CHANGES IN THE STATIC STRUCTURE FACTOR BY VARYING THE PAIR POTENTIAL STUDIED BY COMPUTER SIMULATION

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CHANGES IN THE STATIC STRUCTURE FACTOR

BY VARYING THE PAIR POTENTIAL STUDIED BY

COMPUTER SIMULATION

T. Kinell, I. Ebbsjö

To cite this version:

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JOURNAL DE PHYSIQUE CoZZoque C8, suppl6ment a u n08, Tome 41, a o e t 1980, page

~ 8 - 3 0 1

CHANGES I N THE S T A T I C STRUCTURE FACTOR

BY VARYING THE P A I R P O T E N T I A L STUDIED

BY COMPUTER

SIMULATION

T. Kinell and I . ~ b b s j s *

Department o f P h y s i c s and MQasurements T e c h n o l o g y L i n k a p i n g U n i v e r s i t y , 5-581 83 Linkiiping, Sue'de

h he

S t u d s v i k S c i e n c e R e s e a r c h L a b o r a t o r y , S-611 82 Nykiiping, Sue'de

Introduction

The purpose of this molecular dynamics

(MD) study is to gain some insight into

how the part outside the repulsive core

(hereafter called attractive part) of a

pair potential influences static proper-

ties. The simulations are done at liquid

densities for a few pair potentials, all

with the same repulsive core. Here we pre-

sent results for the pair distribution

function, the static structure factor,

the isothermal compressibility and the

self-diffusion coefficient. To our know-

ledge no systematic investigation of the

effects of the attractive part in the po-

tential has been published earlier.

Potentials and description of the

MD-

systems

The four pair potentials (POT1, POT2, POT3,

and REP-POT), shown in Fig. 1,

are des-

cribed by the analytical formula

2

@

(r)

=Ae

-ar2

+

~ e - ~ ~ c o s

(kor+a)

+ ~ e - ~ ( ~ - ~ o )

/r.

Over the region of interest for the simu-

lation, the potential POTl is very similar

to one of the pair potentials used in an

earlier study [l]. The reason for this

choice is to allow further analysis of

the results of this earlier work, espe-

cially to extend the investigations

to dy-

namical properties. POT2 is obtained by

varying the parameters in the gaussian

term centered at ro, POT 3 neglects this

term and REP-POT, finally, is truncated at

the first

-minimum.

@('I

I

I I I I I I

0 2

4

6

rlA)

Fig. 1. Pair potentials used in the MD-ex-

periments. POT1

-( ) ,

POT2

(-

-

) ,

POT3

( - - - ) .

REP-POT

(--- )

is truncated at the posi-

tion of the first minimum.

Technical details of the MD-simulations

are the same as reported in our earlier

work [I]. The number density is

n

=

5.2763-10

2 8

m-3

,

the temperature is

about 1000

K and the number of time steps

are so far 480.

The radial distribution function

The radial distribution functions, g(r),

for the four systems are shown in Fig.

2.

We find that

1. The position and the height of the

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C8-302 JOURNAL DE PHYSIQUE

Maximum estimated

error :

5

0.02

Fig. 2. Pair distribution functions for the four potentials in Fig. 1. Notations

are the same as in Fig. 1.

main peak is nearly the same for all oscillations at large r.

four potentials.

2. The steep rise and the width of the main peak is rather similar but there is a small systematic dependence on the potential.

3. The amplitude of the oscillations at large r is most pronounced for POT2. The potential POT1 and REP-POT yield the smallest amplitude and are very similar at large r.

4.

The second maximum in g(r) for POT2 is

distorted on the right side.

The static structure factor

By Fourier transformation of the pair distribution functions we obtain the static structure factors, S(k), shown in

Fig. 3 . To reduce the errors in the

transform for small k-values we have ex- tended g(r) by a simple analytical formu-

la

[ l l ,

but the errors are still signifi-

cant for

k

< 2.3 Sianificant diffe-

rences in g (r) are nevertheless clearly reflected in S(k). The position of the main peak is about the same but the

Our results are complementary to those of height of the peak as well as the depth

Schiff [ 2 1 and Hansen and Schiff [31. These of the first minimum varies as the ampli-

authors varied the repulsive part of the tude of the oscillations in g(r). The

potential and observed a variation in the amplitude of the oscillations in S(k) at

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Maximum estimated

error at the main peak

:

2

0.1 Fig.

3.

Static structure factors for the four potentials in Fig. 1. Notations

are the same as in Fig. 1.

tials. The results are cemple~.entary

to

lidification at least not

z s

in2nifcsted

what Schiff

[

21 and Hansen and Schiff

[ 31

in the diffusion coefficient.

found, namely that a change in the repul-

sive part only effects the amplitude of

the oscillations in S(k) at large k-values.

An interesting observation is the splitt-

ing

of the second peak in

S

(k) for POT2, it

seems to come from the distortion of the

second peak in

g ( r ) .

Amorphous materials

show a similar structure in S(k).

In or-

der to illucidate this feature we have

calculated the self-diffusion coefficients

for all the potentials, see Table 1. As

can be seen there is no indication of so-

Table 1

POT1

POT2

POT3

REP-POT

D

( 1 0 - ~ m ~ / s )

4.10 1.20

2.66

4:15

Compressibility

The long wave-length limit of S(k3 yields

the compressibility'&,

through the rela-

tion S(o)=nkgTXT, where n is the number

density,

kg

is.3oltzmann's

constant and

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c8-304

JOURNAL D;E PHYSlQUE

We have t h e r e f o r e c a l c u l a t e d S ( k ) d i r e c t - past o f a p a i r p o t e n t i a l . Four d i f f e r e n t l y from t h e MD computer.density-maps 1 s ( k ) = \C exp

( i k

Ri)

I

2 i=l f o r f o u r k - v a l u e s (0.30, 0.66, 0.89 and 1 . 2 2 The r e s u l t s a r e shown i n F i g . 4 where t h e c u r v e s a r e weighted l e a s t s q u a r e f i t s of S ( k ) =so t s 2 k 2 t o t h e MD-results. The d i f f e r e n c e s between t h e p o t e n t i a l s r e f l e c t e d i n g ( r ) and S ( k ) and d i s c u s s e d above i s n o t observed i n t h i s c a s e . The r e s u l t s f o r REP-POT i s now r a t h e r c l o s e t o POT3 and f o r a l l k - v a l u e s t h e r e s u l t f o r P O T l c l e a r l y l i e s above t h e o t h e r c u r v e s . The r e a s o n f o r t h i s i s n o t y e t u n d e r s t o o d . I I

-

_{Maximum e s t i m a t e d}

-

e r r o r :

5

0.002

-

I I F i g . 4 . The s m a l l wave-vectof r e g i o n o f t h e s t a t i c s t r u c t u r e f a c t o r f o r t h e f o u r p o t e n t i a l s i n F i g . 1. MD-results :

+

(POT1)

,

o (POT2)

,

r

(POT3)

,

V (REP-POT)

.

C o n c l u s i o n s

We have used t h e method o f MD t o e s t i m a t e

p o t e n t i a l s w i t h t h e same r e p u l s i v e c o r e have been u s e d . Our r e s u l t s c a n b e summa- r i z e d a s f o l l o w s .

1. The a m p l i t u d e o f t h e o s c i l l a t i o n s i n t h e p a i r d i s t r i b u t i o n f u n c t i o n becomes l a r g e r a s t h e a t t r a c t i v e p a r t o f t h e p o t e n t i a l i n c r e a s e s . The p o s i t i o n and th'e h e i g h t o f t h e main peak remain a l - most unchanged. 2. The peak i n t h e s t a t i c s t r u c t u r e f a c t o r i n c r e a s e s when t h e a t t r a c t i v e p a r t of t h e p o t e n t i a l i s i n c r e a s e d . The ampli- t u d e of t h e o s c i l l a t i o n s a t l a r g e k- v a l u e s i s t h e same f o r a l l p o t e n t i a l s . 3 . The p a i r d i s t r i b u t i o n f u n c t i o n and t h e static s t r u c t u r e f a c t o r a r e b o t h v e r y

s i m i l a r f o r POTl and REP-POT.

4 . The c o m p r e s s i b i l i t y l i m i t and t h e s m a l l k-behaviour of S ( k ) does n o t r e f l e c t t h e b e h a v i o u r f o r t h e p o t e n t i a l s ob- s e r v e d i n g e n e r a l f o r g ( r ) and S ( k )

.

Our r e s u l t s show t h a t c o n t r i b u t i o n s from t h e a t t r a c t i v e p a r t of t h e p a i r p o t e n t i a l a r e i m p o r t a n t t o t h e d e t a i l s o f t h e s t r u c - t u r e o f l i q u i d s . We i n t e n d t o c o n t i n u e o u r s t u d y i n o r d e r t o b e a b l e t o draw more de- f i n i t e and g e n e r a l c o n c l u s i o n s . We w i l l a l - s o e x t e n d t h e i n v e s t i g a t i o n s t o i n c l u d e dy- namical p r o p e r t i e s . S t u d i e s o f t h i s k i n d a r e a l s o v e r y u s e f u l t o t e s t t h e o r e t i c a l schemes f o r t h e c a l c u l a t i o n o f p o t e n t i a l s from s t a t i c s t r u c t u r e f a c t o r s . R e f e r e n c e s 1. I. E b b s j o , T. K i n e l l and I W a l l e r 5. Phys.

g ,

1865 ( 1 9 8 0 ) . 2. D. S+Lff Phys. Rev.

186,

1 5 1 (1969)

3 . J . P . Hansen and D. Schiff Mol. phys.

25, 1281 ( 1 9 7 3 ) .