Basics of molecular dynamics

Equations of motion for MD simulations

The classical MD simulations boil down to numerically integrating Newton’s equations of motion for the particles

Nix

V

xdt

xdm

iji i

iji

02

2

Lennard-Jones potential

One of the most famous pair potentials for van der Waals systems is the Lennard-Jones potential

612

4)(ijij

ij rrr

Lennard-Jones Potential

Dimensionless Units

Advantages of using dimensionless units: the possibility to work with numerical values of the

order of unity, instead of the typically very small values associated with the atomic scale

the simplification of the equations of motion, due to the absorption of the parameters defining the model into the units

the possibility of scaling the results for a whole class of systems described by the same model.

Dimensionless units

When using Lennard-Jones potentials in simulations, the most appropriate system of units adopts σ, m and ε as units of length, mass and energy,respectively, and implies making the replacements:

Integration of the Newtonian Equation

Classes of MD integrators: low-order methods – leapfrog, Verlet, velocity

Verlet – easy implementation, stability predictor-corrector methods – high accuracy

for large time-steps

Boundary Condition

Periodic boundarycondition

Specular boundarycondition

Initial state

Atoms are placed in a BCC, FCC or Diamond lattice structure

Velocities are randomly assigned to the atoms. To achieve faster equilibration atoms can be assigned velocities with the expected equilibrium velocity, i.e. Maxwell distribution.

Temperature adjustment: Bringing the system to required average temperature requires velocity rescaling. Gradual energy drift depends on different factors- integration method, potential function, value of time step and ambient temperature.

Conservation Laws

Momentum and energy are to be conserved throughout the simulation period

Momentum conservation is intrinsic to the algorithm and boundary condition

Energy conservation is sensitive to the choice of integration method and size of the time step

Angular momentum conservation is not taken into account

Equilibration

For small systems whose property fluctuate considerably, characterizing equilibrium becomes difficult

Averaging over a series of timesteps reduces the fluctuation, but different quantities relax to their equilibrium averages at different rates

A simple measure of equilibration is the rate at which the velocity distribution converges to the expected Maxwell distribution

)2/exp()( 21 Tvvvf d

Velocity distribution as a function of time

Interaction computations

All pair method Cell subdivision Neighbor lists

Thermodynamic properties at equilibrium

Thermophysical and Dynamic Properties at equilibrium

Example: Calculation of thermal conductivity at eqilibrium

Following are the equations required to calculate thermal conductivity:

i j

jijjii

iiEt rvFvQ )(

N

jBp ttj

VTkk

12

)()0(1

QQ

iN

j

tjitjiN

ti1

))(()(1

)()0( QQQQ

Where Q is the heat flux

Where k is the thermal conductivity

Nonequilibrium dynamics

Homogeneous system: no presence of physical wall, all atoms perceive a similar environment

Nonhomogeneous system: presence of wall, perturbations to the structure and dynamics inevitable

Nonequlibrium more close to the real life experiments where to measure dynamic properties systems are in non equilibrium states like temperature, pressure or concentration gradient

Example: Calculation of thermal equilibrium at nonequilibrium ( direct measurement)

To measure thermal conductivity of Silicon rod (1-D) heat energy is added at L/4 and heat energy is taken away at 3L/4

After a steady state of heat current was reached, the heat current is given by

Using the Fourier’s law we can calculate the thermal conductivity as follows

Stillinger-Weber potential for Si has been used which takes care of two body and three body potential

tAJ z

2

zT

Jk z

/

Example: continued..

200

220

240

260

280

300

320

0 20 40 60 80 100 120 140 160 180 200

Series1

y = 0.2686x + 233.34

R2 = 0.9384

220

230

240

250

260

270

280

40 50 60 70 80 90 100 110 120 130

Molecular Dynamics Simulation of Thermal Transport at Nanometer Size Point Contact on a Planar Si Substrate

Calculated temperature profile in the Si substrate for a 0.5 nm diameter contact radius.

Schematic diagram of the simulation box. Initial temperature is 300 K. Energy is added at the center of the top wall and removed from the bottom and side walls.

Thermal conductivity of nanofluids at Equilibrium

Schematic diagram

Nanoparticles

Results

Results continued

References

Rapaport D. C., “The Art of Molecular Dynamics Simulation”, 2nd Edition, Cambridge University Press, 2004

W. J. Minkowycz and E. M. Sparrow (Eds), “Advances in Numerical Heat Transfer”,vol. 2, Chap. 6, pp. 189-226, Taylor & Francis, New York, 2000.

Koplik, J., Banavar, J. R. &Willemsen, J. F., “Molecular dynamics of Poiseuille flow and moving contact lines”, Phys. Rev. Lett. 60, 1282–1285 (1988); “Molecular dynamics of fluid flow at solid surfaces”, Phys.Fluids A 1, 781–794 (1989).