Introduction to atomic scale simulations
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1 Powder Technology course Autumn semester 2017 Introduction to atomic scale simulations Peter M Derlet Condensed Matter Theory Paul Scherrer Institut peter.derlet@psi.ch
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3 Outline of lecture The big picture of materials modelling Classical Molecular Dynamics basic definitions Atomic interaction The Molecular Dynamics algorithm Boundary Conditions An MD example equipartition theorem Controlling temperature and load An MD example thermal expansion Concluding remarks Understanding Molecular Simulation (Second Edition): From Algorithms to Applications Daan Frenkel and Berend Smit ( Interatomic Forces in Condensed Matter Mike Finnis, Oxford series on materials modelling 3
4 The big picture of materials modelling 4
5 Many problems, parameters & possibilities Creep and stress-rupture strength Low cycle fatigue Tensile properties Thermal stability Resistance to grain growth Oxidation Nitriding Carbonization Hydrogen embrittlement Aqueous corrosion Fabrication Microstructure Welding Machining 5
6 The need for strong materials 6
7 Material properties is a multi-scale problem From atoms to the quantitative prediction of macroscopic properties. Atoms micro-structure effective material mediums 7
8 It all about atoms! Investigating the Diameter- Dependence of Elastic Moduli of Zinc Oxide Nanowires using Molecular Dynamics Simulations Grain boundary energies and cohesive strength as a function of geometry 8
9 Born-Oppenheimer limit By replacing this term with the classical expression for a particle's kinetic energy, MD treats each atom as a classical deterministic particle Atomistic simulations use an empirical based Hamiltonian R. Laughlin & D. Pines, The theory of everything PNAS 97, 28 (2000) This potential energy can sometimes be accurately calculated using Density Functional Methods, but usually suitably parameterized analytical functions are used 9
10 Classical Molecular Dynamics basic definitions 10
11 What is classical MD? Each atom: MD involves simulating the classical dynamics of N interacting atoms. is defined by it s atom type (mass, atomic number) has a coordinate (position, velocity) 6N-dimensional phase space feels a force that depends on the type and coordinates of other (nearby) atoms N-body coupled non-linear differential equation can only be solved numerically 11
12 What is classical MD? What is needed to perform an MD simulation? An efficient & accurate method to calculate the forces/energy between atoms An efficient numerical integrator to perform an MD iteration An MD code to implement the above that is able to: Extract & control equilibrium material properties (thermodynamics) Allow for observation of non-equilibrium atomic scale processes (visualization) 12
13 Atomic interaction 13
14 Three (very broad) regimes of bonding When the electrons remain localized to an atom, the interatomic interaction is well described by a pair potential. Inert gases, some ionic solids When the valence electrons are de-localized, forming an electron gas, then an additional term must be added to model the binding energy of the electron gas FCC sp-valent metals and some BCC transition metals When the electrons remain localized to the bonds between atoms, then more complex angular dependent interactions are needed Covalent materials such as Si and C. Some FCC and BCC metals 14
15 Classes of potentials There are three classes of potentials that occur frequently in the literature Pair potentials (LJ, Ionic, Morse ) Common examples (Morse potential originally used for diatomic molecules) Many-body potentials (2 nd moment, Embedded Atom Method) Explicit angular interactions (Stillinger-Weber, MEAM ) 15
16 The Lennard-Jones Potential Sir John Edward Lennard-Jones ( ) then now V LJ (r) controls size of atom controls strength of bond An empirical representation of Pauli repulsion Van der Walls (attraction) dipole-dipole interactions due to electron dispersion 16
17 Ionic interactions In strongly ionic systems things are simple - the calculation of the Madelung lattice energy Good for well localized ions in which the ionic charge can be approximated as a structureless point z z In this approximation, ions do not polarize lattice distortions polarize ions z z-y y Shell model (for ceramics) in which the heavy central (z-y) ion interacts with a massless charge (y), and: there is a long range (effective) ion-ion interaction 17 a short range shell-shell interaction
18 Many-body potentials of metals Written as with The first term represents the electro-static interaction and overlap repulsion between neighbouring ions. The second term is an embedding potential representing the energy of embedding an atom in the local charge density coming from the tails of the atomic charge clouds of neighbouring atoms M.S. Daw, M.I. Baskes, Phys. Rev. B 29, 6443 (1984) 18
19 The Stillinger-Weber potential for Si F. H. Stillinger and T. A. Weber, PRB 31, 5262 (1985) 19
20 Accuracy & reliability The analytical form of the pair interactions and the embedding energies are fitted to a variety of known material parameters. It is hoped that the model will work well in environments not considered within the materials property data-base (transferability). Empirical models for inter-atomic forces constitute a drastic approximation to the quantum mechanics of material bonding systematic and transferable accuracy cannot really be expected. It is important to understand the limitations of any potential used. 20
21 Use a data-base of material properties (experimental and DFT) An example of a data base used for a well known Aluminium potential Y. Mishin, D. Farkas, M. J. Mehl and D. A. Papaconstantopoulos, PRB (1999) 21
22 Potential generation: science or art? Some properties cannot be included in direct fit. Indirect fitting is tricky. The choice of weights is more important than the optimization algorithm. The choice of weights is subjective. It is impossible to fit to all desired properties. Finding a reasonable compromise (through weights) requires human intervention (creativity, experience, luck) It is not always good to find the global minimum. Rule of thumb (based on personal experience): one publishable potential a year Potential generation is largely based on experience, intuition, tricks of the trade, and luck. It is currently art rather than science. Yuri Mishin Professor of Physics George Mason University 22
23 Three types of published potentials Quick-fix and ad hoc potentials Fitted to a few numbers, tested for a limited number of properties Specific to a particular application, not tested for others Proof-of-principle potentials Demonstrate a particular fitting method or trends across a series of materials Parametric potentials (e.g. composition-dependent) Not tested carefully for a variety of properties Universal potentials Usually fit to a large database (experimental and first-principles) Successfully tested against a large number of properties Successfully applied in many different types of simulations Identified with a particular element or binary system ( canonized ) 23
24 24
25 About KIM openkim.org The Knowledgebase of Interatomic Models (KIM) project is based on a four-year NSF cyberenabled discovery and innovation (CDI) grant and has the following main objectives: Development of an online open resource for standardized testing and long-term warehousing of interatomic models... Development of an application programming interface (API) standard for atomistic simulations... Fostering the development of a quantitative theory of transferability of interatomic models to provide guidance for selecting application-appropriate models based on rigorous criteria, and error bounds on results. 25
26 The MD algorithm 26
27 Equations of motion numerical integrators To obtain Newton's equations of motion, can use the method of Lagrangian mechanics Lagrangian's equations of motion Newton's equations of motion Such an approach works well when the forces do not vary greatly over the time step, t. 27
28 Equations of motion numerical integrators The integrators developed in the previous slide should never be used, since they are not numerically stable to the conservation of total energy. Much better integrators exist (Verlet & more general predictor/corrector algorithms) Velocity Verlet integrator Note that to determine the new velocities both the old and new atomic positions are needed See for example: Understanding Molecular Simulation, Frenkel & Smit,Chapter 4 28
29 Calculation of instantaneous energy & temperature In equilibrium, each atom has thermal energy which is distributed equally between the kinetic and thermal potential energy The temperature is calculated through the use of the equipartition theorem Average over time Instantaneous temperature 29
30 The basic MD algorithm give atoms initial positions & velocities calculate the force on each atom update positions and velocities according to an integrator calculate properties (energy, temperature, pressure, etc.) Calculate running averages update simulation time (t=t+δt) 30
31 Molecular statics structural relaxation (at 0K) By removing a atom to create a vacancy, the surrounding atoms gain a force which initially either accelerates the atoms away or towards the vacancy defect. Via molecular statics, new positions in which the forces are all zero can be found, giving the minimum energy configuration of a vacancy. Atoms coloured by local energy total kinetic energy (ev) 2.5x x x x x simulation time (psecs) total/potential energy (ev) simulation time (psecs) The MD done here, removes the kinetic energy from the system as soon as it begins to decrease, and continues to do this until the forces on the atoms are all zero 31
32 Molecular statics structural relaxation (at 0K) The new positions, can be defined as a strain field around the vacancy defect Unrelaxed atomic positions Relaxed atomic positions The above picture only shows those atoms with an energy higher than that of the perfect lattice. By removing an atom, all remaining atoms that directly interacted with the removed atom have a higher energy (and a non-zero force). By relaxing this force, new positions are found for the surrounding atoms which can be identified as the strain field of the vacancy defect 32
33 Boundary conditions 33
34 Simulating the bulk: periodic boundary conditions By introducing an array of periodic images of the atomic configuration, a bulk crystalline sample can be simulated. Thus despite the finite size of the configuration, surface effects need not be considered. 34
35 Simulating the bulk: periodic boundary conditions By introducing an array of periodic images of the atomic configuration, a bulk crystalline sample can be simulated. Thus despite the finite size of the configuration, surface effects need not be considered. (but micro-structure becomes repetitive) 35
36 Examples of 3D periodic boundary conditions Nanocrystalline Ni containing 100 grains with a mean grain size of 10 nm (4.8 million atoms) Molecular self-assembly and clustering in nucleation 36 processes: general discussion, S. Price et al, Faraday Discuss., 2015, 179,
37 Periodicity need not be in all directions Open boundary conditions The effect of a surface on the deformation properties of nc Ni Phil. Mag. A 82, 1 (2002). 2D periodicity: surface or slab geometry (here only the black arrows indicate periodic boundary conditions) Nano-indentation simulation on single crystal and nc Au PRB 67, (2003). 37
38 Periodicity need not be in all directions 1D periodicity: wire geometry (here only the black arrows indicate periodic boundary conditions) 38
39 An MD example equipartition theorem 39
40 A perfect FCC lattice at 300K Consider a face-centred-cubic (FCC) lattice of 4000 (Al) atoms. The initial positions of the atoms are at the ideal lattice sites (0K configuration). A lattice of infinite extent (the bulk limit) may be simulated by using periodic boundary conditions. [001] [010] [100] 40
41 A perfect FCC lattice at 300K Give each atom a random velocity such that the temperature is 600K. and start the MD instantaneous temperture (K) simulation time (psecs) 41
42 A perfect FCC lattice at 300K At 300K, atoms undergo vibrational motion about their lattice sites. Initial configuration Snapshot at 5psec, 300K 42
43 A perfect FCC lattice at 300K Give each atom a random velocity such that the temperature is 600K. and start the MD instantaneous temperture (K) simulation time (psecs) 43
44 A perfect FCC lattice at 300K The potential energy responds accordingly, since the total energy is conserved. total potential energy (10000eV) simulation time (psecs) 44
45 A perfect FCC lattice at 300K The potential energy responds accordingly, since the total energy is conserved. total potential energy (10000eV) total kinetic energy simulation time (psecs) 45
46 A perfect FCC lattice at 300K By giving each atom a random velocity such that the temperature of the system is initially 600K, kinetic energy has been introduced into the system. From the MD simulation, the kinetic energy reduces by half and thus the temperature decreases to 300K. By conservation of total energy (kinetic + potential), this other half has been converted to potential energy thermal potential energy. An example of the classical equipartition theorem at NVT thermal equilibrium. 46
47 Controlling temperature and load 47
48 MD Ensembles So far we have considered constant energy simulations at fixed volume, and fixed particle number the so-called NVE (microcanonical) ensemble. MD can also simulate the NVT (canonical) ensemble in which a temperature, rather than the total energy, is fixed MD can also simulate the NPT ensemble in which a stress, rather than a fixed volume, is defined. 48
49 How to control temperature? Temperature can be controlled via: Velocity rescaling The new kinetic energy will be equal to Nosé-Hoover thermostat Can be seen as a temperature dependent drag force Langevin dynamics Here there is a fixed drag (dissipative) parameter (value is optimally chosen) whose effect is balanced by the addition of a random force which is not correlated in time. The ratio of damping coefficient to the magnitude of the stochastic force defines the temperature of the equilibrium system 49
50 The stress tensor and hydrostatic pressure The instantaneous stress tensor of the simulation box can easily be calculated Hydrostatic pressure Thermal component Potential zero Kelvin 50
51 Hydrostatic pressure at 0K isotropic strain Equation of state curve Change volume isotropically (done by changing periodicity length and rescaling the lattice positions 51
52 The stress tensor and distortion of simulation cell How to generally distort the simulation volume under periodic boundary conditions? Introduce a new variable, the box matrix: are the dimensionless atomic coordinates. A strain can be applied by changing the box matrix and the response of the system can be measured by calculating the stress tensor 52
53 Parrinello-Rahman method: applying an external load The equations of motion for the atoms become: Whilst for the simulation cell Effective simulation cell mass 53
54 Parrinello-Rahman method: uniaxial deformation For a constant stress uniaxial deformation use Parrinello-Rahman with For a constant strain rate uni-axial deformation, increment and measure, and use a 2D version (in the gauge plane) of Parrinello-Rahman with: Here, Parrinello-Rahman is used to simulate Poisson contraction 54
55 An MD example thermal expansion 55
56 A perfect FCC lattice revisited instantaneous temperture (K) instantaneous stress (GPa) xx yy zz xy xz yz simulation time (psecs) simulation time (psecs) At constant volume, large internal hydrostatic stresses develop due to thermal vibration Thermal expansion parameter NVT Ensemble 56
57 A perfect FCC lattice revisited instantaneous temperture (K) cell side lengths (Angstrom) simulation time (psecs) simulation time (psecs) By setting the applied stress to zero, thermal expansion can be simulated using the Parrinello-Rahman method: Thermal expansion parameter NPT Ensemble 57
58 A perfect FCC lattice revisited Simulation cell mass increased 10-fold instantaneous temperture (K) cell side lengths (Angstrom) simulation time (psecs) simulation time (psecs) By setting the applied stress to zero, thermal expansion can be simulated using the Parrinello-Rahman method: Thermal expansion parameter NPT Ensemble 58
59 A perfect FCC lattice revisited Simulation cell mass with active damping instantaneous temperture (K) cell side lengths (Angstrom) simulation time (psecs) simulation time (psecs) By setting the applied stress to zero, thermal expansion can be simulated using the Parrinello-Rahman method: Thermal expansion parameter NPT Ensemble 59
60 Concluding remarks Inter-atomic force model Have in mind that you are using a severely uncontrolled approximation to model your material. Read all the literature associated with your potential. If you see something interesting, repeat the simulation using another potential. Molecular dynamics is a wonderful tool to study and classify atomic scale processes but it can be difficult to compare results with experiments because of a very large disparity between experimental and simulated timescales. 60
61 LAMMPS (lammps.sandia.gov/) lamp: a device that generates light, heat, or therapeutic radiation; something that illumines the mind or soul Nanoprojectile impacts - model the impact of nanoparticles up to 55 nm in diameter hitting substrate targets with up to 10 billion particles. Amorphous carbon film growth view of the surface after 36,000 C impacts Jamming and deformation of extremely polydisperse granular materials Phys Rev Lett 108, (2012). Carbon, 50, 404 (2012). Entropy, 15, 4802 (2013) There also exists the ITAP Molecular Dynamics Program (imd.itap.physik.uni-stuttgart.de/) 61
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