# Stability of Cu-Precipitates in Al-Cu Alloys

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## Abstract

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## 1. Introduction

## 2. Methods: Computational Schemes

**k**-point mesh [32]. Employing the 108-atom supercell for face-centered cubic (fcc) Al we compare the results obtained with $4\times 4\times 4$ and $6\times 6\times 6$

**k**-point meshes to check the convergence of the total energy. Differences in the total energy of the systems are less than $5\phantom{\rule{0.166667em}{0ex}}\mathrm{meV}$ per atom. All the calculations have thereafter been performed with the finer MP-mesh. A plane-wave cutoff of $300\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$ is used in the calculation of the pseudo valence wave functions.

## 3. Results

#### 3.1. Reliability of Modeling

**k**-point meshes, we have computed the solubility enthalpy of Cu in Al. It is calculated as

#### 3.2. Impurity-Cluster Binding Energies

**k**-point mesh. All the supercell sizes lead to a binding energy of around $50\phantom{\rule{0.166667em}{0ex}}\mathrm{meV}$. For the 128 atom supercell the binding energy is the smallest one reflecting the small spacing between the adjacent Cu habit planes of the periodic images and the ensuing artificial interaction.

#### 3.2.1. 2D-Cu-Clusters in 108-atom Supercells

#### 3.2.2. 2D-Cu-Clusters in 128- and 192-atom Supercells

#### 3.2.3. Lattice Relaxations

#### 3.2.4. Relaxed Versus Static Configurations

#### 3.2.5. Comparison with Zinc Clusters

#### 3.2.6. Pre-Guinier-Preston Zones in Al-Cu and Al-Zn

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PAS | Positron Annihilation Spectroscopy |

XAFS | X-ray Absorption Fine Structure |

DFT | Density Functional Theory |

LDA | Local Density Approximation |

GGA | General Gradient Approximation |

VASP | Vienna Ab initio Simulation Package |

PAW | Projector Augmented Wave |

MP | Monkhorst-Pack |

SIESTA | Spanish Initiative for Electronic Simulations with Thousands of Atoms |

DZP | Double Zeta Polarized |

1NN | Nearest Neighbors |

2NN | Next Nearest Neighbors |

XRD | X-Ray Diffraction |

TEM | Transmission Electron Microscopy |

HR-TEM | High resolution Transmission Electron Microscopy |

fcc | face-centered cubic |

hcp | hexagonal closed-packed |

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**Figure 1.**Configuration of Cu-atoms in 2D platelets on the $\left\{100\right\}$-plane in fcc Al. The Cu and Al atoms are shown by red and grey spheres, respectively. The numbering indicates how the Cu-platelets were assumed to grow. The left and right patterns show the sequences 1 and 2 used in the calculations, respectively.

**Figure 2.**Energy gain during Cu-cluster growth for a supercell of 108 atoms in sequences 1 and 2. (

**left**) The energy gain due to the last attached Cu atom, (

**right**) the total energy gain per Cu atom in the growing cluster. Sequences 1 and 2 are explained in Figure 1.

**Figure 3.**Energy gain during Cu-cluster growth. Results corresponding to supercells of 108, 128, and 192 atoms are compared. (

**left**) Energy gain due to the last Cu atom attached, and (

**right**) total energy gain per Cu-atom of a growing cluster. Sequence 1 is explained in Figure 1.

**Figure 4.**Energy gain during Cu-cluster growth. The results corresponding to the supercells of 108, 128, and 192 atoms are compared. (

**left**) Energy gain of the last Cu atom attached, (

**right**) total energy gain per Cu-atom of a growing cluster. Sequence 2 is explained in Figure 1.

**Figure 5.**Relaxation patterns of growing clusters. (

**left**) Two copper atoms (red spheres) on 1NN position, (

**right**) five Cu-atoms arranged as a platelet on the $\left\{100\right\}$-plane. The Al atoms having two or more bonds to Cu-atoms are plotted in dark-grey color. The relaxation of the Al-layer above and below towards the Cu-atoms on the $\left\{100\right\}$-plane clearly increases with the number of agglomerated Cu-aotms.

**Figure 6.**Total energies calculated by using static (atoms fixed) and relaxed configurations of a supercell of 108 atoms.

**Figure 7.**Binding energies of Zn atoms in 2D and 3D Zn clusters. (

**left**) The binding energy for the last attached Zn atom. (

**right**) The binding energy per Zn atom in the cluster.

**Table 1.**Comparison with experimental results: Formation energies for mono- ${H}_{\mathrm{V}}^{\mathrm{F}}$ and di-vacancies ${H}_{2\mathrm{V},\mathrm{X}}^{\mathrm{F}}$ in the nearest neighbor (X = 1NN) and next nearest neighbor (X = 2NN) positions in fcc Al. The binding energies ${H}_{2\mathrm{V},\mathrm{X}}^{\mathrm{b}}$ of the two vacancies in the two configurations are also given. Positive and negative binding energies indicate repulsion and binding, respectively. (SIESTA results: see [34]).

Method | Volume Relax | MP-Mesh | Atoms | ${\mathit{H}}_{\mathbf{V}}^{\mathbf{F}}$ | ${\mathit{H}}_{2\mathbf{V},1\mathbf{NN}}^{\mathbf{F}}$ | ${\mathit{H}}_{2\mathbf{V},2\mathbf{NN}}^{\mathbf{F}}$ | ${\mathit{H}}_{2\mathbf{V},1\mathbf{NN}}^{\mathbf{b}}$ | ${\mathit{H}}_{2\mathbf{V},2\mathbf{NN}}^{\mathbf{b}}$ |
---|---|---|---|---|---|---|---|---|

(eV) | (eV) | (eV) | (eV) | (eV) | ||||

VASP-LDA | yes | $6\times 6\times 6$ | 64 | 0.71 | — | — | — | — |

VASP-LDA | no | $6\times 6\times 6$ | 64 | 0.713 | 1.506 | 1.409 | +0.081 | −0.016 |

VASP-LDA | no | $6\times 6\times 6$ | 108 | 0.714 | 1.489 | 1.421 | +0.061 | −0.007 |

VASP-GGA | no | $6\times 6\times 6$ | 108 | 0.66 | — | — | — | — |

SIESTA-DZP | no | $3\times 3\times 3$ | 108 | 0.64 | — | — | — | — |

Exp. [16] | — | — | — | 0.67 |

**Table 2.**Solubility enthalpy $\Delta {H}_{\mathrm{mix}}$ of Cu in Al calculated by using Equation (2) and different supercell sizes.

Supercell Size (atom) | $\mathbf{\Delta}{\mathit{H}}_{\mathbf{mix}}$ (meV) |
---|---|

108 | −50.5 |

128 | −54.2 |

144 | $-53.0$ |

**Table 3.**Binding energy of two Cu solute atoms in Al on nearest neigbor positions in fcc Al. The c-direction is perpendicular to the habit plane of the Cu atoms. Negative signs indicate binding.

Scheme | Number Atoms | Size Unit Cells | k-points | ${\mathit{E}}_{\mathbf{bind}}$ (meV) |
---|---|---|---|---|

LDA | 108 | $3\times 3\times 3$ | $4\times 4\times 4$ | −50.3 |

LDA | 128 | $4\times 4\times 2$ | $4\times 4\times 8$ | −46.3 |

LDA | 144 | $3\times 3\times 4$ | $4\times 4\times 4$ | −56.2 |

LDA | 192 | $4\times 4\times 3$ | $3\times 3\times 6$ | −54.7 |

GGA | 108 | $3\times 3\times 3$ | $6\times 6\times 6$ | −51.5 |

**Table 4.**Binding energies for agglomerates of Cu atoms in 1D, 2D, and 3D configurations. The calculation employed the 108 atom supercell. Given is the total binding energy, the binding energy per Cu-atom, and the binding energy of the ‘last’ Cu atom specified in Figure 1. We give here the energy with an accuracy of $0.1\phantom{\rule{0.166667em}{0ex}}\mathrm{meV}$, which is only of internal numerical relevance. The numbering is according to Figure 1 left.

Agglomerate | atom no. | Spatial | ${\mathit{E}}_{\mathbf{bind}}$ (meV) | ${\mathit{E}}_{\mathbf{bind}}$ (meV) | ${\mathit{E}}_{\mathbf{bind}}$ (meV) |
---|---|---|---|---|---|

structure | extension | cluster | per Cu | last Cu | |

(a) 2 Cu 1NN on $\left(100\right)$-plane | 1, 2 | 2D | −50.3 | −25.1 | −50.2 |

(b) 2 Cu 2NN on $\left(100\right)$-plane | 1, 4 | 2D | −9.6 | −4.8 | −9.6 |

(c) 3 Cu in-line on $\left(100\right)$-plane | 6, 7, 8 | 1D | −95.2 | −31.7 | −45.0 |

(d) 3 Cu triangle on $\left(100\right)$-plane | 1, 2, 3 | 2D | −134.7 | −44.9 | −84.4 |

(e) 3 Cu triangle on $\left(111\right)$-plane | – | 2D | − 97.4 | −32.5 | −47.1 |

(f) 4 Cu rectangle on $\left(100\right)$-plane | 1, 2, 3, 4 | 2D | −344.7 | −86.2 | −210.0 |

(g) 4 Cu triangle on $\left(100\right)$-plane | 1, 2, 3, 5 | 2D | −206.1 | −51.5 | −71.4 |

(h) 4 Cu tetrahedron in space | – | 3D | +53.0 | +13.2 | +150.4 |

Type | Size (atoms) | Size (Unit Cell) | MP-mesh |
---|---|---|---|

standard | 108 | $3\times 3\times 3$ | $4\times 4\times 4$ |

flattened | 128 | $4\times 4\times 2$ | $4\times 4\times 6$ |

elevated | 144 | $3\times 3\times 4$ | $3\times 3\times 2$ |

widened | 192 | $4\times 4\times 3$ | $2\times 2\times 3$ |

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## Share and Cite

**MDPI and ACS Style**

Staab, T.E.M.; Folegati, P.; Wolfertz, I.; Puska, M.J.
Stability of Cu-Precipitates in Al-Cu Alloys. *Appl. Sci.* **2018**, *8*, 1003.
https://doi.org/10.3390/app8061003

**AMA Style**

Staab TEM, Folegati P, Wolfertz I, Puska MJ.
Stability of Cu-Precipitates in Al-Cu Alloys. *Applied Sciences*. 2018; 8(6):1003.
https://doi.org/10.3390/app8061003

**Chicago/Turabian Style**

Staab, Torsten E. M., Paola Folegati, Iris Wolfertz, and Martti J. Puska.
2018. "Stability of Cu-Precipitates in Al-Cu Alloys" *Applied Sciences* 8, no. 6: 1003.
https://doi.org/10.3390/app8061003