Computational materials science

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Computational materials science and engineering uses modeling, simulation, theory, and informatics to understand materials. The main goals include discovering new materials, determining material behavior and mechanisms, explaining experiments, and exploring materials theories. It is analogous to computational chemistry and computational biology as an increasingly important subfield of materials science.

Contents

Introduction

Just as materials science spans all length scales, from electrons to components, so do its computational sub-disciplines. While many methods and variations have been and continue to be developed, seven main simulation techniques, or motifs, have emerged. [1]

These computer simulation methods use underlying models and approximations to understand material behavior in more complex scenarios than pure theory generally allows and with more detail and precision than is often possible from experiments. Each method can be used independently to predict materials properties and mechanisms, to feed information to other simulation methods run separately or concurrently, or to directly compare or contrast with experimental results. [2]

One notable sub-field of computational materials science is integrated computational materials engineering (ICME), which seeks to use computational results and methods in conjunction with experiments, with a focus on industrial and commercial application. [3] Major current themes in the field include uncertainty quantification and propagation throughout simulations for eventual decision making, data infrastructure for sharing simulation inputs and results, [4] high-throughput materials design and discovery, [5] and new approaches given significant increases in computing power and the continuing history of supercomputing.

Materials simulation methods

Electronic structure

Electronic structure methods solve the Schrödinger equation to calculate the energy of a system of electrons and atoms, the fundamental units of condensed matter. Many variations of electronic structure methods exist of varying computational complexity, with a range of trade-offs between speed and accuracy.

Density functional theory

Due to its balance of computational cost and predictive capability density functional theory (DFT) has the most significant use in materials science. DFT most often refers to the calculation of the lowest energy state of the system; however, molecular dynamics (atomic motion through time) can be run with DFT computing forces between atoms.

While DFT and many other electronic structures methods are described as ab initio, there are still approximations and inputs. Within DFT there are increasingly complex, accurate, and slow approximations underlying the simulation because the exact exchange-correlation functional is not known. The simplest model is the Local-density approximation (LDA), becoming more complex with the generalized-gradient approximation (GGA) and beyond. An additional common approximation is to use a pseudopotential in place of core electrons, significantly speeding up simulations.

Atomistic methods

This section discusses the two major atomic simulation methods in materials science. Other particle-based methods include material point method and particle-in-cell, most often used for solid mechanics and plasma physics, respectively.

Molecular dynamics

The term Molecular dynamics (MD) is the historical name used to classify simulations of classical atomic motion through time. Typically, interactions between atoms are defined and fit to both experimental and electronic structure data with a wide variety of models, called interatomic potentials. With the interactions prescribed (forces), Newtonian motion is numerically integrated. The forces for MD can also be calculated using electronic structure methods based on either the Born-Oppenheimer Approximation or Car-Parrinello approaches.

The simplest models include only van der Waals type attractions and steep repulsion to keep atoms apart, the nature of these models are derived from dispersion forces. Increasingly more complex models include effects due to coulomb interactions (e.g. ionic charges in ceramics), covalent bonds and angles (e.g. polymers), and electronic charge density (e.g. metals). Some models use fixed bonds, defined at the start of the simulation, while others have dynamic bonding. More recent efforts strive for robust, transferable models with generic functional forms: spherical harmonics, Gaussian kernels, and neural networks. In addition, MD can be used to simulate groupings of atoms within generic particles, called coarse-grained modeling, e.g. creating one particle per monomer within a polymer.

Kinetic Monte Carlo

Monte Carlo in the context of materials science most often refers to atomistic simulations relying on rates. In kinetic Monte Carlo (kMC) rates for all possible changes within the system are defined and probabilistically evaluated. Because there is no restriction of directly integrating motion (as in molecular dynamics), kMC methods are able to simulate significantly different problems with much longer timescales.

Mesoscale methods

The methods listed here are among the most common and the most directly tied to materials science specifically, where atomistic and electronic structure calculations are also widely used in computational chemistry and computational biology and continuum level simulations are common in a wide array of computational science application domains.

Other methods within materials science include cellular automata for solidification and grain growth, Potts model approaches for grain evolution and other Monte Carlo techniques, as well as direct simulation of grain structures analogous to dislocation dynamics.

Dislocation dynamics

Plastic deformation in metals is dominated by the movement of dislocations, which are crystalline defects in materials with line type character. Rather than simulating the movement of tens of billions of atoms to model plastic deformation, which would be prohibitively computationally expensive, discrete dislocation dynamics (DDD) simulates the movement of dislocation lines. [6] [7] The overall goal of dislocation dynamics is to determine the movement of a set of dislocations given their initial positions, and external load and interacting microstructure. From this, macroscale deformation behavior can be extracted from the movement of individual dislocations by theories of plasticity.

A typical DDD simulation goes as follows. [6] [8] A dislocation line can be modelled as a set of nodes connected by segments. This is similar to a mesh used in finite element modelling. Then, the forces on each of the nodes of the dislocation are calculated. These forces include any externally applied forces, forces due to the dislocation interacting with itself or other dislocations, forces from obstacles such as solutes or precipitates, and the drag force on the dislocation due to its motion, which is proportional to its velocity. The general method behind a DDD simulation is to calculate the forces on a dislocation at each of its nodes, from which the velocity of the dislocation at its nodes can be extracted. Then, the dislocation is moved forward according to this velocity and a given timestep. This procedure is then repeated. Over time, the dislocation may encounter enough obstacles such that it can no longer move and its velocity is near zero, at which point the simulation can be stopped and a new experiment can be conducted with this new dislocation arrangement.

Both small-scale and large-scale dislocation simulations exist. For example, 2D dislocation models have been used to model the glide of a dislocation through a single plane as it interacts with various obstacles, such as precipitates. This further captures phenomena such as shearing and bowing of precipitates. [8] [9] The drawback to 2D DDD simulations is that phenomena involving movement out of a glide plane cannot be captured, such as cross slip and climb, although they are easier to run computationally. [6] Small 3D DDD simulations have been used to simulate phenomena such as dislocation multiplication at Frank-Read sources, and larger simulations can capture work hardening in a metal with many dislocations, which interact with each other and can multiply. A number of 3D DDD codes exist, such as ParaDiS, microMegas, and MDDP, among others. [6] There are other methods for simulating dislocation motion, from full molecular dynamics simulations, continuum dislocation dynamics, and phase field models.

[6] [7] [8] [9]

Phase field

Phase field methods are focused on phenomena dependent on interfaces and interfacial motion. Both the free energy function and the kinetics (mobilities) are defined in order to propagate the interfaces within the system through time.

Crystal plasticity

Crystal plasticity simulates the effects of atomic-based, dislocation motion without directly resolving either. Instead, the crystal orientations are updated through time with elasticity theory, plasticity through yield surfaces, and hardening laws. In this way, the stress-strain behavior of a material can be determined.

Continuum simulation

Finite element method

Finite element methods divide systems in space and solve the relevant physical equations throughout that decomposition. This ranges from thermal, mechanical, electromagnetic, to other physical phenomena. It is important to note from a materials science perspective that continuum methods generally ignore material heterogeneity and assume local materials properties to be identical throughout the system.

Machine Learning Molecular Dynamics [10] [11]

Machine Learning Molecular Dynamics (MLMD) is an emerging field that combines traditional molecular dynamics simulations with machine learning techniques to enhance the accuracy, efficiency, and scope of computational chemistry and materials science research. MLMD leverages various machine learning algorithms, such as neural networks, Gaussian processes, and kernel methods, to learn complex potential energy surfaces and interatomic forces from quantum mechanical calculations or experimental data. This approach allows for more accurate modeling of molecular systems while maintaining computational efficiency comparable to classical force fields. MLMD has enabled simulations of larger systems for longer timescales, improved sampling of rare events, and facilitated the discovery of new materials and drug candidates. The conventional atomic simulation based on density functional theory(DFT) is well known for predicting physical and chemical properties with few hundred atoms. Nevertheless, DLMD offers the potential for DFT-level accuracy with greatly improved computational efficiency, allowing for larger systems and longer timescales. [10] However, DFT remains crucial for providing fundamental physical insights and for generating training data for DLMD models. Key applications include the development of accurate force fields for complex molecules, prediction of chemical reactions, and exploration of protein folding dynamics. As the field advances, MLMD is poised to revolutionize our understanding of molecular processes and accelerate materials design and drug discovery.

Materials modeling methods

All of the simulation methods described above contain models of materials behavior. The exchange-correlation functional for density functional theory, interatomic potential for molecular dynamics, and free energy functional for phase field simulations are examples. The degree to which each simulation method is sensitive to changes in the underlying model can be drastically different. Models themselves are often directly useful for materials science and engineering, not only to run a given simulation.

CALPHAD

Phase diagrams are integral to materials science and the development computational phase diagrams stands as one of the most important and successful examples of ICME. The Calculation of PHase Diagram (CALPHAD) method does not generally speaking constitute a simulation, but the models and optimizations instead result in phase diagrams to predict phase stability, extremely useful in materials design and materials process optimization.

Comparison of methods

For each material simulation method, there is a fundamental unit, characteristic length and time scale, and associated model(s). [1]

MethodFundamental unit(s)Length scaleTime scaleMain model(s)
Quantum Chemistry Electron, atompmps Many-body wavefunction methods, Basis set
Density functional theory Electron, atompmps Exchange-correlation functional, Basis set
Molecular dynamics Atom, Moleculenmps - ns Interatomic potential
Kinetic Monte Carlo Atom, Molecule, Clusternm - μmps - μs Interatomic potential, Rate Coefficients
Dislocation dynamicsDislocationμmns - μs Peach-Koehler Force, Slip System Interactions
Phase field Grain, Interfaceμm - mmns - μs Free energy functional
Crystal plasticity Crystal orientationμm - mmμs - msHardening function and yield surface
Finite element Volume elementmm - mms - s beam equation, heat equation, etc.

Multi-scale simulation

Many of the methods described can be combined, either running simultaneously or separately, feeding information between length scales or accuracy levels.

Concurrent multi-scale

Concurrent simulations in this context means methods used directly together, within the same code, with the same time step, and with direct mapping between the respective fundamental units.

One type of concurrent multiscale simulation is quantum mechanics/molecular mechanics (QM/MM). This involves running a small portion (often a molecule or protein of interest) with a more accurate electronic structure calculation and surrounding it with a larger region of fast running, less accurate classical molecular dynamics. Many other methods exist, such as atomistic-continuum simulations, similar to QM/MM except using molecular dynamics and the finite element method as the fine (high-fidelity) and coarse (low-fidelity), respectively. [2]

Hierarchical multi-scale

Hierarchical simulation refers to those which directly exchange information between methods, but are run in separate codes, with differences in length and/or time scales handled through statistical or interpolative techniques.

A common method of accounting for crystal orientation effects together with geometry embeds crystal plasticity within finite element simulations. [2]

Model development

Building a materials model at one scale often requires information from another, lower scale. Some examples are included here.

The most common scenario for classical molecular dynamics simulations is to develop the interatomic model directly using density functional theory, most often electronic structure calculations. Classical MD can therefore be considered a hierarchical multi-scale technique, as well as a coarse-grained method (ignoring electrons). Similarly, coarse grained molecular dynamics are reduced or simplified particle simulations directly trained from all-atom MD simulations. These particles can represent anything from carbon-hydrogen pseudo-atoms, entire polymer monomers, to powder particles.

Density functional theory is also often used to train and develop CALPHAD-based phase diagrams.

Software and tools

MOOSE / BISON simulation: A piece of a fuel pellet has chipped away (center left) due to a manufacturing defect or damage incurred while it was in transit. The damaged pellet surface induces a high-stress state in the adjacent cladding. As a result, the pellets warm up and densify before swelling back out due to fission products building up inside of them, further stressing the surrounding fuel cladding. Multiphysics Object-Oriented Simulation Environment (MOOSE).jpg
MOOSE / BISON simulation: A piece of a fuel pellet has chipped away (center left) due to a manufacturing defect or damage incurred while it was in transit. The damaged pellet surface induces a high-stress state in the adjacent cladding. As a result, the pellets warm up and densify before swelling back out due to fission products building up inside of them, further stressing the surrounding fuel cladding.

Each modeling and simulation method has a combination of commercial, open-source, and lab-based codes. Open source software is becoming increasingly common, as are community codes which combine development efforts together. Examples include Quantum ESPRESSO (DFT), LAMMPS (MD), ParaDIS (DD), FiPy (phase field), and MOOSE (Continuum). In addition, open software from other communities is often useful for materials science, e.g. GROMACS developed within computational biology.

Conferences

All major materials science conferences include computational research. Focusing entirely on computational efforts, the TMS ICME World Congress meets biannually. The Gordon Research Conference on Computational Materials Science and Engineering began in 2020. Many other method specific smaller conferences are also regularly organized.

Journals

Many materials science journals, as well as those from related disciplines welcome computational materials research. Those dedicated to the field include Computational Materials Science, Modelling and Simulation in Materials Science and Engineering, and npj Computational Materials.

Computational materials science is one sub-discipline of both computational science and computational engineering, containing significant overlap with computational chemistry and computational physics. In addition, many atomistic methods are common between computational chemistry, computational biology, and CMSE; similarly, many continuum methods overlap with many other fields of computational engineering.

See also

Related Research Articles

<span class="mw-page-title-main">Computational chemistry</span> Branch of chemistry

Computational chemistry is a branch of chemistry that uses computer simulations to assist in solving chemical problems. It uses methods of theoretical chemistry incorporated into computer programs to calculate the structures and properties of molecules, groups of molecules, and solids. The importance of this subject stems from the fact that, with the exception of some relatively recent findings related to the hydrogen molecular ion, achieving an accurate quantum mechanical depiction of chemical systems analytically, or in a closed form, is not feasible. The complexity inherent in the many-body problem exacerbates the challenge of providing detailed descriptions of quantum mechanical systems. While computational results normally complement information obtained by chemical experiments, it can occasionally predict unobserved chemical phenomena.

Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of molecules, materials, and solutions at the atomic level. These calculations include systematically applied approximations intended to make calculations computationally feasible while still capturing as much information about important contributions to the computed wave functions as well as to observable properties such as structures, spectra, and thermodynamic properties. Quantum chemistry is also concerned with the computation of quantum effects on molecular dynamics and chemical kinetics.

A discrete element method (DEM), also called a distinct element method, is any of a family of numerical methods for computing the motion and effect of a large number of small particles. Though DEM is very closely related to molecular dynamics, the method is generally distinguished by its inclusion of rotational degrees-of-freedom as well as stateful contact, particle deformation and often complicated geometries. With advances in computing power and numerical algorithms for nearest neighbor sorting, it has become possible to numerically simulate millions of particles on a single processor. Today DEM is becoming widely accepted as an effective method of addressing engineering problems in granular and discontinuous materials, especially in granular flows, powder mechanics, ice and rock mechanics. DEM has been extended into the Extended Discrete Element Method taking heat transfer, chemical reaction and coupling to CFD and FEM into account.

<span class="mw-page-title-main">Computational physics</span> Numerical simulations of physical problems via computers

Computational physics is the study and implementation of numerical analysis to solve problems in physics. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science. It is sometimes regarded as a subdiscipline of theoretical physics, but others consider it an intermediate branch between theoretical and experimental physics — an area of study which supplements both theory and experiment.

<span class="mw-page-title-main">Molecular dynamics</span> Computer simulations to discover and understand chemical properties

Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the system. In the most common version, the trajectories of atoms and molecules are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are often calculated using interatomic potentials or molecular mechanical force fields. The method is applied mostly in chemical physics, materials science, and biophysics.

Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals - that is, functions that accept a function as input and output a single real number as an output. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.

The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. Microscopic here implies that quantum mechanics has to be used to provide an accurate description of the system. Many can be anywhere from three to infinity, although three- and four-body systems can be treated by specific means and are thus sometimes separately classified as few-body systems.

<span class="mw-page-title-main">Molecular mechanics</span> Use of classical mechanics to model molecular systems

Molecular mechanics uses classical mechanics to model molecular systems. The Born–Oppenheimer approximation is assumed valid and the potential energy of all systems is calculated as a function of the nuclear coordinates using force fields. Molecular mechanics can be used to study molecule systems ranging in size and complexity from small to large biological systems or material assemblies with many thousands to millions of atoms.

<span class="mw-page-title-main">SIESTA (computer program)</span>

SIESTA is an original method and its computer program implementation, to efficiently perform electronic structure calculations and ab initio molecular dynamics simulations of molecules and solids. SIESTA uses strictly localized basis sets and the implementation of linear-scaling algorithms. Accuracy and speed can be set in a wide range, from quick exploratory calculations to highly accurate simulations matching the quality of other approaches, such as the plane-wave and all-electron methods.

<span class="mw-page-title-main">Force field (chemistry)</span> Concept on molecular modeling

In the context of chemistry, molecular physics, physical chemistry, and molecular modelling, a force field is a computational model that is used to describe the forces between atoms within molecules or between molecules as well as in crystals. Force fields are a variety of interatomic potentials. More precisely, the force field refers to the functional form and parameter sets used to calculate the potential energy of a system on the atomistic level. Force fields are usually used in molecular dynamics or Monte Carlo simulations. The parameters for a chosen energy function may be derived from classical laboratory experiment data, calculations in quantum mechanics, or both. Force fields utilize the same concept as force fields in classical physics, with the main difference being that the force field parameters in chemistry describe the energy landscape on the atomistic level. From a force field, the acting forces on every particle are derived as a gradient of the potential energy with respect to the particle coordinates.

<span class="mw-page-title-main">Multiscale modeling</span> Mathematical field

Multiscale modeling or multiscale mathematics is the field of solving problems that have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins, nucleic acids as well as various physical and chemical phenomena.

Car–Parrinello molecular dynamics or CPMD refers to either a method used in molecular dynamics or the computational chemistry software package used to implement this method.

Integrated Computational Materials Engineering (ICME) is an approach to design products, the materials that comprise them, and their associated materials processing methods by linking materials models at multiple length scales. Key words are "Integrated", involving integrating models at multiple length scales, and "Engineering", signifying industrial utility. The focus is on the materials, i.e. understanding how processes produce material structures, how those structures give rise to material properties, and how to select materials for a given application. The key links are process-structures-properties-performance. The National Academies report describes the need for using multiscale materials modeling to capture the process-structures-properties-performance of a material.

The material point method (MPM) is a numerical technique used to simulate the behavior of solids, liquids, gases, and any other continuum material. Especially, it is a robust spatial discretization method for simulating multi-phase (solid-fluid-gas) interactions. In the MPM, a continuum body is described by a number of small Lagrangian elements referred to as 'material points'. These material points are surrounded by a background mesh/grid that is used to calculate terms such as the deformation gradient. Unlike other mesh-based methods like the finite element method, finite volume method or finite difference method, the MPM is not a mesh based method and is instead categorized as a meshless/meshfree or continuum-based particle method, examples of which are smoothed particle hydrodynamics and peridynamics. Despite the presence of a background mesh, the MPM does not encounter the drawbacks of mesh-based methods which makes it a promising and powerful tool in computational mechanics.

Methods have been devised to modify the yield strength, ductility, and toughness of both crystalline and amorphous materials. These strengthening mechanisms give engineers the ability to tailor the mechanical properties of materials to suit a variety of different applications. For example, the favorable properties of steel result from interstitial incorporation of carbon into the iron lattice. Brass, a binary alloy of copper and zinc, has superior mechanical properties compared to its constituent metals due to solution strengthening. Work hardening has also been used for centuries by blacksmiths to introduce dislocations into materials, increasing their yield strengths.

<span class="mw-page-title-main">CP2K</span>

CP2K is a freely available (GPL) quantum chemistry and solid state physics program package, written in Fortran 2008, to perform atomistic simulations of solid state, liquid, molecular, periodic, material, crystal, and biological systems. It provides a general framework for different methods: density functional theory (DFT) using a mixed Gaussian and plane waves approach (GPW) via LDA, GGA, MP2, or RPA levels of theory, classical pair and many-body potentials, semi-empirical and tight-binding Hamiltonians, as well as Quantum Mechanics/Molecular Mechanics (QM/MM) hybrid schemes relying on the Gaussian Expansion of the Electrostatic Potential (GEEP). The Gaussian and Augmented Plane Waves method (GAPW) as an extension of the GPW method allows for all-electron calculations. CP2K can do simulations of molecular dynamics, metadynamics, Monte Carlo, Ehrenfest dynamics, vibrational analysis, core level spectroscopy, energy minimization, and transition state optimization using NEB or dimer method.

<span class="mw-page-title-main">Binary collision approximation</span> Heuristic used in simulations of ions passing through solids

In condensed-matter physics, the binary collision approximation (BCA) is a heuristic used to more efficiently simulate the penetration depth and defect production by energetic ions in solids. In the method, the ion is approximated to travel through a material by experiencing a sequence of independent binary collisions with sample atoms (nuclei). Between the collisions, the ion is assumed to travel in a straight path, experiencing electronic stopping power, but losing no energy in collisions with nuclei.

TeraChem is a computational chemistry software program designed for CUDA-enabled Nvidia GPUs. The initial development started at the University of Illinois at Urbana-Champaign and was subsequently commercialized. It is currently distributed by PetaChem, LLC, located in Silicon Valley. As of 2020, the software package is still under active development.

Multiscale Green's function (MSGF) is a generalized and extended version of the classical Green's function (GF) technique for solving mathematical equations. The main application of the MSGF technique is in modeling of nanomaterials. These materials are very small – of the size of few nanometers. Mathematical modeling of nanomaterials requires special techniques and is now recognized to be an independent branch of science. A mathematical model is needed to calculate the displacements of atoms in a crystal in response to an applied static or time dependent force in order to study the mechanical and physical properties of nanomaterials. One specific requirement of a model for nanomaterials is that the model needs to be multiscale and provide seamless linking of different length scales.

References

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  2. 1 2 3 Modeling Across Scales: A Roadmapping Study for Connecting Materials Models and Simulations Across Length and Time Scales (Report). The Minerals, Metals & Materials Society (TMS). 2015. Retrieved 20 August 2019.
  3. ↑ Allison, John; Backman, Dan; Christodoulou, Leo (2006-11-01). "Integrated computational materials engineering: A new paradigm for the global materials profession". JOM. 58 (11): 25–27. doi:10.1007/s11837-006-0223-5. ISSN   1543-1851.
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  5. ↑ Curtarolo, Stefano; Hart, Gus L.W.; Nardelli, Marco Buongiorno; Mingo, Natalio; Sanvito, Stefano; Levy, Ohad (2013). "The high-throughput highway to computational materials design". Nature Materials. 12 (3): 191–201. doi:10.1038/nmat3568. ISSN   1476-1122. PMID   23422720.
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  7. 1 2 D. Raabe. "Don't Trust your Simulation - Computational Materials Science on Its Way to Maturity?" (2002) Advanced Engineering Materials 4, No. 5
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  10. 1 2 Selvaraj, Selva Chandrasekaran; Koverga, Volodymyr; Ngo, Anh T. (2024-05-22). "Exploring Li-Ion Transport Properties of Li3TiCl6: A Machine Learning Molecular Dynamics Study". Journal of The Electrochemical Society. 171 (5): 050544. doi:10.1149/1945-7111/ad4ac9. ISSN   1945-7111.
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