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

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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.

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Fig.1: Lennard-Jones potential for argon dimer.

Note that the L-J potential is approximate and the form of the repulsion term has no theoretical

justification (the repulsion force depends exponentially on the distance). The attractive long-range

potential, however, is derived from dispersion interactions.

The L-J potential is a relatively good approximation and due to its simplicity often used to describe

the properties of gases, and to model dispersion and overlap interactions in molecular models. It is

particularly accurate for noble gas atoms and is a good approximation at long and short distances

for neutral atoms and molecules.

2.2. The Molecular dynamics (M.D) simulation:

The simulation system is typically a cubical cell in which particles are placed either in

a very regular manner, as in modeling a crystal lattice, or in some random manner, as in modeling

a gas or liquid. The number of particles in the simulation cell is chosen to be quite small. The next

step is to choose the system density. Choosing the density is equivalent to choosing the system

volume since density n = N/V, where N is the number of particles and V is the volume. My code

uses dimensionless reduced units, so reduced density (DR) has typical values around 1.0-1.2 for

solids, and 0.6 - 0.85 for liquids. For reduced temperature (TR) the recommend values are 0.4 - 0.7

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for solids, and 0.9 - 1.3 for liquids. Assigning particle velocities in (b) above is equivalent to

setting the system temperature.

For simulation of bulk systems (no free surfaces) it is conventional to use the periodic

boundary condition . This means that the cubical cell is surrounded by 26 identical image cells. For

every particle in the simulation cell, there corresponds an image particle in each image cell. The 26

image particles move in exactly the same manner as the actual particle, so that if the actual particle

should happen to move out of the simulation, the image particle in the image cell opposite to the

exit side will move in (and becomes the actual particle, or the particle in the simulation cell) just as

the original particle moves out. The net effect is that with periodic boundary conditions, particles

cannot be lost or gained. In other words, the particle number is conserved, and if the simulation

cell volume is not allowed to change, the system density remains constant.

Since in the pair potential approximation, the particles interact two at a time, a

procedure is needed to decide which pair to consider among the pairs between actual particles

and between actual and image particles. The minimum image convention is a procedure where

one takes the nearest neighbor to an actual particle, regardless of whether this neighbour is an

actual particle or an image particle. Another approximation that is useful to keep the

computations to a manageable level is to introduce a force cutoff distance beyond which particle

pairs simply do not see each other. In order not to have a particle interact with its own image, it is

necessary to ensure that the cutoff distance is less than half of the simulation cell dimension.

Another device often used in MD simulation is a Neighbor List that keeps track of the

nearest, second nearest, ... neighbors for each particle. This is to save time from checking every

particle in the system every time a force calculation is made. The list can be used for several time

steps before updating. Each update is expensive since it involves NxN operations for an N-particle

system. In low-temperature solids where the particles do not move very much, it is possible to do

an entire simulation without or with only a few updating, whereas in simulation of liquids,

updating every 5 or 10 steps are quite common.

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2.3. Properties which make MD unique:

Classical MD simulation described above (as opposed to quantum MD simulation) is

a useful simulation technique because it follows the atomic motions according to the principles of

classical mechanics as formulated by Newton and Hamilton. Because of this, the results are

physically as meaningful as the potential U that is used. This means that whatever mechanical,

thermodynamic, and statistical mechanical properties that the system of N particles should have,

they are still present in the data. How these properties are extracted from the output of the

simulation – the atomic trajectories – will determine how useful the simulation is. Before any

conclusions can be made, one needs to get in the question of how various properties are to be

obtained from the simulation data. Thus, an MD simulation can be visualized as an ‘atomic video’

of the particle motion (one which we can see as a movie), there is a great deal of realistic details in

the motions themselves, but how to extract the information in a scientifically useful is up to the

viewer. And an experienced viewer can get much more useful information than an inexperienced

one! Besides the above, the following aspects of MD make it very useful and practicable:

(a) Unified study of all physical properties: Using MD one can obtain thermodynamic, structural,

mechanical, dynamic and transport properties of a system of particles which can be a solid,

liquid, or gas. One can even study chemical properties and reactions which are more difficult

and will require using quantum MD.

(b) Several hundred particles are sufficient to simulate bulk matter. While this is not always true, it

is rather surprising that one can get quite accurate thermodynamic properties such as equation

of state in this way. This is an example that the law of large numbers takes over quickly when

one can average over several hundred degrees of freedom.

(c) Direct link between potential model and physical properties. This is really useful from the

standpoint of fundamental understanding of physical matter. It is also very relevant to the

structure-property correlation paradigm in materials science.

(d) Complete control over input, initial and boundary conditions. This is what gives physical

insight into complex system behavior. This is also what makes simulation so useful when

combined with experiment and theory.

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(e) Detailed atomic trajectories. This is what one can get from MD, or other atomistic simulation

techniques, that experiment often cannot provide. This point alone makes it compelling for the

experimentalist to have access to simulation.

2.4 Binary mixtures:

Binary mixtures are well known to show marked departure from the ideal behavior

given by Raoult’s law. For a given property P, the latter predicts the following simple dependence

on the composition,

P = x1P1 + x2P2 (1)

where xis are the mole fractions and Pis are the values of property P of the pure (single component)

liquids more often than not significant deviations from eqn. (1) is observed which is usually

denoted by an excess function,

Pex = P – (x1P1 + x2P2 ) (2)

Considerable literature exists on such behavior, where P can be volume, free energy

or viscosity. The deviation from ideality appears to have a correlation with the solute-solvent

mutual interaction. Despite the importance and the long interest in this problem, there does not

seem to exist a satisfactory explanation of this non-ideality (or non-additivity) in binary mixtures.

In fact, we are not aware of any microscopic study (based on time correlation function approach)

of the anomalous (or non-monotonous) composition dependence of viscosity. This is, however, not

surprising because a microscopic calculation of viscosity is quite difficult. In the absence of any

microscopic theory, the experimental results have often been fitted to several empirical forms.

Prominent among them is Eyring’s theory of viscosity extended to treat binary mixtures. This

theory can correlate (with the help of one adjustable parameter) several aspects of the composition

dependence of viscosity of many liquid mixtures (like benzene + methanol, toluene + methanol

etc). However, the very basis of Eyring’s theory has been questioned, as this theory is based on

creation of holes of the size of the molecules which is energetically unfavorable.

There also exist several other empirical expressions which attempt to explain the anomalous

dependence of viscosity in binary mixtures. On the experimental side there are evidences of the

correlation between excess viscosity and excess volume of the liquid mixture where it has been

observed for many cases that if excess volume is positive then excess viscosity becomes negative

and vice versa.

13

In recent years interesting theoretical and computer simulation studies on Lennard-Jones (LJ)

binary mixtures have been carried out. These studies have mainly concentrated on the glass

transition in binary mixtures which are known to be good glass formers (in contrast to one of the

component LJ liquid which does not form computer glass easily).In addition, these studies have

considered only one particular composition and a unique interaction strength. Considerable

research has also been carried out by using equilibrium molecular dynamic simulation to determine

the transport properties such as self and mutual diffusion coefficients in binary mixtures.

Non-equilibrium molecular dynamic ~MD simulation methods have also been

employed to determine the shear viscosity and thermal conductivity of binary soft-sphere mixtures.

Heyes carried out the extensive equilibrium MD simulations of Lennard- Jones binary mixtures by

using both the microcanonical ~N V E and canonical ~N V T ensemble methods to study the partial

properties of mixing and transport coefficients by adopting the time correlation function approach.

Apart from the bulk viscosity, these simulations seem to have satisfactorily reproduced in the

experimentally determined transport coefficients for the Ar–Kr mixture.

However, the strong non-ideality in the composition dependence of viscosity,

observed in many experiments, has not been addressed to in the work of Heyes or by others.

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Fig.2: Composition dependence of excess viscosity and excess volume for a binary mixture with

weak solute-solvent interactions(Model-2).

The non-ideality in the case of inert gas mixtures is small, since their mutual

interaction strength follows the Berthelot mixing rule. To capture this strong non-ideality we

introduce and study two models referred to as model I and model II of binary mixtures in which

the solute–solvent interaction strength is varied by keeping all the other parameters unchanged.

15

All the three interactions solute–solute, solvent–solvent, and solute–solvent are described by the

Lennard-Jones potential. Among the two models, model I promotes the structure formation

between solute and solvent molecules due to strong solute–solvent attractive interaction. The

second model model leads to the opposite scenario by promoting the structure breaking, because of

weak solute–solvent interaction. These two models are perhaps the simplest models to mimic the

structure making and structure breaking in binary mixtures. For convenience, we denote the

solvent molecules as A, and the solute molecules as B.

2.5: Room-temperature ionic liquids:

Room temperature ionic liquids are solvents with many potential applications, as they

have vanishingly low vapour pressures and can be recycled after use in organic reactions. Many

types of chemical reaction can be carried out successfully in these solvents, and solvent recovery

and the work-up of products often depends on their relative solubility in different phases. Thus it is

important to understand the solvation properties of simple solutes in this unusual environment.

Atomistic simulation has proved to be a useful technique for helping to understand chemical

reactivity and solvation properties in aqueous solutions strongly solvated, principally by forming

hydrogen bonds with the chloride ion, while the non-hydrogen bonding solutes interact more

strongly with the cation.

Room temperature ionic liquids (RTILs) have emerged as a cleaner alternative to the

conventional solvents. These are a class of organic salts that, in their pure state, are liquids at or

near room temperature. Some of the more common air-stable ILs are composed of heterocyclic

imidazolium or pyridinium cations having alkyl substituent groups and bulky anions such as

[PF6] , [BF4] or [NO3] . − − −

One of the major barriers preventing the adoption of ILs by industry is the dearth of

physical property data for these compounds, as well as a general lack of fundamental

understanding of how these properties depend on the chemical constitution of the IL. Atomistic

simulation of ILs can help resolve this crisis.

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2.6. Limitations of molecular dynamics:

There are also significant limitations to MD. The two most important ones are:

(a) Need for sufficiently realistic interatomic potential functions U: This depends on what is

known fundamentally about the chemical binding of the system under study. Progress is being

made in quantum and solid-state chemistry, and condensed-matter physics; these advances

will make MD more and more useful in understanding and predicting the properties and

behavior of physical systems.

(b) Computational capabilities constraints. No computers will ever be big enough and fast

enough. On the other hand, things will keep on improving as far as we can tell. Current limits

on how big and how long are a billion atoms and about a microsecond in brute force

simulation.

3. Methodology

3.1 The simulation process and outputs:

1. Initialisation: To start the simulation we must assign initial positions,

velocities and accelerations to all the particles in the system. I have assumed

the atoms to exist in an f.c.c lattice and accordingly generated the coordinates. I

have also assumed a static system with all atoms at rest initially.

2. Evolving the sample from zero time: The system is evolved over a period

of time (1000 time steps in this case) .First from the L.J potential, we generate

corresponding values for force after each time step. The integration of the

equations of motion is carried out using the Velocity Verlet integration

algorithm.

The Velocity Verlet Algorithm: Positions and velocities at time t t t= + are

given by

( ) ( ) ( ) ( )2

i ihr t t r t hv t a t2+ = + + i --------------------- (1)

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( ) ( ) ( ) (i i itv t t v t a t t a t2⎡⎣+ = + + + )i

⎤⎦ ----------------(2)

Fig.2: Algorithm for

an MD simulation

It can be shown that the errors in this algorithm are of ( )4t , and that it is very stable

in MD applications and in particular conserves energy very well.

The Verlet algorithm reduces the level of errors introduced into the integration by

calculating the position at the next time step from the positions at the previous and current time

steps, without using the velocity. Velocities, though not required for calculating trajectories are

useful for estimating Kinetic Energy.

This can create technical challenges in molecular dynamics simulations, because

kinetic energy and instantaneous temperatures at time zero cannot be calculated for a system until

the positions are known at time t t+ ∂ .

3. Equilibration: A time step ‘t’ is chosen, and the equations of motion are solved

iteratively for a sufficient number of steps to allow the system to come to equilibrium.

4. Simulation: The iterations are continued for a specified period. Physical quantities are measured

at each time step, and their thermal averages are computed as time averages.

This approximation works well for a wide range of materials. Only when we consider

translational or rotational motions of light atoms of molecules (He, Hydrogen, etc) or vibrational

motion with a particular frequency , quantum effects must be considered.

Using the above mentioned algorithm, the velocities and coordinates and of each

particle can be calculated at each time step.

5. Outputs: With the above process, the following physical quantities were derived as functions of

time.

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Table 1: Equations used for calculating various physical quantities

Physical quantity

Equation used

1. Temperature Law of Equipartition of energy:

N2

B ii 1

3 mN k T v2 2 =

= ∑

2. Kinetic Energy K.E =

N2i

i 1

m v2 =∑

3. Potential Energy P.E = ( )i j

ij

U | r r |−∑

4. Total Energy

Sum of potential and Kinetic energy:

( )n

2i i

i 1 ij

mE v U | r2 =

= + −∑ ∑ jr |

5. Pressure The Virial theorem:

N

B ii 1

1PV NK T r .F3 =

= − ∑ i

6. Velocity Autocorrelation

Function

( ) ( ) ( )( )N

0 0i 1

1Cv t n t * Vi t t .Vi t t n tN =

= = = = +∑

Velocity autocorrelation function( VAF): The velocity autocorrelation function (VAF) is a time

dependent correlation function, and is important because it reveals the underlying nature of the

dynamical processes operating in a molecular system. It is defined as

( ) ( ) ( )( )N

0 0i 1

1Cv t n t * Vi t t .Vi t t n tN =

= = = = +∑

Consider a single atom at time zero. At that instant the atom ‘i’ will have a specific

velocity vi. If the atoms in the system did not interact with each other, the Newton's Laws of

motion tell us that the atom would retain this velocity for all time. This means that all the points

19

Cv(t) have the same value, and if all the atoms behaved like this, the plot would be a horizontal

line. It follows that a Velocity autocorrelation plot that is almost horizontal. This implies very

weak forces are acting in the system.

However if the forces are small but not negligible, then the magnitude and direction

of velocity change gradually under the influence of these weak forces. In this case we expect the

scalar product of ( )0Vi t t= with ( )0Vi t t n t= + to decrease on average, as the velocity is

changed. (i.e. the velocity decorrelates with time, which is the same as saying the atom 'forgets'

what its initial velocity was.) In such a system, the VAF plot is a simple exponential decay,

revealing the presence of weak forces slowly destroying the velocity correlation. Such a result is

typical of the molecules in a gas.

Strong interatomic forces are most evident in high-density systems, such as solids

and liquids, where atoms are packed closely together. In these circumstances, the atoms tend to

seek out locations where there is a near balance between repulsive forces and attractive forces,

since this is where the atoms are most energetically stable. In solids these locations are extremely

stable, and the atoms cannot escape easily from their positions. Their motion is therefore an

oscillation; the atom vibrates backwards and forwards, reversing their velocity at the end of each

oscillation. If we now calculate the VAF, we will obtain a function that oscillates strongly from

positive to negative values and back again. The oscillations will not be of equal magnitude

however, but will decay in time, because there are still perturbative forces acting on the atoms to

disrupt the perfection of their oscillatory motion. So what we see is a function resembling a

damped harmonic motion.

Liquids behave similarly to solids, but now the atoms do not have fixed regular

positions. A diffusive motion is present to destroy rapidly any oscillatory motion. The VAF

therefore may perhaps show one very damped oscillation (a function with only one minimum)

before decaying to zero. In simple terms this may be considered a collision between two atoms

before they rebound from one another and diffuse away.

Uses of VAF: The VAF may be Fourier transformed to project out the underlying frequencies of

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the molecular processes. This is closely related to the infra- red spectrum of the system, which is

also concerned with vibration on the molecular scale. Also, provided the VAF decays to zero at

long time, the function may be integrated mathematically to calculate the diffusion coefficient D0

given by

( ) ( )0 i i0

1D v 0 .v t3 dtα

= ∫

In addition to the above, the ensemble averages, and the fluctuations were also assessed

Ensemble Averages:

The ensemble is a central concept in statistical mechanics. Imagine that a given

molecular system is replicated many times over, so that we have an enormous number of copies,

each possessing the same physical characteristics of temperature, density, number of atoms and so

on. Since we are interested in the bulk properties of the system, it is not necessary for these

replicas to have exactly the same atomic positions and velocities. In other words the replicas are

allowed to differ microscopically, while retaining the same general properties. Such a collection of

replicated systems is called an ensemble.

Because of the way the ensemble is constructed, if a snapshot of all the replicas is

taken at the same instant, we will find that they differ in the instantaneous values of their bulk

properties. This phenomenon is called fluctuation. Thus the true value of any particular bulk

property must be calculated as an average over all the replicas. This is what is meant by an

ensemble average, and the instantaneous values are said to fluctuate about the mean value.

Molecular dynamics proceeds by a numerical integration of the equations of motion.

Each time step generates a new arrangement of the atoms (called a configuration) and new

instantaneous values for bulk properties such as temperature, pressure, configuration energy etc.

To determine the true or thermodynamic values of these variables requires an ensemble average. In

molecular dynamics this is achieved by performing the average over successive configurations

21

generated by the simulation. In doing this we are making an implicit assumption that an ensemble

average (which relates to many replicas of the system) is the same as an average over time of one

replica (the system we are simulating). This assumption is known as the Ergodic Hypothesis.

Fortunately it seems to be generally true, provided a long enough time is taken in the average.

However it has not yet been rigorously proved mathematically.

Fluctuations

Most of the properties that we calculate for a molecular system are averages. All

averages are obtained by summing over many numbers. Thus in practice we expect the average to

show some dispersion - individual contributions are scattered about the mean value. In statistical

thermodynamics this dispersion about the average value is known as fluctuation and it is both a

subtle and important property of all physical systems.

When calculating an ensemble average (of say, pressure at fixed temperature and

density), we take an instantaneous snapshot of a very large set of replicas of the system concerned

and compute the average from the sum of the individual values taken from each replica. Even

though each replica represents the same system at the same pressure, their individual,

instantaneous values differ slightly, because the molecules that bombard the vessel surfaces to

create the pressure are not in synchronisation between each replica and cannot possibly give rise to

precisely the same surface forces at the same instant. Thus, with pressure, we expect some

fluctuation about the mean value and indeed, similar arguments can be made for all the bulk

properties of the system.

Fluctuations are of fundamental importance in statistical mechanics because they

provide the means by which many physical properties of a molecular system can happen. For

instance, the density of a liquid at equilibrium is a fixed, uniform quantity and we feel justified in

considering the system to be isotropic - the same at all points within its bulk. Yet we know that the

molecules in the system are undergoing diffusion and can easily travel throughout the bulk of the

liquid. It is difficult to imagine how this diffusion can take place if the environment each molecule

is in is completely isotropic. If however we consider the density to be fluctuating minutely from

the mean value at different points in the bulk, we can readily see that such fluctuations would

provide a means by which the diffusion may take place. It is a surprising fact, but most of the

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physical properties of a bulk system are driven by fluctuations, and indeed can be calculated

directly from them. For this reason it is possible to view fluctuations as even more fundamental

than the average value.

3.2 Systems simulated:

Two kinds of systems were studied, Neat Lennard-Jones Fluid and Binary mixtures

1. Neat Lennard-Jones fluid:

(i) A 256 atom Lennard-Jones system was considered and was evolved over a period of 1000 time

steps. Each time step was equivalent to 0.0032 picoseconds.

The units of mass, length and energy were chosen as m = 1, ∈ = 1, and = 1. σ

All plots are for a temperature of 30K.

(ii). The codes for each program were written in FORTRAN 90. Values for the quantities (in table

1) were calculated after every time step using the equations mentioned. The plot was made using

xmgrace software.

(iii). For plotting velocity autocorrelation, the velocities (of each atom in x, y and z directions)

were read after the system attained equilibrium .i.e. the fluctuations were small. Then these were

read in as arrays and manipulated to get Cv.

2. Lennard-Jones Binary mixture:

A series of molecular dynamic simulations at constant pressure (P), temperature (T),

and total number of particles (N) of binary mixtures were carried out by varying the solute mole

fraction from 0 to 1. Temperature and pressure were kept constant using Berendsen NPT ensemble

with thermostat and barostat relaxation times 1.0 and 0.2 respectively. Our model binary system

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consists of a total 500 (solute(A)+solvent(B)) particles. We used simple Lennard-Jones (lj)

potential, as pair potential of interaction between any two particles, which is given by the Lennard-

Jones potential function, which sets a cutoff radius rc outside which the potential energy is zero.

The particular form of the potential is given by

U(r) = 4ε [(σ/r)12 – (σ/r)6]

where the cutoff distance rc in this particular case has been taken as equal to 2.5σ. Use of above

potential form takes care of the fact that both potential and force are continuous at the cutoff

distance. A and B denote two different particles. We set the diameter (σ) and molar mass (M) of

the solute as that of Ar, and the solvent as that of Xe, for simplicity, i.e σAA = 3.405 Å, σBB = 4.1 Å

and the molar mass of A = 39.95 gm (that of Argon) and molar mass of B= 131.29 gm (that of

Xenon). The solute-solute interaction strength, εAA = 0.99768 KJ/mol, solvent-solvent interaction

strength, εBB = 1.837394 KJ/mol where A and B represent the solute and solvent particles,

respectively. The variation of total energy and volume of the system with change in composition in

the binary mixture i.e. by changing the mole fractions of particles was studied.

The simulations were carried out in two sets :

(1) The interaction strengths εAB (solute-solvent) were varied in three ways (a) εAB = 1.353931

KJ/mol (Lorentz-Berthelot mixing value), (b) εAB =2.707863 KJ/mol (twice the Lorentz-Berthelot

mixing value), and (c) εAB =0.676966 KJ/mol (half the Lorentz-Berthelot mixing value), keeping

the σAB =3.7525 Å (which is equal to the Lorentz-Berthelot mixing value).

(2) The solute-solvent zero potential diameter σAB was varied in three ways (a) σAB = 3.7525 Å

(Lorentz-Berthelot mixing value), (b) σAB = 4.6 Å, (c) σAB = 3.0 Å, keeping the interaction

strength εAB = 1.353931 KJ/mol (Lorentz-Berthelot mixing value).In set (1), there are three cases,

in (a) εAA< εAB < εBB where the value of solute-solvent interaction is in between the solute-solute

and solvent-solvent interaction strengths, in (b) εAA< εBB < εAB where the solute-solvent interaction

strength is bigger than both solute-solute and solvent-solvent interactions, and in (c) εAB< εAA<

εBB, where the solute-solvent interaction strength is lesser than both solute-solute and solvent-

solvent interaction strengths. In set (2), there are also three cases, in (a) σAA< σAB < σBB where the

solute-solvent zero potential diameter is in between the solvent-solvent and solute-solute zero

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potential diameter, in (b) σAA< σBB< σAB where the solute-solvent zero potential diameter is

greater than both the solute-solute and solvent-solvent zero potential diameters, and in (c) σAB <

σAA< σBB , where the solute-solvent zero potential diameter is lesser than both solute-solute and

solvent-solvent zero potential diameters.

For both the sets, the pressure P equal to 1 atm and the temperature T as 100 K. After

many trial runs to verify the existing results on total energy and volume of one component liquids,

time step of Δt* = 0.001 ps was found suitable. We have dealt with six different solute

compositions, namely 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0, for each case in each set. For each solute

composition, we have equilibrated the system for 500 ps, i.e the number of equilibration steps was

50,000,and performed the production run for another 50,000 steps, which made the total number of

steps 100,000 3. The plot for average volume of this mixture Vs mole fraction was studied.

(iv) Plots:

The following plots were obtained for the simple system considered:

- The plots of all the physical quantities mentioned in Table 1 were obtained with respect to

time.(figures 1 to 5 )

- The velocity autocorrelation function was plotted for the Lennard Jones fluid considered.

(figure 6 )

- All plots are attached at the end of the report.

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4. Results

4.1 Neat Lennard-Jones fluid:

- The plots of all the physical quantities mentioned in Table 1 were obtained with respect to

time.

- The velocity autocorrelation function was plotted for the Lennard Jones fluid considered.

4.2 Lennard-Jones Binary mixture:

- A number of trial runs for a binary mixture of Argon and Xenon were carried out using the

parameters mentioned in the methodology. However convincing results were not obtained and

further work is required to refine the simulation. This work is expected to provide very useful

insights into the anomalous composition dependence of viscosity, diffusion, and excess volume of

binary mixtures for which a suitable explanation is not available.

5. Conclusion

The current work is aimed at eventually studying various properties of Room-

Temperature ionic liquids using Molecular Dynamics simulation.

- During the tenure of the fellowship, an exhaustive review was carried out on the basics of

molecular dynamics and a simple Lennard-Jones system was simulated. All codes were written in

FORTRAN 90.

- The variation of various physical quantities including velocity autocorrelation was studied with

respect to time step.

- The use of dl_poly 2 software in molecular dynamics simulation was learnt and applied for

studying non-ideal behaviour of binary mixtures.

- The work on binary liquids is expected to provide very useful insights into the anomalous

composition dependence of viscosity, diffusion, and excess volume of binary mixtures.

- The topic of ‘Room-Temperature ionic liquids’ was reviewed.

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References: Textbooks:

1. ‘Computer Simulations of Liquids’ - M.P.Allen and D.J.Tidesley

2. ‘Statistical Mechanics’ - Donald.A.McQuarrie

3. ‘Statistical Mechanics’ - Terell L.Hill

4. ‘Physical Chemical Kinetics’ - RS Berry, SA Rice, J. Ross

5. ‘ Theory of Simple Liquids’ - J.P.Hansen and I.R.McDonald

6. ‘Understanding Molecular Simulation’ - D.Frenkel and B.Smit

7. ‘Computer Programming in FORTRAN 90 and 95’ - V.Rajaraman

8. ‘Physical Chemistry’ – P.W.Atkins

Research papers:

9. Welton.T ; Chem.Rev., 99 (8), 2071 -2084, 1999. 10.1021

10. Jindal.K.Shah et al ; Green Chem., 2002, 4, (2), 112-118

11. Peter Vassilev et al ; J.Chem.Phys, Vol. 115, No. 21, pp. 9815-9820, 2001

12. A.Rahman ; Phys. Rev.136, A405-A411 (1964)

13. A.Rahman and F.Stillinger ; J. Chem. Phys. 55, 3336-3359 (1971).

14. B.Bagchi and Arnab Mukherjee; J.Phys.Chem B 2001, 105, 9581-9585

15. D.M.Heyes; J.Chem.Phys. 1992, 96, 2217-2227

Web links:

16. http://www.ccl.net/cca/software/SOURCES/FORTRAN/allen-tildesley -

book/README.shtml

17. http://www.mse.ncsu.edu/CompMatSci/Tutorial/index.html

18. http://lem.ch.unito.it/didattica/infochimica/Liquidi%20Ionici/Definition.html#note

19. http://www.fisica.uniud.it/%7Eercolessi/md/md/node1.html

20. http://www.ch.embnet.org/MD_tutorial/pages/MD.Part1.html

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