Ionic condensation theories and the liquidlike structures observed in colloidal dispersions
M. Quesada-Pe
´
rez, J. Callejas-Ferna
´
ndez, and R. Hidalgo-A
´
lvarez*
Grupo de Fı ´sica de Fluidos y Biocoloides, Departamento de Fı ´sica Aplicada, Facultad de Ciencias, Universidad de Granada,
Granada 18071, Spain
Received 4 June 1999
Though the notion of effective charge has been widely used to fit experimental data, the possibility of
predicting this adjustable parameter through a model remains unclear. A likely reason for this is the complexity
involved in the theoretical approaches in the case of fluids with large asymmetries between their components.
This paper deals with several condensation theories for spherical colloids, developed to provide effective
charge values from simple models. Liquidlike structures are formed in colloidal dispersions for a set of latexes
with different properties charge, size, and polymeric composition. Effective charges are determined from
experimental structure factors using a Derjaguin-Landau-Verwey-Overbeek potential and an Ornstein-Zernike
scheme. The numerical coincidence between effective and post-condensation charges is fairly acceptable only
for latexes with small size and charge. A simple approach based on the Manning condensation theory for linear
polyelectrolytes is also discussed.
PACS numbers: 82.70.Dd
I. INTRODUCTION
Colloidal dispersions of charged particles exhibit a wide
variety of commercial, technological, and scientific applica-
tions. Glues, paints, or pharmaceuticals are all colloids. Nev-
ertheless, the control of these complex systems depends
strongly on a theoretical understanding of their constituents
and the interaction between them. For instance, latex suspen-
sions are useful in many cases if particles do not aggregate.
Therefore, a theory of colloidal stability, in which electro-
static forces play an extremely important role, is essential.
The electrostatic interactions between charged particles can
cause a certain spatial ordering observed in many different
ways.
For a long time, many workers have tried to determine an
effective charge characterizing such phenomena from optical
techniques 1–3 and others directly related to u ( r ), e.g.,
small angle neutron scattering, shear modulus titration, and
torsional resonance digital video microscopy 4–7. The
Derjaguin-Landau-Verwey-Overbeek DLVO potential 8
has been widely used for this task. One of the most puzzling
findings is that the obtained effective charges are consider-
ably smaller than the total number of elementary charges Z
on the particle surface. The strong accumulation of counte-
rions in the vicinity of the macroion surface due to electro-
static coupling between opposite charges could be respon-
sible for this noticeable reduction. To gain insight into this
phenomenon, the concept of ionic condensation might be a
useful tool. Accordingly, the colloid and the condensed
counterions would be considered as a whole carrying a post-
condensation charge Z * that will be considerably reduced
as compared to Z, since the condensed counterions would
neutralize rather than screen a great amount of surface sites
on the particle.
This concept was initially developed by Oosawa and
Manning for linear polyelectrolytes three decades ago 9,10.
According to the condensation theory, if the dimensionless
charge density L
B
is the linear charge density, L
B
=e
2
/4
0
r
k
B
T ,
r
the solvent dielectric constant, k
B
the
Boltzmann’s constant, and T the temperature exceeds a criti-
cal value 1/v ( v is the counterion valence a condensed layer
emerges see a recent review by Manning 11. The number
of condensed counterions increases in such a way that the net
density of the polymer and the counterions combined de-
creases up to the critical value. But can the Manning theory
be extended to spherical colloidal particles? The answer to
this question does not seem to be so clear. According to
Belloni 12, simple laws for spherical colloids analogous to
the laws for linear polyelectrolytes can be deduced from a
Poisson-Boltzmann PB approach if a certain definition of
which ions can be considered condensed is applied. It should
be emphasized, however, that this one presents some differ-
ent features when compared to the Manning condensation
see Sec. II for further details. Moreover, other models have
been proposed recently in the attempt to predict effective
charges. In the same spirit as the above-mentioned conden-
sation model, the authors of Ref. 13 applied a PB cell
model as well, but assuming a different criterion to say
which counterions neutralize anionic sites on particles. The
authors of Ref. 14 also developed a simple theory for
charged spherical colloids. In this case, however, the number
of counterions condensed on the macroions is calculated us-
ing a thermodynamic approach. However, these theories are
so recent that experimental validation tests are quite scarce
practically nonexistent in the case of the models by Belloni
and Levin, and a comparison between them in the view of
data obtained from real systems has not been carried out yet.
At this point, the renormalization procedure proposed by
Alexander et al. 15 must also be quoted. Although the
strong accumulation of counterions close to the surface and
the nonlinear screening is said to be responsible for this ef-
fect, there exist some differences between the renormaliza-
tion and condensation approaches. The former consists of
matching the nonlinear and linearized solutions of the PB
equation at the edge of the spherical cell. Conversely, the *Author to whom correspondence should be addressed.
PHYSICAL REVIEW E JANUARY 2000 VOLUME 61, NUMBER 1
PRE 61 1063-651X/2000/611/5749/$15.00 574 ©2000 The American Physical Society