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Submitted on 1 Jan 1985
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MOLECULAR DYNAMICS SIMULATION OF THE
GLASS TRANSITION IN BINARY MIXTURES OF
SOFT SPHERES
B. Bernu, Y. Hiwatari, J. Hansen
To cite this version:
MOLECULAR DYNAMICS SIMULATION OF THE GLASS TRANSITION IN BINARY
MIXTURES OF SOFT SPHERES
B. Bernu, Y. ~iwatari+ and J.P. Hansen
Laboratoire de Physique l'k'he'orique des Liquides, Universite' Pierre e t Marie Curie, 75230 Paris Cecklc 05, France++
RQsumk - Des calculs de simulations des ktats vitreux et surfondus en dyna- mique molQculaire ont kt6 effectuks pour des mklanges binaires de ItsphSres molles". La transition vitreuse est observQe pour une valeur d'un parametre effectif de couplage qui dkpend peu de la composition.
Abstract - Kolecular dynamics simulations of supercooled and glassy states of binary "soft sphere" mixtures were carried out, and both of the static and dynamical properties have been studied. The glass transition takes place at a roughly concentration independent value of an effective coupling parameter derived from a simple "one fluid" model.
; - INTRODUCTION
For many metallic alloys the crystallization can be by-passed to form a glassy state, while for pure metals it is very hard. For this reason and for the study of structural properties of metallic glasses we have carried out molecular dynamics simulations(VD) of binary mixture of soft spheres, which is regarded as a crude model for glass-forming alloys.
I1 - MODEL OF MOLECULAR DYNAMICS SINULATIONS
We consider binary mixture of N atoms of mass ml and diameter al and l% atoms of
2
mass m2 and diameter a 2 in a volume V, interacting through the purely repulsive soft-sphere potential:
where 19, and f3 (=1,2) are species indices. The advantage of simple inverse power potential is their scaling property, according to which all excess thermodynamic quantities depend only on two independent variables, which are conveniently chosen to be the number concentration of species 1, xl=N1/N
,
(with N=N +N ) and thedimensionless coupling constant: 1 2
+permanent address : Department of Physics, Kanazawa University, Kanazawa,
++
Ishikawa, 920, Japan.Equipe associse au CNRS.
C8-324 JOURNAL
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PHYSIQUE3
where pX=Nr; /V is the reduced number density and T*=k T / E is the reduced temperature.
1 B
The one component (x =I) version of the model has been extensively studied through, 1
Wonte Carlo or computer simulations, both in the stable fluid and crystalline phases /1,2/ and in supercooled and glassy states /3,4/. The freezing point of the one-component system is at r~1.15, while the glass transition is situated around r~1.5. Our MI simulations were carried out for binary mixtures with a diameter ratio o /a =1.4 and mass ratios m /m =2 and 4, using a very accurate finite differ-
2 1 2 1
ence algor~thm derived recently by one of us /5/ to integrate the coupled equations of motion of the N atoms nu ericaIj3 over typically 8000 time-steps of length
h
ht*=ht/~~0.05, where r=(m a / 1 2 ~ ) 1s our microscopic time scale. Systems of N=1 1
500 and 4000 atoms were simulated for concentrations x =0.9, 0.75, 0.5 and 0.25. We quenched an initial fluid configuration by rapid co&ling achieved by sharply reducing the velocities. In none of our runs did we observe any signs of nucleation confirming thus that mixtures (like alloys) are more easily quenched into a glass than one-component systems. This is obviously linked to the difficulty which mix- tu~esexperience during their evolution in phase space to find a statble lattice
structure compatible with the ratio of atomic volumes. I11
-
RESULTSComparison with an "equivalent" one-component system can be made by defining an effective diameter via
-
-
as suggested by conformal solution (or effective mediumA theory /6/. The corre- sponding effective coupling constant is reff=r(a /a )
.
We found that the equa- tion of state of the mixture in the supercooleds%
regimes is well re- produced at all concentrations ifr
is substituted in the very simple equation of state proposed by one of us /3/,effe.,P/(pk T) - 1 B = 6
+
6.848r
4 eff' (4)
Nerely perfect agreement is achieved if
reff
is systematically increased by about 2 % above its theoretical value deduced from eqn.(3) / 7 / .For sufficiently strong coupling
( r
>1.3), three partial pair di,stribution functions g (r) exhibit the familiar spffgting of the second peak which has been observed inUgnexomponentglasses (see fig. 1). The chracteristic feature of mixtures appears to be that the splitting occurs already in the supercooled liquids.We have calculated the four coordination numbers according to the formula
It is found that the v is nearly constant for each value of xl, independen of the value of
r;
the ne@ly constant values of the coordination numbers are vll=4.8, v =v i6.2 and ~ ~ ~ = 8 . 5 for x 1 =0.5; v11=8.5, v12=3:5! v2:=l0.6 and v22-5.0 for X:~O.%. For the case of x =0.5 these numbers slgnlflcan ly differ from those of any cubic lattice (like ~ a E l or CsC1) structure, while for x =0.75 these coordina-1
tion numbers are quite near to those derived from fcc structure with species 2 at the corner and with species 1 on the face.
Fourier transformation of the computed g (r) in principle yields the partial structure factors s (k). However the resultsalt small k are spoiled by the usual trimcation error ofa!he pair distribution function at large r. In order to estimate the small angle scattering amplitudes we used a recently proposed fluid integral equation / 8 , 9 / , which appears to yield fairly accurate pair structure, even in the supercooled liquid and glassy states. An example of the pair distribution function thus obtained from analytical equation is shown in fig. 2 with comparison with the
MI result.
(ko1730), before vanishing at last (ko1>40). This behavior is perfectly reproducible and 1s observed for equimolar mixtures over the whole range of temperatures typical of the glass
(ref
>1.5). Although less pronounc~d, this behavior is also observed at higher temperagures, .in the supercooled liquid so that it cannot be looked upon as a characteristic of the amorphous state. This interference effect was not ob- served at the other concentrations which we have studied, and may be attributed to an interplay of the relative phases of the oscillations in the S (k), which depend sensitively on the concentration and the size ratio (see fig. 3':)In order to locate the glass transition, we have calculated the two self diffu- sion .constants D (interdiffusion constant D12) from the velocity (interdiffusion current) autocor8elation functions. The reduced diffusion constants DT and D; drop below the noise level around reff=1.5, roughly speaking independtly of the concentration. Thus, within statistical uncertainities, atoms cease to diffuse beyond
r
-1 5 at all concentrations. It is found that values of DT and D$ which differ sf'igificantly from zero gives rise to an experimental formula llkewith A =0.21, A =0.11 and 5=10.6 for equimolar mixtures
,
so that our data are not 1 .incompatible wizh a simple Arrhenius law of the formula. The statistical uncertaini- ties are even larger for the interdiffusion constant D 2, which is a collective transport coefficient, but the trends are similar to tkose for D*.
Ergodicity problems linked to metastability has been investygated, both by varying N and by varying the mass ratio m /m which should have no effect on the
1'.
.
static properties. This is reasonably wel? verlfled both in the supercooled 1ic;uid and glassy states, but different mass ratios affect the velocity autocorrelation functions considerably. It is found from the Fourier transform (i.e. the power spectra) of the velocity autocorrelation functions and interdiffusion current auto- correlation functions that the observed power spectra are significantly sharper than in the supercooled liquid. Not surprisingly the'spectra are very sensitive to the mass ratio /7/. We have no obvious relation between the peak of the power spectrum and the Eistein frequency definded by
in the case of the mass ratio m /m =2.0, while for m /m =4.0 the power spectra for 2 1.
the species 1 exhibits a sigle $ea& at exactly the Elstem frequency. This behavior can be qualitatively attributed to a cage effect, which is more clear-cut when the larger atoms are much heavier than the lighter ones. The Einstein frequencies both for the mass ratios m /m =2 and 4 at each
reff
has been shown in fig. 4.2 1 ACKNOWLEDGEMENTS
The computations for the present work were carried out with the support of DRET, under contract N o 84/132 and of the Conseil Scientifique du Centre de Calcul Vectoriel pour la Recherche. One of us (Y.H.) would like to express his sincere gratitude to CNRS, JSPS, YAMADA ZAIDAN and Grand in-Aid for Scientific Research for their financial support.
REFERENCES
/1/ Hansen, J. P., Phys. Rev. A
2
(1970) 221./2/ Hoover, W. G., Ross, M., Johnson, K. W., Henderson, D, Baker, J. A. and Brown, B. C., J. Chem. Phys.
52
(1970) 4931./3/ Hiwatari, Y., J. Phys. Soc. Japan
47
(1979) 733./4/ Cape, J. N. and Woodcock, L. V., J. Chem. Phys.
72
(1980) 976. /5/ Bernu, B., Physica A (1983) 129./6/ Henderson, D. and Leonard, J . P., in "Physical Chemistry, an advanced Treatise" (Henderson D., ed.), p. 414 (Academic Press, New York, 1971).
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PHYSIQUE/8/ Rogers, F. J. and Young, D. A . , Phys. Rev. A s (1984) 999.
/9/ Kansen, J. P. and Zerah, G, to be published.
/lo/ Bhatia, A . B. and Thornton, D. E., Phys. Rev. B
2
(1970) 3004.G I R I R-fl ' R-B 0-0
'.57-
Fig. 1
-
Three partial pair distribution functions for x =0.5 (epuimolar mixtures) at reff=1.47. The abscissa isr / q
(4000-particle systemj.1.71) (4000-particle system).
Fig. 4 - Einstein frequencies calculated from eqn.(7) versus the effective coupling constant
r
for x ~ 0 . 5 . The ordinate is reduced Einstein frequencyef 1
