Anomalous stretching in a simple glass-forming liquid
Sudha Srivastava, Upendra Harbola, and Shankar P. Das
School of Physical Sciences, Jawaharlal Nehru University, New Delhi, 110067, India
Received 13 June 2001; published 21 May 2002
The frequency dependent shear modulus G( ) for a simple liquid shows strongly stretched behavior and the
stretching exponent increases with decrease of temperature. This unconventional behavior was reported earlier
in Phys. Rev. Lett. 73, 963 1994 from experiments on simple liquids. We demonstrate here that this is a
feature of the characteristic two-step relaxation process of the self-consistent mode coupling theory of super-
cooled liquids.
DOI: 10.1103/PhysRevE.65.051506 PACS numbers: 64.70.Pf, 05.60.-k, 64.60.Cn
The extremely slow relaxation of shear in the glassy sys-
tems is one of the most important signatures of the amor-
phous state. Dynamic shear viscosity and the shear modulus
were measured experimentally for di-n -butylphthalate DBP
in Ref. 1 in the vicinity of the glass transition region. In the
glassy state the relaxation behavior follows the stretched be-
havior slower than the Debye or exponential relaxation. In
general, the relaxation gets more stretched as the temperature
is lowered. However, in Ref. 1, it was reported that for the
dynamic shear modulus one sees a reverse trend—namely
the stretching exponent increasing with lowering of tempera-
ture. In the present paper we report a similar behavior ob-
served for the same quantity G ( ) computed from the self-
consistent mode coupling models. We show that this
unconventional behavior seen in the stretching exponent is a
consequence of the two-step relaxation process, which is a
generic feature of the self-consistent mode coupling theory
MCT. The glassy behavior observed in supercooling the
liquid has been modeled from various approaches, such as
using classical statistical mechanics of many particle systems
2–4 as well as making analogy to models for spin systems
5. The solidlike behavior of the supercooled liquid state is
usually described in terms of the phenomenological vis-
coelastic models. A microscopic mechanism that explains
viscoelasticity comes from the MCT. In the present work we
compute the frequency dependent shear modulus G ( ) of
the liquid from the MCT. The main input in the theory comes
from the structure factor of the liquid. Our main result shows
that the frequency dependence of G ( ) follows the Cole-
Davidson form with an exponent
CD
that decreases with
increasing temperature, similar to the anomalous behavior
seen in the work of Menon et al.
The shear relaxation in a fluid is studied by analyzing the
transverse autocorrelation function ( q , t ), which is ex-
pressed in the Laplace transformed form
q , z =
1
z +i
R
q , z
1
in terms of the memory function or the generalized shear
viscosity
R
( q , z ) =
0
+
mc
( q , z ), where
0
is the short
time or bare part arising from uncorrelated motion of the
fluid particles. The mode coupling contribution for
mc
takes
into account the cooperative effects in the dense fluids. In the
supercooled liquid the density fluctuations are assumed to be
dominant and
mc
is expressed self-consistently in terms of
the density autocorrelation function:
mc
q , t =
n
2 m
dk
2
3
c k -c | q
-k
|
2
k
2
1 -u
2
S | q
-k
| S k | q
-k
| , t k , t ,
2
where u =q
ˆ
• k
ˆ
, the dot product of two corresponding unit
vectors, m is the mass of the fluid particles and =1/k
B
T .
( q , t ) is the Fourier transform of the density autocorrelation
function normalized with respect to its equal time value. The
direct correlation function c ( k ) and the static structure factor
S ( k ) are related through the Ornstein-Zernike 6 relation
S ( k ) = 1 -nc ( k )
-1
, where n is the equilibrium density of
the liquid. For the dense fluid at small enough length scales
i.e., large enough q, the transverse current correlation de-
cays through a damped oscillatory mode termed as the shear
wave 6,7. We focus in this work mainly on the dynamic
response of the system. Here the elastic response of the su-
percooled liquid is defined over a time scale with the corre-
sponding frequency dependent modulus G ( ) defined in
terms of the dynamic viscosity ( ) as
G
=
. 3
The high frequency limit of G ( ) denoted by G
then gives
the elastic response even in the normal liquid state. Follow-
ing Menon et al. we also study a related quantity from our
model, defined as 1
R =
p
=0
T
, 4
which is representative of the cooperation in the dynamics
8. Here
p
is the peak frequency in the imaginary part of
( ). For Debye relaxation the memory function has a
simple exponential decay, i.e., e
-t /
, where is the relax-
ation time. In this case the two quantities R and G
/ T are
identical. In our calculation, the variation of the controlling
thermodynamic parameters are followed as in Ref. 9. The
critical values of these parameters, at which the transition
takes place, depend on the interaction potential. In order to
study the temperature dependence of the stretching we con-
PHYSICAL REVIEW E, VOLUME 65, 051506
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