Bridging Ab Initio and Atomistic Computations of Chromium using Density Functional Theory and Modified Embedded Atom Method
In this work, we present Density Functional Theory and Modified Embedded Atom Method simulation results for pure chromium. We used Quantum Espresso to generate energy versus volume, energy versus atomic separation, and generalized stacking fault energy curves along with the lattice parameter, bulk modulus, and cohesive energy at equilibrium. We used the MEAM Parameter Calibration tool along with the DFT data to create a MEAM potential for chromium. Both sets of simulations were verified by comparison to values from literature, and a sensitivity analysis was conducted for the MEAM potential. These simulations are a valuable first step in generating an effective material model for chromium and may be upscaled to other simulations, such as dislocation dynamics.
Keywords: Chromium (Cr), Modified Embedded Atom Method (MEAM), Density Functional Theory (DFT), Integrated Computational Materials Engineering (ICME), Generalized Stacking Fault Energy (GSFE)
Author(s): Cagle, M. S., Fonville, T. R., Kazandjian, S. L., and Sprow, B. T.
Horstemeyer  proposed the Integrated Computational Materials Engineering (ICME) paradigm as a means to capture the multi-scale phenomenon of a material by bridging information between length scales from the electron scale to the structural (continuum) scale. Multiscale modeling with the ICME paradigm offers the accuracy required to predict failure, reduce design costs, and reduce time-to-market with the simulation-based design paradigm. The fundamental steps required to achieve multiscale modeling with the ICME paradigm are as follows:
- Step 1: Downscaling, where we define the desired higher-level “effects,” or information needed.
- Step 2: Modeling and simulation at lower length scales (including calibration and verification)
- Step 3: Upscaling, where we pass information up to meet the requirements set at the higher length scale in step 1. Iteration is required to ensure a proper connection.
- Step 4: Validation at the higher length scale through experiment or simulation.
In this work, we select pure chromium (Cr) as our material of interest where we wish to model the plastic behavior at the structural scale. First, we downscaled (Step 1) the end-goal of modeling plastic behavior to the electronic scale, where we identified energy-volume (EV) and generalized stacking fault energy (GSFE) curves as the quantities necessary to bridge information up (upscaling) to the nanoscale. We calculated the EV and GSFE curves with ab initio Density Functional Theory (DFT) simulations that we ran in Quantum Espresso (QE). At the nanoscale, we calibrated the Modified Embedded Atom Method (MEAM) potential, derived from the Embedded Atom Method (EAM), using the EV and GSFE curves. The MEAM potential accounts for the angular forces between atoms and provides electronic structure and dislocation mobility information that we will need to conduct dislocation dynamics simulations at the microscale. MEAM potentials are applicable with many kinds of crystal lattices including the primary crystal structure of chromium that is body centered cubic (BCC). We used the MEAM Parameter Calibration (MPC) MATLAB program created by Horstemeyer et al.  in combination with the molecular dynamics (MD) code known as large-scale atomistic/molecular massively parallel simulator (LAMMPS) to calibrate our MEAM potential. For both the DFT and MEAM simulations, we verify the results with values from the literature. In the next section, we give an overview of the theoretical models used in this work.
We use atomistic analysis methods, suggested by Lee et al. , to describe the energy of a system related to interatomic forces and the relative position of atoms. According to the EAM developed by Baskes et al. , the total energy of a system is composed of the embedding function F_i and pair interaction ϕ terms shown in Equation 1 below:
where E is energy, ρ_i is background electron density, ϕ is atomic pair interaction, and R_ij is distance. We describe the atomic pair interactions (ϕ) at a distance R in Equation 2 below:
where Z is the number of first neighbors, E^u (R) is the background electronic density, and ρ ̅^0 (R) is energy per atom for the reference structure. We use the universal equation of state (UEOS) or first principles computations to obtain the atomic energy (E^u (R)). We calculate the UEOS parameters of cohesive energy (E_c), nearest neighbor distance (r_e), atomic volume (Ω), and bulk modulus (B) at equilibrium in the reference structure in Equations 3, 4, and 5, respectively.
The MEAM embedding function comprises the aforementioned terms and the adjustable parameter A as shown in Equation 6 below:
The background electron density is the combination of the spherically symmetric (ρ^(0)) and angular (ρ^(1)),ρ^(2),ρ^(3)) partial electron density terms:
where the terms α,β,γ represent summations over the x, y, and z coordinate directions. Atomic electron density ρ^a(h) decreases exponentially with increasing distance r^i between an atom i and the site of interest as illustrated in Equation 8 that contains constant decay lengths, β^h.
The equations for the partial electron densities are coordinate invariant and equal to zero for cubic symmetric crystals. We represent the angular contributions from the partial electron densities in simplified form in Equation 9 (with constants t^h) and the equation for background electron density in Equation 10.
In order to obtain accurate material properties and avoid imaginary electron densities for Г<-1, we use the functional form of G (shown in Equation 11) where G(0)=1/2. When G(0)=1, the background electron density is non angular dependent and is equivalent to the EAM expression for ρ .
In the next section, we describe the DFT calculations to generate the EV curves.
Density Functional Theory Calculations for Energy versus Lattice Parameter (E-A) and Energy versus Volume (E-V)
Density Functional Theory Calculations for Generalized Stacking Fault Energy Curves
Modified Embedded Atom Method Potential Calibration
MEAM potential calibration has been a manual process since the creation of the MEAM model. Horstemeyer et al.  presented a simple MEAM potential calibration methodology with the MPC software tool. In this work, we follow a similar approach where we calibrated BCC, FCC, and HCP energy versus lattice spacing using energy versus atomic volume (E-V) data produced in QE. After loading the E-V curve results from DFT, we calibrate the MEAM model parameter values of equilibrium lattice constant (alat), energy per unit volume (esub), and the constant related to the bulk modulus (alpha) for LAMMPS to match the minimum BCC energy as shown in Figure 25 below.
We calculate values far away from equilibrium using the attract and repuls parameters proposed by Hughes et al.  that represent the cubic attraction and repulsion terms in the UEOS. We use single crystal elastic moduli experimental values from Yang et al.  (Table 3) to calibrate the MEAM potential elastic constants by varying the screening parameter, Cmin, scaling factor for embedding energy, asub, and exponential decay factor for atomic density, b0. These variables were shown to have the largest effect on the E-V curve calibration based on parametric sensitivity studies.
Furthermore, calibration to the GSFE curve from DFT is achieved by adjusting the t3 parameter (weighting factor for atomic density). In Figure 26 below, we compare the GSFE curves obtained from DFT and the MPC tool and in Table 4, we summarize the MEAM potential constants after calibration to the DFT results.
In this work, we present DFT and MEAM simulations for pure chromium. We started by generating energy versus lattice parameter (E-A) and energy versus volume (EV) curves and computing the bulk modulus, lattice parameter, and minimum cohesive energy. These values converged at 8 K point grid and energy cutoff of 60 Rydberg. A bulk modulus of 251 GPa, lattice parameter of 2.845 Å, and minimum cohesive energy of -4.10 eV was found for BCC chromium. DFT simulation data is in good agreement with existing literature. First principles computations were linked to atomistics through the calibration of MEAM model constants to DFT results and elastic moduli found in literature. Information obtained using atomistic methods will be passed on to dislocation dynamics simulations at the microscale to ultimately model the plastic behavior of chromium.
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