Modeling the aggregation of partially covered particles: Theory and simulation
A. Moncho-Jorda
´
,
1,
* G. Odriozola,
2,²
M. Tirado-Miranda,
3,‡
A. Schmitt,
1,§
and R. Hidalgo-A
´
lvarez
1,
1
Departamento de Fı ´sica Aplicada, Universidad de Granada, Campus Fuentenueva, E-18071 Granada, Spain
2
Programa de Ingenierı ´a Molecular, Instituto Mexicano del Petro ´leo, La ´zaro Ca ´rdenas 152, 07730 Me ´xico, Distrito Federal, Mexico
3
Departamento de Fı ´sica, Escuela Polite ´cnica, Universidad de Extremadura, Avenida de la Universidad, 10071 Ca ´ceres, Spain
Received 2 February 2003; published 15 July 2003
A theoretical model for describing the initial stages of the aggregation of partially covered colloidal particles
is presented. It is based on the assumption of short-range interactions that may be modeled by a sticking
probability on contact. Three types of sticking probabilities are distinguished depending on the collision type,
i.e., for bare-bare, bare-covered, and covered-covered collisions. Hence, the model allows an analytical ex-
pression for the dimer-formation rate constant k
11
, to be deduced as a function of the degree of surface
coverage and the three sticking probabilities. The theoretical predictions are contrasted with simulated data.
The observed agreement between theory and simulations shows the usefulness of the model for predicting the
initial stages of this kind of aggregation processes.
DOI: 10.1103/PhysRevE.68.011404 PACS numbers: 61.43.Hv, 02.50.-r, 82.70.Dd, 05.40.Jc
I. INTRODUCTION
Macromolecules adsorbed onto colloidal particle surfaces
may either stabilize or destabilize the dispersions. This
makes the employment of macromolecules as additives for
suspensions a much extended practice for industrial pur-
poses. Several applications can be found in mineral and
waste water treatments, such as water treatments for human
consumption, paper industry, drilling fluids, ceramics, agro-
chemical formulation, and in immunoassay diagnostic test
design 1. However, such processes are so highly complex
in nature that they have not been completely understood, yet.
Given any particular situation where macromolecules and
colloidal particles are taking part, the process will depend on
the degree of surface coverage with macromolecules and on
the macromolecule-macromolecule and macromolecule-
particle interactions 2,3. When the particle surface is fully
covered by the macromolecules, the observed result is gen-
erally a stabilized suspension 4,5. For partially covered sur-
faces, however, the already adsorbed macromolecules on a
given particle may attach to the bare patch of another one
forming a particle-particle bridge bridging flocculation
6,7.
It is well known that the bridging flocculation rate de-
pends on the degree of surface coverage. The classical work
of La Mer and Healy 8 predicts a maximum of the floccu-
lation rate when half the total surface is covered by macro-
molecules. When additional factors contribute to destabiliza-
tion, the optimum degree of surface coverage usually
becomes smaller 9.
In spite of the large amount of experimental work that has
been performed for studying different aspects of these types
of systems 10–18, there is still a lack of theoretical models
for explaining their flocculation kinetics. In this work we
attempt to fill this gap proposing a model based on La Mer’s
idea of a surface coverage dependent aggregation rate ca-
pable of describing the initial stages of an aggregation pro-
cess. Additionally, the concepts of sticking probability and
consecutive collisions recently employed for modeling the
transition from diffusion to reaction limited cluster aggrega-
tion are also included 19,20. The obtained results are then
compared with Brownian dynamics simulations.
The paper is organized as follows. Section II reports the
theoretical background. Section III briefly describes the
simulations, presents some simulation results, and confirms
that the models found in the literature are not capable of
matching the data. In Sec. IV an alternative model is pro-
posed and its predictions are compared with the simulated
data. Finally, Sec. V tackles the conclusions.
II. THEORETICAL BACKGROUND
Colloidal aggregation processes may be monitored by the
time evolution of the cluster concentrations, c
i
( t )
=n
i
( t )/ V , where n
i
( t ) is defined as the number of clusters
of size i at time t, and V is the whole volume where the
aggregation takes place. For dilute systems the time evolu-
tion of the cluster concentrations is given by the Smolu-
chowski equation 21,22:
dc
i
dt
=
1
2
j =1
i -1
k
j , i - j
c
j
t c
i - j
t -c
i
t
j =1
k
ij
c
j
t . 1
The kinetic rate constants, or the aggregation kernel, k
ij
,
represent the mean rate at which two i- and j-size clusters
stick to form a ( i + j )-size cluster. It contains all physical
information about the kinetics of the aggregating system.
The cluster concentrations c
i
( t ) are average quantities that
do not consider the internal cluster structure. Nevertheless,
this information is implicitly included in the size dependence
of the kernel k
ij
. It should be noted that the kernel is an
orientational and morphological average of all particular
cluster formation possibilities.
*Email address: moncho@ugr.es
²
Email address: godriozo@imp.mx
‡
Email address: mtirado@ugr.es
§
Email address: schmitt@ugr.es
Email address: rhidalgo@ugr.es
PHYSICAL REVIEW E 68, 011404 2003
1063-651X/2003/681/01140412/$20.00 ©2003 The American Physical Society 68 011404-1