The Molecular Dynamics Simulation Method

1 ChE/MSE 557 Lecture 3 Fall 20061 Computational Nanoscience of Soft Molecular DynamicsSimulation MethodRecommended reading:Leach Chapter 6and/orFrenkel & Smit Chapter 4 Lecture 3 ChE/MSE 557 Lecture 3 Fall 20062 Computational Nanoscience of Soft Dynamics Simulation Molecular Dynamics (MD) is a Method that simulates the real Dynamics of a collection of atoms, molecules, particles, or otherextended objects. We ll use the word particle to denote atom, molecule, orcolloidal particle, as appropriate. MD is one of the most commonly used methods for materialssimulations. Same Method useful for all types of materials. Metals, ceramics, polymers, biological matter.

2 Solids (crystals and glasses), liquids, and gases In principle, best Method for investigating dynamical behaviorof materials, if system size not too large. ChE/MSE 557 Lecture 3 Fall 20063 Computational Nanoscience of Soft Dynamics SimulationIn an MD Simulation , particle positions are generated in asequence by solving Newton s equations of motion for everyparticle in the system:F = matimeMD requiresa force field ChE/MSE 557 Lecture 3 Fall 20064 Computational Nanoscience of Soft Simulation : How it works Result: a trajectory that specifies how the positions andvelocities of the atoms change with time. From this trajectory, any structural, dynamical,thermodynamic and statistical properties may ChE/MSE 557 Lecture 3 Fall 20065 Computational Nanoscience of Soft Dynamics SimulationThus MD is a deterministic Method :The state of the system at any future time can bepredicted from its current state (in principle, atleast).

3 MD simulationof a LJ liquid. ChE/MSE 557 Lecture 3 Fall 20066 Computational Nanoscience of Soft Dynamics Simulation MD simulations are similar to real experiments. In an experiment, we Prepare a sample of the material Connect the sample to a measuring instrument Measure some property of the material during some time interval If measurements are subject to statistical noise, then the longer we average, the more accurate our measurement. ChE/MSE 557 Lecture 3 Fall 20067 Computational Nanoscience of Soft Dynamics Simulation In an MD Simulation , we follow the same approach. Prepare a sample of the material Select a model system containing N particles Choose a force field for the system Solve Newton s equations of motion for every particle until the system properties no longer change with time (equilibrate the system) Calculate an instantaneous property of the system Average that property over some time interval time averaged If measurements are subject to statistical noise, then the longer we average, the more accurate our measurement.

4 ChE/MSE 557 Lecture 3 Fall 20068 Computational Nanoscience of Soft Dynamics Simulation Basic MD Simulation program: Read in run parameters ( initial energy ortemperature; N) Initialize positions and velocities Compute forces on all particles Integrate Newton s equations (F = ma) Update particle positions and velocities Calculate instantaneous properties Stop after iterating tmax stepsrepeat ChE/MSE 557 Lecture 3 Fall 20069 Computational Nanoscience of Soft Four basic partsInitializationForce CalculationIntegrationUpdating of xi,viRecording of data Calculation of properties Setting up initial conditions. Boundary conditions. Dealing with different kinds of forces.

5 Tricks tracking particles. Different integration schemes. Different thermodynamic ensembles. What to calculate? Averaging data ChE/MSE 557 Lecture 3 Fall 200610 Computational Nanoscience of Soft Starting up In starting up an MD Simulation , each particle isinitially assigned a position and a velocity. The positions may be chosen arbitrarily, as long as theparticles don t overlap . Easiest: periodic array. ChE/MSE 557 Lecture 3 Fall 200611 Computational Nanoscience of Soft Simulation : How it worksNewton s laws of motion :1. A body continues to move in a straight line unlessthere s a force acting on Force equals the time rate of change of To every action there is an equal and opposite reaction.

6 ! m r i= p i=Fi="#Ui! Fij="Fji! r"x x +y y +z z ChE/MSE 557 Lecture 3 Fall 200612 Computational Nanoscience of Soft Simulation : How it works The force on a particle changes whenever the position ofthe particle or surrounding particles with which it interactschanges. Most force fields involve continuous potentials, and so thischange is a continuous one. Under those circumstances, Newton s equations of motionmay be integrated using a finite difference ChE/MSE 557 Lecture 3 Fall 200613 Computational Nanoscience of Soft Simulation : Finite Difference methods Essential idea: the integration is broken down into manysmall steps, each separated in time by a fixed time t.

7 The total force on particle i at time t, Fi(t), is calculated asthe vector sum of the individual forces Fij(t) on i due toevery particle j within its range of interaction at time t:! Fi=Fijj" ChE/MSE 557 Lecture 3 Fall 200614 Computational Nanoscience of Soft Simulation : Finite Difference methods From the total force acting on particle i, we can find itsacceleration ai(t), which -- combined with xi(t) and vi(t) --gives the new positions and velocities at time t+ t . The force is assumed to be constant during the time step(which means we must always choose our time step to besmall enough that this is true.) After all the particles positions and velocities have beenupdated to their new values at t+ t, the force on everyparticle is again evaluated to determine the subsequentparticle positions and velocities at t+2 t, and so on.

8 ChE/MSE 557 Lecture 3 Fall 200615 Computational Nanoscience of Soft Simulation : Integration Schemes There are many finite difference algorithms, and severalare commonly used in MD simulations (many are related). Each has trade-offs: accuracy stability time reversibility area preserving memory requirements complexity Many research papers just on algorithm design for solvingF = ma. ChE/MSE 557 Lecture 3 Fall 200616 Computational Nanoscience of Soft Simulation : Integration Schemes All algorithms start from assumption that the positions,velocities, and accelerations can be approximated by aTaylor series expansion:! r(t+"t)=r(t)+v(t)"t+12a(t)("t)2+16b(t)(" t)3+124c(t)("t)4+.

9 V(t+"t)=v(t)+a(t)"t+12b(t)("t)2+16c(t)(" t)3+..a(t+"t)=a(t)+b(t)"t+12c(t)("t)2+.. b(t+"t)=b(t)+c(t)"t+..! vi= r iai= v i= r ibi= a i= v i= r iwhereetc. ChE/MSE 557 Lecture 3 Fall 200617 Computational Nanoscience of Soft Simulation : Euler The Euler algorithm uses the positions, velocities andaccelerations at time t to calculate the new positions r(t+ t): Simplest integration scheme. Bad: Not very accurate, so suffers from catastrophicenergy drift. Not area preserving or time reversible. NOT RECOMMENDED; never used. Problem: uses only values at one time to estimate newvalues. All algorithms in use today interpolate .! ri(t+"t)=ri(t)+vi(t)"t+12ai(t)"t2vi(t+"t )=vi(t)+ai(t)"t!

10 Ai(t)=Fi(t)mi ChE/MSE 557 Lecture 3 Fall 200618 Computational Nanoscience of Soft Simulation : Verlet The Verlet algorithm uses the positions and accelerationsat time t, and the positions from the previous step r(t- t),to calculate the new positions r(t+ t): The velocities do not appear in the Verlet integrationscheme: velocities are not necessary for generatingparticle trajectories.! ri(t+"t)=ri(t)+vi(t)"t+12ai(t)("t)2+..ri (t#"t)=ri(t)#vi(t)"t+12ai(t)("t)2#..! ri(t+"t)=2ri(t)#ri(t#"t)+ai(t)("t)2+! ai(t)=Fi(t)mi ChE/MSE 557 Lecture 3 Fall 200619 Computational Nanoscience of Soft Simulation : Verlet To obtain the new velocities, we can calculatethem from the difference between the positionsat two different times:!