Physics 270: Introduction to Computational Physics
Problem 12-Molecular dynamics simulations, part I.
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Introduction and Goals
To introduce the concept of the pair potential, and learn how to calculate
a potential energy "surface" for a two dimensional system.
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The Lennard-Jones potential
A cube of matter 1 cm on a side contains about 1021 atoms.
In principle, the energy of this cube is the result of the interaction
of every single atom with every other atom of that ensemble of 1021.
If all of these interactions had to be tracked in a simulation, no computer
program would be able to simulate the behavior of matter. In reality, the
forces between the atoms in a piece of matter (gas, solid or liquid) are
finite in range. For simple atoms (like Argon), the interactions between
atoms can be modelled by a potential energy which depends only on the distance
between any two atoms. This simple approximation can, in many cases, be
applied to more complicated systems, like solids. The "pair potential"
method, although fraught with problems and difficulties, is still one of
the more widely used methods of simulating the behavior of matter. In this
problem, and the one that follows, we look at some of the basic ideas behind
"molecular dynamics" simulations.
The model potential we will use is the Lennard-Jones potential, which
gives excellent results in modelling rare gas interactions, which are primarily
due to induced electrostatic dipole effects (van der Waals forces). The
total potential energy of an ensemble of atoms is proposed to be the sum
over pair-wise interactions, in the form
UTOT = V(r12) + V(r13)
+ ..... + V(r23) + V(r24) + .... = SUM(i<j=1 to
N) V(rij),
where care has been taken to avoid "double counting" the pair-wise energy.
Each of the pair-wise potential interactions is given by the L-J potential,
which has the form
V(r) = 4E[(s/r)12 - (s/r)6],
where r is the inter-atomic distance, E is the scaling factor for the energy,
and s is the scaling factor for the distance. For Argon, the values are
(approximately) s=.34 nanometer, E = .012 eV .
In this first problem involving molecular dynamics, we make the further
assumption that we are working at T=0, so that we can safely ignore an
energy of motion (kinetic energy).
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Tasks
Use the form of the Lennard-Jones potential given above for these projects.
Do your calculations using a FORTRAN program, and use spreadsheets for
plotting and analysis of the results.
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Plot the L-J potential for a pair of Argon atoms. Identify the location
of the point at which V=0, and the point at which the force between the
atoms is zero. How large a separation is needed to reduce the potential
interaction to a negligible value?
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Create a two dimensional "lattice" of Argon atoms that interact via the
L-J potential. The number of atoms will have to be variable, since this
is one of the parameters that you will change. A "lattice" is a regularly
spaced array of particles.
Place your atoms on a square grid, which can be characterized by a
"lattice parameter" or "lattice constant", a. For values of a
in the vicinity of the value that minimizes the L-J potential for two atoms,
calculate the total energy, per atom, of the square lattice
at zero temperature (meaning you only consider the potential energy). First
plot the energy as a function of the number of atoms in your lattice, to
determine the size of the lattice needed to get a "converged" (asymptotic)
value of the energy per atom. Then, keeping the number of atoms fixed at
the value you have found gives a converged result, plot the energy as a
function of the lattice parameter a.
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Next, do the same thing for your atoms placed on a triangular regular grid.
Again, plot the energy as a function of lattice parameter a.
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Compare the energy curves for the two lattices. For this comparison, you
need to convert your lattice parameters to an atomic density. That is,
you can compare the energy for each lattice, but only for equivalent densities,
which is the number of atoms per unit area. This is not the same as comparing
them for the same value of inter-atomic spacing, a. Which of the
two lattices is the stable (lowest energy) configuration?
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Compare the force curves at the equilibrium lattice spacing for the square
and triangular lattice, with the force curve for two isolated Argon atoms.
What are the differences?
NOTE ON THE CALCULATION:
Since this model assumes an infinite extent of atoms in two directions,
every single atom is assumed to have the same amount of potential energy.
This means you can do the calculation for an atom in the center of your
lattice, and sum over the rest of the atoms in the lattice, to get the
value of the energy per atom.