1. Introduction
Al-Cu alloys have been under active scientific research and technological development for more than 100 years because of their applications in light weight constructions [
1,
2]. Nowadays, they are especially important in aviation and automotive industry. Aluminum alloys show a rich variety of metastable and stable phases from which a few are ordered compounds. Since usually the surface energy is too large to form directly thermodynamically stable phases, alloying elements precipitate in a sequence of clusters, zones and metastable phases. Clusters are non-ordered, locally increased concentrations of solute atoms, zones are locally ordered but do not have a long-range ordering, while stable and metastable phases possess the latter [
3]. Zones and some metastable phases are typically fully coherent with the matrix [
3].
Precipitation and clustering phenomena of solute atoms in a light metal matrix are the reason for the superior properties of aluminum alloys, i.e., this results in a high strength at a small specific weight. The obtained mechanical properties arise from a suitable thermal treatment of these alloys [
2]. Typically, after casting these age hardenable alloys are extruded or rolled to their final form. Thereafter they undergo a solution heat treatment at about
below the melting point of aluminum in order to obtain the maximum solubility of the chosen alloying elements [
3]. After the heat treatment the materials are quenched to room temperature to freeze-in the finely distributed solute atoms. Storing these alloys then at room temperatures causes the solute atoms to diffuse by the help of quenched-in vacancies [
4]. Via this process the solute atoms form agglomerates, which grow subsequently in size. After storing Al-Cu alloys for some hours at room temperature the agglomerates become visible in X-ray diffraction patterns and they are called Guinier-Preston zones according to Guinier and Preston, who discovered them independently in 1938 [
5,
6,
7]. Further storage at elevated temperature causes the growth of meta-stable Al-Cu phases:
and
[
3]. However, in new generation Al-Cu-Li alloys like AA2198, AA2050 or AA2199 besides Al-Cu-Li precipitates these Al-Cu-phases are detected as well [
8,
9]. Those Al-Cu-Li alloys are considered for the fuselage of new generation aircrafts due to their high strength and good welding behavior and have been, thus, subject to intense research in recent years (see e.g., [
10,
11]), while also Al-Cu alloys are still a matter of active research [
12,
13].
However, the understanding of the precipitation process in metallic alloys on the atomic level is still one of the main problems in materials science. It hampers a purposeful improvement of alloys, i.e., an alloy design as a bottom-up approach. Since the atomic structure of small, i.e., sub-nanometer, precipitates is difficult to access experimentally, numerical ab initio simulations are often the only way to obtain data on the geometry of atomic arrangements and their binding properties. Up to now, just a few numerical results on vacancy formation energies and di-vacancy binding energies in aluminum are available [
14,
15], which can be compared with accurate experimental data on vacancy formation energies in pure Al (see [
16] for an overview). Only recently, research on vacancy binding with different isolated solute atoms has been published for Al [
17,
18] and Mg [
19].
Results from ab initio calculations can be compared to experiments probing, e.g., vacancies by positron annihilation spectroscopy (PAS) [
20,
21,
22,
23] or individual elements by X-ray absorption [
24,
25] or small solute atom clusters employing the atom probe methods [
26]. Moreover, the recently re-discovered X-ray absorption fine structure (XAFS) spectroscopy is sensitive for the atomic environment of, e.g., Cu-atoms [
24,
27]. For the two spectroscopic methods, PAS and XAFS, spectra can be calculated from first principles (For PAS see, e.g., ref. [
28] and for XAFS [
29]). The resulting spectra, which are related to trapping of positrons to vacancies or to the excitation of solute atoms like Cu and Zn by X-rays, depend strongly on the atomic positions around those defects. A comparison of the simulations with existing experimental data can be effectively used to search for an explanation of the clustering phenomena on the atomic level in different sample compositions and conditions. Thus, it can provide guidelines to metallurgists to perform thermal and mechanical treatments on Al-alloys in order to obtain the desired materials properties.
Specifically, the results of the present work clearly give an ab initio explanation, why in Al-Cu alloys copper precipitates on the -planes, while for Al-Zn alloys three-dimensional (3D) agglomerates of Zn-atoms are formed. The reasons for these findings are easy to understand in the named simple two-component systems. However, his understanding will also pave the way for controlling processes taking place in actual technical alloys composed often of more than five elements.
The present paper is organized as follows. The computational schemes employed are presented in
Section 2. Results on the formation energies of vacancies and di-vacancies are given in
Section 3. Then we present vacancy-solute and solute-solute binding energies for clusters containing up to nine copper atoms.
Section 4 contains a discussion—in particular, a comparison between the different employed calculation schemes is presented.
2. Methods: Computational Schemes
All our calculations are based on density functional theory (DFT) within the local density approximation (LDA). In some cases a comparison with the generalized gradient approximation (GGA) of DFT has been performed. The computations are carried out using the plane-wave code VASP [
30,
31], implemented with the projector augmented-wave (PAW) method to account for the valence electron-ion core interaction.
In our VASP calculations, we have employed supercells of different sizes—namely 64, 108, 128, 144 and 192 atoms per supercell are used to check the influence of finite size effects on the relaxation of the atoms and the derived total energies. In all calculations the first Brillouin zone of the superlattice is sampled using a Monkhorst-Pack (MP)
k-point mesh [
32]. Employing the 108-atom supercell for face-centered cubic (fcc) Al we compare the results obtained with
and
k-point meshes to check the convergence of the total energy. Differences in the total energy of the systems are less than
per atom. All the calculations have thereafter been performed with the finer MP-mesh. A plane-wave cutoff of
is used in the calculation of the pseudo valence wave functions.
In the defect calculations, atomic positions are relaxed and the total energy is minimized until the forces acting on atoms are less than 0.04 eV/Å. The volume relaxation is not performed systematically, because we found that the use of a larger (108 and more atoms) supercells gives well-converged results without volume relaxation. However, results of test calculations employing the smallest (64 atom) supercell are given below. In the plane wave calculations we have used lattice constants optimized for each set of computational parameters, i.e., the cut-off energy and MP-mesh, used. For error cancellation, the total energy differences and relative ionic relaxations between defect and perfect bulk systems are calculated from results for supercells of the same size and obtained with the same computational parameters.
4. Discussion and Conclusions
Our computational results employing DFT-LDA are consistent and in sufficient agreement with previously calculated and experimental values. Especially, our results on vacancies in aluminum agree well with other ab initio calculations [
15] and experiments [
16] giving credence to our approach. Moreover, the inward relaxation of nearest-neighbor Al-atoms surrounding a single Cu atom is in accordance with experimental results from X-ray absorption [
47]. Furthermore, we find a binding between Cu atoms situated on the {100}-habit planes in fcc aluminum, while, e.g., a 3D tetrahedron even shows a repulsive interaction. Thus, 2D structures are favored compared to 3D ones, which is in agreement with experimental observations finding mono-atomic copper platelet on the {100}-planes in the fcc Al-lattice when segregation of the super saturated solution is starting [
41].
The calculated total energies for a 2D copper cluster vary slightly with used supercell sizes and shapes. However, the dependence of the total energy on the number of Cu atoms shows a similar trend in all cases. Nevertheless, it turns out that there have to be at least three layers of Al atoms in the supercell perpendicular to the Cu-habit plane separating periodic images of the copper platelets. Obviously, the relaxation of aluminum planes parallel to the copper platelet, which is important for energy lowering, cannot be reliably described with wide but flat supercells. This type of cells is not large enough in the z-direction perpendicular to the copper platelet.
In contrast to magic number arrangement of defects in semiconductors [
48], for metallic materials it seems to be difficult to state that distinguished structures are existing at all. While in general the binding energies for different geometric configurations differ very little (about 50–100 meV), we also find some special platelet structures of Cu atoms with quite high binding energies between the copper species. However, the generally small differences in binding energies may be responsible for not finding any special configurations in experiments. Remarkable is only that some closed configurations of Cu atoms on the {100}-planes show higher binding energies per Cu atom (four Cu arranged in a square). Obviously, their geometries allow electronic structures, which are energetically more favorable. However, it seems to be experimentally demanding to confirm this finding.
Concerning 3D instead of 2D plate-like structures, such as the 3D-tetrahedron, we find that these close-packed structures are the least favorable ones at all, since the Cu atoms inside their habit plane have a tendency to move away from each other. Hence, close-packed 3D-structures are unlikely to be energetically favorable in Al-Cu alloys.
According to our calculations the formation of platelets during the aging of Al-Cu alloys is in contrast to Al-Zn alloys, where obviously 3D-Clusters are preferred. Again, this is in accordance with experimental results for pure Al-Zn alloys, where clusters grow in early stages of decomposition in a spherical manner [
41,
43,
45].