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COMPUTER SIMULATION OF THE LIQUID Li4Pb
ALLOY
G. Jacucci, M. Ronchetti, W. Schirmacher
To cite this version:
G. Jacucci, M. Ronchetti, W. Schirmacher.
COMPUTER SIMULATION OF THE
J O U R N A L DE PHYSIQUE
Colloque C8, suppliment au n012, Tome 46, d i c e m b r e 1985 page C8-385
COMPUTER SIMULATION OF THE LIQUID Li4Pb ALLOY
+*
G. J a c u c c i , M. R o n c h e t t i and W. Schirmacher
Dipartimento d i Fisica, Universita' d i l'rento, 38050 Povo, Trento, I t a l y + I n s t i t u t e Max von Laue
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Paul Langevin, 156 X, 38042 GrenobZe Cedex, FranceResume - Une s i m u l a t i o n s u r o r d i n a t e u r de l ' a l l i a g e l i q u i d e Lig P b d 1085°K €?St p r e s e n t e e . Le c a l c u l e s t fond6 s u r l e s p o t e n t i e l s de p a i r e u t i l i s 6 s p a r
Copestake e t a l . pour d 6 c r i r e l a s t r u c t u r e de c e systeme : i n t e r a c t i o n s de Coulomb 6 c r a n t 6 e s p l u s une r e p u l s i o n de coeur mou. Les c o e f f i c i e n t s de d i f - f u s i o n a i n s i que l e s f o n c t i o n s de s t r u c t u r e s t a t i q u e s e t dynamiques s o n t 6va- l u d s e t compares aux r e s u l t a t s expdrimentaux. Les f o n c t i o n s de c o r r e l a t i o n s de p a i r e s s o n t a u s s i c a l c u l 6 e s . Les f a c t e u r s de s t r u c t u r e dynarnique m e t t e n t e n evidence l ' e x i s t e n c e d ' u n mode p r o p a g a t i f , s u r t o u t v i s i b l e s u r l a f o n c t i o n de Van Hove p a r t i e l l e L i
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L i , e t q u i e s t probablement un mode o p t i q u e dans Le l i q u i d e .Abstract - We present a computer simulation of the partially ionic liquid alloy Li4Pb at 1085K. The calculation is based on the pairwise potentials used by Copestake et al, to describe the structure of this system: screened Coulomb interact ions plus a soft core repulsion. Diffusion coefficients and static and dynamical structure functions are evaluated and compared with experimental results. Pair correlation functions are also presented. The dynamical structure factors show in addition to the interdiffusion mode a propagating mode which is most strongly visible in the Li-Li partial van Hove function. We interpret this mode as an optical-like mode in the liquid.
I
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IntroductionAn important and well studied binary metallic fluid is .the liquid alloy Li-Pb. Many of its properties have been measured over the last ten years: all of them vary in a non ideal way with the composition, and show a maximum deviation close to the composition Li0,80Pb0,20 iLi4Pb for short). {For a review of the experimental data, see ref .l). In particular, the system deviates strongly from the ideal mixing behaviour. The particles in the melt are not distributed at random: unlike atoms are preferred a s nearest neighbours.
The nature of the bonding in Li4Pb has been investigated by several authors, following mainly two different approaches. In the first approach the existence of chemical complexes with finite lifetime has been conjectured /2-5/; in the second one it has been suggested that bonding might be partially salt-1 ike ./6,7.,'. While direct evidence for the existence of chemical complexes has not emerged, many experiments indicate that there is charge transfer.
It has been suggested /i/ that the experimental concentration
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concentration structure factor Scc(q) can provide informations about the intermediate-range p a r t o f the " o r d e r i n g " o r " i n t e r c h a n g e " potential d e f i n e d a s v(r) = where U i j is the pairwise potential. Applying this procedure to L14Pb Copestake et al. /8/ found by using the mean spherical {MSA! and hypernetted-chain iHNC) approximations that the structure can be well decribed using a screened Coulomb iYukawa) potential {plus a short range *permanent Address : Phys. Dept. E 1 3 , Techn. Univ. Miinchen, D-8046 Garching, F.R.G.C8-386 J O U R N A L D E PHYSIQUE
repulsive part). This corresponds to a model in which negative charge is transferred from Li to Pb.
At the composition Li4Pb the Li and Pb atoms are expected to have similar diameters d (chosen equal to 2.658 from the experimental Scc data). The pair potential used by Copestake et al. /8/ in their HNC calculation which forms the basis of our simulation /9/ is of the form
where a and p are soft core parameters that were arbitrarily fixed to values similar to those used for molten alkali halides (a=2, p3.38). The values of A , Ql and Q2 were directly estimated from the experimental Scc(q) data to be
X=i.lft-',
Qi=QLi=.533 and Q2=Qpb=-2.i34s For a detailed descrrptron of how this potential was obtained we refer to /8/.In this work we adopted this potential to simulate the model on the computer using the method of Molecular Dynamics (MD). The double aim is to check the analytical results obtained by Copestake et al. against the computer simulation data, and to extend the calculation to the dynamical structure factors, to compare with recent neutron scattering data /lo/.
I1
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The comvuter simulationWe used the MD method which consists in the integration of the classical equation of motion for all the particles in the system. A constant force approximation for very small time intervals ~ 0 . 5 x 1 0 - ~ ~ s e c ) is assumed, and the trajectories of the particles in phase space are accurately followed.
We started our calculation using a cubic box composed by 250 particles: 200 Li atoms and 50 Pb atoms. The temperature was 1085K and the number density .04558A-~. The system was subjected to periodic boundary conditions: in such conditions the maximum possible wavelenght is equal to the box lenght. Correspondingly the only a1 lowed values of the wavevector are <nx ,ny ,n,)qmin, where n,, ny and n, are integers and q,in is equal to 2 N/lmax.wavelenght). In
our case q m i n was ,368-I.
After equilibration we started collecting data. In order to get the dynamical structure factor it is n,ecessary to run a long job (at least 10000 steps) recording on tape the position of all the particles every few steps. These data are then used to evaluate the space-time Fourier transform for each desired value of q. We run the system for 10000, 20000 and 50000 steps.
In order to check the influence of the finite size of our system we doubled the 1 inear dimension of the box: our "large" system was composed of ZOO0 particles, and it took one hour to run
it
for 500 time steps. We used this system only to get more extended and accurate glr) and S(q).To investigate the dynamical behaviour of the system for smaller values of the wavevector q without increasing too much the number of particles we used a rectangular box. In arbitrary program units this "long" box was 4 x 4 ~ 1 6 , while the "regular" box was 5 x 5 ~ 5 and the "large" one was 10x10~10. So we were able to gain a factor 3 in lenght (and hence to decrease by a factor 3 our smallest q) by only doubling the number of particles: the long box contained 510 atoms. Data were collected during a 20000 steps run after equilibration.
I 1 1
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Results and discussionIn fig. 1 we have plotted the static partial radial distribution functions g i jtr).
nearest neighbours: although P b h a s the same soft-core r a d i u s a s Li, the first peak of the gpb_pb(r) is at 4 . 9 8 .
Fiq.l:
partial radial distribution f u n c t i o n s g i j(r)In fig. 2 w e have plotted the Bhatia-Thornton / 1 1 / structure f a c t o r s which describe the c o r r e l a t i o n s of concentration a n d density fluctuations. 0 T h e first peak of the Scc 1 ies at q-1 .68-1 whilst that of the S n n is at q.-2.6~-1. T h e s e r e s u l t s are typical f o r s y s t e m s w i t h s t r o n g hetero-coordination and are of the kind o b t a i n e d strongly charge-ordered s y s t e m s /12/.
Fio.2:
T h e Bhatia-Thornton structure f a c t o r s I . . , a J0
2
4
6
8
10
/A-'
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C8-388 JOURNAL
DE
PHYSIQUEW e turn n o w to the discussion of our r e s u l t s for the dynamical structure factors. T h e s e can be c o m p a r e d with the experimental r e s u l t s obtained by Soltwisch et a1 ./lo/.
w:
Dynamic concentration-concen- Fip.4: Partia! Li-Li dynamictration structure factors. structure factors.
T h e propagating m o d e is present in Li, a n d completely absent in Pb, w h i l e the low frequency interdiffusion m o d e is present in both species. T h i s unforeseen fact c o m e s probably from the great difference between the m a s s e s of the two species. T h e Scc(q,w) (fig. 3) is dominated by a central peak. Beside t h i s diffusive m o d e we s e e clearly a l s o a propagating m o d e , w h i c h w a s not observed in the experiment: experimental r e s u l t s are r e p o r t e d for Q up to . 1 5 ~ 1 0 ~ ~ s e c - ~ w h i l e this m o d e
o c c u r s at f a r higher energy.
T o investigate the origin of t h i s peak it is very interesting to look at the partial structure f a c t o r s S L i L i ( q , w ) (fig.4) a n d Spbpb(q,w) ( t h i s last is shcwn in fig.5 for only low v a l u e s of 12 because f o r larger w it becomes indistinguishable from zero).
P b is in fact 3 0 times heavier than Li, and therefore the atomic motion time s c a l e s of the two e l e m e n t s are separated in a Born-Oppenheimer like fashion: short-period short-wavelenght density w a v e s are supported by Li a t o m s alone. T h e "sound velocity" extracted from the dispersion curve given by the m a x i m a of Scc(q,w) or S L i ~ i ( q , ~ ) . is 7 5 0 0 m/sec: almost four times the experimental value ( 1 3 1 , which is 2000 m/sec. T o explain this discrepancy w e decided t o check whether the behaviour of the dispersion curve w a s linear a l s o f o r lower v a l u e s of q. A bending in the curve could have revealed t h i s mode a s an optical one, due t o the light, fast and "positivelycharged" Li a t o m s vibrating against the heavy, still and "negativel)chargedU P b atoms. S o w e devised our "lon box a n d w e evaluated the S ( q , d for three lower v a l u e s of q (.11, .22 and . 3 4 s 1 , not shown in the figures). For all the new values of q w e observed a 1 inear behaviour of the dispersion curve: w e are therefore not able to either prove or disprove t h i s conjecture a s f a r a s the q dependence of the dispersion curve is concerned (we cannot see what h a p p e n s f o r still lower values of q).
On the other hand, it is interesting to look m o r e closely at the Spbpb(q,w) for low values of q: it p r e s e n t s in fact an outstanding deviation from a simple Lorentzian spectrum. T h e bumpy s h a p e of this curve is a l s o observable in the Scc:q,o) (when plotted on a scale different from that in fig.3), and a similar d e v ~ a t i o n h a s a l s o been reported in the experimental r e s u l t s (see the discussion in terms of "fast a n d s l o w relaxation channels" in /lo/). T h i s could be the signature of the "real" sound mode. T h e dispersion curve obtained by this shoulder g i v e s a sound velocity m u c h nearer to the experimental one.
W e used our simulation data to evaluate a l s o the self diffusion coefficients, calculated a s ~ = < x ~ > / 6 t . W e obtained ~ ~ ~ = 2 0 ~ 1 0 - ~ c m ~ / s a n d ~ ~ ~ = 3 . 5 x 1 0 - ~ c m ~ / s , which are in good agr,..:ent with the experimental r e s u l t s obtained
/lo/
for theLi f r o m incoherent scattering intensity and for P b from an extension of Darken's re1 at i on ( D L i , e x p = 2 ~ x i ~ - 5 c m 2 . / s at 1098K a n d ~ ~ ~ = 3 . 3 x i 0 - 5 c m 2 / s ) .
IV
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Summary and conclusionT h e usefulness of the model potential proposed by Copestake et al. /8/ h a s been shown. Static structure f a c t o r s which are in qualitative accord with the experiments have been obtained. Good agreement with the experiments is obtained a l s o for diffusion data. Dynamical structure f a c t o r s have been shown, presenting a rather peculiar behaviour: a propagating mode to w h i c h only Li a t o m s contribute h a s been found. T h i s is due to the vast difference between the atomic m a s s e s of Li and Pb. A possible interpretation of this mode a s an optic or a plasma mode h a s been suggested.
References
/1/ Ruppersberg, H. a n d Reiter, H., J.Phys.F:Met.Phys.
12
11982) 1311-25 /2/ Bhatia, A.B. and Ratti, U.K. J.Phys.F:Met.Phys.4
(1976) 927-41 / 3 / Bhatia, A.B. and Singh, R.N. Ph~s.Chem.Liq. fi (1982) 285-313 / 4 / Predel, 8. and Oehme, G., Z.Metallk. 7 0 11979) 450-3./5/ Hoshino, K. a n d Y o u n g , W.H., J . P h y s . ~ : K t . ~ h y s . (1980) 1365-74
/6/ Ruppersberg, H. a n d Egger, H., J.Chem.Phys.
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(1975) 4095-103 /7/ Nguyen, V.T. and Enderby, J.E., Phi1.Mag.3
(1977:) 1013-19 '/8/ Copestake, A.P., Evans, R., Ruppersberg, H. and Schirmacher, W.,J.Phys.F:Met.Phys.
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(1983) 1993-2010./9/ Jacucci,G., Ronchetti, M., Schirmacher, W., in "Condensed Matter Research Using Neutrons: T o d a y and T o m o r r o w " , ed. S.W.Lovesey and R.Scherm, Plenum Press, N e w Y o r k 11984) 139-61
/lo/ Soltwisch, M., Quitmann, D., Ruppersberg, H. a n d Suck, J.B., Phys.Rev. ( 1983) 5583-98
./11/Bhatia, A.B., T h o r n t o n , D.E., Phys.Rev. 21970) 3004-12
