Monte-Carlo modelling of Au nanocluster growth on graphene-oxide
Peter Dawson,1 Gavin Bell,2 and Paul Mulheran3
1 Centre
for Complexity Science, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL.
of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL.
3 Department of Chemical and Process Engineering,
University of Strathclyde, 75 Montrose Street, Glasgow, G1 1XJ.
2 Department
The growth of Au particles on a graphene oxide substrate is examined through the analysis
of experimental samples imaged by transmission electron microscopy, and modelling by
kinetic Monte Carlo simulation. In order to recover the experimental observed bimodal
distributions for island size s, point-island and two-dimensional models were simulated with
nucleated islands being allowed to diffuse once they reached a certain size. The model
successfully reproduces the bimodal distribution. The diffusion coefficient for these islands
was found to be ∼ s−1.75 .
I.
INTRODUCTION
tal TEM data to a kinetic Monte Carlo model of
growth. We begin by analysing further the experimental samples of Pandey et al. In particular
we generate probability density functions using
kernel density estimation for the island sizes and
capture zone areas. A simulation mode is developed to try and capture the experimental distributions. Whilst traditional island growth models consider nucleated islands to be static we explore the inclusion of a critical island island size
at which islands can diffuse and various rules
that govern this diffusion.
Graphene is known to posses a range of
unique properties1 . Graphene being of a two
dimensional nature and having a low atomic
number is highly electron transparent and therefore easily imaged using transmission electron
microscopy (TEM) . Not only can graphene
be studied using TEM but it also acts as
perfect support for the imaging of nanoparticles and macromolecules due to its chemical
inertness2–4 . However, it has been shown that
nucleation does not occur homogeneously on a
In 2011 Pandey et al6 , using physical vapour
graphene substrate. Graphene is chemically in- deposition (PVD) deposited various metals onto
ert so adsorbates only stick at surface defects5 . a graphene oxide substrate. The deposition rate
Graphene oxide (GO) is a form of chemically F was kept constant throughout the experiment
modified graphene (CMG) which retains many as was the temperature, presumably giving a
of graphene’s interesting properties such as high constant adatom diffusion rate D. Using TEM
electron transparency but is less chemically in- they observed various different morphologies
ert. Unlike graphene, GO is known to be a good depending on the relative binding energies and
substrate for the growth of nanoclusters and thin migration barriers of the adatoms on graphene.
films as functional groups increase the energy
We greatly extend their analysis but focus
of adatoms/molecules making GO more ‘sticky’ only on the deposition of Au monomers. Rather
than graphene. Though GO is heterogeneous in than forming a thin film across the surface of the
nature it would appear that for the case of Au substrate, the Au monomers nucleate to form iscluster growth the growth process is not domi- lands. The size and shape of the distribution of
nated by nucleation at defects6 . An application island sizes is dependent on the monomer covof metal nanoclusters grown on either graphene erage, θ defines as: θ = 0.75 nm means that the
or GO is in single molecule gas detectors7 .
amount of Au deposited would be equivalent to
In this study, we examine Au nanocluster covering the substrate in a film of Au 0.75 nm
growth on GO in detail, connecting experimen- thick. Fig. 1 shows bright-field TEM images for
2
coverages of θ = a) 0.15, b) 0.30, c) 0.75 and d)
1.5 nm.
b
FIG. 2: TEM image of Au deposited at 0.75 nm
thickness on GO a). On the right b) is the processed
image with Voronoi polygons imposed on top (blue
lines). The blue dots are the centroids of the Au
clusters from which the Voronoi polygons are constructed
gons is of the form
H(x; c) =
cc c−1 −cx
x e
Γ(c)
(1)
FIG. 1: TEM images of nominally θ = a) 0.15,
b) 0.30, c) 0.75 and d) 1.5 nm thick Au films on where c is a free parameter suggested by Kiang
to be c = 4 but has since been calculated as c =
graphene oxide.
The focus of the analysis shall be on islands size distributions (ISDs) and on the island’s ‘capture zone’ distributions. An island’s
capture zone (CZ) is the area surrounding an
island from which a diffusing particle is more
likely to diffuse to that particular island than
any other island; assuming an isotropic diffusion process deposited monomers are most
likely to encounter their geometrically closest
island. These regions are closely approximated
by the Voronoi polygons corresponding to the
nucleation sites for each island (see Fig. 2 b).
For the case of heterogeneous two dimensional
island growth it has been observed that the individual islands grow at a rate proportional to the
area of their capture zones, thus the ISDs may
be estimated from the cell area in the Voronoi
network8 .
For the growth of Au on GO we have assumed a homogeneous nucleation process so the
ISDs in general may not follow the CZ distributions. It was postulated by Kiang9 that the
scaled distribution of cell areas of Voronoi poly-
3.6110 for the case of random fragmentation.
Though while one would expect to see scaling
for different coverages in the CZ distributions,
Mulheran and Blackman11 observed that in the
case of homogeneous thin-film growth island
scaling with substrate coverage is only approximate and coincidental.
Monte Carlo simulations have been used to
model surface growth since the early 1990s.
They offer a distinct advantage over the numerical solving of rate equations, as rate equations
implicitly require mean field approximations
and assume that islands of the same size grow at
the same rate. An initially empty lattice is used
to represent the substrate. Monomers are deposited at a constant rate to random lattice sites.
The monomers diffuse by hopping randomly
to nearest neighbour sites. If two monomers
meet they nucleate and an island begins to grow
around them. The density of islands nucleated
in the simulations depends on the ratio of monolayer deposition rate to the monomer hopping
rate12 . This ratio, generally known as R = D/F,
alone dictates the Monte Carlo simulation procedure, by changing the probabilities of the next
3
where θ is the coverage. Traditionally these
islands are ’point-like’, but extended-island
growth has also been studied11,14–16 .
II.
ANALYSIS OF EXPERIMENTAL DATA
We chose to concentrate our analysis on the
lower coverages as islands begin to coalesce and
cease to be circular at higher coverages (Fig.
3d). The island size distributions (ISDs) can
be extracted from the bright-field images. The
images were processed using tools available in
Matlab; thresholding (converting the grey-scale
image to black (substrate) and white (island))
was performed using Otsu’s method17 .It should
be noted that a blanket thresholding of the image was not sufficient as have background gradients which cause shading. Thus the images were
thresholded in sections. The islands areas were
extracted from the thresholded images. The islands were assumed to be circular and their diameter was calculated. The island size d was
then defined as the number of atomic diameters corresponding to the islands diameter, with
0.288 nm being taken as the covalent diameter
of Au. The ISDs for samples at coverages of
0.15 nm and 0.75 nm at are shown in Fig. 3. The
ISDs are bimodal with a lower peak for small
islands and a higher peak for larger islands. We
note here that all distributions in this paper are
depicted as probability density functions calculated using Gaussian kernel density estimation.
Standard irreversible aggregation of
monomers produces uni-modal ISDs12 . Therefore, there must be some additional underlying
Experiment
0.25
θ= 0.15nm
θ= 0.75nm
0.2
probability density function
step step being a deposition event or a monomer
diffusion event. In typical growth experiments
R ∼ 105 − 1010 . The higher the value of R the
more a monomer can diffuse before the next deposition event, monomers can travel further increasing their likelihood of joining an existing
island12 . The simulated island density N therefore decreases with increasing R. It has been
shown13 that N at fixed monomer deposition
varies as
1
N ∼ θ 1/3 R− /3 ,
(2)
0.15
0.1
0.05
0
0
10
20
d − atomic diameters
30
FIG. 3: Island size distributions from Gaussian kernel density estimation obtained from images of Au
deposited on GO at coverages of 0.15 nm (dashed
blue line) and 0.75 nm (dot-dashed green line).
process for the formation of a second peak in
the bimodal ISDs. We hypothesise that there
is a critical island size at which static islands
begin to diffuse and coalesce with monomers
and small islands to form this second peak.
The scaled CZ distributions for samples at
coverages of 0.15 nm and 0.75 nm are shown
in Fig. 4. The scaled form of the distributions
is shown, partly as it makes fitting curves of
the form H(x; c) easier but also to identify if
scaling behaviour is exhibited. If the CZ distributions for different coverages were to fall on
top of one another this would indicate that the
growth processes were scale free, i.e. there is
no special length during growth. We however
find ’pseudo-scaling’, the distributions are similar but upon fitting H(x; c) we note that c = 9.5
for the lower coverage and c = 12.1 for the
higher coverage. The H; c = 3.61 curve is also
plotted to emphasise that the islands are not randomly distributed in a Poisson process, but have
4
that enough monomers have been deposited to
cover the lattice with a one monomer thick film
of monomers. The average thickness of a gold
monolayer relates these coverages to the experimental values: 0.15 nm coverage gives θ ∼ 50%
and 0.75 nm coverage gives θ ∼ 250%.
Experiment
θ= 0.15nm
θ= 0.75nm
H(x; c=9.5)
H(x; c=12.1)
H(x; c=3.61)
1.6
probability density function
1.4
1.2
For point-island simulations the islands have
no diameter so their size s is measured as the
number of monomers they possess. Plots of the
ISDs (Fig. 13) and the CZ areas (Fig. 12) are
appended. The CZ areas are scale free and follow a distribution of the form H(x : c) though
with c higher than the case of random fragmentation. The ISDs are distinctly uni-modal. The
peak can be seen to shorten and shift to the right
at higher coverage.
1
0.8
0.6
0.4
0.2
0
0
1
2
3
x=CZ area / mean CZ area
4
FIG. 4: Capture zone distributions from Gaussian
kernel density estimation obtained from images of
Au deposited on GO at coverages of 0.15 nm and
0.75 nm. Curves of the form of H(x; c) are fitted
to each distribution. A plot of the expected distribution for random fragmentation is also shown,
H(x; c = 3.61).
a degree of self organisation which comes about
through the surface diffusion process.
A linear correlation between island area and
CZ area was also observed (Fig. 15 appended).
These plots lend further weight to the idea that
there is a critical islands size at which larger islands begin to diffuse.
III.
SIMULATION
To begin with simple point-island simulations were carried out. These simulations were
run for varying values of R. It was found
that the island density does indeed scale as ∼
θ 1/3 R−1/3 (see Fig. 10 appended). It should
be noted here that the coverage θ here is expressed as a percentage where θ = 100% means
To further develop the model of island
growth two directions were explored. The first
was to change from modelling point islands to
modelling two dimensional growth. The second change was to allow larger islands to diffuse with some given probability. The inclusion
of this diffusion was justified as operators of the
TEM had noted that under high electron flux,
which increases the effective temperature of the
substrate, large islands have been seen to move.
Also previous studies have reported diffusion of
large (100-720 atoms) extended-isalnd Ag clusters on Ag16 and Voter18 in 1986 found the diffusion coefficient for Rh clusters (larger than 15
atoms) on Rh to scale as s−1.76±0.06 .
Allowing larger islands to diffuse created
new parameters for the simulation. The ratio
of diffusion to deposition R would still have to
be optimised but now a critical island size scrit
at which islands began to diffuse would have to
be found, a rule for the probability of diffusion
would also have to be established. Also when an
island reached scrit it would change from a being two-dimensional circular island to a threedimensional hemispherical island as it was assumed that a transition of this type was causing
larger islands to diffuse. The probability p of an
5
island moving was given by
(3)
IV.
RESULTS
The effects of the parameters scrit , R and α
are discussed below for the case of extendedisland islands growth incorporating a transition
from two to three-dimensions at scrit . This
transition is necessary in order to still have
islands at higher coverages, if only purely
two-dimensional islands were grown then at a
100% coverage there would be only a film of
monomers one monomer thick.
Using the locations of the first peak and first
trough of Fig. 3 an estimate for the size at which
large islands become mobile scrit . The number
of atomic diameters is roughly between 2.5 .
dcrit . 3.5 converting to a hemispherical volume
gives 4 . scrit . 11. Within this range the choice
of scrit does not have a great impact on the ISDs
provided it is near the high end of this range.
This is discussed in the appendix, scrit = 10 has
been used for the results below.
probability density function
where s = 1 corresponded to a monomer. The
exponent α determines how much diffusion the
larger islands undergo, if it were too small p
would be come top large and smaller islands
would be swept up by large islands which would
coalesce, becoming very large. If α was too big
p would be very small and the effects of movement would be lost.
As we know from the static point-island case,
island density increases as the amount of diffusion monomers can undergo decreases, dictated by R. We can safely assume that this relation also holds for island diffusion, increasing
α, decreasing the probability of islands diffusing should, therefore yield higher island density.
Higher island density should reduce the amount
of small static islands as they get swept up by
larger moving islands.
0.25
θ=0.15 nm, experiment
θ=50%, simulation
0.2
0.15
0.1
0.05
0
0
5
10
15
20
d − number of atomic diameters
25
0.25
probability density function


1 if s = 1
p(s) = 0 if s < scrit

1/sα if s ≥ s .
crit
θ=0.75 nm, experiment
θ=250%, simulation
0.2
0.15
0.1
0.05
0
0
10
20
d − number of atomic diameters
30
FIG. 5: Comparison of experimental ISDs (dashed
blue lines) with simulation (dot dashed green lines)
using the optimised parameter values, scrit = 10, R =
7.5 × 105 , α = 1.75.
Fig. 5 compares the experimental data with
the simulation results using the optimised parameter values of parameter values, scrit = 10,
R = 7.5 × 105 , α = 1.75. For both the 0.15
nm data and the 0.75 nm data the position and
height of the first peak in the bimodal distributions fits very well. The second peak has the
right height, but is shifted to the right. The analysis leading to these choices for the parameters
6
is discussed below.
The island density, N (Fig. 6) decreases with
increasing α this is what one would expect as
high α implies low island diffusion and was discussed for the static point-island case N is inversely proportional to monomer diffusion rate.
Indeed as α increases the island density curve
begins to approach the α = ∞ case corresponding to static islands. Graphical analysis can then
be used to determine the qualitative effect that R
and α have on the ISDs.
−3
N − island density [per (lattice site)2]
4
x 10
α=∞
α=1.25
α=1.5
α=1.75
α=2.5
α=3
3.5
3
2.5
2
1.5
1
0.5
0
50
100
150
200
250
θ − percentage coverage
300
FIG. 6: Island density plots from extended-island
simulation for different values of R with α = 1.75
and scrit = 10.
Fig. 7 shows the effect that varying α has
on the the ISDs. As expected the island density decreases as α goes up. The shape of the
distributions is changed quite dramatically. For
α = 1.25 the small island peak is larger than
the big island peak, with increasing α this begins to change and at α = 1.75 the small island
peak is now below the large island peak, a situation more reminiscent of the experimental ISDs.
When α = 2.5 the small island peak has practically disappeared, island density at θ = 100%
is around three times higher than the α = 1.25
case, leaving us with a distribution almost made
up entirely of large moving islands. The static
island case of α = ∞ is almost recovered for
α = 3, the distribution is distinctly uni-modal.
As coverage is increased the larger peak
moves to the right, lowers in height and broadens whilst the smaller peak stays reasonably
fixed though its peak does also lower. This is
interpreted as the large islands moving around
merging with one another and with small islands
thus becoming even larger.
R increases the amount of diffusion for
monomers and large islands. The effects of
changing R can be seen by comparing in Fig. 8.
As before increasing R reduces the island density it also shifts the large island peak to the right
and lowers it down. This is a similar effect to increasing α but without such a pronounced effect
on the overall shape of the distribution. A value
of R = 7.5 × 105 provides the best comparison
with experiment.
As well as comparing the ISDs the capture
zone distribution is also a feature we would like
to recover from simulation. The CZ area distributions shown in Fig. 9 for α = 1.75 and
R = 7.5 × 105 are reminiscent of the experimental distributions of Fig. 4. Indeed the curve fitted to the 0.75 nm coverage (corresponding to
θ = 150%) data can also be successfully fitted to
the simulation data at high coverages. However
we observed that 0.15 nm (corresponding to
θ = 50%) coverage data had a slightly shorter,
wider distribution with c = 9.0. This is not recovered by simulations at θ = 50% but wider,
shorter distribution can be seen for low coverages.
The correlation between CZ area and island
area is discussed in detail in the appendix. It
gives further evidence for the correctness of the
model and the existence of scrit .
7
0.2
0.1
0
α=1.5
0.25
0.2
θ=10%
θ=25%
θ=50%
θ=150%
0.15
0.1
0.05
0
0
0.3
0.25
0.15
0.1
0.05
0
5
10
15
20
25
d − number of atomic diameters
θ=10%
θ=25%
θ=50%
θ=150%
0.15
0.1
0.05
0
0
5
10
15
20
25
d − number of atomic diameters
α=1.75
0.25
0.2
θ=10%
θ=25%
θ=50%
θ=150%
0.15
0.1
0.05
0
0
0.4
θ=10%
θ=25%
θ=50%
θ=150%
0.2
0
0.2
5
10
15
20
25
d − number of atomic diameters
α=2.5
α=1.25
0.25
5
10
15
20
25
d − number of atomic diameters
probability density function
probability density function
0.3
θ=10%
θ=25%
θ=50%
θ=150%
probability density function
0.3
0
probability density function
α=∞
probability density function
probability density function
0.4
5
10
15
20
25
d − number of atomic diameters
α=3
0.3
θ=10%
θ=25%
θ=50%
θ=150%
0.2
0.1
0
0
5
10
15
20
25
d − number of atomic diameters
FIG. 7: Plots for ISDs from extended-island simulation for different values of α with R = 7.5 × 105 . The
multiple curves on each plot show the peak of large islands moving to the right as coverage increases.
Extended−island Simulation
θ=50%
0.2
1.6
1.5 × 106
1.4
2.25 × 106
0.15
0.1
0.05
0
0
5
10
15
20
d − number of atomic diameters
θ=150%
0.2
25
7.5 × 105
6
1.5 × 10
2.25 × 106
0.15
θ=10%
θ=50%
θ=150%
θ=250%
H(x; c=12.1)
7.5 × 105
probability density function
probability density function
probability density function
8
1.2
1
0.8
0.6
0.4
0.1
0.2
0.05
0
0
5
10
15
20
d − number of atomic diameters
25
FIG. 8: ISDs from the extended=island simulation
for varying R with α = 1.75 and scrit = 10 for θ = 50
% and θ = 150 %. As R increases the second peak
in the distributions shifts to the left.
V.
DISCUSSIONS AND CONCLUSIONS
We have shown that static point-island simulations are insufficient to reproduce the bimodal experimental ISDs. Indeed it has been
established that not only is it necessary to use
extended-island models but that the inclusion of
large island diffusion is essential to produce the
bimodal distribution. The form of this diffusion
coefficient is the most important parameter governing the distribution shape. Our choice of a
s−α rule seems to have captured the qualitative
form of the experimental ISDs with α = 1.75.
However the distributions we have do not perfectly match those of experiment.
The distributions of small and large islands
in the experiment are far wider than those of the
simulation; the simulation has no intermediate
regime, islands are either clearly in the large island peak or the small island peak. This probably is a result of the diffusion rule which does
0
0
1
2
3
x=CZ area / mean CZ area
4
FIG. 9: Capture zone distributions from extendedisland simulation with α = 1.75, scrit = 10 and R =
7.5×105 for different values of coverage θ . A distribution of the form H(x; c = 12.1) is fitted for the high
coverage covers, this compare with the experimental
distribution from the 0.75 nm coverage sample.
not allow small islands to move until they have
reached scrit , making this quite a sharp transition. Perhaps a softer diffusion rule, possibly
allowing small islands to diffuse with a probability drawn from a Gaussian distribution would
yield results more akin to those of the experiment.
The pseudo-scaling observed in the experimental CZ distributions does appear to have
been captured by the simulations, though at an
earlier coverage. While the correlation between
CZ area and island size area does lend further
support for our hypothesis of a critical island
size, this analysis seemed to over estimate the
magnitude of scrit .
An obvious extension of this work would
be to try large island diffusion rules of different forms which would not result in such a
hard cut-off between mobile and immobile is-
9
VI. ACKNOWLEDGEMENTS
lands. The morphology of the Au islands could
also be examined as it is unlikely the islands
are true hemispheres but probably shallower
I would like to thank EPSRC for funding
domes. More experiments could be carried out
with varying temperature and deposition rate, to and Dr. Paul Mulheran and Dr. Gavin Bell
change R and see if the ISDs vary in different for their supervision and guidance over the past
three months.
growth environments.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007).
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Nano 5, 608 .
H. Zhou et al., Journal of the American Chemical
Society 132, 944 (2010).
P. A. Pandey et al., Small (Weinheim an der
Bergstrasse, Germany) 7, 3202 (2011).
G. Lu, L. E. Ocola, and J. Chen, Nanotechnology
20, 445502 (2009).
P. A. Mulheran and J. A. Blackman, Philosophical Magazine Letters 71, 55 (1995).
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(1966).
D. Weaire, J. P. Kermode, and J. Weichert, Philosophical Magazine B 53, L101 (1986).
P. A. Mulheran and J. A. Blackman, Physical review. B, Condensed matter 53, 10261 (1996).
P. Mulheran, in Metallic Nano Particles, Volume 5 (Handbook of Metal Physics), edited by
J. Blackman (Elsevier B.V., Amsterdam, 2009),
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10
APPENDIX A: POINT-ISLAND
SIMULATIONS
−3
16
x 10
−3
N − island density [per (lattice site)2]
As mentioned in the main body of the text,
point-island simulations were the simplest simulations that could be implemented. Many of
the results found for these simulations could
be assumed true for more complex simulations
later. For instance, in the case of static island
growth the result that increased probability of
monomer diffusion (higher R) lowers the island
density, N is shown in Fig. 10. From this result
one would then safely assume that once islands
are also allowed to diffuse with some probability that increasing this probability, which in our
case manifests itself in decrease in α would also
cause the islands density to lower, Fig. 11. Indeed, the island density at high coverage is far
lower for lower values of α, as α is increased
the density begins to converge on the static island case of α = ∞.
16
x 10
α=∞
α=1
α=1.5
α=1.75
α=2.25
α=3
14
12
10
8
6
4
2
0
50
100
150
θ − percentage coverage
200
FIG. 11: Island density plots for different values of
α with R = 7.5×105 and scrit = 10 from point-island
simulation. N increases with increasing α, and will
eventually converge on the static case of α = ∞.
R= 7.5 × 105
14
R= 1.5 × 106
N − island density [per (lattice site)2]
To compliment the analysis of the extendedisland
simulations, results for the point-island
R= 3.0 × 10
12
simulations are shown with the same parameters as those used for the extended-island simulations, though the ISDs for static point islands
10
are also included (α = ∞), Fig. 13. For ease of
comparison the average island size savg is shown
8
in the legend.
In the static case there is only one peak and
6
savg is low. As coverage is increased this peak
drops
and moves to the right as expected. As
4
α is increased there is only one noticeably high
peak for small islands, there is then a wide range
2
0
50
100
150
200
of large islands of different sizes, this shape
θ − percentage coverage
tends to preserved as coverage increases. The
average island size drops with increasing α, this
FIG. 10: Island density plots for different values of is consistent with an increase in island density.
R from static point-island simulation. Lines of the The probability density for α = 1.75, the result
form N = θ 1/3 R−1/3 have been fitted. N decreases found to best reproduce experimental data for
with increasing R.
the extended-isalnd case, in no way resembles
its extended-island counterpart. Finally as α is
R= 2.25 × 106
6
11
θ=10%
θ=25%
θ=50%
θ=150%
H(x;c=7)
probability density function
1
0.8
0.6
0.4
0.2
0
0
1
2
3
x=CZ area / mean CZ area
4
FIG. 12: Capture zone distributions from pointisland simulation with α = 1.75, scrit = 10 and R =
7.5 × 105 for different values of coverage θ . A distribution of the form H(x; c = 7) is fitted for the data.
increased even further the ISD for the static case
is recovered.
The effect of increasing R on the ISDs is
not shown here but it is the same as for the the
extended-island case, the peaks move to the left
and fall in height. The island density also decreases as expected.
The inclusion of island diffusion does not
seem to have a noticeable affect on the CZ distributions. Fig. 12 shows the scaled CZ area distributions at difference coverages collapse onto
each other indicating there is no characteristic
length scale that is dictating their form. A curve
of the form H(x; c = 7) seems to fit reasonably
well, this is a shorter wider distribution than
that found for the experimental data or in the
extended-island simulations.
12
probability density function
θ=25% ,savg−33
θ=50% ,savg−52
θ=150% ,savg−108
0.03
0.02
0.01
100
200
300
s −number of monomers
α=1.5
θ=50% ,savg−122
θ=150% ,savg−340
0.03
0.02
0.01
0
0
0.05
100
200
300
s −number of monomers
α=2.25
θ=50% ,savg−76
θ=150% ,savg−193
0.03
0.02
0.01
0
0
θ=150% ,savg−512
0.03
0.02
0.01
0
100
200
300
s −number of monomers
400
100
200
300
s −number of monomers
α=1.75
400
θ=10% ,savg−26
θ=25% ,savg−56
0.04
θ=50% ,savg−105
θ=150% ,savg−286
0.03
0.02
0.01
0
0.05
θ=10% ,savg−21
θ=25% ,savg−43
0.04
θ=50% ,savg−173
0
400
θ=10% ,savg−37
θ=25% ,savg−89
0.05
θ=10% ,savg−29
θ=25% ,savg−66
0.04
α=1
0.04
0
400
probability density function
0
0.05
probability density function
θ=10% ,savg−18
0.04
0
probability density function
0.05
α=∞
probability density function
probability density function
0.05
100
200
300
s −number of monomers
α=3
400
θ=10% ,savg−18
θ=25% ,savg−34
0.04
θ=50% ,savg−56
θ=150% ,savg−128
0.03
0.02
0.01
0
0
100
200
300
s −number of monomers
400
FIG. 13: Plots for ISDs for different values of α with R = 7.5 × 105 and scrit = 10 from point-island
simulation. The multiple curves on each plot show the peak of large islands moving to the right as coverage
increases.
13
APPENDIX B: CRITICAL ISLAND SIZE
Varying the critical island size scrit has some
obvious consequences. When scrit is low the
first of the bimodal peaks will be small as islands will reach scrit quickly and begin to diffuse, they will diffuse faster than at high scrit
as the rate of island diffusion is inversely proportional to island size. This in turn will make
the second peak increase faster. From the ex-
scrit=1
scrit=8
probability density function
APPENDIX C: CAPTURE ZONE AREA ISLAND AREA CORRELATION
From experiment (Fig. 15) there is a correlation between CZ area and island area. This implies a degree of ‘smooth’ growth occurs inside
what must therefore be fairly static CZs. This
is evidence for larger islands being effectively
immobile enforcing the need for α to be high
enough to kill off the diffusion of larger islands.
These plots also lend further weight to the idea
of a critical island size. This is most noticeable
in the 0.75 nm coverage data where there is a
thinning out of the density of island areas from
≈2-3 nm2 which then thickens again with larger
area. This would suggest that the islands stay
static until they reach an area of 2 nm2 which
corresponds to a width of ≈ 5 atomic diameters,
which is slightly higher than the estimates taken
from the ISDs.
0.25
scrit=10
0.2
perimental data the range of scrit is limited to
4 & scrit . 11. The effect of varying scrit within
this range is shown in Fig. 14 as well as an
example of low scrit = 1 and high scrit = 25.
There is not much difference in the results for
scrit ≈ 10 so this value was chosen for the rest
of the simulations.
scrit=12
scrit=25
0.15
0.1
0.05
The simulation appears to capture the CZ
area - island size correlation , Fig. 16. The effect of having a critical isalnd size is also evident at higher coverages .The bulk of the large
FIG. 14: Plots for ISDs for different values of scrit islands appear to grow at the same rate as the
with R = 7.5 × 105 and α = 1.75 from extended- large bunch moves to the right with increasing
island simulation.
coverage, this is equivalent to the large island
peak moving to the right in the ISDs of Fig. 7 .
0
0
5
10
15
20
d − number of atomic diameters
25
14
Experiment
50
θ =0.15 nm
CZ area [nm2]
40
30
20
10
0
0
1
2
3
4
5
6
2
Island area [nm ]
7
8
CZ area [nm2]
80
9
10
θ =0.75 nm
60
40
20
0
0
5
10
15
20
25
Island area [nm2]
30
35
40
45
FIG. 15: Plot of CZ area against island area for θ = 0.15 nm and θ = 0.75 nm experimental data. There is
a correlation between CZ area and island area.
15
120
θ=10%
100
100
80
80
CZ area [nm]2
CZ area [nm]2
120
60
40
20
0
θ=50%
60
40
20
0
1
2
3
4
5
6
0
7
0
5
Island area [nm]2
160
10
15
20
25
Island area [nm]2
200
θ=150%
θ=250%
150
120
CZ area [nm]2
CZ area [nm]2
140
100
80
60
100
50
40
20
0
10
20
30
Island area [nm]2
40
50
0
0
20
40
60
80
100
Island area [nm]2
FIG. 16: Plot of CZ area against island area for θ with R = 7.5 × 105 , α = 1.75 and scrit = 10 from
extended-island simulations.