Modeling Dislocation in Binary Magnesium-Based Alloys Using Atomistic Method

Abstract

In the wake of developing biodegradable metallic implants for orthopedic practice or lightweight structural components for the automotive industry, both fundamental and applied research on magnesium and its alloys regained a high interest in the last decade. As of today, the major issues delaying the integration of the magnesium technology in the medical and automotive industries are (i) a lack of ductility and (ii) a poor corrosion resistance. Alloying is a common strategy used to improve the ductility and the corrosion resistance. Although density functional theory is a powerful method that allows one to quantify material parameters to be used later in a theoretical model, atomistic methods in the framework of semi-empirical potentials are complementary to density functional theory. While the data obtained from semi-empirical potentials are more qualitative than quantitative, it does not prevent atomistic calculations in the framework of semi-empirical potentials to validate/disprove/enrich an existing theoretical model or even to provide insights for the development of a new theoretical model. The validity of the data derived from atomistic calculations in the framework of semi-empirical potentials depends on the accuracy and transferability of the potentials to capture the physics involved in the problem. In view of modeling the mechanical properties of a binary magnesium-based alloy using semi-empirical potentials, one has to validate the ability of the potentials to capture the physics governing the interactions between the alloying element and the micromechanisms carrying the inelastic behavior. In this chapter, we are reviewing the interaction between alloying elements and (i) stacking faults and (ii) <a> dislocations from the basal and prismatic slip systems.

Appendices

Annex A Generalized Stacking Fault Energy Calculation

The energy-displacement curve, also referred in the literature as generalized stacking fault energy curve (GSF), cannot be measured experimentally except for a single point known as the intrinsic stacking fault energy, γ_s. Since the GSF energy curve is an important property that controls the slip behavior, it is common to use the energy-displacement curve predicted by density functional theory to validate the ability of a semi-empirical potential in modeling dislocation core properties. As two sets of atomic planes glide rigidly with respect to each other, the energy per area of the faulted plane increases to γ_us and then decreases. For slip along the <$10\overline{1}0$> direction in the basal plane, it exists a position at which is the lattice is stable although the crystal is not its bulk equilibrium structure. Such stable configuration is an intrinsic stacking fault characterized by its energy γ_s. A representative energy-displacement curve obtained for slip along <$10\overline{1}0$> in the basal plane is given in Fig. 9 with the corresponding packing of basal planes and γ_s and γ_us labeled.

Fig. 9

The GSF energy curve for the <$10\overline{1}0$> direction in the basal plane in a hcp crystal. The packing of basal planes is given in the perfect crystal and with displacement of 0.17 and 0.33

Although several methods are available in the literature to calculate the generalized stacking fault energy, since the stable and unstable stacking fault energies in the basal and prismatic planes are not sensitive to the method, the slab method was used in this work. Based on the slab method, two rigid blocks of atoms are displaced with respect to each other in the plane of the fault. After each increment of displacement, the forces in the direction normal to the fault are relaxed using a conjugate gradient method. The atomic configuration is relaxed when the forces are lower than 10⁻⁵ eV/Å.

Annex B Interaction Energy Between Dislocation and Solute Element

The interaction energy between a solute element A and a dislocation is calculated as the difference between the total energy of a system with a solute element located close to the dislocation and far from the dislocation given by the following formula:

$$ {E}_{b,x}^A={E}_x^{A,\mathrm{dislo}}-{E}_f $$

where ${E}_x^{A,\mathrm{dislo}}$ represents the substitutional formation energy at the location x in the vicinity of the dislocation, and E_f is the bulk substitutional energy. In the limiting case, when the substitution occurs far away from the dislocation core, ${E}_x^{A,\mathrm{dislo}}$ = E_f, and thus the binding energy tends to 0. The substitutional formation energy at the location X in the vicinity of the dislocation is given by

$$ {E}_x^{A,\mathrm{dislo}}={E}_{\mathrm{total}}^{Mg_{n-1}{A}_1,\mathrm{dislo}}-{E}_{\mathrm{total}}^{Mg}+\left({E}_c^{Mg}-{E}_c^A\right) $$

where ${E}_{\mathrm{total}}^{Mg_{n-1}{A}_1,\mathrm{dislo}}$ is the total energy of the ensemble when one magnesium atom is substituted by a solute element A in a dislocated crystal, ${E}_{\mathrm{total}}^{Mg}$ is the total energy of a pure and dislocated Mg crystal, and ${E}_c^{Mg}\mathrm{and}\ {E}_c^A$ are the cohesive energies for Mg and the solute element A, respectively.

The bulk formation energy is calculated using the following equation:

$$ {E}_f^A={E}_{\mathrm{total}}^{Mg_{n-1}{X}_1}-{E}_{\mathrm{total}}^{Mg}+\left({E}_c^{Mg}-{E}_c^X\right) $$

where ${E}_{\mathrm{total}}^{Mg_{n-1}{A}_1}$ is the total energy of the ensemble when one magnesium atom is substituted by a solute element A in a perfect crystal and ${E}_{\mathrm{total}}^{Mg}$ is the total energy of a pure Mg perfect crystal.

The resistive force is given as the absolute value of the binding energy gradient along the dislocation slip direction. Assuming the slip direction is along the x-direction, the resistive force is obtained as

$$ F=\left|\frac{\Delta {E}_b}{\Delta x}\right| $$

In this work, a dislocation was introduced in a crystal by displacing the atomic locations according to the dislocation displacement field derived from the isotropic linear elasticity theory. A cylinder of radius 80 Å was considered. Periodic boundary conditions were imposed along the dislocation line direction, whereas nonperiodic boundary conditions were applied in the other two directions. Atoms located in an inner cylinder with 65 Å of radius were allowed to relax in all three directions to minimize the potential energy of the system. The minimization of the potential energy was performed using a conjugate gradient algorithm. The atomic configuration is relaxed when the forces are lower than 10⁻⁵ eV/Å.

A process window around the dislocation core was defined, in which substitution of Mg atom by solute element A occurs. Energy minimization was performed using the (M)EAM implementation of the LAMMPS package [50]. The Mg-Ca, Mg-Li, Mg-Sn, Mg-Y, and Mg-Al of Lee and coworkers were provided by B.J. Lee. The Mg-Li potential proposed by Karewar et al. [11] was provided by S. Karewar. The Mg-Al potentials proposed by Liu and Adams [25] and Mendelev et al. [12] were downloaded from Interatomic Potential Repository Project of the NIST.