2.4.1. Commensurate Phases and Surface Structures
Among the low index ZrC surfaces, the (100) stoichiometric and non-polar surface is found to be the most stable [
4,
5]. However, even though the (111) surface is polar upon cleaving from the bulk phase by terminating on one side with a carbon layer and the other side with a zirconium layer, surface reconstruction revealed a more stable surface terminated on both sides with Zr atom layers [
4]. With a lattice parameter of 6.698 Å,
b of 5.801 Å, and
β = 60°, the exposed surface area of the ZrC(111) surface is 38.854 Å
2. On accounts of several studies made on the
c-ZrO
2 surfaces, the (111) surface is found to be the most stable [
33,
34]. Surface energies are calculated for one layer up to six layers of ZrC to ascertain the effect of the layer thickness on the surface energy. The surface energies are calculated as
where
Eslab is the total energy of the surface slab, Ebulk is the energy per formula unit of ZrC in the corresponding bulk,
A is the surface area, and
n is the number of formula units in the surface slab.
Surface energies are also computed for the (111) terminations of
c-ZrO
2. The surface energies were calculated for different numbers of layers, starting from one to six layers of ZrO
2. Upon cleaving the
c-ZrO
2 along the [111] direction, a polar slab is obtained with an OO layer terminating on one side and a Zr layer terminating on the other side. The slab can, however, be terminated in three different arrangements as OO|Zr|OO|Zr|OO-, Zr|OO|Zr|OO|Zr–, and O|Zr|OO|Zr|O– (
Figure 1). Only symmetric slabs were used (slabs with mirror symmetry) in the calculation of the surface energy to eliminate the net dipole moment. The calculation of the interface tension defined in a subsequent section requires these surface energies. Thus, surface energies of three different terminations were calculated: Zr-termination, O-termination, and OO-termination.
To access surface energies of both stoichiometric and non-stoichiometric slabs, the surface grand potential is defined as
Ωi which implies contact of the Zr and O reservoirs with the surface:
NZr and
NO are the numbers of Zr and O atoms in the slab with
µZr and
µO being the chemical potential of Zr and O, respectively.
is the total energy of the surface slab and
A is the surface area. The chemical potentials of Zr and O are related by bulk ZrO
2 in the expression:
with
being the total energy per bulk ZrO
2 unit. Rearranging this expression and substituting it in Equation (1), we obtain the following:
Defining the chemical potential of O in relation to the chemical potential of the reference state, O
2 which is defined as half the total energy of O
2 gas as ∆
µO =
µO −
and substituting it in Equation (2) with further rearrangements, we obtain:
If we make the following definition:
where
γi is the surface energy of the stoichiometric part of the selected slab. Substituting Equation (4) into Equation (3), we come to the following expression for the surface grand potential:
Thus, the surface grand potential is defined in terms of the surface energy arising from the stoichiometric part of the slab, and another part correcting for the extra number of Zr or O atoms.
From Equation (5), a range of
values can be accessed if we define the lower and upper limits. In defining the upper limit of the O chemical potential, we assume that the chemical potential of O must be lower than the energy of O in its reference stable gaseous state. Thus comes the upper limit of the O chemical potential:
For the lower limit of the O chemical potential, if we combine the expressions
with
and
, and make rearrangements, the lower limit of the O chemical potential is obtained:
is the formation energy of ZrO
2 defined as
and we calculated it as −9.97 eV. Thus, the range of potential accessible chemical values of O is:
A plot of the surface grand potential Ωi against the accessible range of O chemical potential is obtained for both stoichiometric and non-stoichiometric slabs for easy comparison of surface energies.
Phase commensurability is one major problem that is encountered when forming interfaces. The two surfaces used in forming the interface must be coherent due to the periodic boundary condition imposed in the calculation. The surface misfit parameter ϒ can be used to obtain highly coherent interfaces [
35]. This parameter is defined as:
Thus, a unit cell of
c-ZrO
2 with a surface area of
SB is forced into coherency onto a substrate ZrC(111) with surface area
SA, and the resulting overlap area between the two surfaces is
SA–B. The misfit parameter measures the average length scale misfit between the two-unit cells [
21] rather than an area misfit. In
Table 1, the calculated misfit parameters between ZrC(111) substrate surface and all
c-ZrO
2 surfaces are summarized. It is apparent from this table that the ZrC(111)||
c-ZrO
2(111) interface combination has the lowest misfit parameter of 8.2% and is acceptable. The resulting interface unit cell defined by the substrate ZrC(111) is 6.698 Å × 6.698 Å which is small and can be easily managed by the DFT calculation.
The misfit parameter, being a geometrical measure, cannot be used alone in building the interface. It must be combined with other models. Two models are widely known for ensuring the commensurability of two different phases when forming an interface. Within the first approach, the unit cells of the two phases are multiplied by a factor corresponding to the other unit cell until both cells are commensurate with each other. The resulting supercell is usually large and unbearable for ab initio calculations. However, the resulting interface is coherent with very small mismatch parameters [
36].
The second method is widely used [
37,
38,
39,
40,
41,
42] as it results in small and manageable interface supercells (a single unit cell), suitable for ab initio calculations. In this model, the lattice parameters of the phase considered as the substrate are considered for the interface, with the lattice parameter of the other phase scaled until a perfect match with the substrate lattice is obtained.
2.4.2. Geometrical Models for Interface
Within the slab model described for studying the interface, a thickness of 10.945 Å of ZrC (nine layers) was used. This thickness was enough to mimic the electronic structure when ionic positions in the bulk are relaxed. The
c-ZrO
2(111) units were then pinned into the registry, layer by layer on the exposed ZrC(111) surface. Thus, in straining the
c-ZrO
2 to match the dimensions of the ZrC surface, coherent interfaces are ensured. The interface unit cell is therefore determined by the bulk and surface parameters of the ZrC(111). In this manner, the unit cell lattice parameter of the
c-ZrO
2(111) is shrunk by about 8%. After fixing the geometries of the two surfaces at the interface, the remaining degrees of freedom in the resulting interface structure are perpendicular to the interface and the interface chemical composition [
43]. From one to five layers of the
c-ZrO
2(111) units were built on the ZrC(111) surface.
In
Figure 2, we provide side views of the interface models used with the different numbers of ZrO
2 layers. Each
c-ZrO
2 bilayer is approximately 3.5 Å thick. All interface models used were symmetric with respect to the center of the interface slab to remove any long-range dipole-dipole interaction between exposed surfaces. A total of 14 Å of vacuum was applied between two subsequent interface slabs to avoid any physical interactions between the slabs. The interface slab used thus has a configuration of: —ZrC(111)|
c-ZrO
2(111)|vacuum|ZrC(111)|
c-ZrO
2(111)|vacuum|ZrC(111)|
c-ZrO
2(111)|vacuum|ZrC(111)— Since the ZrC(111) slab is a reconstructed structure with four extra Zr atoms, the interface chemical composition is dependent on the number of Zr atoms (ZrC side) and the terminating layer of the
c-ZrO
2(111) phase. In constructing the interface, three different terminations along the
c-ZrO
2[111] direction were considered: Zr|OO|Zr|OO|Zr|OO—, O|Zr|OO|Zr|OO|Zr|O—, and OO|Zr|OO|Zr|OO|Zr—. We also considered the OO|Zr|OO|Zr|OO|Zr— on a ZrC(111) surface with an oxidized layer. A total of four different interface models were built as shown in
Figure 2.
2.4.3. Mechanics and Cohesion at the Interface
In defining the interface cohesion and stability, one important parameter used is the interface tension ɣ
int, defined as the reversible work needed to separate the interface into two free surfaces [
44]. According to this definition, an assumption made is that both diffusional and plastic degrees of freedom are suppressed and hence negligible. The greater the ɣ
int value, the higher the energy needed to separate the interface into two surfaces.
The interface tension can be defined according to the Dupre equation in terms of the interface and free surface energies as [
45,
46]:
is the interface energy also known as the adiabatic work of adhesion,
Wad > 0,
, and
are the relaxed surface energies of the ZrC(111) and
c-ZrO
2(111) surfaces, respectively. In this definition, the relative strength of the interface versus the bulk bonds decides the preference for the formation of either the interface or the open surfaces [
22].
A measure of whether there is the formation of an interface or the free surfaces can be determined by the interface tension. The magnitude and sign of ɣ
int (Equation (10)) provide a measure of whether the interface bonds are stronger than the internal bonds in the separate phases [
22]. The criteria are that 0 < ɣ
int <
corresponds to weakly coupled interfaces and ɣ
int < 0 to strongly coupled interfaces. The calculated values of
used here are obtained from their respective relaxed bulk equilibrium phases (i.e., strain-free surface slabs).
In parallel, the adiabatic work of adhesion
Wad is defined as:
is the total energy of the fully relaxed interface slab,
A is the interface area, and
are the total energies of the fully relaxed isolated ZrC(111) and
c-ZrO
2(111) slabs, respectively. Usually, the calculated Wad value is a lower bound as compared to values obtained in cleavage experiments due to dissipative processes in physically separating the interface [
44] There is no relation between characterizing the interfacial strength and the bulk strain when depositing the
c-ZrO
2. Hence, the
value used is the total energy of the strained
c-ZrO
2 for commensurability with the ZrC surface. In this manner, the strain energy component between
and
is canceled out since the
c-ZrO
2 is in the strain state [
22].
Aside from the relaxed work of adhesion, the rigid work of adhesion
can be used in characterizing the interface cohesion and stability. In this definition, the same strained state is ensured to exist in both the interface and the free surfaces. This provides maximum cancelation for the strain energy in the calculated interface energy [
43]. This quantity gives information purely on the bonds formed at the interface irrespective of the free surfaces. It is calculated by separating the optimized interface structure into the different phases and rigidly calculating their energies without allowing the phases to fully relax. Equation (11) is finally applied in calculating the rigid work of adhesion.