Molecular Dynamics Simulations

Fes, December 2008

MD of the enzyme Catechol-O-Methyl

Transferase Catechol-O-

methyltransferase (COMT) enzyme

www.apmaths.uwo.ca/~mkarttu/gallery-movie.shtml

Trimer manipulation on Cu(111) at 100K

Standard MD SimulationYildirim et al, 2006

A nanotube with a single vacancy has been stretched at a strain using the parallel-replica method. A REBO interatomic potential has been employed and the temperature is T=300K. A.F. Voter, et al, 2002

Molecular dynamics simulation of the deposition of a single copper atom with a kinetic energy of 1 eV on a copper surface.

INTRODUCTION

MD is a computer simulation technique at which the time evolution of a set of interacting atoms is achieved by integrating their equation of motionsIt follows the Newton’s law:

iii amF

In contrast to MC method, MD is a deterministic technique: given an initial set of positions and velocities, the time evolution is in principle determined. The end trajectory will deviate from the true trajectory because of the finiteness of the time step and rounding errors.

During the simulation, atoms will do what happens in real systems, moving, jumping, oscillating so on.. A trajectory of a 6N-dimensional trajectory in phase space will be calculated. MD is a statistical method, it provides a way to obtain a set of configurations distributedaccording to some statistical distribution function, or an ensemble (nve).

The first MD paper: J. Chem. Phys. 27, 1208 (1957) to investigate the phase diagram of hard sphere system Phys. Rev. 120, 1229 (1960), Phys. Rev. 136, A405 (1964), Phys. Rev. 159, 98 (1967)

Application Areas of MD Simulations

1. Liquids2. Defects3. Fracture4. Surfaces5. Friction6. Clusters7. Bio molecules8. Dynamics

Limitations of MD Simulations

Usage of Classical Forces: Systems at the atomic level obey quantum mechanical laws rather than classical ones. How realistic is to use the Newton’s law to treat the system and move the atoms?

The validity of classical approximation is based on De Broglie wavelength:

TkM

h

B

2

2

If << a (lattice parameter), nuclei follow Newton equation of motion

How realistic are the forces: Atoms move under the action of the forces, so all the physics is introduced comes from the forces. The simulation is being realistic depend on the ability of the potential to reproduce the behavior of the material under the given conditions

Time and size limitations: A typical MD simulation can be performed from few picoseconds to hundreds nanoseconds. To consider a simulation is safe, the simulation time has to be much longer than the relaxation time of the quantities

Be careful with

•The classical approximation is poor for very light systems such as He, Ne, H2

•Quantum effects become important in any system when T is quite low

Classical molecular dynamics (MD)

1) Choose an appropriate form for potential2) Solve numerically Newton’s equations of motion

Modeling a system

•For any simulation, we need to choose an appropriate form of a potential describing the Interactions between the entities. It is a function of positions of nuclei, representing the potential energy of the system.

•The forces will be derived as gradient of the potential with respect to atomic displacement

•Till early 1980’s the most potentials were described through pair wise interactions such asLennard Jones (mostly used for rare gases where no electrons are available for bonding and atoms are attracted to each other through weak van der Waals forces).

•For systems with defects, such as surfaces, and especially for metals, these pair potentialsare proven to be inaccurate to describe the interactions for such systems for which the environment is a key factor.

•To describe the properties of such systems, many body effects has to be introduced to the description of the potentials energy function.

Construction of potentials

The goal is to select functional forms which will mimic the behavior of the potential.

1.) Selecting an analytical form for the potential. A typical analytical form consists of number of functions, depending on geometrical quantities and variablessuch as atom coordination

2.) Finding an actual parameterizations for the functions

•Some of the studies focused on a first principle description and determine an expression for the energy as a function of nuclei positions •Others tried to fit the potential on to experimental data, and some used both.

It is always necessary to keep in mind that the potentials are designed for specific applicability in mind.

A good potential for metals should bring a better description of the role played by the electrons in the bonding.

Development of many-body potentials based on density or coordination:

The physical point is: The cohesive energy should not be decreasing linearly, it should decrease faster when coordination is low, and slower as coordination increases.

the energy is proportional to the square root of the coordination not the coordination itself.

ii ZE Finnis and Sinclair (1984)

• Many semi-empirical potentials have been proposed, based on ideas of adding a many-body term to the decsription of the potential:

• Embedded Atom Method potentials Daw and Baskes (1984); modified EAM: Baskes (1992)• RGL Potentials Rosato, Guillopé, and Legrand (1989)• Glue modelErcolessi, Parrinello, and Tosatti (1988)• Effective medium potentialsJacobsen, Nørskov, and Puska (1987); Corrected effective medium method: Kress and DePristo (1987) • Finnis-Sinclair potentialsFinnis and Sinclair (1984)

Embedded Atom Method: EAM potential

In the EAM, the energy of each atom is computed from the energy needed to embedthe atom in the local-electron density as provided by the other atoms of the metal.

i

itot EE )()(2

1ii

ijijiji FRVE

ij

ijji R )(

Many-body interactions

The total electron density is approximated by the linear superposition of contributionsfrom the individual atoms. The electron density in the vicinity of each atom can be expressed as sum of the density contributed by the atom plus the density from allthe surroundings.

PRB, 29, 6443 (1984)PRB, 33, 7983 (1986)

EAM successEAM success

Excellent potentials exist for several FCC metals (e.g. Cu, Ag, Ni and Al)

Accurately reproduce E0, a0, cij, phonon dispersion curves, thermal expansion, Evf,

Evm, interstitials, structural energies, etc.

Melting properties:

Cu Tm = 1327 K (experiment 1357 K)

Ag Tm = 1265 K (experiment 1235 K) Thermal expansion:

Cu Ag

Cu Ag

Phonon dispersion curvesPhonon dispersion curves

MD Simulations

Cutoff: To save time in a computer simulation, the potential should have a cutoff. There is a cutoff radius say rc, beyond that the interactions are ignored.

Potentials for metals and semiconductors are designed with a cutoff, and it goes to zero at this cutoff along with their first two derivatives.

The number of atoms interacting will Increase with the increase of the cutoffFrom 4, 8, …

Periodic boundary conditions

How about the atoms at the boundaries?

It is always the case that how large the system we simulate, the number of atoms in a simulation is negligible compared to the number of atoms in a real piece of matter (1023)

The ratio of the surface atoms to the all atoms will be larger that causes the surface effects.The way to solve this is to introduce Periodic boundary conditions.

how to: The particles are put in a box, and the box is replicated to infinity by translation in all three Cartesian directions, filling the whole space. If one of the particle is at the position r, it represent a set of particles located at

r+la+mb+nc, l,m,n are integer numbers

So each particle will be interacting with the others in the box, also with their images in nearby

Integrating the Newton’s equations of motion

),,,( 21

..

Nrii rrrVrmi

Using time integration algorithms which are based on finite differences methods (timeis discretized on a grid), and time step is being t

The problem is to start with a known positions and derivatives at time t, using the integration scheme, we reach to the same quantities at later time t+ Δt. That way, we get the time evolution of the system under specific conditions.

Configuration at time tConfiguration at t+ Δt

Algorithm

Integration Algorithms

Verlet Algorithm: The basic idea is to write the third order Taylor expansion for position r(t), one forward and one backward in time.

)()(6

1)(

2

1)()()(

)()(6

1)(

2

1)()()(

432

432

tOttbttattvtrttr

tOttbttattvtrttr

)()()()(2)( 42 tOttattrtrttr This is the standard Verlet algorithm [Ref].

)(2

)()()( 2terror

t

ttrttrtv

The Velocity Verlet algorithm is an equivalent algorithm which handles better the velocities.

Several other algorithms can be used (predictor-corrector, Runge-Kutta, etc.)

Simulations Start the simulations by defining the MD box with the initial set of positions and velocities. Positions are most stable crystal structure at T=0K produced using the given potential and velocities initially can be taken to zero, or from Maxwell distribution.

Bringing the system in to a desired state: Bringing the system into a desired temperaturecan be achieved by rescaling the velocities. So the definition of the velocities from theVelocity Verlet algorithm will be replaced by

ttatvttv )()2/1()()2/(

)(/0 tTT ttatv )()2/1()(

At this point, the total energy is no longer conserved. This procedure is to get the system into a desired stage. One should wait till the system reaches to the equilibrium to collect data!

Time-stepLong time-steps allow to reach long time-scales, but

-If the “characteristic period” of the system is t, then we must impose:

vibrations in solids:

t

s)1010( 1312

Typical time steps for simulating realistic systems are usually of the order of femtoseconds

What information can we extract from MD simulations?

Total Energy: It is a conserved quantity in the simulations. There could be fluctuationsin the energy (caused by the errors in time integration), and can be reduced by reducing the time step.

The Caloric Curve: Using the total energy and the temperature in different runs for differentstates, we can construct the caloric curve. It helps to monitor any phase transitions whichimply a jump of E(T) if it is first order so on..

Mean Square Displacement: Mean square displacement is calculated using the equationbelow. It contains information about the atomic diffusivity. Using the slope of MSD vs timeone can extract the diffusion coefficient.

MSD=<(r(t)-r(0))2>

Melting Temperature: To estimate the location of melting temperature, one can set up a sample consisting of %50 solid and %50 liquid. It will have an interface between solid and liquid. Established equilibrium state where solid and liquid exist where the temperature of state is Tm. Dynamical Analysis: In MD simulation, information on the dynamics of a system is presentThe way to extract is to construct what is called a dynamic structure factor. S(k,w)=integral dtexp(-iwt)F(k,t)F(k,t)=1/N <rho(k,t+to)rho(-k,to)> time dependent density-density correlation functionS(k,w) will have peaks in the k,w plane corresponding to the dispersion of propagating modes.

Some Examples of Applications

Diffusion of 2D IslandsDiffusion of Cu 7-atom Cluster on Cu(111) at 400K for 200ns

H. Yildirim et al, 2008

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Trajectories of Center of Mass Motion

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7-atom Ag cluster Ag(111)

7-atom Cu cluster Cu(111)

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7-atom Cu cluster Ag(111)

D(T) (cm2/s)700K

D(T) (cm2/s)550K

D(T) (cm2/s)400K

D0(T) (cm2/s)System

1.82x10-60.41x10-65.66x10-80.18x10-3 Cu/Ag(111)-MD

3.16x10-60.67x10-61.71x10-972.0x10-3 Ag/Ag(111)-MD

3.05x 10-61.19x 10-61.45x10-957.0x10-3 Cu/Cu(111)-MD

Effective Diffusion Barrier, Diffusion Coefficients and Pre-exponential Factors

MD – inclusion of all dynamics, for all the processes occurs during the simulation

Effective Diffusion Barrier

(MD-EAM Simulation)

Cu/Cu(111) = 0.615 eV

Ag/Ag(111) = 0.605 eV

Cu/Ag(111) = 0.279 eV

0.382 (fcc-hcp)

0.392 (hcp-fcc)

0.372 (fcc-hcp)

0.336 (hcp-fcc)

Cu/Ag(111)

0.408 (fcc-hcp)

0.407 (hcp-fcc)

0.406 (fcc-hcp)

0.370 (hcp-fcc)

Ag/Ag(111)

0.319 (fcc-hcp)

0.303 (hcp-fcc)

0.450 (fcc-hcp)

0.385 (hcp-fcc)

Cu/Cu(111)

Barrier (eV)

EAM-FBD

Barrier (eV)

DFT-GGA-PW91

System

Activation Barriers for Concerted Motion

sharp tip (35 atom)

tip apex

adatom

3D island

cluster

• substrate= 6 atomic layers in fcc (111) orientation and 8x10 atoms in each layer

• 3D island= 2D pad (25 atoms) on top of which a 3-atom cluster is adsorbed.

substrate

• adatom is placed in the 3-fold site on top of the 3- atom cluster.

Model System

Manipulation/Extraction of 3D Islands

A. Deshpande, H. Yildirim, A. Kara, D. P. Acharya, J. Vaughn, T. S. Rahman, and S.-W. Hla

Phys. Rev. Lett. 98, 028304 (2007)

Cu mound on Cu(111)Ag mound on Ag(111)

Handan Yildirim, Abdelkader Kara, and Talat S. Rahman Phys. Rev. B 75, 205409 (2007)

Movies from MD Simulations at 100K for 200ps

Trajectory of the adatom and the tip apex

b) Cu system at 1.93 Å

Cu mound on Cu(111)Ag mound on Ag(111)

Growth on a metal surface

A temperature-accelerated dynamics (TAD) simulation of deposition of copper onto a silver (100) surface at T=77K and 0.04 monolayers per second, matching the deposition conditions of a 1989 experiment by Egelhoff and Jacob.

In the simulation, each deposition event was performed with direct molecular dynamics, allowing steering and a thermal reactive events. In the movie, the deposition "event" is simply an animation that that moves the atom towards the surface site in which it came to rest in the actual MD.

J. A. Sprague, F. Montalenti, B. P. Uberuaga, J. D. Kress, and A. F. Voter Phys. Rev. B 66, 205415 (2002)

Cu atoms on Ag(100)

Temperature-accelerated dynamics (TAD)