Computer simulation study of irreversible adsorption: Coverage fluctuations
Jordi Faraudo* and Javier Bafaluy
†
Departament de Fı ´sica, Grup de Fisica Estadistica, Facultat de Ciencies Edifici Cc, Universitat Auto `noma de Barcelona,
E-08193 Bellaterra, Barcelona, Spain
Received 04 August 2001; published 7 February 2002
In this paper, we develop a cellular automata model to study the coverage fluctuations in monolayers of
irreversible adsorbed particles. The effect of bulk diffusion and excluded volume interactions between ad-
sorbed and incoming particles on coverage fluctuations is analyzed by simulations and analytically. We also
show that the macroscopic boundary and initial conditions imposed at the system open or closed cell deter-
mine the effect of these factors on coverage fluctuations. In fact, under certain conditions, the excluded volume
interactions only influence fluctuations near the jamming limit.
DOI: 10.1103/PhysRevE.65.037101 PACS numbers: 05.10.-a, 82.70.Dd, 68.43.Mn, 05.40.-a
The irreversible adsorption of colloidal particles macro-
molecules, latexes, bacteria, etc. from fluid suspensions to
solid surfaces is a complex phenomenon of great interest for
example, in filtration, chromatography, . . . ). Much effort
has been devoted to the study of the effect of transport
mechanisms on the adsorption kinetics and on the structure
of the adsorbed monolayer 1,2. Recently, both experimental
3–5 and theoretical 6 studies have analyzed also the fluc-
tuations in the number of adsorbed particles. It is expected
that coverage fluctuations reveal valuable information about
the adsorption process. However, the experimental results
3–5 are difficult to interpret without a theory that can take
into account the effect of bulk diffusion. Up to now, theoret-
ical results concerning fluctuations have been developed in
the framework of geometrical models based on the surface
excluded by adsorbed particles 6. These models do not con-
sider the transport of the particles form the bulk towards the
surface. Thus, the influence of bulk diffusion on coverage
fluctuations is not known.
In this article, we develop a cellular automata model CA
in order to analyze coverage fluctuations in irreversible ad-
sorption driven by diffusion. Two main reasons justify the
convenience of CA models in diffusion problems 7: a it is
possible to develop computer simulations with a reasonable
effort they require less computational resources than other
techniques and b their analytical tractability. Our goal in
this paper is to determine, within this CA model, the role on
coverage fluctuations of a bulk diffusion and b the ex-
cluded volume interactions between incoming and adsorbed
particles. Also, we show that the relative effect of each of
these factors strongly depends on the boundary and initial
conditions imposed on the system. This important effect was
not predicted in previous studies and should be taken into
account in order to interpret properly the experimental re-
sults.
The CA model consists of a square adsorbing surface la-
beled as j =0) with N
max
adsorbing sites and a bulk phase
( j =1, . . . , L
z
) with V =N
max
L
z
sites. Each site can allocate
only one particle. At each time step, all diffusing particles
randomly select with equal probability p an adjacent node
( p =1/6). If the selected node is free, a move to this node is
performed, but if it is occupied, the particle remains at its
initial position. When a particle reaches a free site at the
adsorbing surface, it is irreversibly adsorbed and remains
immobilized at this site. The process ends when the jammed
state is reached all the sites at the adsorbing surface are
occupied or when all particles are adsorbed. We consider
periodic boundary conditions on axis x and y. On the z axis
we consider two kinds of conditions: a a reflecting barrier
at j =L
z
closed cell conditions, and b an equilibrium res-
ervoir with a fixed number of particles N
R
maintained at j
=L
z
open cell conditions. This reservoir is maintained by
removing or adding particles if necessary at each time step.
The initial condition ( t =0) is a uniform distribution of N
B
particles in the case of closed cell conditions and an empty
system in the case of open cell conditions.
The number of adsorbed particles N
0
( t ) increases mono-
tonically with time due to the irreversible nature of the ad-
sorption process. However, N
0
( t ) presents statistical fluctua-
tions: at a given time t, identical surfaces with the same
boundary and initial conditions may have different number
of adsorbed particles. Thus, we define N
¯
0
( t ) as the mean
number of adsorbed particles averaged over an ensemble of
realizations of the adsorption process with the same macro-
scopic boundary and initial conditions. The coverage is
defined as the mean fraction of the surface covered by par-
ticles ( t ) N
¯
0
( t )/ N
max
. It increases monotonically with
time from (0) =0 to its maximum value ( t → ) =1 satu-
ration due to irreversible adsorption. We also define N
¯
j
( t ) as
the mean number of diffusing particles at the plane j at time
t. The mean fraction of occupied sites at slab j is n
j
( t )
=N
¯
j
( t )/ N
max
. The mean flux of adsorbing particles towards
the surface is defined as J
S
( t ) ( t +1) - ( t ). Typically, in
adsorption studies, one characterizes the adsorbed particle
number fluctuations by the reduced variance V
r
defined as
V
r
t
N
¯
0
2
-N
¯
0
2
N
¯
0
=
2
N
¯
0
. 1 *Electronic address: jordi@circe.uab.es
†
Electronic address: Javier.Bafaluy@uab.es
PHYSICAL REVIEW E, VOLUME 65, 037101
1063-651X/2002/653/0371014/$20.00 ©2002 The American Physical Society 65 037101-1