The MEAM parameter calibration tool: an explicit methodology for hierarchical bridging between ab initio and atomistic scales

The concept of integrated computational materials engineering (ICME) has garnered extensive interest for its potential to improve understanding of material structures and thereby enhance manufactured materials [1, 2]. This concept is important both as a research topic and, pragmatically, as a manufacturing approach to accelerate novel materials development and understanding. The central concept of ICME is hierarchical bridging of spacial and time scales wherein different computational paradigms are employed to understand material behavior [3]. This bridging may be split into two parts. Downscaling is the methodical movement down through computational length/time scales by first defining the required material properties at the upper scale and then identifying what lower scale properties and structures are necessary for the material to fulfill those requirements. Once this process is completed, the movement through the scales is reversed and upscaling is employed by simulating and analyzing lower scales to define their effect on properties at higher length scales. ICME is a complex method of studying materials which was envisioned some time ago, [4], but has become more and more prominent as the computational methods and understanding required to use it effectively for various materials has matured [5, 6].

This complex methodology represents the most effective method known to assess how the physics of electron interactions is applied to macroscopic structures. It requires experimental validation to ensure that the approximations at each scale do not lead to errors dominating the response. At each scale, the microstructure and material arrangement have a critical effect on the material properties. These microstructural effects must be incorporated manually at higher scales. In contrast, at lower scales, the microstructural effects would be naturally captured if the scale were not too confining for them to become noticeable. Thus, a detailed focus on microstructural effects at each scale is necessary to fully optimize material design.

Currently, there are three major areas where ICME would benefit from further improvements:

The purpose of this work is to advance this third area of development for metals between the ab initio and atomistic scales. At the ab initio scale, density functional theory (DFT) is employed, and at the atomistic scale, the modified embedded atomic method (MEAM) is employed. Both of these methods contain significant approximations but have been previously noted for their usefulness and are popular today. DFT has several notable strengths: its fundamental equations are derived from the exact quantum Hamiltonian of the system, hence it is called an ab initio, i.e., first principles, method [7, 8]. Typically, for any input system, DFT produces a relaxed structure and the groundstate energy [9]. This information is applied in various ways to produce other properties such as the stiffness tensor. DFT is less accurate or less efficient for computing properties which depend on electron excitations [10]. Also, DFT does not provide an indication of the local distribution of system energy. The most notable approximation made in DFT is in the functional form given to the exchange-correlation energy functional which describes electron-electron repulsions, the exact form of which is unknown [8]. DFT also only produces the groundstate energy and system wavefunction and neglects dynamic effects of the nuclei [11]. Thus behavior which occurs far from the material equilibrium will not be captured. Nevertheless, DFT is the most exact method known which scales by N ³ where N is the number of electrons. This scaling generally limits DFT to simulating only thousands or hundreds of atoms. Psuedopotentials are generally employed to describe the combined behavior of some of the most bound electrons and the nuclei, only the valence electrons are separately considered [12, 13]. This greatly increases the efficiency due to DFT’s N ³ scaling with relatively minor sacrifices to accuracy. Parallelization of traditional DFT structures is fairly effective but requires a high communication bandwidth since the computation is inherently nonlocal.

The MEAM is described by an atomistic potential which consists of a pairwise form and an embedding function with angular screening which improves structure accuracy [14‐16]. The MEAM formulation is empirical rather than ab initio, but calibration of its 13 free parameters for each element to match ab initio results strengthens the physical grounding while allowing the development of an efficient model. The MEAM has the disadvantage that each alloy combination must be separately calibrated as the introduction of new elements alters the electron densities, directly altering certain calibration parameters in ways that are not obvious at the atomistic scale [17]. The MEAM formalism was developed to describe metallic bonding [18]. Many aspects of covalent bonding cannot be adequately captured by MEAM, restricting it to primarily describe metal alloys (although recently some modification of the formalism to handle covalent bonds has been developed [19]). MEAM scales linearly in the number of atoms, enabling computations with greater than tens of millions of atoms on modern computational hardware. In the MEAM formulation, each atom’s potential depends only upon its nearest neighbors. Therefore, parallel computation with spacial partitioning of MEAM simulations requires relatively little information to be passed from each processor’s partition to the others. This contrasts strongly to DFT computations. Efficient spacial partitioning allows nearly linear scaling as the number of processors is increased until each spacial partition only includes a few atoms.

At the ab initio scale of DFT, which is usually described as the lowest scale useful for deriving bulk material properties, alloying is naturally captured because the electrons and nuclei are each separately tracked in the computational framework. Atomistic scales do not have this advantage because the individual electrons are replaced by an energy potential associated with each atom, and thus the potentials require substantial calibration to artificially define electron densities of alloy structures. At the atomistic scale, deformation properties such as crack initiation and growth, dislocations, grain boundary sliding, grain growth, and twinning are all naturally captured because these properties all follow naturally from the discrete nature of the ordered atomic lattice under stress. These properties all must be defined artificially at higher scales where the material is approximated as continuous. Without a good understanding of these properties, the strengths of new alloys after various processing steps cannot be clearly derived. Thus a rigorous and efficient method of developing atomistic potentials and measuring properties with them which may be used for upscaling is of paramount importance for the development of better materials via the ICME framework.

Efforts have been made recently to make atomistic potential development with MEAM more rigorous. A design framework for atomic potentials which implements sensitivity analysis to tailor potential accuracy was introduced by [20]. Other authors have focused on uncertainty analysis methods to predict and manage potential reliability [21]. On the other hand, recent efforts have focused on making upscaling from atomistic potentials to course grain or continuum models more rigorous using similar uncertainty quantification [22‐25]. The bridging process thus is usually complex and difficult to precisely quantify, but current progress suggests that it is possible to reduce user mediation in the bridging process to merely the specification of desired primary applications.

As a means to streamline this process, we have developed a novel heuristic software package, the MEAM Parameter Calibration tool (MPC tool), for development and calibration of MEAM alloy potentials from target data obtained from a database. The database is filled from experimental measurements and DFT simulations. This software package semi-automates the fitting process for MEAM alloys. It allows a rigorous algorithm to be used to link target data and calibration parameters but also enables the user to specify fitting methods, manually define parameters, or emphasize accuracy of certain properties when fitting for a particular application. Previously, we introduced this software methodology in a more primitive form for Al as a test material [26] and performed a sensitivity and uncertainty analysis on our calibration approach [27]. Now, this software tool has been redesigned in a generalized form to work appropriately for all elements which the MEAM is applicable, regardless of their cohesive structure and to calibrate alloy parameters using binary and ternary data for all element combinations.

Many different target properties have been used in the past for the fitting of MEAM alloys to experimental and first principles data. The targets discussed in detail below are based primarily on the objective of will fitting plasticity properties of the material. Some of them, such as the energy-volume curves for various lattices and the elastic constants, are closely ingrained in the MEAM formalism and should always be included as targets. Others, such as the generalized stacking fault energies, ensure reasonable results for specific aspects of plasticity. The MPC tool is designed with a modular approach so that various targets may be used and others ignored. The modular approach also ensures that new target data may be added later for calibration or validation with minimal effort.

Calibration of MEAM parameters for metal alloy plasticity was based first on the following properties of single elements determined by DFT or experiments:

Calibration targets were computed from the raw simulation data as follows. The energy in the energy-volume curve was divided by the number of atoms to get the energy-per-atom vs. volume. Then the cohesive energy and relaxed volume were obtained by interpolation using a Murnighan equation of state [28]. The lattice parameter and atomic radius were derived from the relaxed volume using the basic lattice-dependent formulas which relate lattice parameter, atomic radius, and volume.

The elastic constants were obtained from the following formula [29]:

Here Einstein notation has been used, e and C are the strain tensor and the stiffness tensor in Voigt notation. E ₀ is the cohesive energy and E(e _i) is the energy per atom for a given strain. V is the volume per atom, and P(V) is the pressure, which can be expressed as a trace of the stress and thus as a contraction of the stiffness tensor and the strain. As the strain components approach zero, the higher order terms (\(O\left ({e_{i}^{3}}\right)\)) become negligible, and thus the stiffnesses may be computed from the cell energy for a certain strain. The energies with small positive and negative strain values were used along with the equilibrium energy to correctly produce the curvature proportional to each elastic constant. The equilibrium volume was obtained from the Murnighan equation of state, but the energies for both equilibrium and strained configurations were calculated directly rather than obtained from the equation of state. The strain values are adjustable to fit the needs of various materials.

The vacancy formation energy is given by:

where E _t is the total simulation energy, N is the number of atoms, and E _c is the cohesive energy per atom. Since periodic conditions are used, a significant number of unit cells, and hence atoms, must be replicated in the simulation to avoid interactions between the vacancy and its periodic images. The exact size of our simulations varies with the lattice type, but typically over 100 atoms are simulated for vacancy energy.

Interstitial energies are computed using the same formula as the vacancy energy. Again, a large cell must be used and interstitial energies differ depending upon the substitution site for the extra atom.

The surface formation energy is given by

where E _s is the surface formation energy and A is the surface area. The factor of two appears because of the periodic conditions enforcing that two free surfaces are simulated. The box dimension normal to the surface is twice the thickness of the material so that the spacing of the surfaces through the vacuum and through the material is identical. A material thickness of at least 10 Angstroms was employed to minimize interactions between the two surfaces.

The GSFE is given by:

The shift is a displacement along the slip direction of all the material above the slip plane which slices the simulation box in half. The GSFE simulation contains free surfaces on the top and bottom of the box. The energy of these surfaces is subtracted from E _GSFE by the E(0)_t term which also subtracts the cohesive energy for all atoms. This setup is used rather than having stacking faults on the top/bottom of the cell because not all GSFE curves are symmetric with respect to the sign of the shift, leading to incorrect stacking fault averaging in non-symmetric cases with a setup including two stacking faults.

The DFT target values for these properties were computed using VASP [13]. VASP uses periodic boundary conditions. In all simulations, ion positions were relaxed, but box shape and size was unaltered. (Energy-volume curves were generated by running a series of simulations in which the volume was held constant, but varying the volume from one simulation to the next.) The relaxed atomic radius from the energy-volume curve was used for the ion positions in all other calculations. For lower symmetry structures such as the hexagonal close-packed structure, there are two independent lattice parameters. Ideal values were obtained for each of these, and then the energy volume curve was obtained keeping the c/a ratio constant. Because of the periodic boundary conditions, surfaces and generalized stacking faults were produced in sets of two, one in the center of the simulation cell and a second one on the top/bottom face. LAMMPS was used for atomistic scale simulations [30]. At the atomistic scale, identical simulations were used to those employed for calibration targets with DFT. LAMMPS was compiled as a shared library allowing it to be directly integrated into our calibration algorithm.

Alloy properties must also be fit in order to develop a meaningful atomistic potential. The following properties of alloys are used for multi-element calibration:

Heat of formation is computed per mole as follows:

Here i represents atoms and energies of one element, and j represents those of another. l represents the binary lattice in question. It should be noted that for lattices which are not composed of equal numbers of each type of atom, H _l is not symmetric with respect to switching i and j. The heat of formation is computed for a fully relaxed lattice found using energy volume curves, as described for single element lattices. E _c(i) is the cohesive energy per atom of the ith element.

Binary elastic constants are computed in the same manner as those for single elements.

The foreign element substitution and foreign interstitial energies are computed using the following equation:

where E _c(i) is the cohesive energy of the matrix and E _c(j) is the cohesive energy of the foreign element.

Binding energies are computed using a similar formula:

The MPC tool has been tested with various single elements and alloying combinations, but most of the new potentials developed using the tool have not been validated enough for publication. Validation external to the MPC tool will always be necessary as MEAM is an empirical formulation and may lead to unexpected errors if adequate validation is skipped. However, some validation of the tool within this publication is necessary and so we here illustrate its use to develop a new Mg potential with plasticity property predictions which are superior to previous potentials. These results should be considered preliminary as the predicted potential has not been subjected to extensive finite temperature testing to ensure its accuracy.