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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. Lennard-Jones potential.

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Basics of molecular dynamics

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Basics of molecular dynamics l.jpg

Basics of molecular dynamics


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Equations of motion for MD simulations

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


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Lennard-Jones potential

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


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Lennard-Jones Potential


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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.


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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:


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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


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Boundary Condition


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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.


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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


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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


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Velocity distribution as a function of time


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Interaction computations

  • All pair method

  • Cell subdivision

  • Neighbor lists


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Thermodynamic properties at equilibrium


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Thermophysical and Dynamic Properties at equilibrium


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Example: Calculation of thermal conductivity at eqilibrium

Following are the equations required to calculate thermal conductivity:

Where Q is the heat flux

Where k is the thermal

conductivity


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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


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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


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Example: continued..


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

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.

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


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Thermal conductivity of nanofluids at Equilibrium

Schematic diagram


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Results


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Results continued


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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).


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