THE TRANSPORT COEFFICIENT FOR NON-TRANSITION ELEMENTS

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THE TRANSPORT COEFFICIENT FOR

NON-TRANSITION ELEMENTS

Dj. Jovik, M. Davidovik, D. Chaturvedi

To cite this version:

Dj.

Jovik, M. Davidovik, D. Chaturvedi.

THE TRANSPORT COEFFICIENT FOR

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JOURNAL DE PHYSIQUE CoZZoque C8, suppZ6ment au n08, Tome 41, aoiit 1980, page

Cg-276

THE

T R A N S P O R T C O E F F I C I E N T F O R N O N - T R A N S I T I O N

ELEMENTS

Dj. JoviE, M. DavidoviE and D.K. Chaturvedi X

Boris ~ i d r i ; I n s t i t u t e o f NucZear Sciences, vinza, Belgrade, YugosZavia

Abstract.- Using the recent experimental data for liquid aluminum, copper, lead, tin, gallium and bismuth, the effective interatomic potential has been calculated by inverting the Week-Chand- ler-Anderson (WCA) perturbation theory.Also, first and second derivatives of interatomic poten- tial y(r), along with different correlation functions as g(r), c(r), have been calculated for the mentioned liquid metals. With knowledge of the above quantities and their products, different trans- port coefficients, such as self-diffusion D, viscosity r7 and conductivity &have been studied, for the group of transition and non- transition elements, and their results compared with classical measurements of the macroscopic quantities.

Our aim in this work was t o study the transport processes i n the transition and non-transition elements from the aspects of their microscopic sta-

tic properties. The transport coefficients D can be defined in t e r m s of the intermolecular forces through the relations of Rice and Kirkwood /1-3/,

. .

Here

\Q

( r ) and ( r ) denote the derivatives of the effective pair potential

'4

( r )

,

g ( r )

,

m, n a r e r e s - pectively the pair correlation function, m a s s and density of the system. In a similar manner the s h e a r viscosity coefficient

,

the bulk viscosity

2,

and the coefficient of the conductivity

a'

can be expressed a s / 3 k / 00 2 T n z n

'2-%

Tfi4[?~r't$i(ri]9(.,d.

O ( 3 )

%=

9 n&fa

jr9[-i+)

+

~ ( r d

g

(r) dr ( 4 ) and

J

a _a

-

y

4 - 1 4 r ( 5 ) 0

From the above equations, i t can be shown that a l l

the transport properties can be obtained if the shape of the effective interatomic poteritial

\P

( r ) is known.

In the present work we extract

'f

( r ) from the '

.. . .

measured data of the structure factor s ( ~ ) . The pro-

blem is of fundamental importance and many different theoretical approaches were used in order t o study this problem. In this paper we use an inverted perturbation scheme of weekPs

,

Chandler and Anderson

/ 5 - 6 / (recently used by Mitra /7/ f o r liquid Ne and Pb) to derive the effective pair potential

Y

( r ) for s i x liquid metals, namely copper /8/, aluminum /9/,

gallium

/lo/,

tin /11/, lead /12/ and bismuth /13/

since we had a recent structure data of those metals f o r several temperatures. However, due t o limitati- ons of space we shall present h e r e only the results f o r copper and aluminum.

In WCA approximation,

\P

( r ) is divided into a short-range repulsive

\p

(r) and long-range attrac-

S tive part (

r

) a s

where r is the distance a t which

'f

( r ) has a mini-

0

mum. Then instead of calculating the actual system directly, one s t a r t s with a t r i a l ( T ) system with hard spheres fluid a s the reference system and t r e a t s the effect of attractive interaction a s a small perturbation t o this reference system. Thus the t r i a l pair potential can be written a s

'f,

0)

=

Y'd

(r

)

+

Ye(')

( 7 )

where

\PA

( r ) is the h a r d sphere potential, d being the

hard c o r e diameter. The t r i a l pair potential i s made

, X) I

.C

.T.P.

,P.

O.B. 586 Miramare, 341000 Trieste, Italy

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close to the actual pair potential by choosing the ha- r d c o r e diameter through a n optimized method. There- f o r e this method is a l s o applicable t o systems

-

like liquid metals, where long range part of the potential

is a large one.

Our approach differs from that of Mitra /14/ who used random phase approximation (RPA) for the structure factor of the t r i a l system. As i t is well known, RPA does not include the effect of correlation between atoms which is important in determination of the structure factor a t small wave vectors. The shape of S ( q ) and the location of the f i r s t peak ( q ) have

0

very different effects on the pair potential of a liquid state. For example, shape of S( q ) f o r q

>

q a s well

0

a s the position of f i r s t peak (but not i t s height) determinate the repulsive part of the potential, especially f o r high densities. On the contrary, values of S( q ) f o r q ( qo and the height of the f i r s t peak (but not i t s af- fect t o the long-range p a r t of the pair potential

\PQ(r),

Therefore, we consider a t r i a l structure S ( q )

T

where S ( q ) is the hard sphere s t r u c t u r e factor .and d

yL

( q ) is the Fourier transform of the effective atrac- tive field VC(r). Recently, Emns and Schirmacher /15/ and a l s o Gaskell /16/ have tried t o calculate

Vt(q) through a perturbation procedure. I n this work we use the expression due to Gaskell /16/

IP

p'yttd

=F~+&.F$~Y~($)

S:($)[SO+Z$-~]

( 9 0

where Sd(q) was replaced by S ( q ) . exP

In o r d e r to calculate ST( q ) from Eqs. ( 8 ) and ( 9 ) the hard sphere structure factor and the attractive p a r t of the pair potential is needed. For S ( q ) , we u s e

d

Ashcroft and L e h e r /17-18/ solution of the Percus- Yevick equation of the hard sphere s t r u c t u r e factor and for\pe(r) we make the ansatz

where c ( r ) is the direct correlation function. In the physically inaccessibIe region, r

<

d, q ( r ) is assu- med of the form /19/

fiYe(rj =

A + (

(-u)B+(!-u~

Cn

pn

(2"

- 0

n=o (11)

r

where P a r e the Legendse Polynomials and

n

=i.

The physical reasons for choosing such a form and the determination of the parameters A , B, Cn, etc. have already been discussed by WCA and Ailawadi e t al. /19-20/. The pair correlation function of the r e a l system

-

liquid metal is now obtained by the approximation

9

(Y)

*

I-Q

Y

(r)] YT(r)

(12)

where YT(r) is related to the pair correlation function of the t r i a l system

Finally, we obtained an optimum value of the hard c o r e diameter d, using the condition

Numerical procedure

Due t o experimental limits the structure factor can not be precisely measured both for small and l a r g e wave vectors. Therefore, we extrapolate S ( q ) t o the exact value of S ( o ) , determined by

exP

the isothermal compressibility

K,

a s

n

S [ o )

9;p.

Kt (15)

For q

>

qmax (max.measured q ) , we used the expression

Parameters C ' n

=

1,.

. .

,

5 a r e determined by the n

'

least square fit of eq. ( 16) with S ( q ) for q grea- exp.

t e r than the f i r s t minimum position. We a l s o defined S ( q ) f i r obqC20

2-',

which is sufficient t o ob-

exP

tain q(

r

) by the Fourier transform.

For numerical calculations of a transport coefficients we started with the Aschcroft-Lekner expression f o r the hard sphere structure factor. The initial values of the hard c o r e diameters, now denoted a s d"), a r e obtained for a l l liquid metals by comparing the

S

( q ) with the rkspective

S

( q ) through the

d eXP

last square method. The next s t e p is t o obtain Yc(q) from Eqs. (10) and (11). This requires the data of c ( r ) , f o r which we u s e the relation

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C8-278

JOURNAL DE PHYSIQUE

Then we tested i t s accuracy through another compressibility equation

By means of Eqs. ( 8 ) and ( 9 ) we obtained S ( q ) , T and i t s Fourier transform g m ( r ) . By substitution of

1

g ( r ) and 'PT(r) in (13) and ' P T ( r ) in (12) one gets T

( 2 )

a pair potential, denoted a s ( r )

,

corresponding t o g ( r )

*.

Finally, the condition ( 14) determi-

exp

nates an improved value of the hard-core diameter d ( 2 ) , where ~ ( ~ ) ( r ) is again split up a s described

( 2 )

in eq.

( 6 )

t o obtain a new

yL

( r ) . With new values of s , ( ~ ) ( ~ ) (corresponding t o d ( 2 ) and \ P f 2 ) ( q ) ) a s input, we calculate

S

( ( q ) and repeated the

T

whole procedure until a satisfactory convergence f o r \ P ( r ) is achieved.

After this, the f i r s t and the second derivatives of the potential with the pair correlation function determinate the transport coefficients a s given in Eqs. ( 1 )

-

( 5 ) .

Results and Discussion

By the above procedure we calculated the effective interaction potential in liquid copper and aluminum. The experimental s t r u c t u r e factor data S

( 9 ) for liquid copper have been measured by O.Eder, B.

Kunsch e t al. (19791, a t two temperatures : 1393 K

and 1833 K. The structure factor s ( ~ ) for liquid aluminum has a l s o been measured by Dj. JoviC et al. (1977) a t two temperatures : 939 K and 1073

K.

The structure factor data both for Cu and Al w e r e extend- ed up to 20

a

-

'

i n order t o get more accurate Fourier transform t o calculate the correlation functions. For long wave vectors of momentum transfer the variati- on of S ( q ) is small and less than 0.1%. According to this procedure the ripples in g ( r ) f o r small r almost disappear. The calculated direct correlation function c ( r ) shows very small fluctuations f o r l a r g e

r

but c ( r ) exhibits small positive values when

r

increases. This means that a long-range attractive p a r t of y t ( r ) is a l s o small; pronounced bumps disappear through iteration process.To get the short-range component

of y b ( r ) one has t o solve a linear integral equation of the Fredholm type-This equation has t o be convert- ed into a system of linear equations /21/

where

fii

and

A .

a r e variables proportional t o the J

pair potential \Pe(r) in RPA and

GRPA

approximation respectively. The matrix element G..

,

i - e . the ker-

1J

ma1 of Fredholm"~ equation ~ ( q ' , q ) is expressed a s

p1+2

i

G ( P : ~ I = = S ~ ~ ~ ~ ~ )

J ~ ~ s e ~ ( x ) - g d x (20)

ly-pi

where h . J i s a variable of the integration interva1.A special program was used for solving the matrix equation ( 19 )

.

The details of this program and the results will be presented in the next paper.

In the Figs. 1 and 2 is shown, for two different temperatures the pair interaction p t e n t i a l Y ( r ) . Both figures show the expected behaviour f o r data a t higher temperature Ghere potential

\P

( r ) is softer and the steepness is smaller than near the melting point.

As i t can be seen f r o m Eqs. 1-5 the transport

.

coefficients generally depend on the shape of

..

( r ) and

\Q

(

r

) which again a r e sensitive on the profile. Table 1 contains numerically calculated transport coefficients and compared with experimental ones for two metals and given temperatures.

Since we have data for two temperatures only, the value of the thermal conductivity coefficient should be considered a s a preliminary one.

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Fig. 1

I t should be pointed out that transport coefficients a r e v e r y sensitive t o the values of S f q ) a t low

q, what h a s been improved i n calculation by GRPA

.

The differences i n Table 1 between calculated and experimental values a r e caused by neglection of electron contribution.

References

1 Rice, A .S., Kirkwood, G. J., J.Chem.Phys., 31 (1959) 901.

2 Bansal, R., J.Phys. C . 6 (1973) 1201 and 3071

3 Eder, J.O., Kunsch, B., Presse, G., Sonderdruck

a u s 52. Heft 6 (1976) 319.

4 Croxton, C., Introduction t o Liquid State Physics, John Wiley and Sons, London, 1975.

5 Weeks, D.J., Chandler, D. and Afidersen, C.H., J.Chem.Phys., 55 (1971) 5422.

6 Andersen, C.H., Chandler,D.J. and Weeks, D. J.

J. Chem. Phys., 57 (1972) 2626.

7 Mitra, K.S., 3.Phys. C. 1 0 (1977) 2033. 8 Kunsch, 0. ( 1979) (private communication). 9 Jovik, Dj., Padureanu, I. and Rapeanu, S., Inst.

Phys.Conf,No. 30, Chap. 1, Part

1,

l ( 1 9 7 7 ) 120. 10 Bellissent, R. ( 1979) (private communication).

11 JoviC, Dj. and Padureanu, I., (1978) (unpublished data)

Fig. 2 Acknowledgement

The authors wish t o thank t o Drs. T. Gaskell,

U. Dahlborg and I. Ebbsja f o r the useful discussions and suggestions. Also, Dr. M .Davidovik e x p r e s s e s his gratitude t o ICTP a t T r i e s t e f o r the hospitality during his work a t the Computer Centre,

12 Dahlborg, U., ~ a v i d o v i d ,

M.

and Larsson, K.E., Phys-Chem, Liquid 6 (1977) 149.

1 3 Lampater, P. ( 1979) ( ~ r i v a t e communication). 14 Mitra, K.S. and Gillan,

J.

M., Phys. C. 9

(1976) 1515.

15 Evans, R. and Schirmacher, W., J.Phys. C 11

(1978) 2437.

16 Gaskell, T., Phys. Lett. 65A (1978) 421. 17 Ashcroft, W.N. and Langreth,

C.D.,

Phys. Rev.

159 (1967) 500.

18 Davidovi;

,

M

.

,

PhD Thesis, Belgrade University, (19781 (unpublished).

19 Ailawadi, K.N.

,

Miller, E.D. and Naghizadeh, J., Phys.Rev. Lett. 36 (1976) 1494.