Surface Science 454–456 (2000) 598–601
www.elsevier.nl/locate/susc
Diffusive spreading of rodlike molecules on surfaces
J.M. Lahtinen a, *, T. Hjelt a, T. Ala-Nissila a,b
a Helsinki Institute of Physics and Laboratory of Physics Helsinki University of Technology, P.O. Box 1100, 02015 HUT, Espoo, Finland
b Department of Physics, Box 1843, Brown University, Providence, RI 02912, USA
Abstract
We study the diffusion and submonolayer spreading of rigid rodlike molecules on smooth surfaces. The molecules
interact through a short-range repulsive potential and their motion is described through molecular dynamics
simulations. We concentrate on the behavior of the collective diffusion coefficient D (h) as a function of the surface
C
coverage h. Through analysis of spreading profiles of circular droplets, and rectangular density profiles we determine
D (h) for rods of varying lengths. We demonstrate that in analogy to experiments and simulations on chainlike
C
flexible molecules, D (h) initially increases with h due to entropic repulsion and displays a peak at intermediate
C
coverages. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Computer simulations; Surface diffusion
1. Introduction
The diffusion and spreading of polymers on
solid surfaces is an interesting theoretical problem
with important applications related to thin surface
films. Many experimental studies have been carried
out on the diffusive dynamics of small molecules
and polymers in such systems [1–6 ]. However,
most theoretical investigations up to date deal with
the (tracer) diffusion of single atoms or molecules
[7–12], and there have been only a few theoretical
studies concerning the collective diffusion properties of polymers on surfaces [13–16 ]. These studies
have nevertheless shown that even in the case of
athermal polymers, there are strong entropic interactions present for finite surface coverages that
strongly influence the collective diffusion proper* Corresponding author. Fax: +358-9-4513116.
E-mail address: jukka.lahtinen@hut.fi (J.M. Lahtinen)
ties of the system. In this paper we present results
of a computer simulation study of a system of
rigid rodlike molecules on smooth surfaces. The
present study thus continues and extends the work
done previously in the case of flexible chainlike
molecules [14,16 ]. Here, we concentrate on the
behavior of the collective diffusion coefficient
D (h) as a function of the submonolayer surface
C
coverage h. We simulate two different spreading
processes in our system. The first is the spreading
of a circular droplet which yields qualitative information about D (h). The second is the spreading
C
of a density front in a rectangular geometry which
is amenable to Boltzmann–Matano analysis [11],
from which (almost) the whole function D (h) can
C
be obtained. By carrying out this latter procedure
for rods of various lengths, we demonstrate that
initially D (h) is an increasing function of h, and
C
it displays a strong maximum for intermediate
coverages. This is similar to the case of chainlike
molecules where strong entropic repulsion is gener-
0039-6028/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.
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J.M. Lahtinen et al. / Surface Science 454–456 (2000) 598–601
ated for finite coverages, and the interplay between
the increasing compressibility and rapidly decaying
mobility determines the position of the peak
[14,16 ].
599
with respect to the center of mass. Following this,
the molecule was rotated accordingly and its position updated.
3. Results
2. The model
3.1. Spreading of droplets
The molecules in the present case are modelled
by a chain of N particles in continuum which are
forced to stay in fixed positions with respect to
each other along a straight line. Each particle in a
given chain interacts with all the other particles in
other chains through a strongly repulsive shortrange interaction potential of the type V=e/r12,
to prevent chain crossings and overlaps. Here we
present results for rods consisting of two particles
(dimers) and rods of six particles. For reference,
we also show results for the single particle case
N=1. In the present case, the molecules are close
to the athermal limit of hard core interactions.
The surface is taken to be smooth and structureless. The coverage h of the system is in the submonolayer regime so that the molecules are always
lying flat on the surface and in contact with it1
[17]. The number of molecules and the temperature
are held constant during the simulations.
The equation of motion of each particle in the
system is given by the Langevin equation [18]:
d
dt
p(t)=−gp(t)+f (t)+F(t),
(1)
where p(t) is the two-dimensional momentum of
the particle, t is time and f(t) is the total interaction
force with the other particles. The remaining
random term F(t) is the driving force for the
motion of the Brownian particles, and it obeys the
standard fluctuation–dissipation relation [18]. In
addition to Eq. (1) we have the constraint that
the molecules always remain rodlike. This constraint was implemented by first computing the
forces acting on all the N separate particles comprising the molecule, and then calculating from
these the torque and the translational force acting
1 Here the coverage h is defined simply as the coverage relative
to filling the system with hard objects of the same size as used
here. In our case e/(k T )=28.53, see for example, Löwen [17].
B
The collective diffusion coefficient D can be
C
defined through the diffusion equation:
∂h(r, t)
∂t
=V [D Vh(r, t)].
C
(2)
If the collective diffusion coefficient were a
constant, independent of the coverage, the solution
of Eq. (2) for a circular droplet would be a
Gaussian curve. Experiments [2] and simulations
[14,16 ] have shown, however, that this is not the
case for chainlike molecules. From the shape of
the spreading profiles, qualitative information
about D (h) can be obtained.
C
During the simulation of the spreading of a
circular droplet the coverage data as a function of
radius is collected. We have used the following
Ansatz to deduce the qualitative behavior of the
collective diffusion coefficient [16 ]:
D (h)=c +c tanh[c (h−c )].
(3)
C
1
2
3
4
We have solved the diffusion Eq. (2) using this
Ansatz and chosen the coefficients c in such a way
i
that the solution matches the simulated spreading
profiles as accurately as possible. The simulated
spreading profiles and the corresponding solution
of the diffusion equation along with the resulting
diffusion coefficient for the six-particle rods are
shown in Fig. 1. It can be seen that our Ansatz
which leads to an increasing function of coverage
reproduces the simulated spreading profiles reasonably well.
3.2. Boltzmann–Matano analysis
To calculate D (h) with quantitative accuracy
C
we have performed a Boltzmann–Matano analysis
of the spreading of initially steplike density profiles
in a rectangular geometry, where the spreading
takes place in one direction x only and in the
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J.M. Lahtinen et al. / Surface Science 454–456 (2000) 598–601
(a)
Fig. 1. Simulated droplet spreading profile for rods with N=6
(#). The solid line shows the solution of Eq. 2 using D (h)
C
shown in the inset.
direction perpendicular to the spreading direction
we have periodic boundary conditions. We can use
the Boltzmann–Matano formula to extract D (h)
C
as a function of coverage from the spreading
profiles [11]:
P
A
B
dh
h1
g dh= D
,
(4)
C dg
h=h1
0
where g=x/2t1/2.
In Fig. 2a we show a series of simulated spreading
profiles for the case N=6 which display excellent
scaling. From these, we obtain the diffusion coefficient using Eq. (4). In Fig. 2b we show D (h) for
C
three different cases, namely N=1, N=2 and N=
6. It is seen that in each case the diffusion coefficient
is an increasing function of coverage at low and
intermediate coverages, just like the qualitative
results from the droplet spreading indicated. There
is a peak in D (h) at intermediate coverages that
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increases with the length of the rod. This result is
completely analogous to the case of flexible chainlike
molecules, where the initial increase of D (h) was
C
shown to be due to strong entropic repulsion generated in the system [14,16 ]. Also, the peak in D (h)
C
was shown to be due to the competition between
an increasing compressibility of the adlayer, and
rapidly decreasing mobility of the molecules.
−2
(b)
Fig. 2. (a) Boltzmann–Matano spreading profiles for N=6 rods.
(b) The diffusion coefficient D (h) as obtained from the
C
Boltzmann–Matano analysis for the cases N=1, N=2 and
N=6.
4. Summary and discussion
In this paper we have investigated the coverage
dependence of the collective diffusion coefficient
of rodlike molecules that diffuse on a smooth
surface. We found that the D (h) is an increasing
C
function of the coverage at low and intermediate
coverages, and displays a maximum before eventually going to zero in the limit h1.
J.M. Lahtinen et al. / Surface Science 454–456 (2000) 598–601
It is interesting to compare the present results
to the previous studies of diffusion of flexible
chainlike molecules on smooth surfaces [14,16 ].
These studies were done using the fluctuating bond
lattice model combined with Monte Carlo
dynamics. In the model, the chains move by single
segment fluctuations only, and thus the rigid rod
limit cannot be defined. The present continuum
molecular dynamics model correctly includes the
translational modes of the molecules and gives
more realistic dynamics. Despite these differences,
the qualitative behavior of D (h) remains the same
C
for both flexible and completely rigid molecules.
As far as comparison with experiments is concerned the results for rigid rods are in agreement
with those of flexible chains [2] where the entropic
repulsion leads to a strong peak in D (h). We are
C
not aware of any measurements for rodlike molecules on smooth surfaces. The experiments of Mak
et al. [19] on the surface diffusion of cycloalkanes
on Ru(001) have demonstrated that D (h) for
C
coverages =0.7 is an increasing function of h in
the case of cyclopentane. However, since these
molecules are rather small and surface is strongly
corrugated a direct comparison is difficult.
601
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