Effective pair potential for liquid iron near the melting point from low-angle structure factors

104

Journal of Non-Crystalline Solids 106 (1988) 104 107 North-Holland, Amsterdam

EFFECTIVE PAIR POTENTIAL FOR LIQUID IRON NEAR THE MELTING POINT FROM LOW-ANGLE STRUCTURE FACTORS Takashi ARAI, Isao Yokoyama and Yoshio WASEDA* Department of Mathematics and Physics, The National Defense Academy, Yokosuka 239, Japan *The Research I n s t i t u t e of Mineral Dressing and Metallurgy( SENKEN ), Tohoku University, Sendai 980, Japan An e f f e c t i v e pair potential for l i q u i d iron at 1833K has been estimated from the low-angle structure factor data. This potential indicates a marked difference in the position of the f i r s t minimum from the conventional Johnson potential which has been widely used in the analysis on various prope r t i e s of solid and amorphous iron. A molecular dynamics calculation using two potentials has also been carried out. 1. INTRODUCTION

2. MODEL

Since the pioneering works of Johnson and March I and Johnson et al 2 much e f f o r t has been

I t is supposed that a meff(r) can be divided into two parts:

devoted to determine an e f f e c t i v e pair potential Veff(r) from measured structure factor a(k) of a l i q u i d metal. An inversion scheme3 that has been proposed by one of the present authors appears to be competitive in accuracy with those employed hitherto. This scheme is based on the

v,ss(r)

= v .... (r)

(1)

+ vt=,l(r)

where Vcore(r) is a positive repulsive potential which is mainly responsible for the a(k) at

random phase approximation and the Monte-Carlo

large-wavenumber k, while, V t a i l ( r ) is the t a i l potential for modifications at small k. The

(MC) simulation data obtained by Hansen and

V t a i l ( r ) is known to be approximated as 8

Schiff". However, the lack of elabolate lowangle-scattering experiments

.

.

k BT

f ho/

1

i

\minkr4~k2~

seriously prevents (2)

us from the use of the method. The low-angle a(k)'s

(k:from O.IA - I to 1.35A- I ) were recently

where ko=31/3ko(=2.38~ "1) with kD being the ra-

determined by a high temperature X-ray d i f f r a c -

dius of the Debye sphere and n is the number den-

tion technique for the present purpose s . The

s i t y of ions. In this work we employ the r -6

a ( k ) ' s in the low-wave vector region have been

potential as Vcore(r) and the corresponding

evaluated by superimposing on the conventional

acore(k)4. As is seen from figure i , the observed a(k)

large-angle measurements6. Thus, i t has now become possible to extend the method of Ref.3 to

is moderately well reproduced by the MC struc-

l i q u i d iron for obtaining information of V e f f ( r ) . The aim of t h i s paper is as f o l l o w s : f i r s t , to

ture factor corresponding to the r -6 potential.

present the potential VAYW(r) newly estimated including a comparison with the Johnson potent i a l 7 v j ( r ) ; second, to examine the v a l i d i t y of VAYW(r) in terms of a molecular dynamics( MD ) calculation.

The f i t

because we have a sum rule: ~o(a,x~, (k)-a

. . . . (le) ) 4 ~ k 2 d k = O .

(S)

However, we shall neglect any imperfection in the f i t

0022 3093/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

in the region for k>k0 is not perfect

in this region and concentrate exclu-

105

T. Arai et al. / Effectiue pair potential for liquid iron

oo8 F

oo61 A °'°'I tl o.o

/

7

..../

10

u ~vw(r)

= o

k0

~,

• ""

u J(r)

,<

~"

o

10

A

r/A

v ~3

-10

2 k0

4

6~ k /A-l

Figure 1 Structure factors : MC4(

8

i0

Figure 2 E f f e c t i v e pair p o t e n t i a l s f o r i r o n . s,~

) and experiment f i r s t - n e i g h b o u r distance is not to allow to give any d e f i n i t e comment on the v a l i d i t y of a metal-

s i v e l y on the low-k reglon up to kO.

l i c potential I°

3. EFFECTIVE PAIR POTENTIAL

4. MD CALCULATION

The e f f e c t i v e pair potential was calculated

A MD study has been carried out f o r both the

f o r two cases; using the low-angle structure da-

VAYW(r) and v j ( r ) using a manner s i m i l a r to the

ta up to k0 and up to k=4.0A - I . The l a t t e r re-

previous works1~, 12 f o r N(=216) iron atoms in a

gion includes the f i r s t

cubic box( the side length is L ).

minimum of a(k). Surpris-

i n g l y , the two sets of the V e f f ( r ) ' s are found to be much the same, although there are small differences in d e t a i l near the f i r s t the f i r s t

minimum and

4.1 Pair d i s t r i b u t i o n function The pair d i s t r i b u t i o n function

g(r)

is com-

puted by the d e f i n i t i o n

hump in these p o t e n t i a l s . This suggests

that the information on the V t a i l ( r ) s u f f i c i e n t l y

g(r)

l i e s in the low-wavenumber region up to kO. Lanqon et al 9 used the modified Johnson pot-

= (~]_) 4~r~Ar

(4)

where is the mean number of p a r t i c l e s

t e n t i a l 7 f o r s t r u c t u r a l description of a metal-

situated at a distance between r and r+~r from

l i c glass model. Figure 2 shows the present pot-

an a r b i t r a r i l y

t e n t i a l VAYW(r) together with the Johnson potent i a l v j ( r ) f o r comparison. As is seen from f i g u r e

Ar=O.O2A.

2, there is a marked difference in the p o s i t i o n

VAYW(r) and v j ( r ) are given in f i g u r e 3 together

of the f i r s t

with the experimental one ~. The g ( r ) using the

minimum. Thus, our i n t e n t i o n is

chosen centered p a r t i c l e with

The two sets of the ~ r ) ' s

computed using the

focused on the question: which potential is s u i t -

v j ( r ) has a rather high f i r s t - p e a k and the dis-

able for p r e d i c t i n g physical properties of l i q -

t i n c t o s c i l l a t i o n s in the large distance of r.

uid iron? Because of the presence of a large

4.2 Specific heat at constant volume

energy in a metal, the coincidence between the

The s p e c i f i c heat at constant volume due to i o n i c motion Cv ion may be calculated from the

position of the f i r s t

f l u c t u a t i o n s in the k i n e t i c temperature of the

volume( structure-independent ) part of the t o t a l minimum in Veff(r) and the

T. Arai et al.

106

/

Effective pair potential for liquid iron The two p o t e n t i a l s provide almost the same re-

for VAYW(r)

s u l t , although the agreement between theory and experiment should be considered at the semi-

i

-• -. -, expt6 for vj(r)

quantitative level.

"i

4.4 S e l f - d i f f u s i o n constant ( i ) Mean-square displacement

v

To obtain the s e l f - d i f f u s i o n constant D we have computed the mean-square displacement r 2 ( t ) as a function of time from i t s d e f i n i t i o n 0

--

0

I

I

3

6

2

r (t)

1

=

where ~ i ( t )

Figure 3 The experimental g ( r ) is obtained by Fourier transformation of the aexpt(k)6._ _

N

-N<,Z l ~ ' ( t ) - r ' ( O ) 1 2 > = l

(7)

is the p o s i t i o n of p a r t i c l e i at

time t . The values of the D are evaluated from the l i n e a r part of r 2 ( t ) by a least-squares f i t to a s t r a i g h t l i n e and given in table 2. (ii)

Velocity a u t o c o r r e l a t i o n function

A more detailed description of the d i f f u s i v e system through

behaviour can be obtained from the v e l o c i t y autocorrelation f u n c t i o n ~ ( t )

C~°~ = 3 [ 1

3N -2~-I J

(5)

i

defined as

N ~

g~( t ) : -~< [lv~(O ) . v { ( t ) >

(8)

where the angular brackets denote a time averwhere ~ i ( t )

age. Table i shows the calculated r e s u l t s together

is the v e l o c i t y of p a r t i c l e i at

time t . The values of the D computed from t h i s

with the experimental data. The standard devia-

equation are also given in table 2 together

tions

with those obtained from other t h e o r e t i c a l mod-

are of the order of 2% of these values.

The values of C 's, i n c l u d i n g the e l e c t r o n i c v c o n t r i b u t i o n , are in reasonable agreement with

els z~. As is seen from table 2, the d i f f u s i v e

the experimental value.

t i a l employed.

4.3 Vacancy formation energy

phenomenon is quite s e n s i t i v e to the pair poten4.5 Shear v i s c o s i t y c o e f f i c i e n t

The r e l a t i o n between vacancy formation energy Ev and l i q u i d structure theory has been re-

The shear v i s c o s i t y c o e f f i c i e n t ~ is calculated by the Kubo-Green formulaZ2:

ported by Bernasconi et al ~4. Their formula, r e s t r i c t e d to hot close-packed

metals because

of i t s neglect of atomic r e l a x a t i o n round the

Table i

vacancy s i t e , can be w r i t t e n in terms of the

Cv( Nk B unit ) and Ev( eV unit )

Veff

C ion v

C el v

ccalc v

ceXpt v

ECalc v

Eexpt v

uAy W vj

3.26 3.11

1.08

4.34 4.19

4 38 "

1,0 I .i

1.6

g(r) and V e f f ( r ) as

Fv =

(6)

Using equation(6), we worked out Ev. Calculated and observed values of Ev are given in table 1.

c: I, c: xpt (Ref.13) E expt v

(Ref.15)

71 Arai et al. / Ef/ectiee pair potential for liquid iron

REFERENCES 1. M.D. Johnson and N.H. March, Phys. Lett.

Table 2 D( 10-5cm2/s ) and ~( cP ) Vef f r2(t)

@(t)

DLT

DSS

calc

expt

vAy W 1~j

4.66 1.52

8,12

8.14

5.13 14.9

5.82

4.51 1.45

107

3 (1963) 313 2. M.D. Johnson, P. Hutchinson and N.H. March, proc. R. Soc. A282 (1964) 283

DLT and DSS denote the value calculated by the linear trajectory theory and the small-stepdiffusion theory, respectively. (Ref.16) expt (Ref.17)

3. I. Yokoyama and S. Ono, Phys. Chem. Liq. 14 (1984) 83 4. J.P. Hansen and D. Schiff, Mol. Phys. 25 (1973) 1281 5. Y. Waseda and S. Ueno, Sci. Rep. Res. Inst.

=

__CI

~< j

(9)

where Jxy is the component of the tension tensor.

Tohoku univ. 34A (1988) 15 6. Y. Waseda, The structure of Non-Crystalline Materials(McGraw-Hill, New York, 1980)

The calculated value of q are shown in table 2.

7. R.A. Johnson, Phys. Rev. 134 (1964) A1329

The values of n for the VAYW(r) is consistent with the experimental one 17, although the error

8, S. Naito and I. Yokoyama, J. Phys. F: Metal

in the n is rather d i f f i c u l t

9, F. LanGon, L. B i l l a r d and A. Chamberod,

to estimate correct-

Phys.

15 (1985) L295

l y and i t should be kept in mind that the shear

J. Phys. F: Metal Phys. 14 (1984) 579

v i s c o s i t y is also very sensitive to the Ueff(r)

IO.V. Heine and D. Weaire, Solid State Phys.

used.

24 (Academic, New York, 1970) pp.249 !

II.J.M. Gonzalez Miranda and V. Torra, 5. CONCLUSION The following conclusions could be made. ( i ) The g(r) calculated from the VAYw(r) is in reasonable agreement with that from experimental data. ( i i ) The thermodynamic properties such as C v and Ev are rather i n s e n s i t i v e to the V e f f ( r ) used, while the transport properties are quite sensitive to the V e f f ( r ) . ( i i i ) The shear v i s c o s i t y c o e f f i c i e n t is well predicted by the VAYW(r). ( i v ) Accurate experiments on the s e l f - d i f f u sion are highly desirable to c r e d i t the f u l l potential of the VAYW(r).

J. Phys. F: Metal Phys. 13 (1983) 281 12.L.M. Berezhkovsky, A.N. Drozdov, V.Y. Z i t s e r man, A.N. Lagarkov and S.A. Triger, J. Phys. F: Metal Phys. 14 (1984) 2315 13.H. Itoh, I. Yokoyama and Y. Waseda, J. Phys. F: Metal Phys. 16 (1986) Ll13 14.J.M. Bernasconi, N.H. March and M.P. Tosi, Phys. Chem. Liquid 16 (1986) 39 15.H.E. Schaefer, in: Positron Annihilation, eds, P.G. Coleman, S.C. Sharma and L.M. Diana, (North-Holland, Amsterdam, 1982) pp.369 16.Y. Waseda and K, Suzuki, Scripta Metall. 7 (1973) i157 17.M. Shimoji and T. Itami, Atomic Transport in Liquid Metals (Trans. Tech. Publications,

ACKNOWLEDGMENTS Authors would l i k e to thank Mr,Naito for helpful

advice in the MD calculation. I.Y. and Y.W.

are grateful for p a r t i a l support from the Kawasaki Steel Corporation, Technical Research Division.

Switzerland, 1986) pp.192