I am the first author of the following material; however, this work would not have been possible without Dr. Christopher Barrett and Dr. Doyl Dickel. This research builds off of their work here
Pure crystalline Yttrium (Y) has been a material of great interest for solid-solution hardening of Magnesium (Mg) alloys due to its enhancement of Mg’s high strength-to-weight ratio. Molecular dynamics simulations have been a cornerstone of material science research for systems like Mg-Y for characterizing the behavior of various configurations on the atomic level. However, one challenge of modern molecular dynamics is the inflexibility of conventional interatomic potential formulations such as Modified Embedded Atom Methods (MEAM) and Embedded Atom Methods (EAM) due to their limited number of characterizing parameters. The use of neural network architectures for interatomic potential fitting has recently become a computation strategy of great interest in material science due to its superior abilities to train over large, memory-intensive datasets in reasonable time and to tailor the number of characterizing parameters to the needs of the dataset. In this study, an interatomic potential of pure Y was trained using a machine-learned rapid artificial neural network (RANN). The performance of the RANN model was characterized by testing the machine-learned interatomic potential (MLIP) with the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). The model was validated based on two metrics of the potential – energy-volume relation, and elastic constants. For the energy-volume relation, the model achieved accuracies on the order of meV/atom compared to Density Functional Theory (DFT) energies. However, the predicted elastic constants showed significant deviations of up to 50% difference from both DFT and MEAM-based calculations, suggesting the need for further refinement in training datasets and metaparameters.
Pure-Crystalline Yttrium, or more commonly referred to as ”pure Y” or ”Y”, has had a growing body of research literature due to its novelty enhancing strength-to-weight properties of popular Mg-Al ternaries and binaries[]. Namely, MG-3 wt%Y has been proven to have improved strength and corrosion properties with the solid-solution hardening of pure Y, making MG-Y systems of varying Y weight percentages a material of interest for applications demanding high strength, lightweight applications. More specifically, MG-Y systems have been used in artificial knee joints, aircraft components, and various high-performance automotive components. Thus, it is important that highly accurate, well-defined energy potentials of pure Y are published to support the development of more robust, complex molecular alloys for more conditional applications.
The ever-growing study of elements like pure Y on the structural level has introduced new challenges in material science. The study of larger structures has inherently required processing of larger volumes of data. On the small scale( for lattice structures less than a few hundred atoms), electronic structure calculations such as Density Functional Theory(DFT) and first principle energy have proven to produce high quality, accurate energy potentials when compared to published empirical databases. However, the performance of DFT simulations taper off as larger structures consisting of several hundred atoms are modelled.
To deal with this issue, Many-body interatomic potential calculations such as Modified Embedded Atom Methods (MEAM) and Embedded Atom Methods (EAM) formalisms have traditionally been used to scale electronic structure data to molecular dynamics and molecular statics simulations. Despite EAM and MEAM formalisms semi-empirical premise, these existing formalisms do not represent the breadth or complexity of the interatomic energy potentials. Subsequently, MEAM and EAM potentials produce inflexible inputs for molecular dynamics simulations due to a limiting number of 13 free parameters for characterization of the structure.
Given the need for a more accurate energy potential, we sought to evaluate a machine learning framework to approximate metaparameters of pure Y’s elastic constants and energy- volume relation. Machine learning is an excellent solution for big data processing, as a well-architected framework can interpolate complex data trends with thousands of parameters. Our model uses energy and force DFT databases to construct a high-resolution energy potential. Prior research literature in machine learning fitted potentials have not only demonstrated ML-enabled potentials to fit normal material behavior metrics (i.e. elastic constants, energy-volume curves), but also shed light onto more subtle behaviors such as temperature at phase transformation or atomic structure during dislocation compared to MEAM or EAM potentials. In this paper, we test and validate the fidelity of a pre-trained machine learning model for fitting DFT datasets for MD simulation in Large-scale Atomic/Molecular Massively Parallel Simulator(LAMMPS).
Machine learning techniques have propelled many fields to extract information from large volumes of data. Of these techniques, feed-forward multi-layer perceptron (MLP) neural networks have demonstrated impressive benchmarks of computing speed and accuracy. MLP architectures are biologically inspired frameworks. The perceptrons of the model are mathematical functions that simulate neurons, the most fundamental cell responsible for “thinking” in the human brain.
Previous approaches using neural networks for potentials, such as the widely used Behler-Parrinello atom-centered symmetry function, have been demonstrated as an effective fingerprint for an artificial neural network (ANN) architecture by several authors. Artrith and Urban approximated the potential of titanium oxide using the Behler-Parrinello basis function for an ANN consisting of 2 hidden layers and 10 neurons each []. Additionally, the Behler-Parrinello basis functions were utilized in Singraber, Behler, and Dellago authored n2p2 neural network package which has been used to publish various potentials of Al-Cu and Mg.
In the instance of pure Y, careful design must be considered to make the structural fingerprint concise. Our model architecture uses an MLP rapid artificial neural network (RANN) with embedded structural fingerprints based on the MEAM-formalism in the input layer. The MEAM formalism is the best current method of characterizing the atomic environment as it makes use of the angular screening between neighboring atoms. Therefore, we evaluated a model that exploits the underlying structure of data using an amenable number of parameters for faster computation speeds.
The role of the neural network is to predict the energy of each atom given its respective local environment; therefore, two MEAM-based structural fingerprints are used. The foundation of the fingerprints consists of the vectors of radii between neighboring atoms in a 3-D space. The function of the pair interaction fingerprint, (F_{n}), is described as:
Where (\alpha_{n}) is based on the bulk modulus defined in MEAM. The neighbor cutoff distance, (r_{c}), and equilibrium nearest neighbor distance, (r_{e}), are displacements affecting the energy of the atom (i) and atom (j). The 3-body fingerprint, (G_{m,k}), is described below by:
Where (\theta_{i,j,k}) describes the interatomic angle between atom (i), atom (j), and atom (k). The pair interaction and 3-body fingerprints are collected as a vector that will serve as the input layer of the first neuron.
In this section, we present numerical results of the elastic constants obtained using the RANN architecture. Due to the hexagonal lattice structure of pure Y, the selected tensors, (C_{11}), (C_{12}), (C_{13}), (C_{33}), (C_{44}) were used to capture its full stress and strain behaviors. In particular, (C_{33}) is a crucial tensor for understanding the material behaviors in the hexagonal axis of an anisotropic system.
| Property | RANN (this work) | Exp. | 2NN MEAM (ref.) | MEAM (ref.) |
|---|---|---|---|---|
| (C_{11}) | 80.76 | 83.40(^{a}) | 77.74(^{b}) | 86.27(^{c}) |
| (C_{12}) | 20.52 | 29.10(^{a}) | 30.00(^{b}) | 25.73(^{c}) |
| (C_{13}) | 21.74 | 19.00(^{a}) | 28.21(^{b}) | 19.41(^{c}) |
| (C_{33}) | 80.59 | 80.10(^{a}) | 75.32(^{b}) | 69.51(^{c}) |
| (C_{44}) | 25.96 | 26.90(^{a}) | 21.71(^{b}) | 24.10(^{c}) |
References:
Dickel, D. E., Baskes, M. I., Aslam, I., & Barrett, C. D. (2018). New interatomic potential for Mg–Al–Zn alloys with specific application to dilute Mg-based alloys. Modelling and Simulation in Materials Science and Engineering, 26(4), 045010
Dickel, D., Barrett, C. D., Carino, R. L., Baskes, M. I., & Horstemeyer, M. F. (2018). Mechanical instabilities in the modeling of phase transitions of titanium. Modelling and Simulation in Materials Science and Engineering, 26(6), 065002
Dickel, D., Nitol, M., & Barrett, C. D. (2021). LAMMPS implementation of rapid artificial neural network derived interatomic potentials. Computational Materials Science, 196, 110481