# The Energetics and Topology of Grain Boundaries in Magnesium: An Ab Initio Study

^{*}

## Abstract

**:**

_{SEP}) was very sensitive to grain boundary chemistry. Boundaries of higher disorder were found to be more sensitive to boundary chemistry and showed higher values of W

_{SEP}and in the case of Gd, were more sensitive to solute concentration at the boundary. No correlation between the boundary behaviour and crystallography could be found, apart from the over-riding conclusion that all six boundaries showed markedly different behaviours, and the effect of solute on each were unique.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Grain Boundary Crystallography

#### 2.2. Simulation Cell Construction

#### 2.3. Calculation Details

^{−6}eV [28]. The mesh of $\mathsf{\Gamma}$-centered k-points to sample the Brillouin zone were chosen such that their density per reciprocal space is at least 50,000 Å

^{−3}. The atomic configuration is optimized using conjugate gradient method until mean atomic forces are less than 0.02 eV/Å

^{−1}.

_{1}and G

_{2}) represented by two slabs. As a consequence, the simulation cell total energy is the contribution of the bulk energy of grains (${\mathrm{E}}_{\mathrm{Bulk},\mathrm{G}1}$ and ${\mathrm{E}}_{\mathrm{Bulk},\mathrm{G}2}$), two surface energies (${\mathsf{\sigma}}_{\mathrm{G}1+}{\mathsf{\sigma}}_{\mathrm{G}2}$) and the grain boundary energy (${\mathsf{\gamma}}_{\mathrm{GB}}$) itself. Thus, the latter can be expressed as ${\mathsf{\gamma}}_{\mathrm{GB}}=$ ${\mathrm{E}}_{\mathrm{GB}}/\mathrm{S}-({\mathsf{\sigma}}_{\mathrm{G}1}+{\mathsf{\sigma}}_{\mathrm{G}2})-\left({\mathrm{E}}_{\mathrm{Bulk},\mathrm{G}1}+{\mathrm{E}}_{\mathrm{Bulk},\mathrm{G}2}\right)/\mathrm{S}$ and rearranging the right hand side of this equation will give [30]:

_{SEP}and S are the work of separation defined as the reversible work needed to separate the grain boundary into two free surfaces [31] and boundary area, respectively.

#### 2.4. Addition of Solutes

- (1)
- One solute atom at five different locations along the boundary
- (2)
- Five solute atoms all located at the boundary
- (3)
- Grain boundary behaviour at a concentration of 5% (+/0.3%)

- (1)
- 1 Zn and 1 Gd atom at the boundary
- (2)
- 2 Zn and 2 Gd atoms located at the boundary

## 3. Results

#### 3.1. Pur Magnesium

^{2}, which is much larger than the energy of the commonly observed twin boundary $\left\{10\overline{1}2\right\}$ which is reported to be 0.125 J/m

^{2}[28]. The work of separation (W

_{SEP}) for the general boundaries of pure magnesium ranges from 0.904 to 1.208 J/m

^{2}, which is smaller than that of the $\left\{10\overline{1}2\right\}$ twin boundary and several other special boundaries studied in Ref [28].

#### 3.2. Effect of Solutes

_{seg}, is a parameter which indicates the preference of a solute to be at a boundary compared to the bulk and is shown in Figure 5. The E

_{seg}had a spread of values depending on the individual location of the solute, and the simulations with multiple solutes tended to fall somewhere in the middle of this spread.

_{SEP}. This parameter has traditionally been used [29,36] to examine the cohesion of a boundary, as it indicates the work required to separate two neighbouring grains. However, in the present case we use W

_{SEP}as a tool to identify the changes that occur when solute is located at the boundary. If we begin by comparing the W

_{SEP}to the grain boundary energy, we can see that there is some correlation between these two parameters, and that small decreases in the grain boundary energy led to large increases in the W

_{SEP}and vice versa, Figure 7.

_{SEP}is shown in Figure 8. There is some variation in the W

_{SEP}for the single solute simulations, the addition of a single Zn solute could either increase or decrease W

_{SEP}depending on its location, while the addition of a single Gd increased the W

_{SEP}compared to pure Mg, but the magnitude varied markedly. The addition of 5 Zn solutes or 5 Gd solutes increase the W

_{SEP}, and Gd has a larger effect than Zn. The effect of co-segregated Gd and Zn had a variable effect on W

_{SEP}.

_{SEP}for boundaries B, D and E, while it increases W

_{SEP}for boundaries C and F. For the case of Gd, the W

_{SEP}is always increased by the addition of Gd solutes, but the magnitude changes markedly between boundaries. Clearly, the structure of the boundary creates a significant change in the effect of solute on W

_{SEP}.

## 4. Discussion

#### 4.1. Effct of Topology

_{SEP}for lower values of coordination number, where lower CN values correlate with more disorder at the boundary.

_{SEP}. We can see from Figure 11 that the addition of Zn to the boundary increased the W

_{SEP}value, but this did not correlate with CN. However, Gd shows a tendency to increase the value of W

_{SEP}more for the more disordered boundaries with lower coordination numbers. It must also be concluded from this data that the effect of solutes on the boundary is not easily defined and shows significant variability across many energetic parameters examined here, such as the ones shown in Figure 5, Figure 6 and Figure 8.

#### 4.2. Effect of Solute on GB Energetics

_{SEP}data from all simulations for all boundaries, plotted as a function of grain boundary concentration. Note that the concentration scale in Figure 13 (a) is much larger than the other boundaries due to the relatively smaller size of that simulation cell. It can be seen that there is a generalized trend towards increasing W

_{SEP}with increasing solute concentration. This can be understood by considering the solute bonding behaviours. Previous work quantified the strength of the bond between magnesium and the solutes Gd and Zn using the crystal orbital Hamiltonian population method [19] where it was demonstrated that Gd had a stronger bond to Mg than Zn did, but the bonding of both solutes with the Mg matrix was stronger than the bonds between Mg matrix. With increasing concentration at the boundary there will be more solutes, and these provide more bonds, leading to the general trend of increasing W

_{SEP}with increasing solute concentration. This also explains why Gd tends to have a larger W

_{SEP}than Zn, Gd has been found to bond more strongly to the grain boundary than zinc. In addition, worth mentioning is the rate of increase in W

_{SEP}with concentration. Boundaries A and D show a much lower increase in W

_{SEP}per atomic percent increase in solute concentration. These boundaries also show high values of CN and show narrow lattice strain distributions (Figure 4), and it may be the case that the W

_{SEP}is more sensitive to solute in disordered boundaries, resulting in showing a steeper increase in W

_{SEP}with increasing solute concentration. This is consistent with the data shown in Figure 10.

## 5. Conclusions

- The grain boundaries were found to have an intrinsic grain boundary energy, and this energy was not markedly affected by the solute concentration or chemistry at the boundary.
- In contrast with grain boundary energy, the work of separation (W
_{SEP}) was very sensitive to grain boundary chemistry. This parameter was therefore used to interrogate the effect of solute on grain boundary behaviour. It was found that for a boundary solute concentration of 5 at%, the effect of Zn and Gd on the W_{SEP}was markedly different for the different boundaries. This is indicative that the two solutes will have a different effect on different boundary types. - The topology of the grain boundary was correlated with the boundary energetics using the coordination number. For pure magnesium, the work of separation was found to correlate with coordination number, with more disordered boundaries of low CN showing higher values of W
_{SEP}. The effect of solute was not directly correlated with the CN, but there was a general trend for Gd to increase the W_{SEP}more in those boundaries of low CN (high disorder). - The W
_{SEP}was found to increase with increasing boundary solute concentration, with the rate of change being markedly different between the different boundaries. The increase in W_{SEP}with solute concentration was typically higher for Gd compared to Zn. - No correlation between the boundary behaviour and crystallography could be found, apart from the over-riding conclusion that all six boundaries showed different behaviours, and the effect of solute on their W
_{SEP}were unique.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Site specific atom probe tomography of fully recrystallized Mg-0.5 at%Gd-0.5 at%Zn after annealing for 1 h at 500 °C. (

**a**,

**b**) Mass spectra for the specimen volume shown in (

**c**). (

**d**) Iso-surfaces showing Gd segregation to the grain boundaries. (

**e**) Concentration profile for the cylindrical region of interest shown in pink in (

**d**).

**Figure 2.**Pole figures of the eight orientations used to construct the 6 grain boundaries examined in this work.

**Figure 3.**The simulation cells for the six grain boundaries examined here. The locations used for the solute are shown in purple. Note that all 5 locations cannot be seen due to the simulation cells being 3-dimensional. Labels (

**A**–

**F**) correspond to the boundaries listed in Table 2.

**Figure 4.**Lattice strain at the six different boundaries examined. Boundaries A to F are shown in histograms (

**a**–

**f**) respectively.

**Figure 5.**A summary of the segregation energy for all simulations carried out in this study. Note that a negative value of E

_{seg}indicates that the solute has a preference for the boundary, whilst positive values indicate a preference for being located in the bulk. More negative values of E

_{seg}indicate a preference to segregate to the boundary.

**Figure 6.**Grain boundary energy of the six grain boundaries, calculated for (

**a**) pure magnesium, (

**b**) a single Zn atom at five different locations, as well as five Zn in the one boundary, (

**c**) a single Gd atom at five different locations, as well as five Gd in the one boundary, (

**d**) simulations with one Zn and one Gd, as well as 2 Zn and 2 Gd.

**Figure 8.**The work of separation (W

_{SEP}) of the six grain boundaries, calculated for (

**a**) a single Zn atom at five different locations, as well as five Zn in the one boundary, (

**b**) a single Gd atom at five different locations, as well as five Gd in the one boundary, (

**c**) five solutes at the boundary, (

**d**) one Zn and one Gd, as well as 2 Zn and 2 Gd at the boundary.

**Figure 9.**W

_{SEP}for the five boundaries in which a solute concentration of 5% (+/−0.3%) could be achieved.

**Figure 10.**Local topology of the 6 boundaries quantified by pair distribution functions (PDF). The data corresponding to boundaries (

**A**–

**F**) have been labelled individually in the figure.

**Figure 12.**pDOS of the d-orbital of Gd as a solute in the bulk Mg and at grain boundaries A, B and C. Fermi level is adjusted to zero eV.

**Figure 13.**Summary of the work of separation (W

_{SEP}) measurements made on the six different boundaries examined in this study. Data is plotted as a function of the boundary concentration in atomic percentage. Boundaries A to F are shown individually in (

**a**–

**f**) respectively. Note larger x-axis in (

**a**,

**b**).

**Table 1.**The crystallography of the six grain boundaries chosen for study. These are shown stereographically in Figure 2.

Grain Number | Boundary Plane and Parallel Directions | Axis and Angle of Rotation (Three Digit Indices) | Equivalent Four Digit Rotation Axis If Rational | |
---|---|---|---|---|

Boundary A | Grain A1 | $\left(0001\right)\langle 11\overline{2}0\rangle $ | 58.31° $\langle \frac{-\sqrt{3}}{2},0.5,0\rangle $ | $\left[10\overline{1}0\right]$ tilt |

Grain A2 | $\left(11\overline{2}2\right)\langle \overline{11}21\rangle $ | |||

Boundary B | Grain B1 | $\left(10\overline{1}0\right)\langle 0001\rangle $ | 43.37° $\langle 0.52,-0.52,0.67\rangle $ | irrational |

Grain B2 | $\left(11\overline{2}2\right)\langle \overline{11}21\rangle $ | |||

Boundary C | Grain C1 | $\left(10\overline{1}0\right)\langle 0001\rangle $ | 61.59° $\langle \frac{\sqrt{3}}{2},0.5,0\rangle $ | $\left[10\overline{1}0\right]$ twist |

Grain C2 | $\left(10\overline{1}0\right)\langle 1\overline{2}11\rangle $ | |||

Boundary D | Grain D1 | $\left(01\overline{1}2\right)\langle 0\overline{1}11\rangle $ | 18.79° $\langle -1,0,0\rangle $ | $\left[11\overline{2}0\right]$ tilt |

Grain D2 | $\left(10\overline{1}1\right)\langle 0\overline{1}12\rangle $ | |||

Boundary E | Grain E1 | $\left(0001\right)\langle 11\overline{2}0\rangle $ | 10.9° $\langle 0,0,1\rangle $ | $\left[0001\right]$ twist |

Grain E2 | $\left(0001\right)\langle 3\overline{21}0\rangle $ | |||

Boundary F | Grain F1 | $\left(11\overline{2}2\right)\langle \overline{11}21\rangle $ | 74.8° < 1, −0.4, −0.25> | irrational |

Grain F2 | $\left(10\overline{1}5\right)\langle 1\overline{2}10\rangle $ |

Boundary | Cell Dimension (Å) | Number of Atoms in Simulation | Number of Atoms at GB | Solute Conc. (1 Atom) | Solute Conc. (5 Atom) | Additional Simulation |
---|---|---|---|---|---|---|

Boundary A | $5.56\times $$6.42$ × 99.51 | 128 | 8 | 12.5% | 62.5% | none |

Boundary B | $5.21\times $$57.78\times $ 52.96 | 536 | 38 | 2.6% | 13.2% | 2 solute atoms = 5.3% conc. |

Boundary C | $46.90\times $$25.68\times $ 32.10 | 977 | 140 | 0.7% | 3.6% | 7 solute atoms = 5.0% conc. |

Boundary D | $15.24\times $$35.310\times $ 49.43 | 890 | 106 | 0.9% | 4.7% | None |

Boundary E | $16.05\times $$27.79\times $ 46.54 | 742 | 106 | 0.9% | 4.7% | None |

Boundary F | $30.60$ × $22.23\times $ 43.33 | 846 | 148 | 0.7% | 3.4% | 7 solute atoms = 5.0% conc. |

**Table 3.**Summary of grain boundary energy (γ

_{GB}), the work of separation (W

_{SEP}), coordination number (CN), and the width of lattice strain for the six different boundaries for simulations with only magnesium atoms included.

Boundary | ϒ_{GB} (J/m^{2}) | W_{SEP} (J/m^{2}) | CN | Strain Width (%) |
---|---|---|---|---|

Boundary A | 0.584 | 1.049 | 8.4 | 3.0 |

Boundary B | 0.205 | 1.208 | 5.1 | 4.2 |

Boundary C | 0.445 | 1.117 | 5.6 | 7.9 |

Boundary D | 0.465 | 1.025 | 7.3 | 6.6 |

Boundary E | 0.379 | 0.904 | 8.1 | 2.7 |

Boundary F | 0.427 | 0.982 | 7.6 | 7.0 |

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**MDPI and ACS Style**

Mahjoub, R.; Stanford, N.
The Energetics and Topology of Grain Boundaries in Magnesium: An Ab Initio Study. *Alloys* **2022**, *1*, 15-30.
https://doi.org/10.3390/alloys1010003

**AMA Style**

Mahjoub R, Stanford N.
The Energetics and Topology of Grain Boundaries in Magnesium: An Ab Initio Study. *Alloys*. 2022; 1(1):15-30.
https://doi.org/10.3390/alloys1010003

**Chicago/Turabian Style**

Mahjoub, Reza, and Nikki Stanford.
2022. "The Energetics and Topology of Grain Boundaries in Magnesium: An Ab Initio Study" *Alloys* 1, no. 1: 15-30.
https://doi.org/10.3390/alloys1010003