PHYSICAL REVIEW A VOLUME 29, NUMBER 5 MAY 1984

Dynamical model of the liquid-glass transition

E. I.eutheusser
Physik Dep-artment der Technischen Miinchen, D 80-46 Garching, Federal Republic
Uniuersitiit of Germany
(Received 5 December 1983)
Based on a microscopic theory developed recently, a dynamical model of density fluctuations in
simple fluids and glasses is proposed and analyzed analytically and numerically. The model exhibits
a liquid-glass transition, where the glassy phase is characterized by a zero-frequency pole of the
longitudinal and transverse viscosities indicating the systems stability against stress. This also im-
plies an elastic peak in the density-fluctuation spectrum. Approaching the glass transition the slow-
ing down of density fluctuations is controlled by the increasing longitudinal viscosity, which in turn
is coupled via a nonlinear feedback mechanism to the slowly decaying density fluctuations. This
causes a divergence of the structural relaxation time at a certain critical coupling constant A, At
the glass transition density fluctuations decay with a long-time power law 4(t) — t with o;=0. 395
and approaching the transition the viscosity diverges proportional to e " and e ", where
— —
e= 1 A/A, , and p=(1+ai/2a, p'=p 1 below and above the transition, respectively. The
~ ~

long-time tail "paradox" in dense fluids is briefly discussed.

I. INTRODUCTION and line widths.

Theoretically, to describe the behavior of the diffusion
When liquids are cooled down sufficiently the relaxa- constant and thermodynamic quantities near the glass
tion time for structural rearrangements increases drasti- transition the phenomenological f'ree volume theory was
cally. If crystallization can be avoided by large cooling developed by Cohen and Turnbull, where the decrease of
rates most supercooled liquids will enter a metastable the diffusion constant is attributed to the decreasing free
glassy state. The glass transition is characterized experi- volume available to a particle by the rearrangement of its
mentally by relaxation times of the order of minutes or neighborhood upon approaching the transition. Concepts
detours and associated large viscosities of typically 10' of percolation theory were introduced later by Cohen and
poise. At this transition temperature Tg one also observes Grest. In mean-field theories it is argued that the nar-
a gradual drop in various thermodynamic quantities like rowing of the quasielastic peak of the dynamical structure
compressibility, specific heat, and thermal expansion due factor of a supercooled liquid observed in neutron scatter-
to the freezing in of the translational degrees of freedom. ing experiments is due to a soft mode instability caused
In this so-called glass transformation range the various by the divergence of the static structure factor at the wave
measured quantities also depend on the duration of the ex- number corresponding to its main peak. These theories
periments. By reducing the time scale of the measure- were critically discussed by Sjolander and Turski. The
ments, e.g. , in ultrasound experiments, one can observe a authors argued that the narrowing may be caused by the
glasslike transition at temperatures well above Tg, name- decreasing diffusion constant. The important role of the
ly, when the system's relaxation time is of the order of the increasing shear viscosity in the previtrification regime
inverse sound frequency. Thus the glass transition is not was also noted recently. '
well defined experimentally. Theoretically, however, the An interesting question is what the dynamical behavior
ideal glass transition can be defined to occur at that tem- of a liquid would be if it could be cooled down so slowly
perature To or corresponding density where the structural that it always remains in equilibrium and if crystallization
relaxation time becomes infinite assuming that on cooling would not occur. It is this question that will be addressed
the liquid remains in equilibrium and crystallization does in this work using a simple model which, however, is sup-
not occur. posed to represent some essential features of the liquid-
The glass transition can be studied by molecular- glass transition.
dynamic experiments for systems of particles interacting The microscopic theory"' on which the model to be
with various types of pair potentials; for reviews see Ref. discussed here is based and the resulting picture of the
3. Although these simulations are limited by the small liquid-glass transition shall be summarized briefly. The
system size and the short time intervals, they can provide basic quantity of the theory is the dynamical structure
detailed information on the dynamical processes. Trans- factor describing the dynamical properties of density fluc-
port coefficients, thermodynamic quantities, time-de- tuations. Their decay is controlled mainly by the longitu-
pendent correlation functions, and dispersion relations in dinal viscosity. ' Among the various contributions to the
the amorphous state were examined. Neutron scattering dynamical transport coefficients the most important one
is also a valuable tool to investigate the dynamical struc- at high density is that representing configurational relaxa-
ture factor of supercooled liquids and glasses and to ob- tion or, in microscopic terms, dynamically correlated col-
tain various kinds of information like dispersion relations lisions. This part was expressed in terms of bilinear prod-

29 2765 1984 The American Physical Society

2766 E. LEUTHEUSSER

ucts of fluctuations of the slowly decaying conserved vari- the dot denotes the time derivative. Here Qo is the fre-
ables of the fluid. Among those the density fluctuations quency of the free oscillator and y is a damping constant.
were found to yield the main contribution in a dense fluid. The nonlinear term has the form of a memory kernel de-
The relaxation time of density fluctuations is by this non- pending on the past motion of the oscillator and its
linear feedback coupled and increased by the slow decay strength is controlled by the dimensionless coupling con-
of density fluctuations itself. The theory was evaluated stant A, which is assumed to vary between zero and infini-
numerically at a density near the freezing point. The re- ty. The important question concerns the time evolution of
sults' showed an enhancement of the longitudinal and the oscillator, in particular its long-time behavior depend-
shear viscosity compared to the known kinetic theory ing on the coupling constant. Equation (1) is of interest

'
values in qualitative agreement with molecular-dynamic
results. ' Also in agreement with the mo-
also for mathematical reasons as an example of a non-
linear equation of motion with memory effects. Physical-
lecular-dynamic experiments a slowly decaying com- ly the oscillatory coordinate 4(t) is thought to represent
ponent in the density correlation function and propaga- the density correlation function of a classical fluid at a
ting shear waves in the transverse current correlation certain wave number. This interpretation becomes clearer
function were found at intermediate wave numbers. by introducing Laplace transforms
These phenomena can be interpreted as indications of the
nearby transition to the solid state.
4(z)=WI4(t)I =i f dte'"4(t), Imz ~0 (2)
With increasing density the structural rearrangement and rewriting (1) in the form
becomes more difficult and at a certain density the parti-
cles will be arrested in their cages formed by neighboring 4(z) =— (3a)
particles. At this density the relaxation time diverges and Qo
bulk and shear viscosity are infinite, as signaled z+D(z)
mathematically by a zero-frequency pole. This entails
that the system becomes stable against shear stress which D (z) = iy+4AQOW I@'(t)I .
is the main characteristic of a solid body. In the glass Equation (3a) is the well-known representation of the den-
phase the particles execute vibrational motion around sity correlation function in terms of its second frequency
their arrested positions. The configurational contribu- moment Qo and the dynamical longitudinal viscosity D (z)
tions to the various thermodynamic quantities such as the assuming that energy fluctuations can be ignored. ' Ener-
compressibility are frozen in. Transverse sound waves of
gy density fluctuations are not included in our model
arbitrary small wave number can propagate in the glass since the microscopic theory" showed that bilinear prod-
where the sound velocity is determined by the modulus of ucts of density fluctuations do not contribute to the
rigidity of the glass which is the residue of the zero- thermal conductivity. As one expects, this quantity is
frequency pole of the shear viscosity. therefore not singular at the glass transition. Also, com-
The simplifications of the microscopic theory leading puter simulations' show no significant enhancement of
to the model presented below are based on the observation the thermal conductivity near the freezing point.
that the glass transition is neither accompanied by an The wave-number dependence of all quantities in Eqs.
essential change in the short-range order compared to a (3) is not indicated explicitly. For small frequency, as-
dense fluid nor by the divergence of the static structure suming that the zero-frequency limit of D(z) is finite,
factor. Therefore, the static correlations will be ignored
in a first approximation in order to isolate and discuss the lim D(z)=iD,
q, z~O
essential dynamical mechanism of the slowing down of
density fluctuations in its purest form. the spectrum 4"(co) of 4(co+i0) =0&'(co)+i@&"(co),
The paper is organized as follows. In Sec. II the basic
equation of motion for the density fluctuations will be QOD
presented and motivated. In Sec. III it is shown that this
@"(~)=
(a) —Qo) +(coD)
equation exhibits a phase transition at a critical value of
the coupling constant and perturbation theory in the exhibits peaks at the frequencies co=+(Qo — D /2)'~, if
weak- and strong-coupling regimes is discussed. The crit- Qo&D /2. This case is realized in fluids for sufficiently
ical regime is investigated in Sec. IV and the nature of the small wave numbers q, since Qo — -cq, and D=q D&,
phase transition, the divergence of the relaxation time, where c is the sound velocity and D~ —— (g+ —, ri)/p is the
and the viscosity is examined. In Sec. V the dynamical longitudinal viscosity, g and g are bulk and shear viscosi-
shear viscosity is discussed and the results are summa- ty, and p is the mass density. Thus in this small-wave-
rized in Sec. VI. number hydrodynamic regime the spectrum consists of
two sharp sound peaks. The quasielastic heat diffusion
II. THE MODEL peak present in real liquids is not included in our model
for reasons discussed above and to make the model as
Let us consider the following nonlinear equation of simple as possible. One observes that for increasing D&,
motion for a damped oscillator: i.e., approaching the glass transition, the above mentioned
4(t)+yk(r)+Q,'C(t)+4AQ', J dr C'(~)C(r —r)=0 (1) wave-number regime shrinks to zero. Instead, for q &qo,
where qo V2c/D~, there — appears a quasielastic peak in
with the initial condition 4(t =0) =1, 4(t =0) =0, where @"(co ) with a wave-number-independent width
 DYNAMICAL MODEL OF THE LIQUID-GLASS TRANSITION

1/T=QO/D =c /D~ decreasing for increasing D&. Also, @(z)=—

a~ a2
the wave-number-independent zero-frequency value
Z- 1 Z +LV) Z +l V2
4"(0)=D&/c increases indefinitely. Approaching the Z+lf
glass transition the frequency range where D(z) can be as- —[1+y/(y
sumed to be constant is also expected to shrink. Thus the
where v, &2 [y+(y —4)' ]/2
—— and a&zz
—4)'~ ]/2. Inserting this in (3b) the longitudinal viscosi-
dynamical viscosity D(z) is the fundamental quantity to
be calculated. ty in first order is
In the micmscopic theory" D(z) was written as a sum 4A,
of various contributions of different physical origin. One
D(z)=iy— (8a)
2
part results from the transport of momentum by dynami- 2
cally uncorrelated collisions of particles as described by Z+lf- z+2iy
Enskog's equation. This part behaves regularly near the
glass transition and is approximated in (3b) by the con- The zero-frequency value is enhanced compared to zeroth
stant y, vanishing
I',
-q for small wave number. The order
other part describing dynamically correlated processes
was approximately evaluated in terms of bilinear products
D =y+2A(y+1/y), (8b)
of correlation functions of conserved variables of the sys- which suggests that there may be a divergence character-
tem. It was found' that the product of two density ized by an infinite D (z) and @(z) for zero frequency.
modes is the important contribution and this is expressed To examine this possibility we try the following ansatz:
by the second term on the right-hand side (rhs) of (3b) in
our model. @(z)= f/z + (1 ——f)@.(z),
A further simplification in establishing the model equa- where 4(z) is written as a zero-frequency pole contribu-
tion (3b) was achieved by replacing the density correlation tion with weight f to be determined and a remainder
function in the wave-number integral of the microscopic 4„(z) with weight (1 f), so th—at 4, (t =0) =1. Then (3b)
theory" by that at a typical intermediate wave number as- shows that D(z) has a similar structure:
suming that the detailed wave-number dependence is not
essential for the liquid-glass transition. The wave-number D(z) = 4k f'/z+D„—
(z), (10a)
integral can then be included in the coupling constant X,
which is an increasing function of the density. An argu-
D„(z) =i y+ 8k (1 f f)4„(z)—
ment in favor of this approximation is that, as discussed +4k, (1 —f)'W I 4', (t) I . (lob)
above, the wave-number-independent quasielastic peak of
4"(co) is the dominating feature of 4. Also one expects Inserting (10a) into (3) and comparing with the ansatz (9)
that the glass transition is not caused by a small wave- the strength fis given by
number infrared singularity, i.e. , by long-range phenome-
na, but that it is rather a phenomenon where intermediate f =(1+v'I — 1/A, )/2 .
wave numbers are important. The approximation leading
to (3b) is, however, not allowed in two dimensions because
Also, the remaining part 4„(z) can be expressed by D, (z)
in a simple way:
of hydrodynamic singularities.
Equations (3) or, equivalently, (1) is probably the sim-
plest conceivable model for a glasslike transition. Al- 4, (z) = —- 0
(12a)
though it is based on a microscopic theory for the dynam- z+D, (z)
ical correlation functions of a hard-core fluid, the model
is valid and useful for fluids in general. In (3b) the
dynamical longitudinal viscosity D(z) is expressed by the
bilinear product of the density correlation 4(t), which in
0 =1+4Af (12b)
turn is controlled by D(z) in (3a). The equations will be Equation (11) shows that when the coupling constant is
analyzed in subsequent sections. In the following, fre- larger than the critical value I,, = 1 the ansatz (9) leads to
quency and time will be measured in units of Qp or Qp an acceptable solution. This means that for A, & 1 density
respectively, or equivalently Qp ——1 unless stated otherwise fluctuations do not decay to zero for long times as for
in order to simplify the notation. A, & 1 but they decay to the finite value f
which increases
from f f
= —,' at the transition to =1 for A, ~ao. This is
shown in Fig. 1 where the density correlation function
4(t) obtained by numerical solution of (1) is plotted for
III. THE PHASE TRANSITION various coupling constants. Thus the spectrum of density

For small coupling constant A, a weak-coupling expan-

fluctuations exhibits a 5(co) peak with strength which is
characteristic for the glass phase. The nonzero infinite-
f
sion of (3) is straightforward. In zeroth order in A, time limit of @(t) is analogous to the Edwards-
Anderson' order parameter in spin-glasses. In the glass
D(z) =iy
phase the translational motion is frozen in. The vibra-
and the density correlation function 4(z) has two simple tional motion around the arrested positions is described by
poles in the lower complex half-plane: 0
4„(z) where in (12b) is the oscillator frequency, increas-
2768 E. LEUTHEUSSER

0.5

-1 0 1 2 3 0.5
lOg)Ot

FIG. 1. Time dependence of the density correlation function FIG. 2. Inverse longitudinal viscosity vs coupling constant A,
4(t) for various coupling constants A, indicated. Parameters are in the fluid and amorphous phase. Parameters are Qo — 1 and

Qo —1 and y =1. Transition point is at A, , =1. y =1. Solid line represents the numerical solution of the model
and the dashed lines represent the weak- and strong-coupling ex-
pansions.
ing from 0= v 2QO at A, = 1 with increasing coupling con-
stant. D„(z) will be called the longitudinal viscosity of the
glass phase since it describes the damping of the vibra-
Ã(z) =iy (15a)
tional motion. The residue of the zero-frequency pole in
(10a} is related to the bulk and shear moduli of the glass and the vibrational part of the density correlation func-
phase, 8 and 6, respectively, by (8+ —,6)/p tion reads
=4k, AQ /q .
4„(z)=—
1
It is convenient for the following discussion to rewrite (15b}
(10b) with the use of (12a) as 0
2 +lg
z [z+D„(z)]D„(z)+2z—D„(z)(Q' —2)
z [z+D„(z}] 0'— =K(z), In first order (13b) leads to
4A, (1 —f)
where K(z) =iy (16a)
2Q
Ã(z) =iy+4A(1 —f)'W I C „'(t) j . Z
2Q
+lf- z+2lg
In this way the theory in the glass phase is formulated en-
tirely in terms of the vibrational component 4„(z) of 4(z) Thus with decreasing A, the zero-frequency value Ã(z =0}
alone. For a given 4,
(t) the memory kernel K(z) is deter- increases leading with (14) to an increasing viscosity
mined by (13b), which in turn determines D„(z) via (13a)
and finally 4„(z) by (12a). The main advantage of this re- D, = —1/A, 1+ 8yk, +0(1/A, ') . (16b)
1
formulation is that a perturbation expansion in the glass .

~
phase for A, 1 becomes obvious. For example, (13a) sim- The asymptotic expansion (16b) for D„ is compared in
plifies for z~0 to
Fig. 2 with the numerical solution obtained by integrating
D„(z =0) = Ã(z =0)/V'I —I/A, . the differential equation (1). The good agreement down to
values near A, & 1 is remarkable. Also shown is the numer-
Thus, if K(z =0) were regular for A, ~1
the longitudinal ical solution and the weak-coupling expansion (8b) in the
viscosity would diverge with exponent —, approaching the fluid phase. The nature of the singularity near A, = 1 will
transition. However, as will be discussed in Sec. IV, the be discussed in Sec. IV.
feedback mechanism, expressed by the second term in
(13b), leads to a somewhat stronger divergence. A similar IV. CRITICAL REGIME
reformulation of the theory allowing a perturbation ex-
pansion in the fluid phase A, &1 near the glass transition In the following the small-frequency behavior near the
will be presented in Sec. IV. phase transition will be investigated where the inequality
The special value A, , =1 as the glass transition point
was of course achieved in this model by appropriate defi- iz/D, (z) i
((1
nition of the parameters in (1) or equivalently (3). In the is fulfilled. The point A, =l
and the regimes A, ~ 1 and
,
microscopic theory, however, A. is determined by a cer- A, & 1 are considered separately.
tain wave-number integral over static two- and three- 1. A, =l. At the glass transition point (13) simplifies
particle correlations. with the assumption (17) to
~
In the limit A, ao the weight (1 f) of the vibration—al
spectrum which is proportional to the compressibility of D, (z) = I Ã(z)+ [C (z) —8Ã(z)/z]' j /2, (18a)
the glass tends to zero according to (11) and (9), thus also
the vibrations are frozen in. This limit can be interpreted
Ã(z)=iy+WI@„(t)j . (18b)
as the random close-packing density of the glass. Pertur- Assuming for the moment that K(z) is constant for small
bation theory can be performed also in this strong- frequency, (18a) would imply that D„(z) and also (z) 4,
coupling regime. According to (13b) one has in zeroth or- exhibit a square root singularity for small frequency.
der in 1/A, This would imply a long-time power-law decay propor-
 DYNAMICAL MODEL OF THE LIQUID-GLASS TRANSITION 2769

the crossover can be observed is quite small.

3. A, & 2'. In this section it will be shown, starting from
(3), that the viscosity is also diverging when one ap-
-2— proaches the transition point A, , = 1 from below. At first
sight it is not obvious how an expansion of (3) in terms of
@=1— A, can be achieved, but it can be accomplished as
follows. We assume that 4(z) can be written as the sum
-3—
of a pole contribution and a remaining part C&, (z). The
2
lnt pole contribution is expected to evolve continuously into
FIG. 3. Time dependence of the vibrational part 4„(t) of the
density correlation function at the transition A, , =1 and in the
)
the elastic component of 4(z) for A, 1. Then equations
for the pole position, the residue of the pole, and for 4„(z)
amorphous phase for two coupling constants A, . Parameters are can be derived. Thus the ansatz is
Qo — 1 and y=1.
=—
z+i6 + C, (z),
&b(z) b (23)

tional t ' . The singularity of 4,

(z), however, also im- where a +b = 1, so that 4, (t =0) =1. Then according to
plies a singularity of K(z) by the feedback mechanism (3b) the viscosity can also be divided into a pole part and a
manifested in (18b) and so the assumption of a constant remaining part
C(z) is not justified. It is easy to show that a self-
consistent solution of (18) and (12a) for z~O is D(z) = —(4A, a )/(z+2i5)+D„(z), (24a)

D„(z) -z D„(z)=iy+8Aab@„(z+i5)+4kb WI@„(t)I . (24b)

2a —1
g( ) (19b) The pole part in (24a) can be considered as a viscoelastic
component with a Maxwell relaxation' time ~=1/25.
where a=0. 395 is a solution of I (1 — 2a)=21 (1 —a), For the density correlation function 4(z) instead of (3a)
implying a power-law decay 4, (t)-t,
thus a singulari- the simpler representation
ty somewhat stronger than a square root. So the density
correlation function 4(t) decays very slowly with the ex- 4(z) =— (25)
'
ponent a to the time persistent value —, for t ~ ~.
This is 1

shown in Figs. 1 and 3. The spectrum 4"(co) exhibits a D(z)

'
5(co) peak of strength —, and a co ' "
singularity for which is valid for z/D (z) &&1 will be used in order to
~ ~

small frequency. simplify the formulas. The method is applicable, howev-

2. A, )1. —
Defining e=A, 1 the Eq. (13a) for D, (z) for er, also to (3a). Inserting (24a) and (24b) into (25) one
small frequency can be rewritten as finds for the pole position and pole residue two equations
z 2@(z)=0 .
D„(z) 2WeD, (z)+— (20) i5D„( i5) =4a—A, —1, (26a)
In view of (19) for e=O the scaling ansatz a =[2+i5D, ( —i5) —5 D„'( i5)]— (26b)
D„(z)=z 'd(z/s), expressing 5 and a in terms of D, (z), where in (26b)
K(z) =z 'c (z/s), (21b) D,'(z) =dD, (z)/dz. The equation which determines D„(z)
reads
with the critical frequency s =e' and v=1/2o. leads to
the equation for the scaling functions
—Ã(z),
D„(z) 8ab k4„(z +i 5) = (27a)

d (x) —2x d (x)+2c (x) =0 . —

This implies that the zero-frequency limit of the viscosity
bC&„(z) =- zD, (z)(z+2i5)
D„(z)(z+2i5)
4a
4a
Az
A,

z 2i5— — —
scales like
Q
(22) + z+i6 (27b)

where p'=(I — a)/2a=0. 765. This is in agreement with Ã(z) =i y+4Ab'W I 4', (t) } . (27c)
the numerical solution for D shown in Fig. 2. The
density-fluctuation )
spectrum for A, 1 consists of a 5(co) In this way the theory is formulated entirely in terms of
)
peak of strength f) —, and a vibrational part of strength D„(z). Equation (27) is analogous to (13) for A, 1. Once
1 f & —, with a quas— ielastic peak of width 1/D, vanish- D„(z) is known, 5 and a can be evaluated using (26). The
ing proportional (A, —1)" when approaching the transition viscosity is determined by (24a), yielding
point. Correspondingly, the time-dependent correlation D=2ia /6+a, . (28)
function 4(t), after an initial short-time decay, shows a
crossover from the critical power law proportional to t Equations (26) and (27) are a reformulation of the original
to an exponential decay with a relaxation time ~-D„. In problem admitting an expansion for small a= 1 — A, . In

Fig. 3 the numerical solution 4„(t) near the transition the small-frequency regime z &&e one finds, after
~ ~

point is plotted indicating that the critical region where some calculation, in leading order
 E. LEUTHEUSSER

a = '+e/8,
—, (29a)

D„(z) =i e/25, { z ~
&& e (29b)
assuming that 5 vanishes faster than e for e~O T. he
variation of the pole position 5 with e can be determined
in the following way. In the frequency regime z 5«
«e" the set of equations (27) simplifies to
~ ~

0.5
500 oooo

2zi5D„'(z)+z D„(z)+4i5D„(z)+2e+2zÃ(z) =0 . (30)

FIG. 4. Time dependence of the density correlation function
4(t) in the fluid phase near the transition at A, =0.98. Parame-
If one could replace Ã(z) by iy then the choice 5-e ters are Qo —l and @=l. Solid line represents the numerical
would lead to the scaling form D, (z)=e ' d(zie), im- solution of the model and the dashed line represents the visco-
plying that D — e ~ according to (28). However, elastic exponentially decaying component.
4'(z=0) is singular for A, ~l
because of the nonlinear
term in (27c). The scaling ansatz
ponential decay with relaxation time increasing propor-
D, (z) =s 'd (zis), (31a) tional to D, for long time. The typical time separating
power-law decay from exponential decay tends to infinity
K(z) =s 'c (z/s), (31b) when the transition point is approached. At the transition
where s =e'~ is a critical frequency, a=0. 395, and 4(t) decays according to the power law 4(t)-t
with
5=s'+ leads to a Riccati equation for the scaling func- a=0. 395 to the value —,' for t~no.
tions
V. THE SHEAR VISCOSITY
2ixd (x)+xd (x. )+4id (x)+2+2xc(x) =0 . (32)
In this section the implications of the liquid-glass tran-
Thus for zero frequency, according to (31a), D, diverges
proportional to s '=e " '~,
with the same exponent
sition for the transverse current correlation function and
the shear viscosity shall be discussed. The transverse
as for A, & 1. The relaxation rate 5 vanishes as
6-s'+ -e"
+~'~ ~. So one finds with (28) that in leading current correlation function' can be represented in terms
of the generalized dynamical kinematic shear viscosity
order the longitudinal viscosity diverges as
D, (z) by
D-e ", p=(a+1)/2a=1. 765
C&, (z) = —— 1 (34a)
where the exponent p is related to the exponent p' of D, z+D, (z)
by p=p'+1. where in the limit of small wave number q and frequency
Qualitatively, the behavior of the solution for A, & 1 near z for fluids the shear viscosity ri is given by
the transition can be characterized as follows. The spec-
trum @"(co) consists of two contributions. One part is a lim lim D, (z)=iq ri/p . (34b)
' z-+0 q-+0
very sharp quasielastic I.orentzian peak of strength a —, =
and width 5-e"shrinking to zero for A, ~1.
This part In the framework of the microscopic theory" the dynami-
evolves continuously into the 5(co) peak in the glass phase. cal shear viscosity, similar to the longitudinal viscosity in
=
The other part of weight b —, also exhibits a quasielastic (3b), has two contributions of different physical origin,
peak, but its width is proportional 1/D, -e
similar to
D, (z)=iy, +pAQOWIC (t)I . (35)
above the transition. This part diverges at the transition
as co " ' for small
frequency. So the main difference The first part stems from the usual two-particle collision
between the two phases near the transition as it shows up processes described by kinetic theory. It has the form of a
in the density-fluctuation spectrum is that there is a very Lorentzian with a width given by the collision frequency,
sharp quasielastic peak in the fluid phase while it is elastic but for simplicity it is approximated in (35) by the con-
in the glass phase. In both phases an additional broader stant iy, -q for q~O. The second part in (35) arises
quasielastic peak is superposed, whose width shrinks to from the structural rearrangements in a dense fluid. In
zero upon approaching the transition. In Fig. 1 the nu- the microscopic theory" it is represented by bilinear prod-
merical solution for the time-dependent density correla- ucts of correlation functions of conserved variables and it
tion function is shown. For A, & 1 near the transition the was found' that, as in the longitudinal case, the product
slow exponential decay stemming from the quasielastic of two density modes is the dominant contribution at high
Lorentzian dominates the behavior at long times. This is density. Only this contribution is considered in the model
also clearly shown in Fig. 4 where the straight line corre- equation (35) where p is a numerical constant arising from
sponds to —,exp( — t/2D) as discussed above. This part a wave-number integral over static two-particle correla-
evolves into the time persistent component
A, & 1. An exponential
f
[see (11)] for
decay near the fluid-solid transition
tions. Note that this structural rearrangement contribu-
tion to D, (z) is proportional to the corresponding one for
was recently observed in molecular-dynamic experiments the longitudinal viscosity D(z) in (3b) so that the dynami-
for supercooled liquid rubidium. ' The other component cal behavior of D, (z) is governed by that of D(z) which
of 4(t) behaves akin to X&1. It has a crossover from we have already discussed.
power-law decay -t for intermediate times to an ex- In particular, the shear viscosity diverges at the same
 29 DYNAMICAL MODEL OF THE LIQUID-GLASS TRANSITION 2771

critical point A, =1 and with the same exponent as the

, An ergodic-nonergodic phase transition similar to the
longitudinal viscosity in both phases. Furthermore, in the liquid-glass transition occurs also in the Anderson locali-
Quid phase near the phase transition, the time-dependent zation problem' ' and the diffusion-localization problem
transverse stress correlation function which is essentially of a classical tagged particle in a static random poten-
the inverse Laplace transform of D, (z) has a very slowly tial. ' In these theories, ' ' however, the relation between
exponentially decaying component, stemming from the the current relaxation kernel and the tagged-particle den-
pole contribution of sity correlation function, which corresponds to (3b) in this
model, is a linear one entailing enormous simplifications
D, (z) = —PAQou'/(z+2i6)+D (z) . since the frequency is simply a parameter. The physical
In (36} D~(z) is the remaining part of D, (z), which is reason is that the tagged particle is scattered elastically by
proportional to D„(z) in (24a). The relaxation time the fixed scatterers. In the case of a fluid, however,
r= I/25 of the exponential in the transverse stress corre- scattering is inelastic, and momentum and energy are
lation function is the same as the corresponding one of the transferred via interaction processes into at least two other
longitudinal stress correlation function in (24a). At the modes. It was the main purpose of this paper to study the
phase transition the transverse stress correlation function pure effect of this nonlinearity in (3b) by neglecting the
decays according to a power law with the same critical ex- wave-number dependences in a first approximation.
ponent as the longitudinal stress correlation function. Qualitatively, however, no change is expected when this is
In the amorphous phase D, (z) has a zero-frequency incorporated.
pole For small coupling constant A, the dynamical viscosity
D(z) in (3a) is a constant iy For. increasing A, it becomes
D, (z) = f IIO/z—+D~(z),
PA (37) frequency dependent, D (z =0) increases, the system slows
where the residue is related to the modulus of rigidity G down and becomes more and more rigid and a viscoelastic
of the glass phase by G/p=iai f Qo/q . Thus the system part develops, as described by (24a). The irreversible
has the property of a solid being stable against shear structural rearrangement, as described by the product of
stress. Inserting (37) into (34a), the resulting form of the two density modes in (3b), leads to an enhancement of the
transverse current correlation function longitudinal viscosity, resulting in turn in a slower relaxa-
tion of density fluctuations via (3a). This process may be
@,(z) =- z —q ct +zD~(z} (38) called self-induced slowing down. For increasing A, this
finally leads to the freezing in of the translational motion
at the transition point A. =1 where the relaxation time
,
shows that transverse waves can propagate with the trans- ~= 1/26 and the longitudinal and shear viscosities diverge
verse sound velocity c, =v G/p at arbitrarily small wave proportional to 1 — A, /k, &, @=1. 765. This result is
numbers. Their damping is described by D~(z =0)-q ~ ~

intermediate between the Batchinski-Hildebrand '

law
for small wave number. — —k) ' and Fulcher law -exp[1/(A, —A, ) j.
(i, ,
It is interesting to note that the residue of the viscoelas- At this density the viscosities and the density correla-
tic pole is continuous at the glass transition since tion function acquire a zero-frequency pole, characteristic
f= = =
a —, at it 1. Thus one can observe solidlike of the glass phase, signaling an ergodic-nonergodic transi-
,
behavior already in the fluid phase below A, for co) 25 tion. The particles can only sample a restricted phase
and can determine the bulk and shear moduli of the amor- space since they are arrested by their neighboring particles
phous phase. It should, however, be emphasized that in a cage. At the transition the compressibility drops to
these bulk and shear moduli are not the high-frequency
moduli of the liquid as is often assumed in viscoelastic
(1 f) ', = —, e.g., one-ha—lf of its fluid value, but in any ex-
periment involving nonzero frequencies or finite observa-
models of fluids. ' For example, in the model presented tion times a gradual decrease of the compressibility will be
here as well as in fluids with hard-core interaction, the
,
observed. At A, =A, the density correlation function de-
high frequency elastic moduli, defined in terms of the
second frequency moments of the longitudinal and trans-
'
cays to —, for long times with a power law C&(r) -t ~ with
a =0.395, related to p by p =(1+a)/2a.
verse currents, are infinite. In this model it becomes ap-

f)—
For increasing i the viscoelastic component of the den-
parent, that the viscoelastic component of the stress corre-
sity fluctuations in the fluid evolves continuously into the
lation functions in (24a) and (30) which determines their
elastic peak -rrf5(co) in the glass phase. In addition,
long-time properties is not related to the short-time
there is a vibrational part whose spectral weight (1
behavior of the system. One may conclude that the
approaches 1 for k — +00. Near the transition the vibra-
liquid-glass transition is universal in the sense that short-
tional motion of the particles around their arrested posi-
time properties are irrelevant.
tions is overdamped since the viscosity diverges propor-
VI. SUMMARY AND CONCLUSIONS tional to (A, —1) "' where p'=(I — a)/2a=0. 765.
The time dependence of the longitudinal and. transverse
In this work the liquid-glass transition was studied us- stress correlation function is particularly interesting. As
ing a model derived from a microscopic theory" of densi- was already discussed in connection with (36), the visco-
ty fluctuations in a dense hard-core liquid. It was shown elastic part in (24a) and (36) manifests itself near the tran-
that a system described by (1) or (3) evolves from the fluid sition in a very slow exponential decay where the relaxa-
to the amorphous state when the coupling constant A. re- tion times of the longitudinal and transverse stress corre-
lated to the density is increased beyond a critical value k, . lations are both ~=1/26, approaching infinity for A, — +1.
 E. LEUTHEUSSER 29

This may provide an explanation for the slowly exponen- liquid-solid transition. Therefore, it is understandable
tially decaying component observed in molecular-dynamic that to explain this effect, the theory must be able to
experiments for liquid argon at the triple point performed describe the liquid-solid transition.
by Levesque et a/. They found that the longitudinal as It is also remarkable that the slowly decaying com-
well as the transverse stress correlation functions could be ponent of the stress correlation function found by Erpen-
fitted by two exponentials with relaxation times r& and beck and Wood in the hard-sphere system can be fitted
~& where ~& &&~&. The slower relaxation process corre- equally as well by an exponential as by a t power law.
sponds to the cooperative process of structural rearrange- Note that the time integral of the stress correlation
ments in the fluid and r& can be viewed as the Maxwell function is the viscosity which diverges at the liquid-glass
relaxation time r= 1/25. As in the model presented here, transition. In the present theory this divergency is not re-
was found to be the same in the longitudinal and lated to the hydrodynamic long-time power-law decay
transverse case. The faster initial exponential decay with In fact, this power-law decay due to hydro-
relaxation time ~& probably is due to binary collisions, dynamic singularities is not contained in the present
characterized by the collision frequency. This part is also model because of the approximations leading to the model
present in the microscopic theory" but was replaced in equations (3) and (35), as discussed in Sec. II. The in-
the model equations (3b) and (35) by an instantaneous clusion of the hydrodynamic singularities is expected to
term y5(t) or y, 5(t), respectively, for simplicity. affect the stress correlation functions only at very long
The long-time tail of the stress correlation function is, times. Moreover, approaching the transition the decay
at present, not well understood. The stress correlation -t " disappears and is replaced by the critical power-
function can be divided into a kinetic part, a potential law behavior.
part, and a cross term. Kinetic theory predicts that, of The tagged-particle motion and the self-diffusion coef-
these contributions, only the kinetic part has a long-time ficient do not play a direct role in the formulation of the
power-law decay, with exponent d/2 and well-known am- liquid-glass transition presented here. This is physically
plitude, where d is the space dimension. Molecular- reasonable, since the motion of a tagged particle is not ex-
dynamic experiments' ' ' ' show, however, that the pected to have any influence on the glass transition which
cross term and to a greater extent the potential contribu- is rather a cooperative phenomenon where all particles are
tion are very slowly decaying in dense liquids in the time involved and thus is expressed in terms of the density
regime observable in these experiments. If the slow decay correlation function and the longitudinal viscosity. On
observed in liquid argon is interpreted as the power-law the other hand, the tagged particle is strongly influenced
decay -t one finds an amplitude which is 2 orders by its surroundings and thus the self-motion is strongly
of magnitude larger than predicted for the kinetic part. coupled to the density fluctuations of the system. o'2'3
Similarly, computer simulations for the hard-sphere sys- The effect of the liquid-glass transition on the incoherent
tem' ' near the liquid-solid transition clearly demon- dynamical structure factor and the extension of the
strated that both the cross and the potential part of the present model to include the wave-number dependence
transverse stress correlation function have a slowly decay- will be discussed elsewhere. '

interpreted as a power law -t,

ing component. If the slow decay of the potential part is
its amplitude is 372
times larger than the one of the kinetic part, while the
ACKNOWLEDGMENTS
It is a pleasure to thank Professor Sidney Yip for the
kinetic-theory prediction is zero. hospitality offered by him at the Massachusetts Institute
The theory of the liquid-glass transition may provide an of Technology (MIT) and for interesting discussions.
explanation to this phenomenon. As explained above, the Financial support by the Alexander von Humboldt-
stress correlation function has a very slowly, exponentially Stiftung as a Feodor Lynen Fellow and by MIT is grate-
decaying coinponent, which is a precursor of the nearby fully acknowledged.

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