IC/94/247

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

^-DYNAMICS IN SIMPLE LIQUID METAL

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

S.K. Lai

and

H.C. Chen

MIRAMARE-TRIESTE

IC/94/247

International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

/J-DYNAMICS IN SIMPLE LIQUID METAL

S.K. Lai 'International Centre for Theoretical Physics, Trieste, Italy

and

H.C. ChenDepartment of Physics, National Central University,

Chung-li 32054, Taiwan, Republic of China.

M1RAMARE- TRIESTE

August 1994

'Permanent address: Department of Physics, National Central University, Chung-li32054, Taiwan, Republic of China.

Abstract

The modified hypernetted chain theory is applied to calculate the static liquid structure

factor for simple liquid metals at rapid cooling. With a proper choice of bridge function,

it is found that this integral equation theory is capable of yielding reasonably accurate

liquid structure factor in the supercooled liquid region. To exploit the usefulness of the

calculated liquid structure, we combine these structure data with the mode-coupling theory

to investigate the dynamics of the /3-process. It is demonstrated in this work that the

critical temperature for an ideal glass transition predicted here for liquid metals Na and

K are slightly lower than those determined previously [S.K. Lai and H.C. Chen, J. Phys.

Condens. Matter 5, 4325 (1993)] using computer simulated structure factors. In particular

we show that the material-dependent exponent parameter A which is used widely in the

literature as a fitting parameter in the analysis of light scattering experiments can be given

more physical significance if one correlates the change of A with the microscopic interaction

of particles specifically for the liquid metal, Lennard-Jones fluid and hard-sphere system.

The implication of the present results and the possibility to extract useful information for

more complicated systems will be discussed in the text.

I. Introduction

The .^-relaxation for a glass-forming supercooled liquid has been a subject of current in-

terest. Empirically this phenomenon was studied using different experimental probes. The

dielectric loss spectroscopy is perhaps one of the earlier experiments that provides evidence

for the existence of /3-process. Such an experimental technique was applied extensively by

Goldstein and .Johari [1,2] to understand the /^-resonances for a wide class of rigid molecular

substances and amorphous polymers. Specifically they observed in their measured dielectric

loss spectra a secondary absorption band. To explain its origin they [3,4] proposed a potential

energy barrier picture which is a concept comprising the cooporative and the hindered types

of molecular rearrangements. In an entirely different approach Knaak et al [5] carried out

an inelastic neutron scattering experiment for an ionic glass. This experiment focused on the

scaling behavior of & resonances. The dynamics of/^-relaxation was investigated also by Cum-

mins and coworkers [610] for the ionic and molecular glasses using the Brillouin-scattering

spectroscopy. This study was made feasible through an improved technique on the Sander-

cock tandem multipass Fabry-Perot interferometer which is an experimental setup specifically

designed to circumvent the overlapping order problem. Their Brillouin spectra measurements

for the dynamical structure factor exhibit a fractal law which varies as i j"1 ' "" ' , where a is an

exponent parameter to be defined below. In an attempt to look into the dynamical behavior

of a supercooled liquid, Pusey and Megen [11] prepared suspensions of small, equal size, and

long-hours metastable spherical colloidal particles and used the dynamical light scattering tool

to investigate the structural and dynamical properties of the system. Because of the unique

means the colloidal system was prepared, the interactions of particles resemble that of a fluid

of hard spheres thus permitting a tractable theoretical analysis of the /^-process.

Theoretically Goldstein [3] and Johari [4] attempted to interpret the /^-process as origi-

nated from the mobile clusters of atoms which, due to stringent atomic rearrangements, suffer

the hindered type of motion for encaged atoms. Such an idea has been questioned, however,

by Gotze and Sjogren [12] to be physically unsound since this latter feature has not been

observed generally. In a scries of works [12-19] Gotze, Sjogren and coworkers have studied the

/3-relaxation within the recently developed mode-coupling theory (MCT) of glass transition.

According to the latter works the /3-phenomenon can be understood microscopically by ex-

amining the temporal and spatial changes of the normalized intermediate scattering function

R(r,t). Specifically the MCT predicts the presence of three relaxation regimes for R(r,t): a

microscopic relaxation time to ~ 10 Ti|s corresponding to uncorrelated binary collisions such

as those happened in an equilibrium liquid state, an intermediate mesoscopic time- tg typically

on the order of 10~8-10~ns and a long-lasting time scale t,, ~ l-103s (so-called a-relaxation).

The /3-relaxation belongs to the mesoscopic time scales. It can be shown in the MCT that

near a critical temperature Tc the normalized density correlation function R.(r,l) exhibits a

factorization property which expresses R(r,t) in terms of an universal temporal scaling func-

tion uncorrelated with two material-dependent spatial functions. This factorization property

has been investigated first for a Lennard-Jones system [20], subsequently for a hard-sphere

system [17,18,21], recently for a lattice gas system [22] and Cummins and coworkers [6-10]

have quantitatively examined this property for the ionic and molecular glasses. In this paper

we intend to supplement these works for a metallic system. Our purpose for carrying out this

calculation is threefold.

First of all we plan to perform a realistic calculation for the supercooled liquid metals

Na and K by applying the modified hypernetted-chain integral equation theory to the liquid

structure. This recently proposed approach has been evaluated critically [23] by us to be

highly reliable and accurate. As the static liquid structure factor S[q) is a sole input to MCT

this application is important not only because it offers an alternative means to calculate Tc its

comparison with that Tc determined previously [24] using the molecular dynamics simulated

S(q) further exploits the potentiality of the method to an undercooled liquid. Second, we

intend to compare the scaling properties of the /^-relaxation for the liquid metal, Lennard-

Jones and hard-sphere systems. This comparison is instructive since it enables us to give more

physical significance to the MCT exponent parameter A (to be defined below) which has been

widely used as a fitting parameter in experiments on glass transition. In particular one can

infer from the variation of A the microscopic interactions of more complex systems. Finally,

we shall calculate the lagged particle distribution function P*(r,t) and see if the Ps(r,t) result

for a metallic system can give any hint to the large discrepancies in this quantity [18] between

the hard-sphere and binary soft-sphere systems.

The format of this paper is as follows. In the next section we briefly summarize the for-

mulas used in the modified hypernetted-chain theory and draw attention to a bridge function

that we found l.o yield reasonable S(<;). The interatomic potential used in the calculation is

also outlined briefly. Section Ifl is devoted to an introduction of the mode-coupling theory.

Here the nonlinear integral equation which R(r,t) satisfies will be introduced. The explicit

expressions for the scaling formula R(r,t) and for the distribution function P'(r,t) will be given

in this same section. We discuss our numerical results in section IV and make a concluding

remarks in section V.

II. Static structure factor

In this section we summarize essential formulas that appear in the modified hypernetted-

chain theory and outline briefly the evaluation of the S(g).

The modified hypernetted-chain integral equation begins with the Ornstein-Zernike rela-

tion

-r'ijc(r') (1)

where /> is the number density; c(r) is the direct correlation function; and d(r) = g(r) - 1 is

the total correlation function, g(r) being the pair correlation function. To solve this equation

one must supplement it with a closure. The most frequently used form is

g{r)=exp[d(r)-c(r)-/3<4(r)-B(r)], (2)

where /? = (1<BT) ' is the inverse temperature; <f>(r) is the interatomic potential, and B(r)

is the bridge function. Following our previous applications to liquid metals we construct the

pair potential at each T using the modified generalised non-local model pseudopotential of Li

A

et al [25] and Wang and Lai [26]. According to the latter works <4(r) can be written

• (

where G^c(q) is the normalized energy-wave-number characteristics with the Singwi et al [27]

exchange-correlation factor included and 'L\g = 7? - p\,l and pd being the nominal valence and

the depletion charge, respectively. It should be emphasized that in Eq. (3) proper attention

has been given to the one-electron energy and pseudo-wave-function by carefully incorporating

higher-order perturbative corrections through a parameter in the bare-ion pseudopotential (see

[25] for details). For the bridge function B(r) we have chosen the empirical B(r) of Malijevsky

and Labile [28] which reads

fi + ^(r/ff - 1 )][r/(T - 1 - f3][r/<r - I - v^/iWi) r < I^/T

Aiexp[—(^s(r/c — 1 - f4)]sin[A2{r/CT — 1 - f4)]/r r > i^cr

(r being the hard-sphere diameter and the parameters A, and v, are determined, respectively,

by continuity conditions and by fitting to all known structural and thermodynamic computer

simulation data of hard spheres over the entire fluid range up to the density of freezing.

Equations (l)-(4) constitute a set of equations which are to be solved self-consistently for

g(r). The static structure factor S(</) is then obtained by Fourier-transformation.

III. Mode-coupling theory

Original derivation of the non-linear self-consistent equations has been well documented

in several papers by Gotze and Sjogren [29-32]. Here we review the essential ones sufficient

for our present purpose.

A. Basic Formulas

Central to the MCT is the density-density correlation function F(r,t) = < 6p(r,t) Sp(0,0)>

where 6p(r,t) is the density fluctuation for the position vector r at time t and < ., > denotes

the ensemble average. According to the MCT the normalized correlator R(q,z) = P(q, z)/S(q)

satisfies

Rfa.z) = — —• (5)zM(q, z)

where m is the atomic mass and M(g, z) is the generalised fractional term or the so-called

memory function. Here the Laplace transform of a quantity O(g,t), denoted by a caret, is

defined as

O(q,z) = i / dte"'O(q,t) (6)Jo

Since we shall be concerned with the long-time behavior, we approximate M(q,t) RJ T(q,t) and

ignore contributions from the transient part of M(g,t). P((/,t) has been derived previously by

Sjogren [33] and Bengtzelius, Gotze and Sjolander [34] and is given by

-W) ~ <

r",t). (7)

On the basis of many-body methods and kinetic theory, Bengtzelius et al [34] derived suc-

cessfully n closed non-linear self-consistent dynamical equation for the solidification which

reads

= ;—r(^, t —* oo) = ^(/(fc)) (8)1 - JW r

In Eq. (8) the Debye-Waller form factor f(q) = R(<J,t—» oo) and J",(/(i)) is a functional of

}{k) which appears in Eq. (7). Equations (7) and (8) are basic formulas which have to be

solved iterativcly.

Similar equation for a tagged particle exists and it can be written [34]

where-

dq'q'

(10)

In solving for f'(q) one requires f(q) as input but again it has to be solved self-consistently.

We shall apply these equations in our calculation of the tagged particle motion.

B. 0— relaxation

Following the work of Gotze and Sjogren [30,32] we present in this subsection those equa-

tions in MCT which are essential for an analysis of /^-process. The readers arc referred to

original papers [24,29-32] for technical details.

According to Gotze and Sjogren [30,32] the correlator R(</,t) in the vicinity of Tr can be

expressed in a factorized form

R(</,t) = fc(q)+ hc(q)ii\/l^Ay'\e\g±{k/t,p}. (11)

Here /c(<?) is the Debye-Waller form factor which is the solution of Eqs. (7) and (8); hc(<;)

= [1 - IcMY^l is t l l e c r i t i c a l amplitude [24] where fj (fj) being the right-hand (left-hand)

eigenvector of the stability matrix

ji2 = [ £ ^Tc((),F,/flT)/(t)]/(l-A), A being a material-dependent exponent parameter which

can be calculated from Eq. (8) [24]

1(13)

£=(TC-T)/TC is the separation parameter, and fl±(t/ty3) is a scaling function which satisfies

Tl+C29±(C) + ACi / dr^glir) = 0, (14)

Jo

where f=wt/j and T = ijtp both being rescaled variables. The plus and minus in <7±(t/t,g)

refer to glassy £ >0 and liquid € <0 sides of the critical Tc respectively. Note that the rescaled

time ip = to|/j2(l — A)£|~'''(2'1> where the exponent parameter a can be determined from Eq.

(13)

= r2(l -«)/r(l-2a)

7

(15)

in terms of the Gamma function T(x).

In the same vein the tagged particle correlator can be shown to be

(16)

where P(ij,t) = < <V(q, t)Sp'(O,Q) >, f'(q) is the Lamb-Mossbauer factor, being the solution

of Eqs. (9) and (10), and h'c(q) is given by [29]

KM = (17)k,k'

in which V = (1-C;)"1 is a.n inverse matrix and C'qk = [l - f'(k)?dT'!df'(k) is the tagged

particle stability matrix. The above equations constitute the starting point for our discussion

of the /^-relaxation.

IV. Numerical results and discussion

We have applied Eqs. (l)-(4) to obtain the static liquid structure factor for the liquid

metals Na and K from near freezing point down to the supercooled liquid temperatures. For a

liquid metal these calculations pose a great difficulty since the mass densities of liquid metals

are a priori unknown. To circumvent this difficulty we have resorted to our recently proposed

novel method [24,35] which determined this physical quantity indirectly from the simulated

Wendt-AbraJiam parameter [36], We refer the interested reader to these works for further

details. Now, given an atomic volume at each lower T, it is then a straightforward matter to

evaluate S(</) [23,37,38]. The calculated S(q)s are displayed in Fig. 1 and they are compared

with those obtained from the molecular dynamics simulation. An immediate conclusion can be

drawn from this comparison: it justifies our present choice of the modified hypernetted-chain

theory to the calculation of S{q) for an undercooled liquid.

To proceed further we input the above S(<?) to Eqs. (7) and (8) and iteratively solve for

/(</). Subsequently, the obtained solution /(<?) is used in conjunction with Eqs. (9) and (10) to

solve iteratively for the non-ergodic form factor f'(q). The critical temperatures Tcs obtained

for the Na and K liquid metals are Tc = 211 K and 194.18 K respectively. These temperatures

g

are slightly lower than the ones determined previously using simulated S(q)s (for Na, T<; =

215.3 K; for K, Tc = 200.2 K). In the following we present and discuss some interesting results.

A. Exponent parameter A

Using Eq. (13) we have calculated the exponent parameter A and hence the parameter a

from Eq. (15). These exponent parameters are given in table 1 along with the ones corre-

sponding to the HS calculated using the empirical B(r) given by Eq.(4) and the Lennard-Jones

[20] system. A notable feature in this table is that A shows a systematic change; it decreases

from a large value of 0.772 for hard spheres (characterized by a purely infinite repulsive part),

to a value of 0.714 for a Lennard-Jones system (delineated by a still highly repulsive but a

weakly attractive parts), and to values of 0.712 and 0.710 for liquid metals (described by

a relatively softer repulsive followed by an oscillatorily and weakly attractive parts). This

variation of A is surely intimately related to the details of microscopic interactions. In fact,

it is possible to infer from this change of A that for a simple monatomic system such as the

one-component plasma (having a much softer repulsive and zero attractive parts), it will be

expected to have an exponent parameter that lies within the range 0.772 > A > 0.714. This

prediction is based on the assumption that the attractive part of pair potential does play some

role in the dynamics of liquid structure, and that the Lennard-Jones system has a sufficiently

weak attractive tail. It would be interesting to carry out such kind of calculation to check this

conjecture.

B, Master function for the ^-process

From the As given above, it is numerically straightforward to calculate the master functions

<7±(T). They are given in Fig. 2 for the hard-sphere (HS) and liquid metals Na and K. However,

the calculated results are equally instructive if they are expressed in their Laplace transform

g±{z). For real frequencies u, z —> u)+ iO, one obtains

, 0

= % /

JO

ro

/

JO

and the corresponding susceptibility spectra X±{u) c a n De calculated from x±iljJ) = ^?±(^)-

The A'if'4') f°r lifJLiid metals Na and K are depicted in Fig. ,'} together with (.hat for the

HS system included for comparison. It is interesting to iiote from these figures the following

discernible differences.

(a) For given e, the master function <f±(r) for a metallic system varies more steeply with r

for T -Cl and r > 1 compared with that of hard-spheres.

(b) The susceptibility spectra \"(w) for the HS system has a broader dispersion relative to

those of liquid metals Na and K.

These two points, the validity of one implies the other, can be understood qualitatively by

examining the details of interactions between particles. For HS we note that the 'microscopic

interaction" is purely geometrical being characterized by an infinite repulsive potential. The

structural behavior is therefore determined solely by the excluded volume effect. On the

other hand, the microscopic interaction of a liquid metal is described by a relatively softer

finite repulsive force embellished by an oscillatorily and weakly attractive part. Thus the

structural behavior is determined both by the geometrical factor similar in nature to that

of a HS system, and by an additional Coulomb attraction due to the presence of valence

electrons. When the temperature (density) of an equilibrium liquid is decreased (increased)

rapidly it is anticipated that the non-linear coupling between particles is enhanced by the

attractive force that prevails in a metallic system. Since /?-relaxation is believed to be a

localized excitation describing particles moving dynamically with their surrounding cages,

such enhancement in the attractive interaction with decreasing temperature (or increasing

density) has the consequence of robusting the occurrence of /̂ -process as evident by a longer

characteristic time scale t̂ for the liquid metal than for the HS system (sec table 1). This

would imply that the atomic motion in its associated cage is relatively more stable for a

metallic atom than for a HS particles. We will see more clearly of such a picture below when

we discuss the tagged particle distribution function.

Before proceeding to the latter discussion, we should mention one interesting aspect of

Fig. 3. This figure reminds us of recent theoretical fittings of the MCT to the light scattering

10

experiments on Cao.jKo.etNOa)!^ [8-10]. There the fitted values X = 0.81-0.85 surely imply

that the details of inter-particle interactions should differ considerably from the presently con-

sidered monatomic systems. Indeed for this particular fragile glass which consists of argon-like

ions K+ and Ca+, and optically anisotropic ions (NO3)~, it is believed that the orientational

motion for the planar (NO3)" is probably the most relevant cause for the distinctly large value

of A.

C. Tagged particle distribution function

We turn now to a discussion of the tagged particledistribution function Ps(r,t) = 47rr2F'(r,t)

which can be obtained from Eq. (16) by a back Fourier transformation. This equation, which

(18)

describes the probability density of locating a tagged particle at time t and position r, if it

was found to appear at the origin r = 0 at t = 0. Figures 4 and 5 display the P^(r,r=t/t^) vs.

r/<7 for the liquid metal K and HS system calculated using the <?-(T) and jf+(r) respectively.

(The results for the liquid metal Na is virtually the same and is therefore not given here).

There are two interesting points that merit emphasis.

(a) For given e < 0, the HS tagged particle Ps(r,T-) at different T intercepts at r/V = 0.172

(for K, r/<r = 0.211). Since the probability of finding the HS lagged particle is higher

for increasing T for r/cr > 0.172 (for K, r/cx > 0.211), the HS tagged particle is seen to

push its way through the cage of neighboring particles. Such a picture for an atomic

motion in a cluster of particles appears to persist longer for a metallic atom than for a

HS particle.

(b) For given £ > 0, the tagged particle P*(r,r) at different r displays the characteristic

localized motion and shows a slightly more stable configuration for metallic particles

than HSs.

The first point can be explained by referring to the master function given in Fig. 2. There

we see that the metallic ff-(i") lies above that of HSs throughout the time window of the

11

/3-process. Further evidences for this difference in the diffusive motion can be gleaned also

from table 2 where we give the P'(rmax,T) at the maximum position rmwt for increasing T. AS

regards the second point, it can be attributed physically to the attractive part of <£(r) being

more important when the system is at a lower (higher) temperature (density).

Finally, it is of great theoretical interest to compare our calculated monatomic metallic

H"(r) with that, of the computer simulated soft sphere mixture (see Fig. 1 in Ref. 18). Our

metallic H"(r) shows the same peak-valley behavior similar to that of HSs. Quantitatively,

however, the magnitude of the first peak is lower and the whole curve is shifted to larger

rvalues (see Fig. ft). This latter feature being qualitatively similar to the soft sphere mixture

may be attributed to the more realistic repulsivenessof <#>(r). Comparison between the present

result for K and the soft sphere mixture further implies that the single peak feature followed

by an extensive spatial H*(r) for the latter might be due to the size effect of different soft

spheres.

V. Conclusion

In this paper we first examined the appropriateness of the modified hypernetted-chain

theory applied to the evaluation of trie static liquid structure factor for an undercooled liquid.

We compared the calculated liquid structure factor with MD simulated S(q) and found that

they agree very well for the supercooled liquid region of interest in this work, This prompted

us to apply the calculated S(q) in conjunction with the mode-coupling theory to study the /?-

relaxation. We have chosen the liquid metal as a prototype system since the latter is described

by the simplest realistic interactions which, when compared with the US and Lennard-Jones

systems, permits extraction of fruitful information about the supercooled liquid dynamics.

It was attempted further in this work to establish the connection between the details of

microscopic interactions and the parameters that appeared in the mode-coupling theory. For

simple monatomic systems, we found that the strength of the attractive part of pair potential

(a) varies inversely with the X parameter and (b) robusts the occurrence of /3-process and

hence has the tendency of stabilizing the motion of the tagged particle in its surrounding

neighbours.

Acknowledgments

This work has been supported iti part by the National Science Council, Taiwan, R.O.C. under

contract No.:NSC83-0208-M008-056. We thank the Computer Center of the Department of

Physics, National Central University (NSC82-0208-M008-094) for the support of computing

facilities. One of the authors (SKL) would like to thank Professor Abdus Salain, the Inter-

national Atomic Energy Agency and UNESCO for hospitality at the International Center for

Theoretical Physics, Trieste, Italy.

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15

Table 1: Parameters A and a calculated from the present work, characteristic time t/s scaledby microscopic time t0 for the liquid metals Na and K compared with those of the hard-sphere(HS) calculated at the critical packing ratio 0.5324 and the Lennard-Jones (LJ) [20] systems.

System a t,?/toxl0

IIS 0.772 0.294 6.1fi

LJ 0.711 0.321

Na 0.712 0.322

K 0.710 0.323 20.71

16

Table 2: Tagged particle distribution function Ps(rmax,r,) evaluated at the maximum positionrmax f° r T% — W t f = 5xlO"3+> where i = 0,..4 for the HS and liquid metal K in the e < 0liquid region. In the e > 0 glassy region T ( + 4 = t(+ 4 / t^ = 10"3+* where I = 1,..3.

T2 T5

HS system

rm.x/ir 0.120 0.126 0.132 0.144 0.184 0.120 0.124 0.125

P"(rmM,Tv) 6.876 6.181 5.751 5.309 5.044 6.804 6.379 6.289

K system

r™*/ff 0.152 0.157 0.159 0.164 0.187 0.152 0.157 0.159

P*(rmax,r,) 5.582 5.252 5.066 4.867 4.409 5.507 5.289 5.229

17

FIGURE CAPTIONS

Fig.l - Static liquid structure factor S(q) calculated by the modified hypernetted-chain

theory (dashed curve) with the Malijevsky and Labik empirical bridge function (28)

compared with the Fourier-transforming molecular dynamics g(r) (full curve) for (a)

sodium and (b) potassium liquid metals.

Fig.2 - Master function g±(i") versus T—tji^ for the hard-sphere system (full curve) and

liquid mt'tals Na (light dashed curve) and K (heavy dashed curve).

Fig.3 - Susceptibility spetra x"(^) versus fi = ui/wp (where up = l/t^) for the hard-sphere

system (full curve) and liquid metals Na (light dashed curve) and K (heavy dashed

curve),

Fig.4 - Tagged particle distribution function PI(r) for the (a) hard spheres and (b) potassium

element evaluated at r , /^ = 5xlO~3 (full curve), 5xlO~2 (dotted curve), 5x10"' (short

dashed curve), 5x10° (long dashed curve), 5x10' (chain curve) for e < 0 liquid region.

Fig.5 - Tagged particle distribution function P+(r) for the (a) hard spheres and (b) potassium

element evaluated at r./t/j = xlO"2 (full curve), xlO"1 (dotted curve), xlO° (dashed

curve) for £ > 0 glassy region.

Fig.6 - Spatial functions HJ(r) and p*(r) for the tagged particic distribution for the hard-

sphere system (full curve) and liquid metal K (dashed curve).

18

I 11 I I I I I I I I I I I I I I I I I I I M I I I I I I I I I I I I I I

2 3 4 5q (A"1)

F i g . l

19

2 3 4 5 6q (A"')

15

10

0 r—

-5

-10

-15

' 4 nl ,J>10° 10' 102

F i g . 2

2021

P-(r)

o

(a)

0.50

0

Fig.

22

Fig.523

05 O

24