Summer Project Report “Molecular Dynamic simulation of a neat Lennard-Jones fluid and a Lennard-Jones binary mixture” N. Sridhar Summer Fellow (2006) Indian Academy of Sciences Supervisor: Prof Biman Bagchi Solid State and Structural Chemistry Unit Indian Institute of Science Bangalore May 8 – July 12 1
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Molecular Dynamic simulation of a neat Lennard-Jones
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Summer Project Report
“Molecular Dynamic simulation of a neat Lennard-Jones
fluid and a Lennard-Jones binary mixture”
N. Sridhar
Summer Fellow (2006)
Indian Academy of Sciences
Supervisor: Prof Biman Bagchi
Solid State and Structural Chemistry Unit
Indian Institute of Science
Bangalore
May 8 – July 12
1
Certificate
The work embodied in this report is the result of investigations carried out by Sri N. Sridhar in
Solid State Chemistry Unit, Indian Institute of Science, Bangalore under my supervision
during May 8 to July 15, 2006. Sri Sridhar was a Summer Fellow of the Indian Academy of
Sciences during this period.
Prof Biman Bagchi
Solid State and Structural Chemistry Unit
Indian Institute of Science
Bangalore
2
Acknowledgements
I am grateful to Prof. Biman Bagchi for giving me the opportunity to work in the Solid
State Structural Chemistry Unit of the Indian Institute of Science, Bangalore as a Summer Fellow
of the Indian Academy of Sciences, for introducing me to the exciting area of theoretical chemistry
in general and molecular dynamics in particular, and for overall guidance during the course of this
Fellowship. I am also grateful to Prof S.Yashonath of the Solid State Structural Chemistry Unit for
introducing me to new models and software for simulating molecular dynamics in binary mixtures
and for thought provoking discussions at various stages of the work.
I am grateful to the members of the theoretical and computational chemistry group in the Solid
State and Structural Chemistry Unit for their help and guidance. In particular, I would like to
express my gratitude to Mr.Subrata Pal for his patience in leading me through the initiation phase
and help in subsequent stages in spite of his busy schedule. I also thank Mr.Dwaipayan
Chakrabarti, Ms.Sangeeta Saini, Mr.Bharat Adkar, and Mr.Biman Jana for their cooperation.
I would like to thank my teachers at St. Stephen’s college especially Dr.S.V.Easwaran for
encouraging me to explore science beyond the syllabus.
Finally I express my gratitude to the Indian Academy of Sciences for offering me the Summer
Fellowship and providing me the opportunity to work at an advanced level in the Soild State
Structural Chemistry Unit, Indian Institute of Science, Bangalore.
N.Sridhar
3
Table of Contents
1. Introduction 5
2. Molecular Dynamics 2.1 Conceptual basis 6
2.2 The Molecular dynamics (M.D) simulation 10 2.3 Properties which make MD unique 12 2.4 Binary mixtures 13 2.5 Room Temperature ionic liquids 16 2.6 Limitations of molecular dynamics 17
3. Methodology 3.1 The simulation process and outputs 17 3.2 Systems simulated 23
4. Results 4.1 Neat fluid 26
4.2 Binary mixture 26
5. Conclusion 26
6. References 27
7. Plots 28
4
1. Introduction
Simulation of molecular dynamics allows us to obtain accurate information about the
relationships between the bulk properties of matter and the underlying interactions among the
constituent atoms or molecules in the liquid, solid or gaseous state. Computer simulations provide
a direct route from the microscopic details of the system (the masses of atoms, the interactions
between them, molecular geometry, etc.) to the macroscopic properties of experimental interest
(the equation of state, transport coefficients, structural order parameters, and so on). In addition,
such simulations can be used to simulate critical conditions that are difficult to conduct in real
experiments such as investigations under very high pressure and temperature. Further, the ever-
increasing power of computers also makes it possible to obtain ever more accurate results about
larger and larger systems. As a result, applications of molecular dynamics are to be increasingly
found not only in all branches of chemistry, but also in physics, biophysics, materials science and
engineering, and in the industry as well.
One such area of recent interest is room-temperature ionic liquids (RTILs). RTILs
have attracted tremendous interest in recent years as promising media, which could be an
alternative to the environment polluting, volatile common organic solvents. These are organic
liquids formed solely of ions which, in contrast to their inorganic counterparts like NaCl, exhibit
significantly lower melting temperatures. Besides being liquid at room temperature, they are non-
volatile. This implies that many industrially relevant processes can be influenced significantly if
the dynamics of interfaces between RTILs and other types of liquids are understood. This study
had its motivation in attempting to understand these dynamics through computer simulation.
In view of the limited duration of the Summer Project, and the need to develop a basic
understanding of the molecular simulation processes before proceeding to address the more
complex questions involved in RTILs , the specific objectives of the study were defined as:
(i) develop a basic understanding of simulation of molecular dynamics and related
computer models
5
(ii) apply the principles to molecular dynamics to two simple systems: (a) neat Lennard-
Jones Fluid and (b) a Lennard - Jones binary mixture
The programme code for (a) above was written in FORTRAN 90 and for (b), dl_poly
2 software available in the SSCU was used.
2. Molecular Dynamics
2.1 Conceptual basis:
Molecular dynamics predicts atomic trajectories by direct integration of the equations
of motion – Newton’s second law for classical particles – with appropriate specification of an
inter-atomic potential and suitable initial and boundary conditions.
Treating the problem as the classical many-body problem:
For a system of N particles enclosed in a region of volume V at temperature T, the
positions of the N particles are specified by a set of N vectors, {r(t)} = (r1(t), r2(t),..., rN (t)), with rj
(t) being the position of particle j at time t. Knowing {r(t)} at various time instants means that the
trajectories of the particles are known.
If the system of particles has a certain energy E which is the sum of kinetic and potential energies
of the particles, E = K + U, where K is the sum of individual kinetic energies:
N
2f
j 1
1K m v2 == ∑
and U is the prescribed interatomic potential given by
U = U (r 1, r2,...,rN ).
6
In general, U depends on the positions of all the particles in a complicated fashion.
We will soon introduce a simplifying approximation (assumption of pair wise interaction) which
makes this most important quantity much easier to handle.
To find the particle trajectories requires the solution of Newton's equations of
motion, F = ma which all particles must satisfy. However, the equations for motion for an N-
particle system are more complicated because the equation for one particle is coupled to all the
other equations through the potential energy U.
The equation that needs to be solved is:
( )j
2j
r2
d rm U r , j 1, ....., N
dt= −∇ =
-------(1)
Eq.(1) is a system of second-order, non-linear ordinary differential equations and represents the
famous many-body problem which can be solved only numerically when N is more than 2 to
obtain the atomic trajectories. For this purpose, the time interval is divided into many small
segments, each of Δt. Given the initial condition at time t
1r
o, {r(to)} , integration means advancing
the system by increments of Δt ,
{r(to)} → {r(to +Δt)} → {r(to + 2Δt)} →...{r( to + Nt Δt)} ------(2)
where Nt is the number of time steps making up the interval of integration.
By Taylor series expansion, for a particle j
rj (to +Δt) = rj (to ) + vj (t) Δt + 1
2 aj (t)( Δt)
2 +... ------ (3)
Write a similar expansion for rj (to −Δt ) , then add the two expansions to obtain
rj (to +Δt) =−r
j (to −Δt) + 2 r
j (to ) + a
j ( to )(Δt)
2 +... ------ (4)
7
Notice that the left-hand side represents the position of particle j (the trajectory needed) at the next
time step Δ t , whereas all the terms on the right-hand side are quantities evaluated at time t0 and
are therefore known. Eq.(4) is therefore the integration of (1). The acceleration of particle j at time
t0 is Fj ({r(to )}) / m. The process can be repeated to move another step, etc. and repeated as many
times as one wants to generate a sequence of positions (or trajectories) for as long an interval as
desired. There are more elaborate and standard ways of doing this integration as in formal
numerical methods but the basic idea of marching out in discrete steps is the same. A more
accurate method allows one to take a larger value of Δt, which is certainly desirable, but this also
means one needs more memory relative to the simpler method.
These trajectories (positions and velocities) are therefore the raw output of molecular dynamics
simulation. The flow- chart for a typical MD simulation looks like the following.
(a) → (b) → (c) → (d) → (e) → (f) → (g)
a = set particle positions; b = assign particle velocities; c = calculate force on each particle; d =
move particles by time step Δt; e = save current positions and velocities; f = if preset no. of time
steps is reached, stop, otherwise go back to (c); g = analyze data and print results
The Lennard-Jones Pair Potential
To make the simulation tractable, it is common to assume the inter-atomic potential U can be
represented as the sum of pair wise interactions,
( ) ( )1 Ni j
U r ,....., r V r2 ≠
≅ ∑n
ij1 ------ (5)
where rij is the separation distance between particles i and j. V is the pair potential of interaction; it
is a central force potential, being a function only of the separation distance between the two
particles. A very common pair potential used in atomistic simulations is one which describes the
van der Waals interaction in an insulator. This is of the form, Lennard-Jones (L-J or 6-12)
potential) :
8
( )6 1
V r 4r r
⎛ ⎞σ σ⎛ ⎞ ⎛ ⎞= ∈ −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
2
------(6)
with parameters, ∈=well-depth σ = hard-sphere diameter
These parameters can be fitted to reproduce experimental data or deduce from results of accurate
quantum chemistry calculations. The 121
r⎛ ⎞⎜ ⎟⎝ ⎠
term describes the repulsive force and the 61
r⎛ ⎞⎜ ⎟⎝ ⎠
term
describes the attractive force.
Thus, by (6), neutral atoms and molecules are subject to two distinct forces in the
limit of large distance, and short distance: an attractive van der Waal's force, or dispersion force, at
long ranges and a short range repulsion force. The repulsion arises from overlap of the electron
clouds, the result of overlapping electron orbitals, referred to as Pauli's repulsion. The attraction is
associated with the interaction between the induced dipole in each atom (the London dispersion
interaction). The Lennard-Jones potential (L-J or 6-12 potential) is a simple mathematical model
that represents this behaviour.
The short-range repulsion rises sharply (with inverse power of 12) at close
interatomic separations, and an attraction varying with the inverse power of 6 (Fig 1). The value of
12 for the first exponent has no special significance, as the repulsive term could just as well be
represented by an exponential, whereas the second exponent results from quantum mechanical
calculations and therefore should not be modified. The importance of short-range repulsion is that
this is necessary to give the system a certain size or volume (density), without which the particles
can collapse onto each other, whereas the attraction is necessary for cohesion of the system of
particles, without which the particles will all fly away from each other. Both are necessary for
solids and liquids to have their known physical properties.