A computer simulation of the dynamical properties of diatomic fluids

Advances in Molecular Relaxation and Interaction Processes, 11 (1977) 295-308 0 Elsevier Scientific Publishing Company, Amsterdam - Printedin The Netherlands

A COMPUTER

SIMULATION

Gerard I-l.Wegdad,

Department (Received

OF THE DYNAMICAL

PROPERTIES

Gareth J. Evans and Myron

of Chemistry, 14 March

University

OF DIATOMIC

295

FLUIDS

Evans,

College,

Aberystwyth

SY 23 1NE.

1977)

ABSTRACT The recently

developed

[l] for the motion

of 256 atom-atom

study the behaviour velocity,

interatomic

(iii) increasing

density

T* at constant

The mean

appear

(i).

maxima

or minima

seem linear

in T* at constant

w decay generally

p* and d*.

square torque

is used to

linear and angular

the following

conditions.

temperature

(T*) and

d* at constant

T* and p*;

The following

indications

and force can exhibit

p* and d*.

(ii) Autocorrelation

of the intera omit vector

functions

of time.

early truncation

of the Mori continued

(iii) The effect

of elongation

functions

themselves,

and

These data do not support

fraction

at constant

range,

u or the angular velocity

on the same time scale as the vectors

become more complicated

algorithm

of p* or d*, but over a restricted

as a function

of high derivatives

potentials

(p* ) at constant

(d*); (ii) increasing

dynamics

functions:

torque and force under

number

distance

molecular

Lennard-Jones

of five autocorrelation

orientation,

(i) Increasing

Tildesley/Streett

with a simple function.

p* on dynamical

properties

as the above is much more pronounced

than that of p* at constant

d*

indicating

is the important

the

that hard-core

determination

anisotropy

of, for example

nematic

an

factor

in

such

behaviour.

INTRODUCTION

This paper reports autocorrelation numerically which

the Newton

is represented

of selected

equation

vectors

is a modified

*Physical

Chemistry

version

Laboratory,

fluid dynamics

obtained

for an assembly

by a pair of Lennard-Jones

algorithm

The Netherlands.

recent work on microscopic

functions

of 256 molecules,

each of

atom potentials.

The

of that which Tildesley

Nieuwe

using

by solving

Prinsengracht

and Streett

126, Amsterdam,

used

296 recently

[I1 to calculate

of computer

nitrogen.

were first developed primarily

[2,3]

in extending

relevant

thermodynamic

Atom-atom

these spectra

studies

can be related

frame, usually

this paper we observe

properties

dynamics

algorithms

some three years ago, and here we are interested

[4] or depolarised

molecular

molecular

their use to simulate various

to spectroscopic

microwave

and other equilibrium

potential

in such fields

Rayleigh

of light

of a unit vector

fixed in the dipole moment

the behaviour

properties

as the far infra-red/

scattering

to the motion

dynamic

[5].

Both

u in the

if this exists.

of the 256 molecules

In

under the follow-

ing conditions.

In

(i) Increasing'the interatomic _:!' doing this we effectively change

to a dumbell,

and so measure

tropy on spectral functions)

properties

density'p*

function effect

of p* and d*.

molecular

geometrical

and reduced

temperature

function

angular

the projection

fraction

of a dynamical'variable

1

acceleration approximating

variables"

[7] to

study of the

[8] to follow

the

temperature

p*.

(iii) It is- inherent"in

function

T*.

formalism

mean square torque both as a

It is also of great interest

up to the Mori continued function

shape

aniso-

of autocorrelation

the need for a detailed

averaged

(d* =d/u).

on and , the mean square force, of increasing

at constant

is deemed

acceleration atomic vector

'[Y] leading

such as the

to have near t = o an autocorrelation function

in comparison

or orientation.

of such a "fast variable"

the autocorrelation

formalism

[lo] to the correlation

that at some point a property

to a deita

the validity

operator

approximation

such as the angular velocity

investigate ting

transforms

[63 of memory

spectra has revealed

of'the ensemble

from a pseudospherical

(the Fourier

at 'constant reduced

far infra-red

distance

the effect of increasing

(ii) The recent applications

behaviour

(interpotential)

function

with "slow In this paper we

hypothesis

of the derivative

by calcula-

of the angular

G, and comparing its decay rate with that of u, the inter_., . (normalised to unity), related to o by u = w x u.

_-

---

COMPUTATION

2 The Newton

equations

a time increment u are always

those for nitrogen,

only for d* = 0.3292 dynamic

are solved by the predictor-corrector method with -15 sets. The Lennard-Jones parameters e/k and

of 5 x 10

stability

so we have

- the nitrogen

a "real" molecular

internuclear

of the system is judged

distance.

on criteria

system

The thermo-

such as the constancy

291 of the total pressure, internal

calculated

or configurational

were calculated unstable

found to be constant

of the correlation

ison of some calculations The statistics

an anistropic

essentially

anisotropy.

can be judged

from the compar-

out at 200 and 400 time steps in fig (1).

RESULTS

Using the two-centre

core, and the dispersive

is representative

The computations

U.M.R.C.C.

the first

after about 200.

since most auto correlation

is slight.

repulsion

so that the whole

functions

were

carried

are

in figs (I) to

potential

point

of geometrical

CDC 7600 link in 20 minute

builds

is cut off at 3.2s,

and electrostatic

out via the Aberystwyth

segments

in

-

of real time.

AND DISCUSSION

Some results different reduced

are illustrated

auto correlation

number

Some features popular

models

diffusion

density2

which

momentum

formalism

correlation

function

(5) decays

$(o).;(t)>

with

form

’

[12],

is a single function

models

is a

[~~],for the

cl43 quite easily using

the autocorrelation

function

of

that of ii. Since the auto-

has the units of angular

to the torque autocorrelation

on the e

corrected

function

u can be, derived

[15] wherein

some currently

that rotational

The M and .J diffusion

compared

1'

and vice-versa.

the torque auto correlation

vector

2 and i are slowly decaying

it is related

It is obvious

even in its;inertia

at the origin.

operator

temperature

auto-correlation

[13], and where

of andagainst,

can be used to criticize

of the fluid state.

of the interatomic

projector

and as plots

p* at a constant

is very inadequate,

delta function

in figs (1) to (5) in terms of several

functions

are revealed

the angular

exponential

motion

functions

carried

are adequate

The variation

where

and the total

The andterms

to zero in about 50 to 100 steps, as is illustrated

damped (5).

theorem,

per molecule.

with runs of 1600 time steps after rejecting

few, but were

The stability

by the virial

energy

function,

time scale as .

acceleration,

whic'h in figs (2) to

It seems that
(t)> sometimes decays on a much longer scale (figs (2) to (5)), but more often it does not.

Even the autocorrelation

times takes longer

to decay than ,

cated function fraction

of time than the latter

representation

usefulness

on the hope that autocorrelation

u might have had simpler time dependencies empirically.

some.71.~, and is a much more compli.

(fig. (1)).

[16] of +(0).2(t)>

could have been represented

function



The Mori continued

depended

partly

functions

for its

of derivatives

than that of 2 itself, According

to our results

of

and on

i

298

- 0.5

0.5

-0

0

- -0.5

-0.5

20

40

60

60

I 20

100 1

!

,

I

I

I 40

I

]

I 60 ,

8

I

I 60

b

,

!

100 I

7

-0.5 -

0.5

I 20

I,,

I

40

60

I

I 60

IS

I

I, 20

I 40

I

I 60

I

11’ 60

I-

Figure (1) Left to right, top to bottom: normalized autocorrelation functions. (a) Effect of using 400 and 200 time steps on the calculation of <:(0).2(t)>, d* =O.l, T* =3.6, p* -0.1. (1) 400 time steps, (2) 200 time steps (b) as for (a), d* =O.l, T* =2.32, p* =0.645. (c) Linear velocity autocorrelation function, d* =0.3292 (N2). (d) as for (a), d* =O.l, T* x3.6, p* =0.7. (e) , N2, T*= ,p*= N2,T* = (f) $$o).$(t)>, N2,T* =2.2, p * =0.645. (g)G(o)%(t)>, > p*= (h) , N2,T* =2.0, p* -0.645. Abscissae: time steps. Ordinates: (a) 103 ; (b) .

299 nitrogen hypothesis

are not favoured

this does not mean merely

both

and rough dumbells,

by the present

that Mori's

that the successive

functions

this hope

equations

kernels

of time than hitherto

in

computer

function

its second memory pertaining

to points

However

untenable,

his series are more complicated Of course

envisaged.

knowing

function.

evaluations.

are conceptually

series is still a useful way of generating autocorrelation

and the fast variable

the memory

the whole

only the -short time behaviour

Below we dicusss

function

time dependence

in more detail

of an

of, say,

results

(i) to (iii) of the introduction.

T* = 3.6, d* = 0.1, p* = 0.2 to 0.8 (Fig. 2) This set of data is intended reduced number

dominated,

decaying

that observed rotations

density

rapidly

by Kneubuhl

function

tends to oscillate derivative

and exhibiting

and Keller

of angular

velocity,

with the others

troque

oscillations

increase

ethylene

has suggested

or decrease

with

temperature

The force and linear velocity this pressure

range.

At p*=O.2

former has an extended with

the well-known

This means

velocity

value,

a.c.f. would

negative

a long period, negative

tail

together with

show no

a miximum

[18] and harmonic

with number

a turning

density.

well Recent

gaseous

point with increasing

at constant

that

p*.

a.c.f. 's have maximum

tail.

At p*=O.7

and the velocity [20], indicative

variation

through

becomes

the force a.c.f. a.c.f.

in turn develops

of hydrodynamic

our range.

at the high p* are those of angular

their short time behaviour

since in

the latter decays very slowly and the

the limit of very long times, outside a.c.f.'s

part and

and similar work on Ccl4 and CS2 liquids has revealed

may increase

oscillates

free

of the

range of p*.

of [I91 compressed

that exhibits

to

of which decays very slowly

the whole

work on the f. i-r. induced absorption

pressure,

the dimensions

of hard collisions

a monatomic

similar

The form of the torque auto-

that both and go through

theories

predict

maximum

is small in absolute

the angular

reduced

is inertia

[17] for the exceptionally

the a.c.f.

throughout

It is significant whereas

at constant a.c.f.

a secondary

p*: having

square torque

the limit of vanishing

at p*=0.5,

molecule

is very similar but this has a negative

at the higher

that the overall mean

decay.

spherical

the effect of increasing

this range the orientational

of HF and HCl in SF6 solvents.

correlation

compared

on a roughly

Throughout

temperature.

to simulate

coupling,

The slowly decaying

and linear velocity,

progressively

the fact that there is now apparent

in

different.

althowh

These,

shift in the minimum

Figure (2) Left to right, top to bottom, (T* =3.6, d* =O.l): (a) Plot of 103 vs. p* (b) Plot of vs. p* (c) Autocorrelation functions for p* =0.2: (1)velocity; (2)orientation; (d) As for (c), p* =0.3. (3)angular velocity; (4)force; (5)torque. (e) As for (c), p* =0.4. (f) As for (c), p* =0.5. (g) As for (c), p* =0.6 (h) As for (c), p* =0.7. (i) As for (c), p* =0.8. Ordinates, (a) 103 ; (b) &> Abscissae (a) p*; (b)p*; (c) -(i) time steps.

301 of the orientational

a.c.f.

movement

constraint

and gradual

increases.

The oscillatory

indicative

of the beginnings

a given molecule

upon translational

nature

motion

The changes

of the nature

in the fluid.

The force a.c.f.

is mass dependent

as gradual

of the collision

spectrum

encaging

of

throughout

rather

than any

p* these are not as

At the very,highest

since there is a narrower

preferred,

as p*

in the a.c.f.'s

as at p* =0.5, but at p* =0.8 certain

forceful

freedom of angular

movement

of the force a.c.f. at p* ~0.8 is

of "rattling"

takes place.

their range are indicative "structuring"

shows up the considerable

types of encounter

are

of force and torque.

and the torque a.c.f.

inertia dependent.

T* = 3.6, p* = O.I,d* = 0.2 to 0.7 (Fig. 3) The torque increases the angular absolute density

velocity

by an order of magnitude

a.c.f. simultaneously

value of is small throughout of the fluid is low.

gression

is that the velocity

ively closer

do those of force and torque a.c.f.

decay becomes

slower

expected

translational Elongation

apparent

(zq =sxFJ.

of molecular a degree

how these would motion

of ordering

theories

elongation

movethe

density

fluid.

into the fluid as shown by the a.c.f. with

density

at high d*.

It is

the first studied in this respect

that

at

Here we have the first vague indications

in the formation

of a nematic

phase - the molecular

that the mean square force becomes as increases

dramatically,

[18] would be hard pressed

progressively although

to reproduce

present

this accurately

of the five a.c.f. 's to decay more quickly with increasing

is that of angular moment

intermolecular orientation

in turn affect

to birefringence.

It is significant

analytical

impact

tempertaure.

smaller with elongation

The only-one

It might be

in this low number

decay of the orientational

important

in the and the

would have a large effect upon rotational

than number

log jam leading

maximum

with elongation

at d* =0.7.

of this progression

reduced

get progress-

the same rate, as

The secondary

progressively

has a larger

of factors

molecular

velocity

from a comparison

elongation constant

a.c.f.'s

not be obvious

aspects

number

and angular

and almost exponential

introduces

almost exponential

the molecular

aspect of this pro-

disappears

that elongation

ments but it would

because

and

The

The most interesting

and at d * =0.7 decay at virtually

together

orientational

with elongation,

decays more quickly.

of inertia

effects,

velocity

and this occurs

is increasing

would

even though the

with d*, which

alone cause the torque,

to decay more slowly on the scale of absolute

The mass of the molecule

is of course unaffected

in the absence

angular velocity time (in ps).

by elongation.

of and

302 1

!

G6Q!~

2Q

40

60

80

100

06 04 02

_2 -04 -06

' ~$

'

2o

'

~~o'

~0'4'

4'$~oo

1 Oa

G6 04 02

~6 oN 02 o

_2 -04 -06 i

20I ~

40I

I

~0

' --~80

100 l

'

5 1

20

40

60

80

1~

o.~

Q~

0.6

Q4

0.4

02

0.2

v02

-0.2

0

-04 -Q6

-0.8

-06

201 t 401 ' ~ '

~ ~0

;

-o~

~ ~ ~ ~,,~o

Figure 3 Left to right, top to bottom, (p* =0.], T* =3.6): (a) vs d*; (b) vs d*; (c) Autocorrelation functions, (1)velocity; (2)orientatlon; (3)angular velocity; (4)force; (5)torque d* =O.I. (d) as for (c), d* =0.2. (e) as for (c), @* =0.3. (f) as for (c), d* =0.4. (g) as for (c), d* =0.5. (h) as for (c), d* =0.~ (i) as for (c), d* =0.7. Ordinates: (a); (b) Abscissae: (a) d*; (b) d*; (c)-(i) time steps.

303

P* = 0.643, T* = 2.3, d* = 0.1 to 0.7 (Fig. 4) The mean square

torque behaves

well defined minimum is a pronounced elongation

at the nitrogen

change from gas-like

d* increases

a.c.f. decays

at constant

exponential. factor

Hard-core

which

is $(o)._(t)>

solid the velocity

is characterised

freedom,

a.c.f.

by rotational

is nearly

and, in this limit,

the orientational exponential,

a.c.f. and where

very rapidly.

is a pure cosine, but the nematic

(albeit restricted)

and translational

and the forms of the five a.c.f.' s may well be an extreme version

Unlike

the longer molecules

between

dumbell.

at p* =O.l

there is no

the rates of decay of velocity

particular

and angular

velocity,

torque and force, at p* =0.643, where with d* =O.l the velocity decays much more rapidly

than the angular velocity, The linear velocity

d* ~0.7 both are oscillatory. decays

initially

much more quickly

the orientational

a.c.f.' s at p* =0.643

the linear velocity

density.

lower number

behaviour

decay

is the slower

translational

at p* =0.647,

ively encaged

under

elongation,

and the built-in potential

of our a.c.f.'s.

hard-core

out the orientational

why nematic

behaviour

rather of repulsive

repulsive

density

density. on liquid

(i.e. molecular

oscillations

is much more effective

a.c.f.'s,

seems to be not

part of our double

in promoting

elongation

flattening

reduced number

the

than the effect of p* at constant

seems dominant

Certainly,

are effect-

For the longest molecule

longer at the higher number

at constant

is much more pronounced

at the hi&er

the oscillatory

that the molecules

torsion.

is much

a.c.f. at the

This elongation

much more effectively,

identical,

the higher

for d* =0.7 the velocity

d* =0.7 indicating

and "rattling"

time of the orientation

Lennard-Jones

the torque

and p* =O.I are virtually

to decay.

motion

The effect of elongation structure

a.c.f. at p* ~0.643

a.c.f. decays much faster'at

On the other hand,

density

p* constrains

at

more slowly, but at d* =O.l, the decay time of

a.c.f.'s

number

or

a.c.f.

and thereafter

for the larger molecules,

and orientational

although

so

an

an important

at t=o, oscillates

of ours at d* =0.7 for the Lennard-Jones

coupling

in liquids

where

as the

at d* ~0.7, where

must be, in extremis,

properties,

a

there

the angular velocity

which

[6a, 211 to be slowly decaying

function,

In an Einstein phase

behaviour

For example,

of "structure"

of nematic

observed

the memory

to liquid-like T*.

the orientational,

anistropy

in the determination

in the appearance has been

At this density

slowly at d* =O.l, but is oscillatory

are all the other c.f.' s except

but now .F'>shows

as before,

elongation.

in some at

and our data probably

so much a matter

shape)potentials.

explain

of attractive We will

but

test the

I

1

20

>

I

I

I

40

60

60

,

I

100

.0.5 11

20

I1

II

I

40

60

60

II’. 100

Figure 4 Left to right, top to bottom, (p* -0.643, T* ~2.3). d*; (c) Autocorrelation functions (a) vs.d*; (b) VS. (l)velocity, (2)orientation; (3)angular velocity; (4) force;(5)torque; (d) as for (c)., d* =0.5. (e) as for (c), d* ~0.425. d* =0.7. (f)as for (c), d* =0.3292. (g) as for (c), d* =0.3. (h) as for (c), d* =0.2 (i) as for (c), d* =O.l. Ordinates: (a) ; (b) Abscissae: (a) d*; (b) d*; (c)-(i) time steps.

1.6

1.7

1.6 1.9

2

2.1

1.6

1.7 1.8

1.9

2

2.1

0.60

0.7

0.72

0.66

0.7

0.72

1

0.5

0

-0.5

-0.5

II _^

Figure

^^

^^

^^-

I 20 1

I I

I,, 40 ,

I I

60 II,

I 00

I 1

I 100 ,I

5

Left to right, top to bottom (for nitrogen) 2 (a) 10 vs. T* at p* =0.6964, d* t0.3292. vs. T* at p* =0.6964, d* =0.3292. (b) (c) $O&Tq2' vs. p * at T* =1.75, d* =0.3293 vs. p* at T* =1.75, d* =0.3292 (d) (e) Autocorrelation funcitons, (1) velocity, (2) orientation, (3) angular velocity, (4) force, (5) torque; T* =1.75, p* SO.6964 (f) as for (e); T* =1.75; p* =0.68. (J.4 as for (e); T* =1.9 ; p" =0.6964 (h) as for (e); T* =1.75; p" =0.74. Ordinates: (a)lO; (c)WTq2>; (d) Abscissae:

(a) T*; (b) T*; (c) p*;

(d) ni (e)-(h) time steps.

I

306 effect of quadrupole-quadrupole The mean square over this admittedly well models,

where

(elastic) models,

interaction

torque and force increase restricted

density

work.

linearly with

temperature

This is in accord with harmonic

range.

= c kT, where c is a constant, and hard-core 2 where = (F'/d) kT where F' is a force coefficient

and d is a mean angle of free rotation. number

on forthcoming

In this simulation

the reduced

p* is kept constant.

T* =1.75, d* =0.3292,

p" =0.68 - 0.74

In this range the five a.c.f.' s change very little but and trend upward

in value with increasing

inflexion,

or maybe a slight maximum,

d* =0.3292

corresponds

at constant

reduced

ethylene,

calculated

a minimum

as pressure

isoelectronic

to nitrogen,

temperature.

measurement

to note that for

induced absorption,

temperature

[19].

shows

Ethylene

is

several

function

(such as scattering

of the depolarisation

fluid, s&hat

can be discerned. mation

are for a real molecule

forms of autocorrelation

experiments

and Raman wings) will be more fruitful one selected

of

with nitrogen.

here that further

radiation,

pressure

at constant

It is clear from the varied displayed

so our results

there is an

The elongation

It is interesting

from far infra-red decreases

p*, although

at p* =0.715.

of laser

of fluorescence,

infra-red

when carried out simultaneously

different

aspects

of the motion

on

of

Unfortunately, n.m.r. spin-rotation relaxation infor'6. is virtually impossible to obtain at present Larmor

on

frequencies

in non-viscous

will soon directly

examine



at constant

very scarce

at present.

fluids, nor is it probable

that any technique

the decay of $(o).i(t)>.

Measurements

p* and varying

T* are difficult

Those at a constant

even of

to carry out and

T* and varying

p* are much

easier. Despite formalism

these drawbacks

some recent empirical

usage

[20] of Mori

has shown that does not have a simple experimental

ence on either p* or T*, but can go through a maximum (2) to (5) reveal

or minimum.

dependFigs.

such behaviour.

It is clear that a study of alone by one experimental technique averaging;

in isolation

disposes

of a lot of information

and in fact experiments

below the far infra-red function,

which

difficult

and tedious

on Debye relaxation

look at the long-time

is almost always exponential.

by statistical at frequencies

tail of this autocorrelation Such experiments,

although

to carry out, thus yield close to the minimum

307 information

possible

about the trochilics

even when this happens available

to be dipolar.

large computers

intermolecular

potentials,

which,

of a typical molecular

Now we are fortunate

with admittedly

fluid,

in having

rough and ready

can be used to yield useful

complementary

data.

ACKNOWLEDGEMENTS

G.J.E.

and M.W.E.

respectively.

M.W.E.

We are very grateful Oxford)

thank the S.R.C.

thanks

for a studentship

the Ramsay Memorial

to Prof. J.S. Rowlinson

for invaluable

and fellowship

Trust for a fellowship.

and Dominic

Tildesley

(of

help in the early stages of this work.

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