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\headline={\ifnum\pageno=1 \hfil {Invited Review article for Int.~J.~Modern
Phys.~B}\fi}
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\centerline {\bf KINETICS OF IRREVERSIBLE MONOLAYER}
\centerline {\bf AND MULTILAYER ADSORPTION}
\vskip 0.4in
\centerline{{\bf M.C.~Bartelt} and {\bf V.~Privman}}
\vskip 0.2in
\centerline{\sl Department of Physics, Clarkson University,
Potsdam, NY 13699--5820, USA}
\vskip 0.3in
\NI {\bf PACS:}$\;$ 05.70.Ln, 68.10.Jy, 82.20.Wt.
\vskip 0.4in
\centerline{\bf ABSTRACT}
\vskip 0.20in
We review recent theoretical developments of microscopic
lattice and continuum models of the kinetics of irreversible
monolayer and multilayer surface adsorption. Such models
have been used to describe adhesion and reaction processes
of colloidal particles and proteins at solid surfaces.
Theoretical results surveyed here include the void-filling
rate equation approach, exact results for low dimensionalities,
the mean-field theory, and large-time kinetics arguments.
Numerical simulations serving to test and supplement the
analytical theories, are reviewed as well. We also elucidate
the crossover from the discrete to continuum behavior, analyzed
via scaling arguments in the large-time limit of the deposition
kinetics.
\NP
{
\parskip 0pt
\centerline{TABLE OF CONTENTS}
\vskip 0.20in
\NI Abstract \hfill 1
\NI 1. Introduction \hfill 3
\NI 2. Mean-Field Theory \hfill 6
\NI \hphantom{AAA} 2.1. Survey of Results \hfill 6
\NI \hphantom{AAA} 2.2. Note on Experimental Data Analyses \hfill 8
\NI 3. Void-Filling Rate Equations Method \hfill 10
\NI \hphantom{AAA} 3.1. Definitions \hfill 10
\NI \hphantom{AAA} 3.2. Extension to Multilayers \hfill 12
\NI 4. Numerical Simulation Methods \hfill 16
\NI 5. Late Stage Deposition Kinetics \hfill 20
\NI \hphantom{AAA} 5.1. Large-Time Asymptotic Behavior of the
Coverage \hfill 20
\NI \hphantom{AAA} 5.2. Continuum and Lattice Limits \hfill 24
\NI 6. Miscellaneous Topics \hfill 26
\NI \hphantom{AAA} 6.1. Finite-Size Effects \hfill 26
\NI \hphantom{AAA} 6.2. Unoriented Deposition \hfill 29
\NI 7. Deposition of Mixtures \hfill 31
\NI 8. Summary \hfill 37
\NI References \hfill 38
\NI (Figure Captions --- page 42)
\NI (Figures --- appended)
}
\NP
\centerline{\bf 1. INTRODUCTION}
\vskip 0.20in
Models of monolayer particle deposition, without relaxation
(diffusion or detachment) on the time scales of the deposit
formation, have been investigated extensively,$^{1-28}$
under the terms ``random sequential adsorption''
and ``car parking problem.'' In these models, rigid
particles are
placed at random, sequentially and irreversibly onto solid smooth
surfaces in such a way that the particles do not overlap. If
an incoming particle approaches already covered part of the
substrate, it is rejected. In the simplest theories no redeposition
attempts are allowed. Eventually no more particles fit on the
surface and the
process stops in the so-called {\it jamming\/} (or saturation)
limit. Experimentally, such processes are realized,
e.g., in the adhesion of proteins and colloidal particles on uniform
surfaces,$^{29-33}$ as well as in various other situations$^{11}$
including polymer chain systems, consideration
of which led to interest in the one-dimensional models.$^{1-7}$
Recently, evidence was
reported$^{34-37}$
that in certain colloid experiments {\it multilayer\/}
irreversible deposition processes can be observed and studied
systematically. Due to the complexity of the packed-bed systems
utilized in such experiments, attempts
to describe theoretically the
multilayer deposition processes$^{34}$
were largely based on the mean-field theory.
The selection of the topical coverage for this review
was biased by our recent work, Refs.~34-36, 38-40, etc.,
on both monolayer and multilayer adsorption,
and by the emphasis on topics of relevance in the
interpretation of recent colloid experiments.
Several dynamical mechanisms underline the
formation of the deposit in real systems resulting in two
interesting effects. Firstly, there is the {\it blocking\/} of the
available area for deposition which stops the process at a
certain less than close-packed coverage and causes the formation
of a random deposit morphology. The
blocking will also play a role (to a lesser extent) in higher-layer
deposition (particle-on-particle deposition). Secondly, there is the
{\it screening\/} (or overhang) effect, i.e., the shadowing of
the lower layers by the particles in the higher layers. However,
the overhangs in the $n$th layer can ``stick out'' over voids
diffusion-like so that the overhang size will grow as
$\sqrt n\; $ (possibly some other power of $n$ close to
${1 \over 2}$). Thus, their effect will be significant for
$n >\!> \ell^2 $, where $\ell$ is the linear
size of the depositing objects measured in some microscopic units.
Models without blocking but
with screening allowed fall in the class of, e.g., the {\it
ballistic deposition}$^{41}$ or {\it diffusion-limited
aggregation}$^{42}$
(depending on the mechanism of the particle transport to the surface),
which were studied extensively with the recent focus on the
growing-surface scaling properties after many layers have been
deposited.$^{41-42}$
However, the emphasis in colloid deposition is usually on
phenomena within not too many layers (up to
order 10 to 30) because this seems to correspond to the
experimental situation.$^{30,35-37}$ In this regime
the dominant morphological effects may be expected to
be due to blocking.
The most profound feature of the random sequential adsorption model
is its infinite memory: once a particle is placed on the surface it
affects the geometry of all later nearby placements.
Thus the process is extremely
non-Markovian and many methods developed in equilibrium statistical
mechanics cannot be used without further development,
to describe
the deposit formation. It is therefore
important to identify general, universal characteristics
such as, for instance, the logarithmic
divergence of the pair correlation function at contact$^{8,10}$
and the late stage power-law asymptotics of the coverage, in the
continuum limit.$^{8-10}$ The values and universality
of the associated power-law exponent, the
control of the appropriate continuum limit and the interpretation
of the discrete-to-continuum crossover in terms of the precise
scaling combinations have motivated many of the more recent
studies, both theoretical and experimental, of monolayer or
multilayer formation.
Unlike many other branches of statistical
mechanics, one-dimensional models of irreversible adsorption
(usually exactly solvable) have many non-trivial features
representative of higher-dimensional systems.
We do not survey here in detail several important topics
such as the low density (virial-like) expansions,$^{3-4,28,43}$
percolation properties of the deposited layers,$^{14-15,26}$ etc.
The outline of the topics covered is as follows:
In Section 2 we describe the mean-field
deposition models. In Section 3 we summarize
the void-filling rate equation approach
which is not always exact but which captures many of the
correlation aspects of the deposition processes
beyond the simple mean-field theory. Numerical simulations of the
deposition models are discussed in Section 4.
The large-time (late stage) asymptotic analysis methods are presented
in Section 5. Section 6 surveys
several additional recent results and developments.
Section 7 completes the presentation with some new features
in the monolayer deposition of {\it mixtures}.
Finally, in Section 8 we give a short summary.
\NP
\centerline{\bf 2. MEAN-FIELD THEORY}
\vskip 0.20in
\NI {\bf 2.1. Survey of Results}
\vskip 0.16in
The mean-field theory (see Ref.~34) of irreversible
multilayer adsorption
assumes that the coverage (number of particles per unit area),
$\Gamma_n (t)$,
in the $n$th layer approaches the limiting value
$$ \Gamma_n (\infty ) = \Lambda \, , \eqno(2.1) $$
\NI where $1/\Lambda$ is the average {\it blocked\/} area per particle,
{\it a priori\/} not a simple function of the size of the particles.
One further assumes that, to contribute to the growth of the $n$th
layer, the particles must successfully deposit on top of particles
already forming the $(n-1)$st layer, at some uniform rate $R_n$
(per unit time and area). If at time $t=0$ the substrate is
empty, then
$$ \Gamma_n (0) =0 \; , \qquad n=1,2, \ldots \eqno(2.2) $$
\NI One can then write a set of mean-field equations,
$$ { d \Gamma_n \over d t } = R_n \left( \Gamma_{n-1}
- \Gamma_n \right) / \Lambda \, , \eqno(2.3) $$
\NI where $n \geq 1$, and we conveniently defined
$$ \Gamma_0 (t) \equiv \Lambda \, . \eqno(2.4) $$
\NI The right hand side of (2.3) describes the growth stage
$(n-1) \to n$, the rate of which is proportional
to the density of uncovered sites in the $(n-1)$st
layer, given by $\left( \Gamma_{n-1} - \Gamma_n \right)$.
In the symmetric-rate case, $R_1 = R_2 = \ldots = R$, the solution
can be obtained in a simple form,
$$ \Gamma_n (t) = \Lambda \left[ 1 - {\rm e}^{- R t / \Lambda }
\sum_{m=0}^{n-1} { \left( Rt/ \Lambda \right)^m
\over m! } \right]\, ,
\eqno(2.5) $$
\NI corresponding to the total coverage $\sum\limits_{n=1}^\infty
\Gamma_n = Rt$. The sum in (2.5) is omitted for $n=0$. Another
simple case is $R_2 = R_3 = \ldots = 0$,
which is appropriate for the monolayer deposition, i.e.,
$\Gamma_{n \geq 2} \equiv 0$, $\;\Gamma_1 = \Lambda
\left( 1 - {\rm e}^{- R t / \Lambda } \right)$.
For $R_1\ne R_2\ne R_3\ne \ldots$, the closed form expression for the
coverages is
$$ \Gamma_n(t) / \Lambda =1-{\rm e}^{-R_nt / \Lambda }-
\sum_{m=1}^{n-1}\,{\displaystyle\prod_{i=1 \atop i\ne m}^n\,
R_i\over \displaystyle\prod_{i=1 \atop i\ne m}^n\,
\bigl(R_i-R_m\bigr)}\,\biggl( {\rm e}^{-R_mt / \Lambda }
- {\rm e}^{-R_nt / \Lambda }\biggr)
\, .\eqno(2.6) $$
\NI This relation can be used till the first occurrence of
$R_n = R_{n-1}$ or $R_{n-2}$ or $\ldots$ or $R_1$. One
can still apply it to obtain $\Gamma_n\,$, but the coverages in
layers higher than $n$ are no longer given by (2.6). Particular
choices for the values of the $R_n$'s must be examined separately.
In Ref.~34 it was assumed that $R_1\ne R_2 = R_3 = \ldots$.
In this case the coverage in layer $n$ is given by
$$\Gamma_n(t) /\Lambda = 1- {\rm e}^{-R_1t / \Lambda } \left(\displaystyle{
R_2 \over R_2-R_1} \right)^{n-1} +
{\rm e}^{-R_2t / \Lambda }\displaystyle{\sum_{m=0}^{n-2}}
\displaystyle{(R_2t/\Lambda )^m \over m!} \left[ \left( \displaystyle{
R_2 \over R_2-R_1}\right)^{n-m-1} -1 \right] .
\eqno(2.7) $$
\NI For equivalent mean-field stacking models without overhangs,
scaling analyses of the average height of the ``columns'' in the
deposit as a function of $R_2/R_1$ are available$^{44}$ and
suggest that the average stack height scales as $\left( R_2/R_1
\right )^{{1\over 2}}$.
Finally, note that the general short time mean-field behavior is
$$ \Gamma_n ( t \to 0 ) = { \Lambda^{1-n} \over n! }
\left( \prod_{m=1}^n R_m \right) t^n \, , \eqno(2.8) $$
\NI this being the first nonzero term in the Taylor series solution
of (2.3) around $t=0$.
\vskip 0.20in
\NI {\bf 2.2. Note on Experimental Data Analyses}
\vskip 0.16in
In actual experimental data fits, the surface kinetics description
must be combined with the equations governing the transport of the
particles towards the surface. In colloid particle deposition from
flowing suspensions, the transport is essentially
convective-diffusional. The resulting theory is, not surprisingly,
quite complicated (see Ref.~34 and the literature cited therein)
but the mean-field surface kinetics approach outlined
here$^{34}$ provides a fairly good quantitative fit of the
multilayer deposition data recently reported in Ref.~37.
However, deviations from the mean-field theory have been observed
in monolayer deposition experiments$^{32}$ for coverages above
50\% of the jamming coverage. Other theoretical and experimental
studies$^{45-46}$ of sequential adsorption of {\it soft\/} particles
(e.g., interacting via repulsive screened Coulomb potentials) have
reported slower kinetics and smaller saturation coverages
than for rigid (hard-core) adsorbing particles. In this review we consider the
hard-core case only. Similar differences are also expected to be found in the multilayer
deposition experiments at dense coverages. Geometrical and physical restrictions on the
allowed adsorption sites, effective at these coverages,
are expected to have dramatic effects on the layer morphology
and growth.$^{47}$ Theoretical studies beyond the
simple mean-field theory will be reviewed in the following sections.
\NP
\centerline{\bf 3. VOID-FILLING RATE EQUATIONS METHOD}
\vskip 0.20in
\NI {\bf 3.1. Definitions}
\vskip 0.16in
It is convenient to introduce the void-filling rate equations first for
the case of the one-dimensional models. Indeed, the problem of
the {\it monolayer deposition\/} of $k$-mers on the one-dimensional
lattice of spacing $b$ is exactly solvable.$^{5-7}$
Each $k$-mer covers ``area''
(length) $kb$. The case $k=1$ is trivial because there is no blocking.
Thus, one gets a mean-field like (uncorrelated) result,
$$ \Gamma_1 (t) = \left( 1 - {\rm e}^{-brt} \right) \big / b \;,
\qquad k=1\,, \eqno(3.1) $$
\NI which is the solution of the rate equation
$$\displaystyle{d\Gamma_1 \over dt}=r(1-b\Gamma_1)\, , \eqno(3.2)$$
\NI where we have assumed that the deposition of $r$ objects
($k$-mers) is attempted per unit length and unit time (i.e.,
the attempt rate is $r$).
The solution for $k > 1$ is more complicated. However, it can
be formulated$^{6,12}$ in terms of the rate equations for the
probabilities $P(s,t)$ that sequences of $s = 1, 2, \ldots \;$
lattice sites (to be termed $s$-voids) are empty. An $s$-void
may be blocked at one or both ends or it can be part of a larger
sequence of empty sites. In the rest of this section we review
the solution for the case of dimers ($k=2$). The solution for
higher $k$ values$^{5,7}$ is not more complicated and is, in fact,
of interest in the study of such aspects as the crossover from
the discrete exponential large time asymptotics of $\Gamma_1$
to the power-law asymptotic behavior in the continuous line
case.$^{8-10,40}$ We return to these issues in Sections 5 and 7.
Here only the dimer case will be reviewed.
The {\it exact rate equations\/} for the $s$-void probabilities
$P(s,t)$ are ($k=2, s \geq 1 $)
$$ - { d P(s) \over dt } = br \left[ \, (s-1) P(s) + 2 P (s+1)\,
\right]\, . \eqno(3.3)$$
\NI The terms on the right hand side count the rate of
elimination of the $s$-voids by the deposition of incoming dimers on
the $(s-1)$ possible groups of two consecutive sites inside each void or
in the two end-sites (such that only half of the dimer length,
$2b$, is in the $s$-void). The initial values are $P(s,0) = 1\;$.
The solution of (3.3) is
$$ P(s,t) = \exp \left[ - (s-1) brt -2 + 2 {\rm e}^{-brt}
\right] \, . \eqno(3.4) $$
\NI The monolayer coverage (number of dimers per unit length)
is obtained directly form $P(1,t)\;$,
$$ \Gamma_1 (t) \equiv {1 \over 2b} \left[\,1 - P (1,t) \,\right]
= { 1 - \exp \left[ -2 \left( 1 - {\rm e}^{-brt} \right)
\right] \over 2b}\, . \eqno(3.5) $$
\NI The jamming coverage is thus$^1$
$$ \Gamma_1 (\infty ) = { 1 - {\rm e}^{-2} \over 2b}\, .
\eqno(3.6) $$
Close inspection of (3.4), or any other set of solutions of (3.3)
with uniform initial conditions [the same $P(s,0)$ for all $s$],
confirms that in $1D$ the probability to have an empty site
adjacent to one that is known to be empty is independent of the
number of additional consecutive empty sites.$^{6,12}$ As a result, the
infinite hierarchy of equations for $P(s,t)$ can be exactly decoupled.
However, for a general set of initial conditions even in $1D$, (3.3) yields
$$P(s,t)={\rm e}^{ -(s-1)brt}\,\displaystyle{\sum_{n=0}
^{\infty}}\,\displaystyle{(-2)^n\,P(s+n,0) \over n!}\,
\left( 1-{\rm e}^{-brt} \right)^n \, , \eqno(3.7)$$
\NI which depends on {\it all\/} the initial values,
$P(s,0)$, and the decoupling is no longer possible.
In higher dimensions, the rate equations for the monolayer deposition
can be also formulated in terms of the probabilities of
various voids.$^{11-12,19}$ However, now all types of void shapes
are possible and the rate equations even for the smallest voids rapidly
become very complicated. No exact solution is available for this
hierarchy of equations in $D>1$. However, one can devise
truncation schemes to obtain closed sets of
differential equations for the void probabilities,
guided by the intrinsic structure of the hierarchies and the
pattern of exact results in $1D$.
Extensive studies within this approach have been reported for the $2D$
monolayer deposition.$^{11-12,19}$
\vskip 0.20in
\NI {\bf 3.2. Extension to Multilayers}
\vskip 0.16in
For multilayer deposition one can no longer write down the
exact rate equations in terms of the single-layer void probabilities
only. However, one can develop approximate
rate equations which include fluctuation features of the
deposition beyond the simple mean-field theory. One such
calculation for dimers was reported recently$^{38}$
for a particular choice of deposition rules in $1D$. We will
summarize the results briefly in this subsection.
Let $P_n (s,t)$ denote the probability of finding an $s$-void
in the $n$th layer. For an initially empty lattice we have
$$ P_n (s,0) = 1\; , \qquad n \geq 1\; , \; s \geq 1 \,.
\eqno(3.8) $$
\NI The monolayer probabilities will be now
identified as $P_1 (s,t) $. It is also useful to introduce the
notation
$$ P_0 (s,t) \equiv 0 \; , \qquad s \geq 1 \, .\eqno(3.9) $$
In multilayer deposition we will assume that a dimer can adhere
in the $n$th layer ($n \geq 2$) only if it is ``supported'' by
one or two occupied segments in the $(n-1)$st layer. One can
still choose various rules for adhesion. It is natural to
always allow adhesion on top of a pair of occupied segments.
In the model considered in Ref.~38, we also allowed adhesion
on top of 1-voids {\it provided\/} the unsupported half of
the depositing dimer screens only a single-segment void
(of length $b$). Thus, we {\it eliminate screening effects\/}
by disallowing formation of overhangs over voids of two or
more empty segments. This does not constitute an approximation
but just a particular choice of adhesion rules. Note that the
single-segment voids are anyway unavailable for deposition of
dimers.
For this model we can no longer write
down the {\it exact\/} rate equations. {\it On the average}, the
rate of successful deposition events in the $n$th layer
will be reduced by a factor $\left[ 1-P_{n-1} (2,t)\right] $,
as compared to the deposition at similar coverages if
it were in the first layer. Thus, we can write the rate equations
($n\ge 1$, $s\ge 1$)
$$ - { d P_n (s) \over dt } = br \left[\, (s-1) P_n (s) +
2 P_n (s+1)\, \right]
\left[ \, 1 - P_{n-1} (2) \, \right] \, . \eqno(3.10)$$
\NI These relations are exact for $n=1$ only. For $n > 1$ they
involve a certain degree of a mean-field type averaging due to
the disregard of {\it fluctuations\/} in the deposition rates in
the stage $(n-1) \to n$ of the layer growth, on top of the
{\it finite-size\/} intervals in the $(n-1)$st layer, as compared
to the infinite-length substrate. Still they must capture most
of the correlation/fluctuation aspects of the one-dimensional
model introduced above.
The elimination of the screening effects secures that each void
of two or more lattice spacings will be filled up eventually.
It is important to point out, however, that this property does not
in itself imply the same jamming coverage in each layer. Since
our rate equations do not account for the deposition rate
fluctuations described above they do yield the same jamming values
in each layer,
$$ \Gamma_n (\infty ) = { 1 - {\rm e}^{-2} \over 2b}\, .
\eqno(3.11) $$
\NI The exact values of $\Gamma_n ( \infty ) $ should have a slight
variation with $n$, with the nonzero limiting value $\Gamma_\infty
(\infty )\; $.
Detailed results for the time-dependence of the coverages
were reported in Ref.~38. Here we present the results of
the comparison of the first three ``rate-equation''
layer coverages with the mean-field approximation: see
Figure 1. The solution of (3.10) for the coverage in the $n$th layer
has the form
$$\Gamma_n(t)=\displaystyle{1\over 2b} \left[ 1-\exp\left(
-2+2{\rm e}^{-Q_{n-1}(t)}\right) \right], \eqno(3.12)$$
\NI where
$$Q_{n-1}(t)\equiv \int\limits_0^{brt} \, \left[ 1-P_{n-1}(2,u)
\right ] du \, .\eqno(3.13)$$
\NI The explicit form of $\,\Gamma_n(t)\,$ rapidly become very
cumbersome. The plotted values for $\Gamma_3$ were obtained
by numerical evaluation of $Q_2(t)\,$ in (3.13) with
$$P_1(2,u)=\exp\left[ -{3\over 2} - u - {1\over 2}
\exp\left(-2+2{\rm e}^{-u}\right) +2\exp\left(
{1\over 2} - u - {1\over 2}{\rm e}^{-2+2{\rm e}^{-u}}\right)
\right ] . \eqno(3.14)$$
\NI Explicit expressions for $\Gamma_{1,2} (t) $ are given in Ref.~38.
The trend is
basically the same in the three layers (see Figure 1). As expected, the mean-field
approach overestimates the surface coverage at late times, when strong in-layer
correlations determine the dynamics, but it yields the correct behavior
in the early stages of the deposit formation. The mean-field
approximation seems to get worse for higher layers.
\NP
\centerline{\bf 4. NUMERICAL SIMULATION METHODS}
\vskip 0.20in
Several numerical Monte Carlo studies of {\it monolayer\/}
deposition have been reported, largely in $2D$. These works
are briefly discussed at the end of this section. The
{\it multilayer\/}
deposition is, however, a rather recent topic. The first
systematic Monte Carlo study of irreversible deposition in
multilayers was reported in Ref.~39, for one- and two-dimensional
models without screening (to be defined below). Indeed, the
experimental situation in colloid systems$^{30,35-37}$ seems to
correspond to the regime of sufficiently few layers (of order
10 to 30) in the deposit so that the dominant correlation effects are
due to the blocking. Thus, as a first step, one considers the
extreme case of no screening at all.
The models studied involved deposition of $k$-mers on periodic $1D$
lattices of unit spacing, and deposition of
square-shaped $(k \times k)$-mers on periodic square lattices
of unit spacing in $2D$. In each layer, the landing sites were
chosen at random,
i.e., for a linear lattice of size $N$,
segments of length $k$ were randomly select. For general (substrate) dimension $D$
the target sites were hypercubes formed by $k^D$ lattice
unit-cubes. The time scale, $T=r\ell^D t$, is fixed by having
exactly $(N/k)^D$ deposition attempts per unit time. Here
$\ell^D$ is the volume of the depositing object, $\ell =bk$. The attempt is successful
if the selected target site is empty, and also, for layers $n\ge 2$, the ``support''
criteria are satisfied. Thus, if {\it all\/} the lattice hypercubes in the selected
landing gap are already covered by exactly $(n-1)$ layers, the
arriving object is deposited, increasing the coverage to $n$.
Otherwise, the attempt is rejected. Only
deposition on top of the the fully occupied regions is allowed in this simplest
version of the model without overhangs and screening.
For convenience, we consider the dimensionless coverage,
$\theta_n (T) $, defined as the fraction of the area covered by particles in
$2D$ (fraction of the filled volume in general $D$), in layer $n$. In Sections 2-3,
we used $\Gamma \equiv \theta \ell^{-D}$.
Let us survey various results found for deposition models without
screening. For lattice models (unlike the continuum monolayer
deposition models$^{8-10}$ described in Section 5), the fraction
of occupied area in the $n$th layer, $\theta_n (T)$, approaches
the saturation value exponentially,
$$ \theta_n (T) \approx
\theta_n (\infty ) - B_n {\rm e}^{-\tilde{\sigma}_n \,T}
\, ,\eqno(4.1) $$
\NI where we omit the $k$-dependence of the various quantities.
The first layer shows the usual in-plane correlations due to
blocking, in the process of buildup of the jammed state. However, the
typical jammed configuration in the higher layers {\it in
deposition without overhangs\/} contains more gaps the larger
the value of $n$. The growth in the higher layers proceeds more
and more via uncorrelated ``towers'' (separated by gaps). The
numerical results$^{39}$ indicate that the jamming coverages
decrease according to a power law, with no intrinsic length scale,
reminiscent of critical phenomena,
$$ \theta_n ( \infty ) - \theta_\infty ( \infty ) \approx
{A \over n^\phi }, \qquad n\gg 1\, . \eqno(4.2) $$
\NI Furthermore, within the limits of the numerical accuracy,
the value of the exponent $\phi$ appears universal, for different
$k \geq 2 $ and dimensionality. This asymptotic power law form was
recently explained analytically$^{48}$ by reducing the problem of
calculating the jamming coverages in the $n$th layer to the
calculation, in $1D$, of the probability that a single ``two-k-mer-wide tower'' in the
$n$th layer will decay into a ``one-k-mer-wide tower'' (assuming that, for large $n$,
the approach to the jamming coverage is dominated by this decay mechanism). This
probability is proportional to $\theta_n(\infty)-\theta_{n+1}(\infty)$ and
decays asymptotically as $\displaystyle{
1\over \sqrt{n}}$. For details on the precise form, see Ref.~48. The
``tower argument'' does not depend on the object size $k$ explicitly
hence a universal exponent, $\phi = {1\over 2}$. However, the
amplitude $A$ has a non-trivial $k$-dependence as suggested by
fits of the Monte Carlo data$^{48}$ in the large $n$ regime, for
several $k$ values.
In the computer simulations of Ref.~39, system sizes were as
large as $N=10^5$ in $1D$ and $N \times N =1000^2$ in $2D$.
A comparison with the results for smaller systems suggests that
the finite-size effects were negligible for the largest system
sizes studied (see below for a discussion of size effects). The
data were averaged over as many as 600 runs, which went up to
times $T=150 k^D$. The $k$ values
in Ref.~39 were $k=2,3,4,5,10$ in $1D$, and $k=2,4$ in $2D$.
As an illustration of a Monte Carlo result,
Figure 2 shows the variation of the coverage for the first 15
layers, for the $2D$ system with $k=2$. Results for $1D$ models and
for other values of $k$ have a qualitatively similar behavior.
For small $T$, the $2D$ coverage was found to increase
according to $\theta_{n}(T) \propto T^{n}$, as expected from
the mean-field theory (Section 2).
The approach to the jamming limit was fitted well by the exponential
time dependence (4.1). Numerical semi-logarithmic least-squares
fits yielded decay constant values $\widetilde{\sigma}_{n}
\simeq k^{-D}$. The power
law behavior of the jamming coverage, see (4.2), was also
checked numerically, with the results
$$ \phi(1D) = 0.58 \pm 0.08
\;\qquad {\rm and} \; \qquad
\phi(2D) = 0.48 \pm 0.06 \, , \eqno(4.3) $$
\NI consistent with the analytical prediction,$^{48}$
$\phi = {1 \over 2}$.
As already mentioned, the monolayer deposition has been studied
by numerical Monte Carlo simulations by several authors
(see, e.g., Refs.~14-17, 20-27, 40, 49-50). The most recent
simulations focused on the asymptotic large time behavior of
the coverage,$^{20-27,40}$ which will be reviewed in Section 5, and on
the kinetics of irreversible adsorption of mixtures,$^{49-50}$ which
will be reviewed in Section 7. Recently, methods were
developed$^{40}$ to address both the lattice (discrete) and
the continuum deposition, in a unified way. The numerical results
were interpreted within the framework of a phenomenological
theory describing the crossover from lattice to continuum.
This theory will be outlined in the next section.
The numerical aspects, including three
different algorithms employed in monolayer deposition
simulations, were detailed, e.g., in Ref.~40.
\NP
\centerline{\bf 5. LATE STAGE DEPOSITION KINETICS}
\vskip 0.20in
\NI {\bf 5.1. Large-Time Asymptotic Behavior of the Coverage}
\vskip 0.16in
In this section we address the large time asymptotic behavior
of the coverage which thus far has been studied only in the
case of {\it monolayer\/} deposition, except for the lattice
simulation results confirming relation (4.1), see Section 4.
For continuum monolayer deposition models, the large time
behavior is generally {\it power-law},
$$ \theta (t) = \theta (\infty) - {{\rm const} \, ( \ln t )^q
\over t^p }\, \eqno(5.1) $$
\NI where $q=0$ in most cases. Note that we will use the
dimensionless coverage as described in Section 4, but we
work with the ``real'' time $t$. In $1D$ one has $p=1, q=0$,
see Refs.~8-10. Analytical arguments$^{9-10}$ support the numerical
conjecture$^{8,29}$ that $p=1/D\,$ (and $q=0$) for deposition
of spherical objects in $D$ dimensions. However, there are
analytical$^{10,40,46}$ and numerical$^{13,20-27,40}$ indications
that the precise convergence law depends on the shape and
orientational freedom of the depositing objects, the curvature
of the substrate, and interactions
between the adsorbed and adsorbing particles.
Numerical simulations of continuum deposition are resource
consuming, and even the results of long Monte Carlo runs
are difficult to interpret unambiguously.$^{20-27}$
On the other hand, lattice model simulations are easier to
perform.$^{14-17}$ However, the approach to the jamming coverage
is asymptotically exponential in lattice deposition models,
$$ \theta (t) = \theta (\infty) - {\rm const}
\, {\rm e}^{ - \sigma \, t}\, . \eqno(5.2) $$
\NI The rate constant $\sigma=\widetilde{\sigma}_1 r \ell^D$, see (4.1),
vanishes in the continuum limit, while the slower power-law
(5.1) builds up.
This effect was studied for the deposition of fixed-orientation
squares
on two-dimensional substrates.$^{40}$ For the continuum version of the
deposition of hypercubic objects of fixed orientation in $D$
dimensions, Swendsen$^{10}$ proposed an analytical argument for
the asymptotic law (5.1) with $q=D-1$ and $p=1$. Here we review
an analytical theory$^{40}$ which elucidates the crossover from
the characteristic lattice behavior (5.2) to the continuum
asymptotic form while providing the theoretical framework for
testing various phenomenological predictions by both lattice
and continuum Monte Carlo simulations.
The analytical considerations, reviewed below, generalize the
``continuum'' ideas of Refs.~9-10 to the lattice kinetics.
Predictions of the phenomenological theory have been checked against
the asymptotic
expressions derivable from the exact solution for the deposition of $k$-mers on $1D$
lattices. The Monte Carlo simulations of the $2D$ deposition of oriented squares
were described in Section 4. The $2D$ data confirm most of the
predictions of the theory outlined below.
Consider generally the deposition of (hyper)cubic objects of
fixed orientation and size $\ell^D$ on a $D$-dimensional
substrate. Let us assume that the substrate has dimensions $L^D$
and cubic shape aligned with the orientation of the depositing
$\ell^D$ cubes. It is convenient to visualize $L$ as an integral
multiple of $\ell$, although the limit $L \to \infty$ will
always be taken prior to any other limits, so that we need not
concern ourselves with finite-size effects. The rate of random
deposition attempts will be denoted by $r$ and measured per unit
time and volume. A point in the volume $L^D$ is chosen at random,
with uniform probability density, $1/ L^D$.
The point will mark the location of an hypercube of size $\ell^D$,
e.g., by being the cube's center. If this cube does not overlap
any other cubes already placed, it is added to the substrate;
otherwise, the attempt is discarded.
The lattice approximation is introduced by choosing the cubic
mesh size $b = \displaystyle{\ell \over k}$.
An hypercubic lattice of spacing $b$
is fixed parallel to the axes of the total volume $L^D$. The
lattice deposition is defined by requiring that the objects of
size $\ell^D$ can only deposit on sites consisting of $k^D$
lattice unit-cubes.
Thus, the deposition is no
longer continuous but occurs only in $(L/b)^D = N^D$ sites (we
neglect boundary effects). In order to preserve the overall
deposition rate, the deposition attempt rate at each
lattice site must be $rb^D$, per unit time.
According to Refs.~9-10, the late stage of the deposition in
continuum can be described as filling up of voids small enough
to accommodate only one depositing object. The deposition actually
proceeds in two regimes. In the initial, fast-deposition stage the
large gaps are partially filled by the depositing objects leaving
smaller gaps. Gaps small enough so that only one object can fit
in are also filled up, simultaneously with the first process.
However, there should exist a certain time $\tau$ after which
most of the large gaps have been eliminated and the deposition
process is dominated by the small gaps. At this time $\tau$, the
density of those small gaps (number of gaps per unit volume) will
be denoted by $\rho$, and one can further assume$^{9-10}$ that
gaps of various shapes have roughly equal density.
For lattice models, a similar picture should apply for
sufficiently large $k$ values,$^{40}$ $\,k^D \gg r \ell^D \tau $.
Specifically, for the deposition
of (hyper)cubes, typical small gaps can be assumed$^{10}$ to have
rectangular shapes, with edges along the lattice directions. For
counting purposes, we can classify these ``small voids'' as
rectangular boxes of sizes $\,\left[ (k+n_1) \times (k+n_2)
\times \ldots \times (k+n_D) \right]\,$, measured in lattice
spacings. The integers $n_j$ can take on values $\,n_j = 0,1,
\ldots , k-1$, in order to prevent deposition of more than one
object in a void. In this approximate classification of the gaps,
there are $k^D$ different types of gaps. Each type will have
density $\rho / k^D$ at time $\tau$, and will be filled up at
the rate $\left[ r b^D (n_1+1) (n_2+1) \ldots (n_D+1) \right]$,
per unit time. We will consider the regime of $t\gg \tau$ so that
no new small gaps are created by the elimination of large gaps.
Then the density, $\Omega$, of each type of small gaps
will have the following time dependence:
$$ \Omega (n_j) = {\rho \over k^D} {\rm e}^{-r b^D (n_1+1) \ldots
(n_D+1) (t - \tau )} \, .\eqno(5.3) $$
In each deposition event, the dimensionless coverage, $\theta$,
is increased by $(\ell/L)^D$. The rate of
such events per unit time, for each type of gaps, is just
$\left[ r b^D\Omega (n_j)\prod\limits_{m=1}^D(n_m +1)\right]$.
Thus, we have
$$ {d \theta \over dt} \simeq \sum\limits_{n_1=0}^{k-1} \ldots
\sum\limits_{n_D=0}^{k-1} {r b^D \ell^D \rho \over k^D }
\left[ \prod\limits_{j=1}^D (n_j+1) \right] \exp \left[ - r b^D
(t-\tau)\,\prod_{m=1}^D (n_m+1)\right]\, . \eqno(5.4) $$
\NI This relation yields the asymptotic
($t \gg \tau$) estimate
$$ \theta (t) = \theta (\infty ) - {\rho \ell^D \over k^D}
\sum\limits_{n_1=0}^{k-1} \ldots \sum\limits_{n_D=0}^{k-1}
\exp \left[ - \left( r \ell^D/ k^D \right)(t-\tau)\,
\prod_{m=1}^D (n_m+1) \right] ,\eqno(5.5) $$
\NI where the $k$-dependence of $\theta (\infty)$ for
$k^D \gg r \ell^D \tau $ should be smooth and have no interesting
features. Here, we omit the $k$-dependence of $\theta (t)$.
We further note$^{40}$ that the expressions (5.4) and (5.5) can only
be used as the leading-order estimates. Indeed, the limits of
large $k$ and $t$ have been assumed, and at the present level
of the derivation we have no control of the corrections. Thus,
we can modify these relations as long as the {\it leading\/}
behavior is preserved. The most important such a change consists
of replacing $(t- \tau )$ by $t$, thus neglecting terms of
relative magnitude $r \ell^D \tau / k^D $, as compared to the
leading-order $t$-dependent terms. This suggests that (5.4) and (5.5)
provide in fact a {\it one-parameter}, $\rho$, asymptotic
representation of the coverage. Thus, we replace (5.5) by
$$ \theta (t) = \theta (\infty) - {\rho \ell^D \over k^D}
\sum\limits_{n_1=0}^{k-1} \ldots \sum\limits_{n_D=0}^{k-1}
\exp \left[ - \left( r \ell^D t / k^D \right)
\prod_{m=1}^D (n_m+1) \right] .\eqno(5.6) $$
This expression provides an asymptotic description of the large-time
kinetics, confirmed by the exact $1D$ results and by Monte Carlo
studies in $2D$, see Ref.~40 for details. Similar considerations
for non-cubic shapes were recently discussed$^{49-50}$ and tested
numerically.
\vskip 0.20in
\NI {\bf 5.2. Continuum and Lattice Limits}
\vskip 0.16in
For $k$ fixed, the ``lattice'' large time behavior sets in
for $r \ell^D t \gg k^D$. In this limit the $n_j = 0$ term in
the sums in (5.6) dominates,$^{40}$
$$ \theta (t) \approx \theta (\infty ) - { \rho \ell^D \over k^D }
{\rm e}^{- r \ell^D t / k^D }. \eqno(5.7) $$
\NI Thus, the time decay constant in equation (5.2) behaves according to
$$ \sigma \approx \displaystyle{ r \ell^D \over k^D} \qquad\qquad
(k\;\;{\rm large})\,. \eqno(5.8) $$
\NI The results (5.7)-(5.8) were confirmed in $1D$ where one finds exactly
$\rho \ell = {\rm e}^{-2 \gamma}$, where $\gamma = 0.57721\ldots$
is the Euler's constant, and in $2D$ where numerical
studies$^{40}$ yielded
$\rho \ell^2 \simeq 0.44$.
The continuum limit of (5.6) is obtained for $k^D \gg r \ell^D t$.
In this limit one can convert the sums to integrals:
$$\theta (t) \approx \theta (\infty ) - \rho \ell^D
\int\limits_0^1 dx_1 \ldots \int\limits_0^1 dx_D
\, \exp \left( -r \ell^D t \; x_1 x_2 \ldots x_D \right)
\, .\eqno(5.9) $$
\NI Recall that all the expressions here apply only for $t\gg \tau$
and $k^D \gg r \ell^D \tau$, where $r \ell^D \tau$ is a fixed
quantity of order 1. Thus, the large-$k$ and large-$t$ conditions
are simply $k \gg 1$ and $t \gg \left( r \ell^D \right)^{-1} $.
The latter condition allows us to evaluate the integrals in (5.9)
asymptotically, to the leading order for large $t$, which yields
$$ \theta (t) \approx \theta (\infty ) - {\rho \left[ \ln \left(
r \ell^D t \right) \right]^{D-1} \over
(D-1)! \; r t } \, .\eqno(5.10) $$
\NI The asymptotic $\left( \ln t \right)^{D-1} \over t $ law was
predicted in Ref.~10 for the continuum deposition of cubic objects.
Numerical studies in $2D$, and exact analyses in $1D$, indeed
confirm this conclusion, with the same $\rho$ values. However,
the crossover criterion from the lattice to continuum behavior,
$k^D \sim r \ell^D t$, is not fully consistent$^{40}$
with the $2D$ data (there is no such problem in $1D$).
Generally, the availability of the phenomenological description
of the lattice vs.\ continuum deposition kinetics in terms of the
same parameter ($\rho$) is useful in deriving improved numerical
estimates for jamming coverages and other quantities.$^{40}$
Specifically, recent jamming coverage estimates$^{17,25,40}$ ruled out
conjectures$^{51-52}$ that the $2D$
jamming values are squares of the $1D$ values.
\NP
\centerline{\bf 6. MISCELLANEOUS TOPICS}
\vskip 0.20in
\NI {\bf 6.1. Finite-Size Effects}
\vskip 0.16in
Understanding the leading corrections due to the size of the substrate
and choice of the boundary conditions is generally important for
numerical studies in high dimensions
as well as for testing conjectures or scaling extrapolations
against results from series analyses and hierarchical truncation
schemes.
Exact results on of finite--size and boundary effects in $1D$
monolayer lattice deposition were reported in Refs.~6 and 53.
In $2D$, computer simulations found rather small size corrections,
for systems of typical size larger than 10 lattice spacings,$^{39}$
roughly proportional to the ratio of the size of the objects to the size
of the substrate.$^{15}$
The form of the finite-size corrections in irreversible deposition
is intrinsically related to the structure of the spatial
correlations in the deposit$^{54}$ which, for large separations,
decay superexponentially (factorially), as compared to the exponential decay of
equilibrium-model correlations. This fast decay of spatial correlations (and thereby
small size effects) is characteristic of kinetic
processes without relaxation and holds regardless of the dimensionality. Thus the
one-dimensional models provide a non-trivial insight into the
general form of the finite-size and boundary corrections. To
our knowledge, no analytical treatment of finite-size effects has
been reported for multilayer deposition.
One of the possible descriptions of random sequential adsorption
on a finite line (of length $N$ and initially empty) follows
closely the rate-equation treatment reviewed in Section 3
for infinite systems. The rate equations are now formulated
for the total number of gaps of exactly size $m$, $\,S(m,T)\,$
with $1\le m\le N$, as well as for the total number of
$m$--gaps (possibly part of larger gaps), $W(m,T)$. Both
quantities vanish if $m>N$ and, for a given choice of boundary
conditions, are simply related to each other (see Ref.~53 for details).
The coverage is easily obtained from $S$ or $W$,
$$\theta(T;N)=1-{\displaystyle{\sum_{m =1}^N\, m \,
S(m,T)} \over N}=1-{W(1,T) \over N}\;. \eqno(6.1)$$
\NI The jamming coverages are obtained as usual in the limit
when $T\to \infty$. For deposition of dimers, the
rate-equation solutions can be formulated in closed form, suitable
for asymptotic analysis. The qualitative features of the leading
corrections should not depend on the choice of the particle
size (as long as it is much less than $N$).
Note that for monomers, the coverage shows no finite-size or
boundary effects,
$$\theta(T;N)=\theta(T;\infty)=1-{\rm e}^{-T}\, .\eqno(6.2)$$
\NI As before, the time has been normalized by the attempt rate:
$\,T=r\ell^D t$.
For the dimer coverages we obtain$^{53}$
$$\theta(T;N)=\theta(T;\infty)+{1\over N}\,\displaystyle{\sum_{n=0}
^{N-1}}(-2)^nnI_n(T)+\displaystyle{\sum_{n=N}^\infty}(-2)^nI_n(T)
\; ,\eqno(6.3)$$
\NI for free boundary conditions, and
$$\theta(T;N)=\theta(T;\infty)+(-2)^{N-1}{(1-{\rm e}^{-T})^N
\over N!}+\displaystyle{\sum_{n=N}^\infty}(-2)^nI_n(T)\; ,
\eqno(6.4)$$
\NI for periodic boundary conditions, in both cases recovering
$\theta(T;\infty)=1-{\rm e}^{-2(1-{\rm e}^{-T})}$, when
$N\to \infty$. In (6.3) and (6.4),
$$\displaystyle{I_n(T)=\int\limits_0^T {\rm e}^{-x_1}dx_1\int
\limits_0^{x_1}{\rm e}^{-x_2}dx_2\ldots\int\limits_0^{x_{n-1}}
{\rm e}^{-x_n}dx_n ={(1-{\rm e}^{-T})^n \over n!}}\, .
\eqno(6.5)$$
\NI In Figure 3 we show the coverages for $N=6$ and $N=\infty$.
For {\it short times}, the coverage of dimers satisfies
$$\theta(T;\infty)-\theta(T;N) \approx \displaystyle{2\over N}\, T
\; , \eqno(6.6)$$
\NI for free boundary conditions, whereas
$$\theta(T;\infty)-\theta(T;N) \approx \displaystyle{(-2)^{N-1}
\over N!}\, T^N \; , \eqno(6.7)$$
\NI for periodic boundary conditions. Numerical evidence
supporting the form (6.6) for oriented deposition of squares
in $2D$ can be found in Ref.~15.
One can also obtain$^{53}$ the leading corrections for the {\it large
time\/} behavior. For $N\gg 1$, we have
$$\theta(\infty;\infty)-\theta(\infty;N) \approx \displaystyle{
(-2)^N \over N!} +{\rm e}^{-2}\,\left[\displaystyle{\Upsilon(N,-2)
\over N!}-1 \right] \, , \eqno(6.8)$$
\NI and, as $T\to \infty$,
$$\theta(T;\infty)-\theta(T;N) \approx
\displaystyle{ (-2)^N \over N!}\, (1+{\rm e}^{-T})
+(1+2{\rm e}^{-T}) \,{\rm e}^{-2}\,\left[
\displaystyle{\Upsilon(N,-2) \over N!}-1 \right]
\, , \eqno(6.9)$$
\NI for free or periodic boundary conditions [for $\,N\sim {\cal
O}(1)$, the expression (6.9) in modified by additional prefactors$^{53}$
that differ
for the two types of boundary conditions by terms of order
${(-1)^N \over N}$]. Here $\Upsilon$
is the incomplete gamma function and $\,\displaystyle{{\Upsilon(N,-2)
\over N!}-1 \approx -{e^{-N} \over \sqrt{N}}}\,$, for $\,N\gg 1$.
The overall size effects are milder for periodic boundary
conditions, at all times, a typical situation in statistical mechanics.
\vskip 0.20in
\NI {\bf 6.2. Unoriented Deposition}
\vskip 0.16in
Several recent numerical studies$^{20-27,49-50}$ have
investigated the
properties of $2D$-deposition of non-spherical (elliptical
and rectangular) objects with both position and orientation
sampled from a random distribution. Several new interesting
properties were found when
the orientational restrictions were removed. However,
a full theoretical understanding is still missing.
In these systems, the deposition process evolves in two regimes.
During the first stage objects can fall at random, nearly every
adsorption attempt is successful and relatively large areas are
``wasted'' due to the orientational freedom. This {\it wasting\/}
effect is more pronounced for particles of large eccentricity.
Thereafter only particles of orientation similar to that of the
already deposited particles in the targeted region of the substrate
will successfully adsorb, which of course slows down the kinetics.
This however produces an {\it ordering\/} effect (hence better
packing), more pronounced the higher
the particle aspect ratio: parallel objects tend to cluster
forming large oriented domains in the jamming limit.$^{20,27}$
One of the resulting striking features is the dependence of the
jamming coverage$^{20-27}$ on the particle aspect (or axial) ratio,
$\alpha$. It presents
a maximum at aspect ratios of order 2 (or ${1\over 2}$ depending
on the definition used), independent of the
type of particle, suggesting a complicated
interplay of the two effects described above. As a function of
$\alpha$, the jamming coverage should of course be invariant
under the transformation $\alpha \longleftrightarrow
\displaystyle{1\over \alpha}$.
For aspect ratios near 1, unoriented
deposition is more efficient than the corresponding fixed-orientation version
by a factor proportional to the area that an unoriented particle can
continuously explore (in terms of the actual area covered by a
particle), the so-called {\it packing efficiency}. This factor
is of order $\alpha$, for small $\alpha$,
and decreases as $1/ \alpha$, for large values of $\alpha$.
At high aspect ratios ($\alpha \gg 1)$, much of the deposition
time is spent building
up aligned domains separated by voids wasted during the early
stages. Snapshots of nearly jammed configurations$^{20}$
show aligned domains of rougly the size of a few low-aspect-ratio particles,
therefore containing a number of deposited particles of
order $\alpha$. However, when $\alpha \to \infty$, the actual area
covered by particles vanishes at any finite time and no jamming
limit has ever been observed in the simulations of {\it continuous\/}
deposition. Numerical evidence for the power law behavior
in time is inconclusive; the indications are that
$\,p={1\over 3}\, $ for $\alpha \gg 1$, see (5.1).
Studies of unoriented {\it lattice\/} deposition of high-aspect-ratio
particles$^{27}$
are more recent. Here the objects are deposited along the
lattice axes only and a jammed state exists. According to the
simulations, the process develops fairly long domains of aligned
particles, the approach to jamming is exponentially fast (with
constant rates independent of the particle size) and the
jamming coverage decreases logarithmically with the characteristic
length of the particles. In {\it continuous\/} deposition the
results$^{20}$
are consistent with a weak power law ($p \simeq 0.2$),
although they can not rule out a logarithmic convergence.
\NP
\centerline{\bf 7. DEPOSITION OF MIXTURES}
\vskip 0.20in
Experimentally, it is also possible to study surface deposition
kinetics of well
defined {\it mixtures\/} of different types of particles.
Very little progress has been made in the theoretical
description of such mixture-deposition processes, even for
monolayers. Most results available were obtained numerically
or within approximation schemes,$^{17,49-50,55-59}$ and no systematic
picture has emerged thus far.
However, exact results can be obtained for a class of monolayer
mixture-deposition models in $1D$ within the rate equation
approach outlined in Section 3. The only previous studies of
$1D$-deposition of mixtures$^{55-59}$ focused on the estimation of
jamming coverages and their sensitivity to ratios
of deposition rates and particle sizes (for fixed shape).
Recently, exact results were reported in Ref.~60 for the case of
the deposition of mixtures of fixed-length and
{\it pointlike\/} particles on $1D$ line ``surfaces.'' The choice
of this particular type of mixture is motivated by recent
studies$^{49-50,58}$ suggesting that interesting effects are to
be expected when particle sizes differ significantly.
Indeed, new interesting properties of the deposition kinetics
emerge in the continuum version of the process.
Here again, we introduce first the corresponding lattice model.
Thus we define a lattice of spacing $b=\displaystyle{\ell \over
k}$, and allow for deposition only at sites where the
incoming particle will coincide exactly with the underlying
lattice. The $k$-mer deposition frequency per site will be $rb$.
For fixed $k \ge 2$, we will denote by $f(k)$ the deposition attempt
rate (per site) of monomers. As before, we use the
dimensionless time variable $T \equiv r\ell t$. In the limit
$\, k\to \infty \,$ ($b\to 0$, with $\ell$ fixed), the monomers
become {\it pointlike}.
The rate equation approach is particularly well suited for extending
the treatment of single-species deposition to the case of deposition
of mixtures. For mixtures, the rate equations are expressed in terms
of the same
probabilities $P_m(T)$ that connected groups of $m$ sites are not
covered by particles at time $T$.
The coverage (density of occupied sites) is still given by
$$ \theta (t) = 1 - P_1 (t) \, . \eqno(7.1) $$
The rate equations appropriate for the present problem are$^{60}$
$$-k {dP_m \over dT}=\cases{(k-m+1)P_k+2
\displaystyle{\sum_{j=1}^{m-1}} P_{k+j}
+am P_m \, , & if $m \le k \, ,$
\cr\noalign{\vskip 15pt}
(m-k+1)P_m+2 \displaystyle{\sum_{j=1}^{k-1}} P_{m+j} +am P_m \, ,
& if $m \geq k \, ,$\cr} \eqno(7.2) $$
\NI where the term $amP_m$ on the right-hand side corresponds
to the new process of the monomer adsorption in any of the $m$ empty sites
of the $m$-gap, and we denote
$$ a \equiv {kf \over r\ell} \,. \eqno(7.3) $$
\NI Initially, we take $\,P_m (0) = 1 \,$
(for all $m$). The $m \geq k$ equations are then solved by
the Ansatz
$$P_{m \geq k} (T) = p(T) \exp \left[ -m (1+a) T /k \right] \,
, \eqno(7.4)$$
\NI where $p(T)$ satisfies
$${dp(T) \over dT}=p(T)\left[ {k-1 \over k} -
{2 \over k} \sum_{j=1}^{k-1}
\exp \left( -j(1+a){T\over k} \right) \right] \, , \eqno(7.5)$$
\NI with $p(0)=1$. The solution is
$$ p (T) = \exp \biggl (\, {k-1 \over k}\, T + {2 \over 1+a}
\sum_{j=1}^{k-1} { 1 - \exp [-j(1+a)T/k] \over j}\,\biggr )
\, . \eqno(7.6) $$
\NI Now, note that the equation for $P_1$ involves only $P_k$.
Explicitly, this equation is
$$ -k {dP_1 \over dT} = kP_k + aP_1 \, , \eqno(7.7) $$
\NI which integrates to yield the coverage in the form:
$$\theta=1-{\rm e}^{-aT/k}\left\{ 1-\int\limits_0^Tdw \exp\left(
{aw -(1+a)kw \over k} +\displaystyle{\sum_{j=1}
^{k-1}} \displaystyle{1-{\rm e}^{-j(1+a)w/k} \over j(1+a)/2}
\right)\right\} \,. \eqno(7.8)$$
\NI Using the identity
$$\displaystyle{\sum_{j=1}^{k-1}} \,\displaystyle{1-{\rm e}^{-j(
1+a)w/k} \over j}={1+a \over k}\int\limits_0^w dz_1 \,\displaystyle{
{\rm e}^{-(1+a)z_1/k}-{\rm e}^{-(1+a)z_1} \over 1-{\rm e}^{-
(1+a)z_1/k}} \, , \eqno(7.9)$$
\NI and the following sequence of variable changes,
$$z_1 \;\longrightarrow\; z_2={\rm e}^{-(1+a)z_1/k} \;\longrightarrow
\; z_3=1-z_2 \;\longrightarrow\; v=kz_3 \eqno(7.10)$$
\NI and finally $\;u=k\left[ 1-{\rm e}^{-(1+a)w/k}\right]\;$,
one obtains$^{60}$
$$\theta(T)= 1- {\rm e}^{-aT/k} \left\{ 1 - {1 \over 1+a }
\int\limits_0^{ k \left[ 1- {\rm e}^{-(1+a)T/k} \right] }
du \left( 1-{u \over k} \right)^{(k-2)a \over 1+a} \,
{\cal P}(u)\right\} \, , \eqno (7.11)$$
\NI with
$${\cal P}(u)=\exp \left[ - {2 \over 1+a }
\int\limits_0^u dv {1- \left( 1- {v \over k} \right)^{k-1} \over v}
\right] \, . \eqno(7.12) $$
\NI This form is convenient for the continuum limit (large-$k$)
analysis. For small $k$ values, simpler expressions can be
obtained by direct evaluation of the integral in (7.12).
The rate ratio $a$, defined in (7.3),
is generally a function of $k$, via the $k$-dependence of $f$.
Some of the implications of the result (7.11) can be seen without
the precise specification of this $k$-dependence. For instance,
the jamming value $\theta (t=\infty)$ changes discontinuously
from 1 for all positive $f$ (in which case the monomers eventually cover
all the available sites), to the jammed-state value $\theta
(t=\infty;k) < 1$, for $f = 0$. The latter values were studied,
e.g., in Refs.~5-7. A more detailed analysis of the result (7.11)
requires the consideration of the proper $k$-dependence of
$f(k)$. Here the attention is restricted to the {\it pointlike
limit\/} in which the $k$-mer deposition becomes continuous.
If the monomer deposition rate $f$ is allowed to stay constant
as $k \to \infty$, the monomers completely preclude the
deposition of extended particles. In this limit, (7.11) reduces
to
$$ \theta (t) = 1 - {\rm e}^{-ft} \left[ 1 - {r\ell \over fk}
+ {\cal O} \left( 1 \over k^2 \right) \right] \,.\eqno(7.13) $$
\NI The original time, $t$, was restored
to emphasize that the leading order result is just that
of monomer deposition (uncorrelated growth of the
coverage). This result has a simple explanation. The {\it
space filling\/} capacity of each monomer decreases as $b=
\ell / k$ (in the limit $k \to \infty$).
However, the {\it blocking\/} capacity remains fixed, $\sim \! kb=\ell$.
Indeed, a monomer excludes length $\ell$ for the centers
of extended particles to land. (While each extended particle
excludes twice that length in $1D$.) Keeping $f$ fixed
corresponds to enhancing the monomer deposition attempt rate,
$f / b=kf / \ell$, to keep
their space-filling effect fixed. The overall monomer blocking capacity
then diverges $\sim \! k$. The
fixed-size ($\ell$) particles then do not play any role in the deposition
process (for large $k$).
The above considerations suggest that a more interesting deposition
process is obtained if the large-$k$ limit is defined with
the monomer deposition frequency per site decreasing as
$\displaystyle{1\over k}$ as $k \to \infty$. Effectively, we
then keep $a$ fixed, of order 1, and define $f(k) =
\displaystyle{r\ell a \over k}$. Hence the jamming ability
of monomers is finite. Therefore, for times $T \ll k$, a nontrivial
configuration will build up, by the mixture of particles.
The precise behavior will depend
explicitly on the rate ratio $a$,
$$ \theta (T \ll k) \simeq \Theta(T) \equiv
{1 \over 1+a}\int\limits_0^{(1+a)T}
du \,\exp \left( -\displaystyle{ua \over 1+a}
-\displaystyle{2 \over 1+a }
\int\limits_0^u dv \displaystyle{1- e^{-v}
\over v} \right). \eqno(7.14) $$
\NI This expression follows from (7.11), up to corrections of order
$\displaystyle{1 \over k}$. It is interesting to note that this
``intermediate'' coverage reaches jamming for $1\ll T \ll k$.
The function $\Theta (T)$ is shown in Figure 4 for several $a$
values. For small $T$, $\,\Theta(T)\,$ is linear in $T$,
with slope at $T=0$ independent of $a$. In fact, it expands as
$$\Theta(T)=T-\displaystyle{a+2 \over 2}\,T^2+{\cal O}(T^3) \,.
\eqno(7.15)$$
\NI The jamming values, $\Theta
(\infty ) $, are shown in Figure 5 as a function of $a$;
$\Theta(\infty)$ decreases as $\,\displaystyle{1 \over 1+a}\,$
for large $a$. For $a=0$, (7.11) and (7.14) give, respectively, the
previously known exact results$^{5-7}$ for the discrete and continuum
deposition kinetics.
For $T \gg 1$, the deposition continues, by monomers only,
with further buildup of the coverage on time scales of order
$k$,
$$ \theta(t) \simeq 1- {\rm e}^{-r\ell at/k} \left[1-\Theta (\infty)
\right] \, . \eqno(7.16)$$
\NI On the time scales of order $k$, the monomers fill up the
remaining void length, i.e., the fraction $ \left[1-\Theta (\infty)
\right]$, while the $k$-mer deposition is fully jammed.
The asymptotic convergence to the jamming value {\it in continuum\/}
(single-species) deposition models follows a power law$^{8-10,40}$
with possible logarithmic factors. In $1D$, the convergence is
rather universal, $t^{-1}$. This behavior results from
the distribution of intervals
that fit nearly precisely the particle shape thus making the
probability of the appropriate deposition attempt vanishingly
small. For the mixture considered here, the contribution of the
pointlike particles makes the standard argument inapplicable in
the regime in which the $k$-mer deposition reaches jamming.
This regime is described by (7.14), a direct analysis of which
yields, for $T \gg 1$,
$$ {d \Theta \over dT } \simeq \left[{\rm e}^{\gamma} (1+a) T
\right]^{-{2 \over 1+a}} {\rm e}^{-aT} \, , \eqno(7.17) $$
\NI where $\gamma$ is the Euler's constant. This relation integrates
to the asymptotic convergence rate $\sim T^{-1}$ when $a=0$.
However, for fixed $a>0$ one gets a leading contribution of a
different form,
$$ \Theta (\infty ) - \Theta (T) \simeq { {\rm e}^{-aT} \over
a \left[ {\rm e}^{\gamma}
(1+a) T \right]^{{2 \over 1+a}} } \, . \eqno(7.18) $$
\NI This new nonuniversal asymptotic behavior is due to the fact
that the pointlike particles can ``jam'' with rate of order 1
those narrow gaps of size $( \ell+\delta \ell )$ which
are reached only with probability of order $( \delta \ell
/ \ell )$ by the fixed-size particles.
\NP
\NI {\bf 8. SUMMARY}
\vskip 0.20in
In summary, in this review, we described recent
numerical and analytical trends
in monolayer and multilayer adsorption. We surveyed the mean-field
approach and a more rigorous though simplified rate-equation
treatment of the multilayer effects. Recent Monte Carlo results for
multilayer deposition were discussed. Theoretical developments
reviewed include large-time asymptotics, and finite-size effects.
A consistent theoretical description of the monolayer deposition
has been largely accomplished by these recent studies. For the
multilayer case, the theoretical information and level of
understanding are still in the early stages of development. More
work, within various analytical and numerical approaches, is needed.
\NP
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\NP
\centerline{\bf FIGURE CAPTIONS}
\vskip 0.20in
\NI\hang {\bf Figure 1.} $\;$
The coverages in the first three layers (solid curves) of dimers
deposited according to the rules defined in connection with
Eq. (3.10). The curves are for layers $n=$1, 2 and 3, respectively.
The $n=1$ values are exact while those for $n=2,\;3$ were obtained
within the rate-equation approach. The dotted
curves were calculated using the mean-field
theory, for layers 1, 2 and 3, respectively. The time variable is
$brt$ while the coverages are multiplied by the particle size,
$2b$. Note that as $t \to \infty$, all six curves
approach the same asymptotic value, $1-{\rm e}^{-2} =
0.8646647\ldots$.
\NI\hang {\bf Figure 2.} $\;$
Variation of the coverages in layers
$\,n=1,2, \ldots, 15\,$ as functions of time, for the
deposition of $(2
\times 2)$-mers on the square lattice. The monolayer coverage
is the upper curve, and generally, $\theta_n (T) < \theta_{n-1}
(T)$, for each $T$. These results were obtained$^{39}$ by Monte Carlo
simulation on the $1000 \times 1000$ lattice.
\NI\hang {\bf Figure 3.} $\;$ The time dependence of the coverage, $\theta$, in dimer
deposition on a linear lattice of size $N$. The dotted curve corresponds to $N=
\infty$. The $N=6$ results are shown for periodic (P) and free (F) boundary
conditions.
\NI\hang {\bf Figure 4.} $\;$ The ``intermediate'' coverage,
$\Theta(T)$, defined in (7.14), for $a=$ 0.0, 0.6, 1.2, 1.8, 2.4.
\NI\hang {\bf Figure 5.} $\;$ Jamming coverage values,
$\Theta(\infty)$, for $1\ll T\ll k$, obtained from (7.14),
as a function of $a$.
\bye