Computer Simulation of Shock Waves in Condensed Matter Matthew R. Farrow 2 November 2007
Outline of talk Shock wave theory Results Conclusion Computer simulation of shock waves
Shock Wave Theory
Shock wave theory Shock waves in gases Theory of shock waves has its roots in the field of gas-dynamics. Piston
Shock wave theory Shock waves in gases Piston moving at velocity u compresses gas. Piston
Shock wave theory Shock waves in gases A continuous solution across the shock front yields a meaningless result: Solutions to the gas-dynamic equations are trivial in which all quantities are constant Piston Discontinuity
Shock wave theory Shock waves in gases In order to satisfy the boundary conditions the variables are assumed to be discontinuous. Discontinuity regarded as limiting case of a very large gradient across a region which tends to zero. Shock waves are these discontinuities.
Shock wave theory Shock waves in solids Conservation relations are used to create a set of equations that govern shock wave dynamics - The Rankine-Hugoniot equations:
Shock wave theory The Hugoniot Hugoniot curves are relations between the thermodynamic state variables that describe the possible states of the system during shock compression. A single curve relating say, the volume and pressure, is called a Hugoniot. Hugoniot Hugoniot Pressure Isentrope Rayleigh Rayleigh line Line Us Hugoniot Isentrope V 0 Specific Volume up
Shock wave theory Hugoniots and shocks in solids For moderate temperatures and pressures, Hugoniots are well represented by: Where is usually between 1 and 2.0 for most materials However, some researchers[1] believe that a quadratic term is required: Shock waves in solids are described using the hydrodynamic approximation. If the Shock is sufficiently strong, the effects of yield strength are neglected [1] F. E. Prieto and C. Renero. Equation for shock adiabat. J. Appl. Phys.,41:3876 &, 1970.
Shock wave theory Isentrope and Rayleigh Line The isentrope is the curve of constant entropy Extremely difficult to calculate experimentally; Created from cold compression simulations where the system is maintained at zero kelvin; As 0K, and also adiabatic, the cold compression isotherm curve is also the isentrope The Rayleigh line connects the initial state with the final (shocked) state, and obeys the following relation:
Shock wave theory Equation of state The Equation of State (EOS) is the goal of shock wave simulations The Equations of State (EOS) gives the all the properties of the material in terms of Pressure, P, Volume, V and Energy, E (or Temperature, T); For example, the ideal gas EOS: PV = nrt However, the full EOS for most materials are very difficult to determine experimentally. Hugoniot is a line on the EOS surface
Shock wave theory Advantages of studying shock waves Studying shock waves gives greater understanding in many different areas: Geophysics: Planetary core modelling Earthquakes Astrophysics: Shock waves through stellar material Supernova events Industrial applications: Explosives investigations Laser cutting
Computer Simulation of Shock Waves
Computer simulation of shock waves Computer Simulation Allows for experimentally inaccessible scenarios to be explored; Computer simulation is a powerful tool if used correctly! Requires an accurate model of the system Interatomic Forces Dynamics of system Constraints of system Two approaches considered for shock waves: Atomistic molecular dynamics Ab-initio molecular dynamics
Computer simulation of shock waves Molecular dynamics Molecular dynamics (MD) is a simulation tool used to explore the atomic / molecular interactions of a system. Ideal for the dynamic processes during a shock; Solves Newton s equations of motion Timestep important consideration Macroscopic properties are calculated as space and/or time-averages Forces in MD can be generated by ab-initio or empirical potentials.
Computer simulation of shock waves Ab-initio molecular dynamics In ab-initio MD Schrodinger s equation is solved to find the ground state energy of the system - Forces are then calculated on the ground state energy. Born-Oppenheimer approximation used de-couples motion of electrons and ions; Requires a wavefunction to be stored Computationally expensive to store: 10 points in each dimension on a 3D sample grid scales as 10 3N where N is number of electrons. For example, 2 electrons storage cost is 15Mb RAM whereas 5 electrons storage is ~ 15 Tb RAM! (for complex numbers)
Computer simulation of shock waves Empirical potentials Empirical potentials are approximations to the Potential Energy Surface. Good for shock waves as: Using large systems (>1000 s atoms); Simple to code; Computationally cheap to run; Shown to be accurate (enough!) for shock wave calculations.
Computer simulation of shock waves Atomistic molecular dynamics The key to successful atomistic MD is the choice of the empirically determined potential that describes the energy surface of the system. Many different potentials available All created for a particular system Created by curve fitting to experimental data, ab-initio data, or sometimes both. Choosing the right potential is very important for meaningful results!
Computer simulation of shock waves Lennard-Jones potential(i) Simple two-body potential; Van-der-Waals attraction, with a computationally efficient (but physically meaningless) r 12 repulsive term; Easy to calculate forces; Great for materials such as argon; BUT! Argon is not a very exciting material
Computer simulation of shock waves Lennard-Jones potential(ii) Lennard-Jones Potential 1.75 0.75 U(r) 0.01 1.01 2.01 3.01-0.25-1.25 Interatomic Spacing (r) U(r) = 0, if r > rcutoff
Computer simulation of shock waves BKS potential (I) Where alpha and beta are atomic species, q, their charges and A, b, and C are constants. Named after authors - van Beest, Kramer and van Santen [1] Empirical potential parametrised by ab-initio cluster calculations Good for quartz (SiO 2) and its polymorphs (coesite, stishovite,etc..) [1] Interatomic force fields for silicas,aluminophosphates, and zeolites: Derivation based on ab-initio calculations, G.J.Kramer, N.P.Farringher,B.W.H. van Beest and R.A. van Santen, Phys.Rev B,43,6 (1991)
Computer simulation of shock waves BKS potential (II) The BKS Potential 11 O-O Si-O 9 7 Energy 5 3 1 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5-1 -3 Interatomic distance (angstroms) Care has to be taken with simulations to avoid atoms becoming closer than maximum - they would feel an attractive force!
Computer simulation of shock waves Periodic Boundary Conditions (PBC) Used to simulate large-scale bulk systems Each simulation cell is part of an infinite array of replica cells Particle numbers are conserved as a particle leaving one cell will be replaced by another entering from the other side. System becomes periodic Minimum Image Convention used with PBC: Ensures forces are evaluated between the smallest pair separation between each atom and neighbours
Computer simulation of shock waves Momentum mirror modification to PBC: Shock wave generation PBC during shock compression would result in many shocks! Results would be meaningless as no steady shock would be travelling through the system due to wrap around of atoms Momentum mirror on one of the directions is used: U Momentum Mirror t = 0
Computer simulation of shock waves Momentum mirror modification to PBC: Shock wave generation Position and velocity reversed if passing the mirror during an MD timestep Vacuum region required to eliminate repulsive force from periodic images U Vacuum Region Momentum Mirror t = t1 -U
Computer simulation of shock waves Non-equilibrium molecular dynamics (NEMD) Equilibrium MD can be used to create the isentrope, isotherm or cold compression curves seen earlier Shock waves are dynamic events and so equilibrium MD is not appropriate to capture dynamics Non-equilibrium MD (NEMD) is used instead. Uses NVE ensemble No equilibration period -all important physics in first few nanoseconds! No thermostat or barostat
* When T=0 the system cannot support a steady wave and the results are meaningless. T > 0 to allow for transverse flow as a steadying mechanism Computer simulation of shock waves Simulating a shock wave A shock wave is simulated thus: At time t = 0, all atoms in an equilibrated at finite temperature * and relaxed system are given a velocity up in the direction of the momentum mirror up Vacuum Region Momentum Mirror t = 0
Computer simulation of shock waves Simulating a shock wave A shock wave is simulated thus: When the atoms reach the mirror they reflect and setup a shock propagating away from the mirror at velocity Us Shock front up Vacuum Region Momentum Mirror t = t1 Us
Computer simulation of shock waves Simulating a shock wave A shock wave is simulated thus: Simulation is ended when the shock reaches maximum compression. Shock front Vacuum Region Momentum Mirror t = tmax Us
Results
Results Simulations with argon Lennard-Jones potential 500,1000, 1800, 2000, 3600, 4000 atom system sizes Piston velocities of 1, 2, 4, 6, 8, 16, 32, 64, and 128 kms -1 Simple system to start with - allowed for analysis tools to be written and tested 4000 atom system in equilibrium
Results Static compression (Equilibrum MD) 300 250 Static Compression Hugoniot or f Argon Lennard-Jones 12-6 Potential Static Compression Experimental data 200 Pressure (GPa) 150 100 50 0 0.4 0.5 0.6 0.7 0.8 0.9 1 Volume (V / V0)
Results Simulations with argon
Results Velocity profile along system 2000 Velocity profile along simulation cell 1km/s shock wave after 3ps, adjusted for system frame of reference 1500 Velocity (m/s) 1000 Shock front 500 0 0 50 100 150 200 250 Position (angstroms)
Results Hugoniots of argon Hugoniots of argon for different system sizes NEMD Shock wave calculations 0.8 3600 atoms 4000 atoms Experimental Data 1800 atoms Specific Volume (V/V0) 0.7 0.6 0.5 0.4 0.3 0 500 1000 1500 2000 Pressure (GPa)
Results Hugoniots of argon Hugoniots of argon for different system sizes NEMD Shock wave calculations 0.8 3600 atoms 4000 atoms Experimental data 1800 atoms Specific Volume (V/V0) 0.7 0.6 0.5 0.4 0.3 0 2 4 6 8 Log of Pressure (GPa)
Results Transients at the momentum mirror Pzz Pressure Plots for Ar450 3000 Pzz (GPa) 2000 1000 0 0 500 1000 1500 2000 Timestep
Results Transients at the momentum mirror Transients are an artifact of the momentum mirror Asymmetric, unlike a real shock that has material both sides of the impact region The region directly next to the mirror will have atoms that are trapped; Atoms want to escape but the mirror keeps them in place Heats them up unrealistically Region near mirror should be disregarded when computing averages of system properties
Results Shock waves in quartz Quartz (SiO 2) and its polymorphs are the 2nd most abundant substances in the Earth s crust High-temperature, high-pressure polymorphs: Stishovite found in meteorite craters Coesite (at low-temperatures) and Stishovite BKS potential used to model quartz and stishovite Aim to find conditions in which phase change occurs Care will have to be taken under high-compression
Results Quartz structure -quartz
Results Stishovite structure Stishovite
Results Shock waves in quartz using BKS potential Initial BKS simulations show that an SiO 2 trimer has a linear molecule for its lowest energy configuration Promising - as same structure as CO 2 Further testing still required.
Conclusions
Conclusions General conclusions Studying shock waves has many different fields in which to contribute Simulation is a great tool for predicting experimentally inaccessible states Non-equilibrium molecular dynamics is ideal for computer simulation of shock waves
Conclusions Further conclusions Argon simulated and shown that Lennard- Jones potential is good up to moderate pressures After that the unphysical r12 region is probed and simulation will deviate from true behaviour Transients at start of shock simulation overcome by calculations on steady-state region BKS potential to be used to model quartz - stishovite phase transition