Interfacial interaction and prenucleation at liquid-Al/γ-Al₂O₃{1 1 1} interfaces

The structural model of γ-Al₂O₃ was built based on the cubic spinel with a chemical (ionic) formula Mg²⁺(Al³⁺)₂(O^2-)₄ via replacement of Mg by Al. To maintain charge balance, Al vacancies must be introduced. The recent studies showed that Al vacancies distribute at the octahedral sites preferably in a homogeneous way [18, 20]. The built L-Al/γ-Al₂O₃{1 1 1} interfaces have a hexagonal supercell with a = (3√2/2)a₀, where a₀ is the lattice parameter of the cubic γ-Al₂O₃ phase with consideration of the thermal expansion at the simulation temperature [16, 18]; while the length of the c-axis is determined by the γ-Al₂O₃ slab and the number of Al atoms with the density at the simulation temperature [43]. Thus, a hexagonal supercell with a = 17.06 Å, c = 40.58 Å which contains 144 O and 72 Al atoms in the substrates and 450 liquid Al atoms was used. The selected substrates are O-terminated (figure 1), one with O1-type (denoted as L-Al/γ-Al₂O₃{1 1 1}_O1) and the other with O2-type (denoted as L-Al/γ-Al₂O₃{1 1 1}_O2). We chose the two O-terminating substrates as the previous simulations [39–42] indicated that at thermal equilibrium, different inputs converge into one of the two independent L-Al/γ-Al₂O₃{1 1 1} interfaces. The supercells are deliberately large for avoiding risk of artificial crystallization of the Al liquid.

For the ab initio molecular dynamics simulations and electronic structure calculations, we employed a pseudo-potential plane-wave approach based on the density-functional theory (DFT), which was implanted into the code VASP (Vienna ab initio simulation package) [44, 45]. This code permits variable fractional occupation numbers, working well for insulating/metallic interfaces [22, 24, 39–42]. The molecular dynamics simulation uses the finite-temperature density functional theory of one-electron states, the exact energy minimization and calculation of the exact Hellmann-Feynman forces after each MD step using the preconditioned conjugate techniques, and the Nosé dynamics for generating a canonical NVT ensemble [44]. The Gaussian smearing was employed with the width of smearing, SIGMA = 0.1eV. The code also utilizes the projector augmented-wave (PAW) method [46] within the generalized gradient approximation [47]. The atomic electronic configurations used are Al ([Ne] 3s² 3p¹) and O ([He] 2s² 2p⁴).

For electronic structure calculations, we used cut-off energies of 400.0eV for the wave functions and 550.0 eV for the augmentation functions. The default energies of the potentials are E_nmax/E_aug = 240.3 eV/291.1 eV for Al and 400.0eV/605.4eV for O. Reasonably dense k-meshes were used for sampling the electronic wave functions, e.g. a 2 × 2 × 1 (8 k-points) in the Brillouin zone (BZ) of the supercell of the interfaces [48]. For the ab initio MD simulations of the interfaces, we employed a cut-off energy of 320 eV, and the Γ-point in the BZ with considering the lack of periodicity of the whole system in such liquid/solid interfaces [37–42]. Test simulations using different cut-off energies ranging from 200.0 eV to 400.0 eV demonstrated that the settings are reasonable.

We prepared liquid Al samples by equilibrating at 3000K for 2000 steps (1.5 fs per step). Then the obtained liquid was cooled to the desired temperature. We used the obtained liquid Al samples together with the oxide substrates for building the L-Al/γ-Al₂O₃{1 1 1} interfaces. A two-step approach was applied in our simulations. We first performed ab initio molecular dynamics simulations with the substrate O atoms pinned for about 2ps (1.5fs per step). Then, we equilibrated further the systems with full relaxation of the substrate atoms for another 4,000 to 7,000 steps (figure 2). The two-step approach avoids risk of collective atomic movements occurring often when we relax all atoms from start. The time-averaged method was used to sample the interfaces over 3.0–4.5 ps to ensure statistically meaningful results [39–42, 49].

Zoom In Zoom Out Reset image size

In order to assess the atomic ordering at the equilibrated L-Al/γ-Al₂O₃{1 1 1} interfaces three coefficients are employed as follows.

2.3.1. Atomic density profile

In order to provide a quantitative measure of atomic layering at a liquid/solid interface, we use the atomic density profile, ρ(z) which is defined as [12, 13, 34, 50, 39–42]:

$$\begin{eqnarray}&&\rho \left(z\right)=\left\langle {N}_{z}\left(t\right)\right\rangle /\left({L}_{x}{L}_{y}{\rm{\Delta }}z\right),\end{eqnarray} \tag{ 1 }$$

here, L_x and L_y are the in-plane x and y dimensions of the unit cell, respectively, and z the dimension perpendicular to the interface. Δz is the bin width, and N_z (t) is the number of atoms between z − (Δz/2) and z + (Δz/2) at time t. 〈N_z (t)〉 means a time-averaged number of atoms in the duration. The unit of the atomic density profile is (Å⁻³).

2.3.2. In-plane (atomic) ordering coefficient

In-plane ordering coefficient, S(z) can assess the atomic ordering in an individual layer [34, 50]. It is defined as:

$$\begin{eqnarray}&&S\left(z\right)={\left[\left(\sum \exp \left(iQ\cdot {r}_{i}\right)\right]\right.}^{2}/{N}_{z},\end{eqnarray} \tag{ 2 }$$

here, the summation is over all atoms within a layer with a bin of width, Δz = z − (Δz/2) and z + (Δz/2). Q is the reciprocal lattice vector in the x-y plane, r _i is the Cartesian coordinates of the ith atom, and N_z is the number of atoms in the layer, respectively.

2.3.3. Atomic roughness of an individual layer

Atomic roughness (R) of an individual layer [39, 42] is quantified as:

$$\begin{eqnarray}&&R=\left[\displaystyle \sum \left(\left|\displaystyle \sum {\rm{\Delta }}z\left(i\right)\right|/{d}_{0}\right)\right]/{N}_{z},\end{eqnarray} \tag{ 3 }$$

where ∆z(i) is the deviation of the ith atom from the atomic plane along the z-axis, d₀(>0) is the interlayer spacing of the metal, and N_z is the total number of atoms in the layer. When an atom is located in the lattice of the plane, ∆z(i)/d₀ = 0, when an atomic site is unoccupied, ∣∆z(i)∣/d₀ = 1.0.

Ab initio molecular dynamics simulations are performed at elevated temperature at which atoms move around their equilibrium positions. Therefore, we use the atomic density profiles for estimation of the atomic roughness. The base-plane is set to be the peak/center at the atomic density profile.

During the ab initio molecular dynamics simulations, some liquid Al atoms moved quickly towards the O ions terminating the substrates. Correspondingly, the total energies of the systems decrease sharply at the first 0.4ps, and then level off slowly with time (figure 2). Full relaxation of all atoms in the system causes some changes in atomic rearrangement as indicated by the energy changes (figure 2). After about 1ps, the systems reached thermal equilibrium and the total energy curves of the two systems overlap with each other, being consistent with the fact that these two systems have the same atomic species and numbers in the unit cells of the same dimensions. This also indicates the similar stability of the two γ-Al₂O₃{1 1 1} surfaces in liquid Al. Furthermore, the simulations showed that the two-step approach provided quick and steady convergence and avoided collective movements of the atoms. The latter happened using one step approach [38–41]. The simulations also showed that at thermal equilibrium, the liquid Al atoms at the terminating layer exhibit ordering and are more solid-like (details in next section). Meanwhile, the Al atoms adjacent to the substrates were moving around and even moved to neighboring layers. But, the numbers of Al atoms at each layer keep statistically constant.

Figure 3 shows snapshots of the thermally equilibrated L-Al/γ-Al₂O₃{1 1 1} interfaces at 1000 K. The snapshots provide us with information about the atomic ordering at the interfaces. Clearly, the O and the Al ions in the substrates are positioned orderly, similar to those in the solid (figure 1), and the Al atoms in the liquid away from the interfaces remain disordered. A closer examination at the snapshots (figure 3) shows that: (i) there is an Al layer terminating the substrates; (ii) the liquid Al atoms close to the substrates display density variation perpendicular to the substrates (layering); (iii) the newly formed Al layer is smooth at L-Al/γ-Al₂O₃{1 1 1}_O2, while the Al atoms terminating the substrate form an uneven layer at the L-Al/γ-Al₂O₃{1 1 1}_O1 interface; and (iv) the liquid Al atoms are well separated from the substrate at the L-Al/γ-Al₂O₃{1 1 1}_O2 interface, whereas there is no clear border between the terminating Al and the liquid Al atoms at L-Al/γ-Al₂O₃{1 1 1}_O1.

Zoom In Zoom Out Reset image size

The atomic density profiles for the L-Al/γ-Al₂O₃{1 1 1} interfaces were obtained for the configurations summed over 3.0ps via equation (1). The results are plotted in figure 4.

Zoom In Zoom Out Reset image size

The atomic density profiles (figure 4) confirmed the layering at the L-Al/γ-Al₂O₃{1 1 1} interfaces (figure 3). The terminating Al atoms are close to the outmost O peak of the substrates. The two interfaces exhibit different atomic arrangements at the terminating Al layer. At L-Al/γ-Al₂O₃{1 1 1}_O2, the terminating Al atoms form a peak which is well separated from the 1st Al layer, whereas at the L-Al/γ-Al₂O₃{1 1 1}_O1 interface the terminating Al atoms form an uneven layer consisting of two or three sublayers. There is no clear border between the 1st Al and the substrate (figure 4(a)), which is similar to that at the L-Al/α-Al₂O₃{0 0 0 1} interface, where the terminating Al layer is composed of two Al subpeaks and admixed with the 1st Al layer [36, 40].

Figure 4 shows structural coupling between the terminating Al atoms and those at the subsurface Al layer in the substrate below the outmost O layer. At L-Al/γ-Al₂O₃{1 1 1}_O1 the multiply peaked terminating Al layer is coupled to the single peak at the subsurface Al layer (figure 4(a)), whereas at L-Al/γ-Al₂O₃{1 1 1}_O2 the single-peaked Al terminating layer is accompanied by the multiply peaked subsurface Al layer in the substrate. From another angle, the atomic arrangements of the terminating Al atoms are similar to those at its 3rd Al layer in the substrate at each L-Al/ γ-Al₂O₃{1 1 1} interface, respectively. (figure 4(b)).

The number of recognizable Al layers, n = 3 for L-Al/γ-Al₂O₃{1 1 1}_O1 and n = 4 for L-Al/γ-Al₂O₃{1 1 1}_O2 with a flatted peak at about 9.0Å (figure 4(b)), respectively. Therefore, the layering phenomenon at L-Al/γ-Al₂O₃{1 1 1}_O2 is more pronounced than that at L-Al/γ-Al₂O₃{1 1 1}_O1.

The epitaxial nucleation model suggested that the substrate surface provides a structural template for nucleation of the solid phase [14]. The prenucleation at the interface relates to the capability of the substrate to nucleate the solid phase in liquid. Figure 5 presents snapshots for the terminating Al, the 1st and 2nd Al layers at the L-Al/γ-Al₂O₃{1 1 1} interfaces.

Zoom In Zoom Out Reset image size

The terminating Al layers at both interfaces contain vacancies, and displacements (figure 5). There are moderate atomic ordering at the 1st Al layer but little at the 2nd Al layer at L-Al/γ-Al₂O₃{1 1 1}_O2. At L-Al/γ-Al₂O₃{1 1 1}_O1 even the 1st Al layer seems hardly any atomic ordering.

The in-plane ordering coefficients for the atomic layers near the interfaces were obtained using the configurations summed over 3ps via equation (2). The results are plotted in figure 6.

Zoom In Zoom Out Reset image size

The in-plane atomic ordering coefficients of the terminating Al layer and the 1st Al layer at L-Al/γ-Al₂O₃{1 1 1}_O2 are higher than the corresponding values at the L-Al/γ-Al₂O₃{1 1 1}_O1 interface as shown in figure 6. This agrees with the conclusions from the snapshots (figure 3), the atomic density profiles (figure 4) and the snapshots of the layer-resolved atomic arrangements (figure 5). The prenucleation at L-Al/γ-Al₂O₃{1 1 1}_O2 is more pronounced than that at L-Al/γ-Al₂O₃{1 1 1}_O1. The S(z) values of the terminating Al layer at L-Al/γ-Al₂O₃{1 1 1}_O2 (0.32) and that at L-Al/γ-Al₂O₃{1 1 1}_O2 (0.17) are smaller than that at the L-Al/α-Al₂O₃{1 1 1} interface (0.37) [40, 41].

We analyzed the occupation rates of the octahedral sites at the terminating layer using the statistics over the summed configurations. The occupation rate at the octahedral sites of the terminating layer by Al is 54.0% L-Al/γ-Al₂O₃{1 1 1}_O1 and 58.1% at the L-Al/γ-Al₂O₃{1 1 1}_O2 interface, respectively. These values are lower than that in the bulk of alumina (66.7%), but they are comparable with that for L-Al/α-Al₂O₃{1 1 1} (55.9%) [36, 40, 41].

For the L-Al/γ-Al₂O₃{1 1 1} interfaces, one issue is the occupation of metallic sites at the termination metal layer [42]. Charge balance in bulk Al³⁺ ₂O²⁻ ₃ requires a N_metal/N_O ratio to be 66.7% (two thirds). The triple nature of the metal atoms and the related ordering at the terminating metal layer provides a constant free electron density at the substrate surfaces to interact with the nearby liquid Al atoms. N_z is the same as that in a substrate metal layer.

Using equation (3) with N_z /N₀ = 2/3 (N₀ is the number of sites produced by the outmost O layer), we can estimate R values for the terminating Al layers at the L-Al/alumina interfaces. The terminating Al layer at L-Al/γ-Al₂O₃{1 1 1}_O2 is flat with an occupation of 58.1% and therefore, R = 12.9%. The terminating Al layer at L-Al/γ-Al₂O₃{1 1 1}_O1 has an Al occupation of 54.0% which contributes 19.0% to R. Moreover, the Al atoms also display splitting along the z-axis and analysis produces another 14.5%. Overall, the R = 33.5% for the terminating Al layer at L-Al/γ-Al₂O₃{1 1 1}_O1. Similar analysis produced R = 33.6% for the terminating Al layer at L-Al/α-Al₂O₃{0 0 0 1} [40, 41].

The chemical interaction between the substrate and liquid affects the atomic ordering of the liquid adjacent to the substrate [10, 13]. Here, we employed Bader charge model [51] to analyze the atomic charges at the interfaces. The results for the systems are plotted in figure 7.

Zoom In Zoom Out Reset image size

The oxygen ions in both substrates have an average charge value of −1.3e/O, whereas the Al in the substrates are positively charged with an averaged value of +2.0e/Al. Therefore, the substrates can be described with the formula, (Al^2.0+)₂(O^1.3−)₃. These charge values are smaller than the ionic model with Al³⁺ and O²⁻ at the atomic sites, indicating some covalent nature of alumina. Meanwhile, the Al atoms away from the interface are electronically neutral.

Figure 7 shows that the terminating Al atoms are charged partially. There is charge transfer from the terminating Al atoms to the outmost O atoms, indicating strong chemical bonding between the terminating Al atoms and the substrates. Therefore, the terminating Al atoms are better regarded as part of the substrate. The charge at a terminating Al-layer decreases strongly with the distance from the outmost O atoms, agreeing well with the chemical bonding theory [52].