ${\mathrm{v}}_{\mathrm{O}}^{••}$ and ${\mathrm{Acc}}_{\mathrm{Ce}}^{\prime }$:

$$\begin{array}{c}\hfill n_{\mathrm{v}}^{ }={\displaystyle \frac{n_{\mathrm{v}}^{\mathrm{b}}{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}+2e\varphi )}}{1+n_{\mathrm{v}}^{\mathrm{b}}\left[{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}+2e\varphi )}-1\right]}},\end{array}$$

(5)

$$\begin{array}{c}\hfill n_{\mathrm{a}}^{ }={\displaystyle \frac{n_{\mathrm{a}}^{\mathrm{b}}{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}-e\varphi )}}{1+n_{\mathrm{a}}^{\mathrm{b}}\left[{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}-e\varphi )}-1\right]}},\end{array}$$

(6)

with $\beta :=1/\left(k_{\mathrm{B}}T\right)$. Note that in Equations (5) and (6), we implicitly set the reference value ${\varphi }^{\mathrm{b}}$ of the electric potential to zero. In the donor-doped case, we approximate the bulk defect chemistry with $n_{\mathrm{d}}=n_{\mathrm{e}}$; that is, we ignore the concentration of oxygen interstitials. Here, the electrochemical potentials have a slightly different form to that of Equation (4), to account for electron polarons and donor cations sharing the same sublattice:

$$\begin{array}{c}\hfill {\eta }_{\mathrm{e}}^{ }={\Delta }_{\mathrm{f}}{g}_{\mathrm{e}}+k_{\mathrm{B}}Tln\left({\displaystyle \frac{n_{\mathrm{e}}^{ }}{1-n_{\mathrm{e}}^{ }-n_{\mathrm{d}}^{ }}}\right)-e\varphi ,\end{array}$$

(7)

$$\begin{array}{c}\hfill {\eta }_{\mathrm{d}}^{ }={\Delta }_{\mathrm{f}}{g}_{\mathrm{d}}+k_{\mathrm{B}}Tln\left({\displaystyle \frac{n_{\mathrm{d}}^{ }}{1-n_{\mathrm{e}}^{ }-n_{\mathrm{d}}^{ }}}\right)+e\varphi ,\end{array}$$

(8)

and with that, the site fractions of electron polarons and donor cations in the Gouy–Chapman regime read

$$\begin{array}{c}\hfill n_{\mathrm{e}}^{ }={\displaystyle \frac{n_{\mathrm{e}}^{\mathrm{b}}{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{e}}-e\varphi )}}{1+n_{\mathrm{e}}^{\mathrm{b}}\left[{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{e}}-e\varphi )}-1\right]+n_{\mathrm{d}}^{\mathrm{b}}\left[{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{d}}+e\varphi )}-1\right]}}\end{array}$$

(9)

$$\begin{array}{c}\hfill n_{\mathrm{d}}^{ }={\displaystyle \frac{n_{\mathrm{d}}^{\mathrm{b}}{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{d}}+e\varphi )}}{1+n_{\mathrm{e}}^{\mathrm{b}}\left[{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{e}}-e\varphi )}-1\right]+n_{\mathrm{d}}^{\mathrm{b}}\left[{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{d}}+e\varphi )}-1\right]}}.\end{array}$$

(10)

In the restricted-equilibrium regime (the donor-cation profile being fixed to $n_{\mathrm{d}}^{\mathrm{crit}}$), the expression for the site fraction of polarons reads

$$\begin{array}{c}\hfill n_{\mathrm{e}}^{ }={\displaystyle \frac{n_{\mathrm{e}}^{\mathrm{b}}{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{e}}-e\varphi )}(1-n_{\mathrm{d}}^{\mathrm{crit}})}{1-n_{\mathrm{d}}^{\mathrm{b}}+n_{\mathrm{e}}^{\mathrm{b}}\left[{\mathrm{e}}^{-\beta ({\Delta }_{\mathrm{seg}}{g}_{\mathrm{e}}-e\varphi )}-1\right]}}.\end{array}$$

(11)

We assume ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{def}}\approx {\Delta }_{\mathrm{seg}}{u}_{\mathrm{def}}$, so that the segregation energies from the atomistic simulations can be used as the thermodynamic driving energies. Outside of the grain-boundary region, all sites are symmetry-equivalent, with ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{def}}=0$ by definition. In the acceptor-doped case, the Poisson equation reads

$$\begin{array}{c}\hfill -{ϵ}_{\mathrm{r}}{ϵ}_0{\displaystyle \frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{x}^2}}=2e{\gamma }_{\mathrm{O}}{n}_{\mathrm{v}}-e{\gamma }_{\mathrm{Ce}}{n}_{\mathrm{a}},\end{array}$$

(12)

and in the donor-doped case,

$$\begin{array}{c}\hfill -{ϵ}_{\mathrm{r}}{ϵ}_0{\displaystyle \frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{x}^2}}=e{\gamma }_{\mathrm{Ce}}(n_{\mathrm{d}}-n_{\mathrm{e}}),\end{array}$$

(13)

with the site densities ${\gamma }_{\mathrm{O}}$ and ${\gamma }_{\mathrm{Ce}}$ of oxygen and cerium sites, respectively, being calculated from the lattice constant a as

$$\begin{array}{c}\hfill {\gamma }_{\mathrm{O}}=2{\gamma }_{\mathrm{Ce}}=8a^{-3}.\end{array}$$

(14)

We assume that the variation in the electric potential is small within the grain-boundary core region, so that it is sufficient to treat the grain boundary as a plane rather than a region of finite width, with the amount of charge per unit area being the key quantity. This areal charge density is calculated from the core-site occupancies and the areal grain-boundary site densities, ${\Gamma }_{\mathrm{O}}$ and ${\Gamma }_{\mathrm{Ce}}$. In the given grain-boundary structure, there are two equivalent sites in every structural motif (owing to screw symmetry at the grain-boundary plane). That is, every single type of site has the same density of sites per unit area (but the areal density of oxygen sites differs from that of cerium sites; ${\Gamma }_{\mathrm{O}}=2{\Gamma }_{\mathrm{Ce}}$). The areal charge density in the grain-boundary core, $Q^{\mathrm{c}}$, is taken into account by a Neumann boundary condition at the left edge of the simulation cell ($x=0$), where the grain boundary is located:

$$\begin{array}{c}\hfill -{ϵ}_{\mathrm{r}}{ϵ}_0{\left.{\displaystyle \frac{\partial \varphi }{\partial x}}\right|}_{x=0}={\displaystyle \frac{1}{2}}·Q^{\mathrm{c}}.\end{array}$$

(15)

In the acceptor-doped case, $Q^{\mathrm{c}}$ is calculated as

$$\begin{array}{c}\hfill Q^{\mathrm{c}}=2e{\Gamma }_{\mathrm{O}}\sum _{\mathrm{O}\mathrm{sites}}{n}_{\mathrm{v}}^{ }-e{\Gamma }_{\mathrm{Ce}}\sum _{\mathrm{Ce}\mathrm{sites}}{n}_{\mathrm{a}}^{ },\end{array}$$

(16)

and in the donor-doped case,

$$\begin{array}{c}\hfill Q^{\mathrm{c}}=e{\Gamma }_{\mathrm{Ce}}\sum _{\mathrm{Ce}\mathrm{sites}}(n_{\mathrm{d}}^{ }-n_{\mathrm{e}}^{ }),\end{array}$$

(17)

with $n_{\mathrm{v}}$, $n_{\mathrm{a}}$, $n_{\mathrm{d}}$, and $n_{\mathrm{e}}$ being calculated with Equations (5)–(11). Parameters for the continuum simulations are given in

Table 2. Simulation parameters for the FEM simulations of the space-charge layers.

Symbol	Value	Comment
$n_{\mathrm{a}}^{\mathrm{b}}$	${10}^{-3}$	bulk doping level
$a/\AA$	$5.45$	lattice constant [83]
${ϵ}_{\mathrm{r}}$	60	rel. diel. permittivity [84]
${\Gamma }_{\mathrm{O}}$	4.282 × 10${ }^{18}$	areal core O-site density
${\Gamma }_{\mathrm{Ce}}$	2.141 × 10${ }^{18}$	areal core Ce-site density

.

3. Results

Figure 1 shows the atomistic structure of the grain boundary, obtained through a combination of static and dynamic simulation methods, as described above. The characteristic structural unit has a double-diamond shape with ion positions alternating in all three dimensions. In previous studies of this boundary in CeO₂, UO₂, and cubic ZrO₂, only the single-diamond motive was considered in many cases [40,42,43,45], and in studies that appear to have found the double-diamond repeat unit [34,38,39,44], this more complex repeat unit has not been identified. In our study, we find the double-diamond structure (referring to the highlighted geometry within the box in the middle of Figure 1) to show the lowest energy. Evidently, such a structure may be obtained only in a sufficiently large simulation box, i.e., the box length must be some multiple of the double-diamond repeat unit. The excess energy of this grain boundary is ${\Delta }_{\mathrm{f}}{E}_{\mathrm{gb}}$ = 2.85 $J{m}^{-2}$; and its excess volume is ${\Delta }_{\mathrm{f}}{V}_{\mathrm{gb}}=0.9$ Å${ }^3$ Å${ }^{-2}$. Note that this grain-boundary structure may be specific to CeO₂ and the interatomic potentials used. Literature values [38,45,85] for this boundary are of the same order of magnitude, but are not strictly comparable, since they refer to other fluorite oxides, and to results obtained with other methods.

When a point defect is introduced into a grain-boundary supercell, its formation energy is expected to show different values within the grain-boundary regions, on account of the altered structure there. At sufficiently large distances from the two boundary planes, i.e., in the bulk region, the energy should assume a constant value (unless there is a non-zero electric field within the bulk part of the simulation cell [86]). This behaviour is observed, satisfyingly, for all point defects examined. We show in Figure 2 the segregation energies at different sites within the simulation cell, as well as the nearest-neighbour distances (with regard to the counter ions) and modified effective coordination numbers ${\mathrm{ECoN}}_{\mathrm{mod}}$. More specifically, atomic sites are clearly bulk-like if they are more than $1nm$ away from the grain-boundary planes. It is also evident that, for all considered defects, the energies in the vicinity of the grain boundaries (within $7.5 \AA$ distance) deviate substantially from their bulk values. The precise choice of a cutoff between these regions (i.e., the structural width), however, is to some degree arbitrary. As can be seen in Figure 2b,c, there are only subtle deviations of the interatomic distances and coordination numbers from their bulk values in the transition regions. This indicates that the defect-formation energy reacts quite sensitively even to minor structural distortions. Here, considering all the information together, we take a distance of $7.5 \AA$ on either side of the boundary plane, and thus obtain a structural grain-boundary width of ca. $1.5nm$. Since this way of estimating a structural width is based upon the spatial profiles of the segregation energies, the obtained width may also be identified as the width of the grain-boundary core in a space-charge model. While the extent of the structural perturbation may vary, to some degree, between different grain-boundary types, it may generally be expected to be of the order of a few lattice constants.

As in earlier studies [34,35], both negative and positive segregation energies in the grain-boundary regions are found, that is, driving energies for defect segregation from the bulk to the boundary, and vice-versa. Although both sets of values are important in deriving relationships between structure and energetics (see below), only the negative values are important for space-charge formation [6], since, in a dilute solution, only these favourable energies produce substantial changes in the grain-boundary charge. And here one finds, as the most negative values, ${\Delta }_{\mathrm{seg}}{u}_{\mathrm{v}}=-1.1\mathrm{eV}$, ${\Delta }_{\mathrm{seg}}{u}_{\mathrm{e}}=-1.1\mathrm{eV}$, ${\Delta }_{\mathrm{seg}}{u}_{\mathrm{d}}^{\mathrm{Ta}}=-2.4\mathrm{eV}$, ${\Delta }_{\mathrm{seg}}{u}_{\mathrm{a}}^{\mathrm{Sc}}=-1.7\mathrm{eV}$, and ${\Delta }_{\mathrm{seg}}{u}_{\mathrm{a}}^{\mathrm{Gd}}=-0.8\mathrm{eV}$.

3.1. Analysing the Correlations between Local Structure and Segregation Energies

The defect formation energies differ from their bulk values in an extended region around the grain-boundary plane, with most changes occurring in the region up to $7.5$ Å but bulk behaviour only being attained at a distance of ca. $10$ Å (see Figure 2). That is, grain-boundary segregation to and from the boundary region must be considered to occur for a multitude of inequivalent sites. In Figure 3, we show the trend of the segregation energies on all inequivalent grain-boundary sites, with the nearest-neighbour distances $d_{\min }^{\mathrm{Ce}-\mathrm{O}}$ indicated by a colour code. While some sites show a monotonous increase in ${\Delta }_{\mathrm{seg}}u$ with ionic radius, and others go through a maximum, there are also cases with a steady decrease in ${\Delta }_{\mathrm{seg}}u$.

Two grain-boundary sites exhibit a particularly pronounced increase of ${\Delta }_{\mathrm{seg}}u$ with the ionic radius; as indicated by the colour code in Figure 3, these sites also exhibit the shortest nearest-neighbour distances. The trends on these two sites suggest that, in line with arguments made in earlier studies [46], sites with short interatomic distances are particularly suitable for accommodating a negative size mismatch, i.e., the impurity cation being smaller than the ${\mathrm{Ce}}_{\mathrm{Ce}}^{\times }$ host cation.

In some instances [88,89], the ionic size mismatch has been discussed as though it were interchangeable with the segregation bias. Our results demonstrate that this perspective is unsuitable, both quantitatively and qualitatively: the interatomic distances are generally smaller in the grain boundary than in the bulk. Hence, if the segregation energies were determined by the ionic size mismatch alone, all ${\mathrm{Sc}}_{\mathrm{Ce}}^{\prime }$ segregation energies would need to be negative, and all ${\mathrm{Ce}}_{\mathrm{Ce}}^{\prime }$ segregation energies positive. More than that, the ${\mathrm{Gd}}_{\mathrm{Ce}}^{\prime }$ segregation energies would, on all sites, have to be smaller in magnitude than the corresponding segregation energies for ${\mathrm{Sc}}_{\mathrm{Ce}}^{\prime }$ and ${\mathrm{Ce}}_{\mathrm{Ce}}^{\prime }$. Clearly, this rationale misrepresents the systematics of point-defect segregation observed here.

In an earlier study [88], it was proposed that the systematics of the point-defect segregation energies should be described by a quadratic dependence of the segregation energy on the ionic size mismatch, based on continuum elasticity theory [90]. A similar dependence has been suggested for the average of the segregation energies at a grain boundary [34]. In that regard, it should first be noted that the (arithmetic) average of segregation energies is not a meaningful measure. In a space-charge model, the effects of a very positive and a very negative segregation energy will not cancel each other out. Rather, the negative segregation energy will dominate the behaviour, as we will demonstrate in the following section. Second, it is to be noted that Figure 3 indicates quite clearly that some sites show a qualitatively different behaviour (${\Delta }_{\mathrm{seg}}{u}_{\mathrm{def}}$ exhibiting a monotonous increase or decrease, rather than a maximum). An approach that only takes the ionic size mismatch as a measure disregards the influence of the geometry at the specific grain-boundary site. Not every site will equally well accommodate either a negative or a positive size mismatch.

We therefore show in Figure 4 how the segregation energies of ${\mathrm{v}}_{\mathrm{O}}^{••}$, ${\mathrm{Ta}}_{\mathrm{Ce}}^{•}$, ${\mathrm{Sc}}_{\mathrm{Ce}}^{\prime }$, ${\mathrm{Gd}}_{\mathrm{Ce}}^{\prime }$, and ${\mathrm{Ce}}_{\mathrm{Ce}}^{\prime }$ depend on the nearest-neighbour distance of the counter ions. While, for ${\mathrm{Sc}}_{\mathrm{Ce}}^{\prime }$ and ${\mathrm{Ta}}_{\mathrm{Ce}}^{•}$ ions, there appears to be a weak positive correlation between ${\Delta }_{\mathrm{seg}}u$ and $d_{\min }^{\mathrm{Ce}-\mathrm{O}}$, no clear trends are discernible for ${\mathrm{v}}_{\mathrm{O}}^{••}$, ${\mathrm{Ce}}_{\mathrm{Ce}}^{\prime }$, and ${\mathrm{Gd}}_{\mathrm{Ce}}^{\prime }$. Similarly, the segregation energies show no clear correlation on the ${\mathrm{ECoN}}_{\mathrm{mod}}$ (not shown). This indicates that the segregation energies are not determined simply by the local environment around a site. As we will see in the next section, oxygen vacancies play the key role in determining the space-charge potential. It is, therefore, particularly striking that no clear trend may be formulated for their segregation energies, since it suggests that macroscopic grain-boundary properties cannot be related to these microscopic structural descriptors.

One further influence that must be taken into account is the lower charge number (3+) of the acceptor cations and electron polarons. The interplay of this charge difference with the altered coordination environment in the grain-boundary region will also contribute to the segregation energy [85]. In other words, trying to obtain a quantitative, universal structure–property relationship (based on simple, local descriptors) appears to be impossible, even for this simple, highly symmetric, tilt grain boundary.

3.2. The Effects of Different Segregation Energies on Space-Charge Behaviour

Having looked in more detail at the origin of point-defect segregation, let us now turn to its effects, i.e., the formation of space-charge zones. We calculated the space-charge potential ${\Phi }_0$ as a function of temperature for (i) the case of a dilute solution of acceptor cations and charge-compensating oxygen vacancies and (ii) the case of a dilute solution of donor cations and charge-compensating electron polarons. In both cases, at high temperatures, the cations (and the charge-compensating defects) are assumed to attain electrochemical equilibrium (this is often termed the Gouy–Chapman case [91,92]), but below a critical temperature $T_{\mathrm{crit}}$, the cations are considered to be immobile, while the charge-compensating defects (oxygen vacancies or electron polarons) remain mobile. The cations, being unable to react to the electrostatic potential, retain the accumulated profile from the Gouy–Chapman case (this we term the restricted equilibrium case [6,25,93]).

On the basis of our atomistic results, we constructed a model that takes all vacancy and acceptor segregation energies (within $d_{\mathrm{gb}}\le 7.5 \AA$) into account. In order to determine the importance of taking all sites into account, we also consider a second model, in which all vacancy segregation energies but no acceptor segregation energies were included; a third model, in which only the most negative acceptor and vacancy segregation energies were included; and a fourth model with only the most negative vacancy segregation energy and no acceptor segregation energies. We analyse these cases because, typically, simulations of space-charge layers are set up empirically—the simplest way to describe experimental data in a consistent thermodynamic framework is by taking a single value of ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$ as the driving energy for space-charge layer formation, with ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}$ being set to zero.

In Figure 5, we show the obtained values of ${\Phi }_0$ obtained for the Gd-doped and for the Sc-doped systems. In the case of Gd-doped CeO₂, ${\Phi }_0$ is positive at all temperatures considered, whereas in the case of Sc-doped CeO₂, it changes from negative to positive (at $T\approx 1667$ K). In the Gd-doped case, the third model (with only the most negative ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$ and ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}$) shows the smallest deviations from the first model, especially in the restricted-equilibrium regime. It is noteworthy to say this very strong simplification performs better than the case that takes all ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$ into account while omitting the ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}$ values. Clearly, acceptor segregation energies play a crucial role for the quantitative description of grain-boundary space-charge layers. Likewise, in Sc-doped CeO₂, the description with only the most negative ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$ and ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}$ performs substantially better than the models that disregard ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}$, but the deviations are generally larger than in the Gd-doped case.

In order to understand the performance of the different simplified models, we show in Figure 6 the site occupancies within the grain-boundary core, calculated from the first model. Given a specific segregation energy, the occupancies of Ce sites with acceptor cations are generally larger than those of O sites with oxygen vacancies, owing to the positive space-charge potential. The contributions of ${\mathrm{v}}_{\mathrm{O}}^{••}$ and ${\mathrm{Acc}}_{\mathrm{Ce}}^{\prime }$ to the grain-boundary core charge are, in all considered cases, of the same order of magnitude, and thus, these contributions are much larger than the total grain-boundary core charge. This behaviour may be explained through the levelling effect that the space-charge potential has on both species. The sign of ${\Phi }_0$ resulting from one of the two species being predominant will serve to decrease the concentration of said species in the core and increase the concentration of the oppositely charged species. Bearing this in mind, one can readily understand why the most negative ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}$ must not be disregarded, even when they are smaller in magnitude than the most negative ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$: both defects contribute substantially to the total core charge.

It may appear surprising that, in the Sc-doped case, ${\Phi }_0$ (and $Q^{\mathrm{c}}$) are positive over most of the temperature range, despite the large segregation energies of down to $-1.7\mathrm{eV}$ that by far exceed in magnitude the most negative ${\mathrm{v}}_{\mathrm{O}}^{••}$ segregation energies. In particular, it is surprising that this sign change takes place within the Gouy–Chapman regime, where a redistribution of acceptor cations from the bulk to the grain-boundary core is possible. This may be explained by the impossibility for a favourable type of site to exceed full occupancy. As indicated by the Fermi–Dirac-type systematics displayed in Figure 6, the grain-boundary core charge would barely be affected if the most favourable types of sites were shifted to even lower energies, since these sites are already fully occupied. Since the areal density of oxygen sites within the grain-boundary core is twice that of cerium sites, and because of the two-fold relative charge of ${\mathrm{v}}_{\mathrm{O}}^{••}$, a $100\%$ occupancy of a specific type of Ce site contributes only as much to $Q^{\mathrm{c}}$ as a $25\%$ occupancy of a specific type of O site. When point-defect segregation energies are obtained from atomistic simulations, one may tend to deduce the sign of the space-charge potential directly from a comparison of the most favourable energies. Our analysis shows that such an approach is not justified; the dependence of the space-charge potential on the set of segregation energies is highly non-trivial.

By considering the occupancies shown in Figure 6, it is also clear why, in the simulations with a scandium dopant, the third model (with only the most negative ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$ and ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}$) shows much larger deviations from the simulated behaviour with all sites included than found in the simulations with a gadolinium dopant. At $T=T_{\mathrm{crit}}$ and below, the three most favourable types of Ce sites (with ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}=\{-1.69,-1.47,-1.46\}\mathrm{eV}$) are all fully occupied, and thus, they all contribute strongly to $Q^{\mathrm{c}}$. None of these sites may be neglected. In the Gd-doped case, even the lowest-energy site (at ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}=-0.82\mathrm{eV}$) is not fully occupied at $T_{\mathrm{crit}}$, and the sites that are next in energy (${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}=\{-0.45,-0.25\}\mathrm{eV}$) are mostly unoccupied.

When space-charge models are set up to explain experimental observations, they are typically formulated on the basis of a single driving energy, ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$, and a corresponding site density ${\Gamma }_{\mathrm{O}}$, with a few notable exceptions [9,94]. In such a model, however, ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$ and ${\Gamma }_{\mathrm{O}}$ do not stand for a single type of site—rather, they represent effective values for the multiplicity of inequivalent sites present in real grain boundaries. It is insufficient, therefore, to simply pick a single value of ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$ and ${\Gamma }_{\mathrm{O}}$ from atomistic simulations. Rather, the grain-boundary core parameters must be understood as effective parameters, ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}^{\mathrm{eff}}$ and ${\Gamma }_{\mathrm{O}}^{\mathrm{eff}}$. Depending on the set of ${\mathrm{v}}_{\mathrm{O}}^{••}$ and ${\mathrm{Acc}}_{\mathrm{Ce}}^{\prime }$ segregation energies, however, such a model need not be able to capture the behaviour of a “full-complexity” model with all segregation energies included. In order to assess how well such a simplification describes the complex behaviour, we fitted a model with a single ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}^{\mathrm{eff}}$ value (and ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}^{\mathrm{eff}}=0$) to the ${\Phi }_0\left(T\right)$ data from the full-complexity model, assuming the same critical temperature. In the fit model, both ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$ and ${\Gamma }_{\mathrm{O}}$ were treated as fit parameters. These results are shown in Figure 7.

Clearly, the space-charge potentials in the chosen temperature range can be excellently described by a simplified model with only one ${\mathrm{v}}_{\mathrm{O}}^{••}$ segregation energy (${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}^{\mathrm{eff}}$) and no acceptor segregation energy. In the case of Gd-doped CeO₂, this model extrapolates very well (with deviations $<6\mathrm{mV}$) across the entire considered restricted-equilibrium temperature range ($120\mathrm{K}$ to $1300\mathrm{K}$). In the case of Sc-doped CeO₂, the extrapolation is less accurate (with deviations up to $24\mathrm{mV}$) but still very good. In both cases, an extrapolation from the restricted-equilibrium range to the behaviour in the Gouy–Chapman regime is very inaccurate; in the Sc-doped case, it even leads to a false prediction of the sign of ${\Phi }_0$ at high temperatures. The effective segregation energies obtained from the fit are, in both cases, remarkably close to the most negative segregation energy (${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}=-1.07\mathrm{eV}$) included in the simulations, thus demonstrating again how space-charge behaviour is dominated by the most negative segregation energies, rather than by an average of segregation energies. The effective areal site densities are substantially lower than the individual site density ${\Gamma }_{\mathrm{O}}$ of each symmetry-inequivalent oxygen site. And the effective site density is much lower in the Sc-doped case than in the Gd-doped case ($0.07{\Gamma }_{\mathrm{O}}$ compared to $0.3{\Gamma }_{\mathrm{O}}$). This can be attributed to the much more negative ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}$ for scandium ions than for gadolinium ions: as shown in Figure 6, there are generally more highly occupied grain-boundary cation sites in the Sc-doped case, and thus, a larger negative contribution to the grain-boundary core charge. As a consequence of this large negative counter-charge, the occupancies of oxygen sites with ${\mathrm{v}}_{\mathrm{O}}^{••}$ can also become larger, leaving a much smaller fraction of the oxygen sites to be unoccupied. In a model that disregards the acceptor segregation energies, this smaller fraction of available oxygen sites has the appearance of a smaller oxygen-site density.

An even simpler analysis can be performed by assuming the Mott–Schottky case, that is, ignoring the accumulated dopant profile. In this case, the segregation energies are again obtained quite accurately (not shown), but the effective site densities are even lower (by ∼40% in the Gd-doped case and by ∼20% in the Sc-doped case) than in the fit of the restricted equilibrium model.

Having examined in detail the effects of multiple segregation energies in acceptor-doped CeO₂, we now turn to donor-doped CeO₂. In Figure 8, we show values of ${\Phi }_0\left(T\right)$ obtained for Ta-doped CeO₂, assuming that the ${\mathrm{Ta}}_{\mathrm{Ce}}^{•}$ donor cations are compensated by electron polarons ($n_{\mathrm{e}}=n_{\mathrm{d}}$). At high temperatures, we find positive space-charge potentials, originating in the strongly negative ${\mathrm{Ta}}_{\mathrm{Ce}}^{•}$ segregation energies (down to $-2.4\mathrm{eV}$). In the restricted-equilibrium regime, ${\Phi }_0$ decreases strongly with decreasing temperature and changes sign at $T=698\mathrm{K}$, indicating that ${\mathrm{Ce}}_{\mathrm{Ce}}^{\prime }$ segregation becomes dominant owing to the ${\mathrm{Ta}}_{\mathrm{Ce}}^{•}$ ions’ inability to further segregate to the grain-boundary core. This behaviour suggests that in a sample with the considered defect chemistry, the space-charge contribution to the grain-boundary resistivity should vanish abruptly above a threshold temperature, owing to the space-charge layer’s changing character, from depleted mobile charge carriers (electron polarons) at low temperatures to accumulated mobile charge carriers at high temperatures.

4. Conclusions

In this study, we investigated various aspects concerning the formation of space-charge layers at a grain boundary in fluorite-structured CeO₂. First, at the atomistic level, we examined how to define the structural grain-boundary width, and how the segregation energetics are related to simple structural descriptors. For this, we employed classical Molecular Statics (MS) and Molecular Dynamics (MD) simulations. We then analysed, by means of Finite Element Method (FEM) continuum simulations of a dilute solid solution, how the behaviour of the space-charge potential is affected by the variety of segregation energies. As answers to these questions we obtained the following:

• The segregation energetics of point defects, the interatomic distances, and the coordination numbers all exhibit a similar length scale on which they deviate from bulk values. Although the precise cutoff between bulk and grain-boundary regions is somewhat arbitrary, all three quantities yield for this grain boundary in this system a grain-boundary width of ca. $7.5 \AA$ on either side of the interface, i.e., ca. $1.5nm$ in total.
• Even for a symmetric, tilt grain boundary, there is no clear relationship between simple structural descriptors (such as the ionic size mismatch, nearest-neighbour distances, and effective coordination numbers) and defect energetics.
• In a space-charge model, most segregation energies (especially the positive values) can safely be neglected and the complexity of the model can in this manner be strongly reduced. To avoid oversimplification, however, one must consider that (i) a correct prediction of the ${\Phi }_0\left(T\right)$ behaviour (both quantitatively and qualitatively) can only be achieved if site-exclusion effects are taken into account; (ii) segregation energies for acceptors (${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}$) must not be disregarded, even though the sign of ${\Phi }_0$ is in most cases governed by oxygen-vacancy segregation, rather than by acceptor segregation; (iii) when a simple, empirical space-charge model (with ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{a}}=0$) is fitted to experimental data, the obtained grain-boundary parameters (${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$ and ${\Gamma }_{\mathrm{O}}^{\mathrm{c}}$) are effective values that are not directly comparable to atomistic results since they incorporate the effects of acceptor segregation. Nevertheless, in our model system, we found such a fit to yield good estimates of ${\Delta }_{\mathrm{seg}}{g}_{\mathrm{v}}$ together with good extrapolating capabilities of the ${\Phi }_0\left(T\right)$ behaviour within the restricted-equilibrium regime.

These results highlight that for a rational design of the grain-boundary electrical properties in ceramics, purely structural considerations are hardly informative. More than that, even atomistic calculations of segregation energies will be insufficient, unless they serve as the basis for a detailed space-charge model. In this study, we have laid out the degree of detail that such a model requires in order to yield meaningful results.