Critical behavior of gelation probed by the dynamics of latex spheres
G. C. Fadda and D. Lairez*
Laboratoire Le ´on Brillouin, CEA-CNRS, CEA/Saclay, 91191 Gif-sur-Yvette cedex, France
J. Pelta
ERRMECE, Universite ´ de Cergy-Pontoise, 95302 Cergy-Pontoise cedex, France
Received 21 November 2000; published 23 May 2001
We report a quasielastic light scattering study of the dynamics of large latex probe particles ( R =225 nm) in
gelatin solution undergoing gelation. We show that by focusing on the short-time and long-time behavior of the
autocorrelation function, it is possible to simply interpret out data in terms of the divergence of the viscosity
and emergence of the shear elastic modulus near the gel point. Our crude analysis allows us to grasp the critical
behavior of gelation and to obtain the two critical exponents of the transport properties.
DOI: 10.1103/PhysRevE.63.061405 PACS numbers: 82.70.Gg, 47.50.+d, 83.10.Pp
I. INTRODUCTION
The use of spherical probe particles to study viscoelastic
properties of polymer solutions has been suggested for a long
time 1,2. The basic idea is that in a semidilute solution of
linear chains, sufficiently small particles freely move in the
solution whereas larger particles would be trapped in the
polymer transient network. Thus, the dynamics of probe par-
ticles is expected to be governed either by the solvent vis-
cosity, or by the macroscopic viscosity, depending on their
size R compared to the correlation length
T
of concentration
fluctuations. A suitable technique to study such dynamics of
probe particles is quasielastic light scattering, which has
been extensively used on various physicochemical systems.
According to the wide literature on this subject it appears
that some complications may occur. For instance, a complex
signal is sometime observed due to the contribution of the
matrix and of the probe particles to the total light scattered
intensity 3. In addition, strong interactions between poly-
mers in solution and particles, are sometimes reported which
may cause adsorption of chains on the particles and particles
aggregation 4,5. However, it has been shown that, as soon
as these complications are avoided, the basic idea is well
founded. In particular for R / 1, the long-time dynamics of
probe particles is slowed down and governed by the macro-
scopic viscosity 6–9 rather than the solvent viscosity.
Compared to rheology, the study of thermal fluctuations
of concentration of probe particles for the investigation of
weak structures is very attractive 10, because it guarantees
the structure integrity. In particular, the use of probe particles
for the study of gelation looks very tempting and has been
already reported. Gelation 11 is a critical phenomenon of
connectivity occurring when molecules randomly connect to-
gether leading at the gel point to a giant cluster: the gel. The
understanding of quasielastic light scattering results obtained
on gelling systems is much less clear as those obtained in
semidilute solutions. On the one hand, gelation essentially
produces a wide polydisperse population of clusters, which is
responsible for complex relation processes. Therefore the
analysis of the dynamical structure factor of probe particles
in terms of time distribution is not quite obvious. On the
other hand, above the gel point, fluctuations tend to be frozen
at long-time because the solution becomes solid. This ap-
pears through a decrease in amplitude of the scattered inten-
sity fluctuations. The most often used approach to account
for this behavior invokes a loss of ergodicity 12–14. The
physical meaning of this approach consists in splitting the
probe particles population into two parts: the first being free
to diffuse in the medium, the second being trapped in the
network and motionless. The scattered intensity has thus two
contributions: a time fluctuating or quasielastic part and a
constant or elastic part. Quasielastic light scattering tech-
nique consists in measuring the autocorrelation function of
the scattered intensity. Consequently it also shows the two
contributions. The first corresponds to the autocorrelation of
the fluctuating part of the scattered intensity and the second
to the correlation of this fluctuating part with the elastic scat-
tering that acts as a local oscillator signal. In terms of the
light scattering language, this amounts to mix a self-beating
and a heterodyne contribution. Such an analysis calls for two
remarks. First, the splitting of the probe particles population
into two parts seems somewhat arbitrary and unsatisfactory.
Second, above the gel point as fluctuations become more and
more frozen the increasing heterodyne part leads at short
time to a slowed dynamics while quite the opposite is experi-
mentally observed 15.
In this paper, we present an analysis of quasielastic light
scattering results from probe particles during gelation of a
gelatin solution. We deliberately consider the simplest case,
that is: 1 very large spherical and monodisperse latex par-
ticles radius R =225 nm; 2 negligible thermodynamic in-
teractions of probe particles between them or with the ma-
trix; 3 a negligible contribution of the matrix to the
scattered intensity; 4 measurements performed at qR =1
with R /
T
1. We show that, neglecting the nonergodicity
and focusing on the short-time and long-time behaviors of
the measured relaxation function, it is possible to simply
interpret our data in terms of the divergence of the viscosity
and emergence of a shear elastic modulus near the gel point.
Our crude analysis allows us to grasp the critical behavior of
gelation and to obtain the two critical exponents of the trans-
port properties. *Author to whom correspondence should be addressed.
PHYSICAL REVIEW E, VOLUME 63, 061405
1063-651X/2001/636/0614059/$20.00 ©2001 The American Physical Society 63 061405-1