Computational materials science
Computational materials science and engineering uses modeling, simulation, theory, and informatics to understand materials. 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.
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.These computer simulation methods use underlying models and approximations to understand materials behavior in more complex scenarios than pure theory generally allows and with more detail and precision than is often possible with experiment. 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 experiment.
One notable sub-field of computational materials science is integrated computational materials engineering, which seeks to use computational results and methods in conjunction with experiments, with a focus on industrial and commercial application. 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, high-throughput materials design and discovery, 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 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 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, becoming more complex with the generalized-gradient approximation 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
Molecular dynamics is the name for simulating atomic motion through time. Interactions between atoms are defined and fit to both experimental and electronic structure data with a wide variety of models, called interatomic potentials. With those interactions, Newtonian motion is numerically integrated. The simplest models include only van der Waals type attractions and steep repulsion to keep atoms apart. Increasingly complex models include charges, bonds and angles, approximations of electronic effects, and more. Some models use fixed bonds, defined at the start of the simulation, while others have dynamic bonding. Most 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 rates for all possible changes within the system are defined and probabilistically evaluated. Because there is no restriction of directly integrating motion, 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
are crystalline defects in materials with line type character. Rather than simulating the full atomic detail, discrete dislocation dynamics directly simulates line objects. Through the theories and equations of plasticity, DDD moves dislocations through time and defines the rules describing how dislocations interact when they cross.There are other methods for simulating dislocation motion, from full molecular dynamics simulations, continuum dislocation dynamics, and phase field models.
Phase field
Phase field methods are focused on phenomena dependent on interfaces and interfacial motion. Both the free energy function and the kinetics 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.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 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.| Method | Fundamental unit | Length scale | Time scale | Main model |
| Density functional theory | Electron, atom | pm | ps | Exchange-correlation functional |
| Molecular dynamics | Atom | nm | ps - ns | Interatomic potential |
| kinetic Monte Carlo | Atom | nm | ps - μs | - |
| Dislocation dynamics | Dislocation | μm | ns - μs | - |
| Phase field | Interface | μm - mm | ns - μs | Free energy functional |
| Crystal plasticity | Crystal orientation | μm - mm | μs - ms | Hardening function and yield surface |
| Finite element | Volume element | mm - m | ms - s | beam equation, heat equation, etc. |
Multi-scale simulation
Many of the methods described can be combined together, 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. This involves running a small portion 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 and coarse, respectively.
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.
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. 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.