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Fractal Properties of Thin Film Surfaces
An analysis of the fractal surfaces produced in two dimensional computer simulations of the growth of thin films by ballistic deposition
Abstract
This report looks at the fractal properties of surfaces produced by ballistic deposition simulations of film growth. A background summary of the meaning and implications of the fractal dimension of a surface will be given along with the results and discussion from the simulations performed. The problems encountered while carrying out this project will also be discussed. 1. IntroductionMany
structures in nature exhibit fractal properties. From trees, to
frosted glass, to island coastlines, we are surrounded by shapes
that upon closer inspection reveal more and more detail and yet
retain the same general appearance no matter on what scale they are
observed. For
example, a square contains no more detail than is readily apparent
at a single glance. Yet the coastline of an island will display
seemingly endlessly increasing detail when viewed from space, from a
plane, from the ground or even portions of it through a magnifying
glass. It is self-evident that increases in detail will translate
into increased estimates of the length of such a fractal shape. It
is not quite so clear that this detail can cause the measured length
of a fractal to increase quite so markedly, with decreasing scale,
as it does. For example, the estimates given by The
amount of detail that becomes apparent when viewing a fractal object
at different scales can be described by a single number, the
object's fractal dimension. Knowledge of the fractal dimension of a
shape can therefore provide a guide as to how much the shape's
apparent length will change with different scales of measurement. The motivation behind this project was to discover if the fractal dimensions of similar films are themselves of similar magnitude, or if not, then are they related through the physical dimensions of the film, in terms of the number of film particles or the surface length. If such a relation was found to exist then it could have "real world" implications in that if a certain type of film surface was known to be characterised by a particular fractal dimension, then the way the apparent surface area of that film changed with scale would be known. This knowledge would be useful as "the roughness of thin films is an important factor in physical phenomena such as absorption, catalysis and the dissolution of a fractal object."[2] 2. Theory2.i. Fractal surface length and box countingThe infinite
detail contained within fractal shapes has implications in terms of
how we define the length of such a shape. If we wish to measure the
length of something we have to make a decision as to what scale we
are going to take measurements at. For instance, do we use a metre
rule or a set of dividers whose tips are set 1mm apart? Obviously,
for regular shapes such as a square, so long as our measuring units
fit an integer number of times along the square's perimeter, then it
doesn't matter at what scale the measurements are taken, the
measured perimeter will be the same.A 1x1m square's perimeter will
be measured as 4m by a person with the metre rule as well as by
someone with the closely set dividers. One method
that may be employed to measure the length of a line is to place a
regular grid over that line and count how many boxes contain
portions of that line. An estimation of the line's length, L, will
therefore be the number of boxes needed to cover the line, N,
multiplied by the width of each box, r, ie: L = N x r. If one is to
measure a straight line in this way then the number of boxes needed
to cover the line is directly related to the size of the boxes. For
example if it takes five 10cm x 10cm boxes to cover a 50cm straight
line then it will take ten 5cm x 5cm boxes to do the same (see
Figure 1). FIGURE 1 : MEASUREMENT OF A STRAIGHT LINE AT DIFFERENT SCALES However, this direct inverse proportionality between box size and number does not hold for fractal shapes. As mentioned previously, when the scale of measurement becomes smaller the level of detail that is measured becomes larger and hence proportionately more boxes are needed to cover the shape. In fact, the
number of boxes that are needed, to cover any shape, is related to
the size of the boxes used through the shape's fractal dimension, df,
in the following equation:
Regular shapes have integer values for their fractal dimension. The fractal dimension for a one dimensional line is 1 and, as would be expected, the above equation predicts that if we halve the size of the boxes that we use to cover the line then we would need twice as many boxes to do so. Hence the measured length of the straight line is constant, regardless of the scale of measurement. Fractal
shapes, however, have a non-integer fractal dimension and this has a
rather startling implication for the actual length of a fractal
line. If we substitute the relation in Equation
1 into the equation for a line's length as determined
using box-counting methods, L = N x r, we get the following relation
between a line's measured length and the size of the boxes used to
measure that length:
To determine a
line's actual length the smallest possible boxes need to be used, to
give infinite precision boxes of size r approaching 0 are required.
However, Equation
2 suggests that, if df > 1, then as r approaches 3. Method3.i. Film and Film Surface ProductionA modified
version of a ballistic deposition simulation program written in C by
physics students at The horizontal
co-ordinate at which a particle is deposited is determined by a
random number generator, in this case the built-in drand48()
function. Particles "stick" at a certain height according
to the positions of the most recently deposited particles at that
and neighbouring horizontal co-ordinates. The horizontal and
vertical co-ordinates of the deposited particles are then output to
a data file which contains the co-ordinates of all the particles
within the film. The original ballistic deposition code, film.c, was
modified to allow the run-time selection of the number of particles
that are to be deposited and the length of substrate (in terms of
number of particle widths) on which the film is to be deposited. It
should be noted that in this simulation all the particles are
uniformly of size 1x1 unit. In order to
produce the actual surface of the film created by the ballistic
deposition simulation, the code was further modified to include a
routine which would "walk" along the surface of the film
in order to produce an output of the co-ordinates of all particles
located on the film's surface (the modified code, filmcoast.3.c,
used in this project . To achieve this, the film is first stored as
a two dimensional array consisting of 1's and 0's, 1's designating
that those array co-ordinates are occupied by a particle. (NB: this
requirement of a two dimensional array limited the maximum tested
film size to a substrate width of 2000 particles, a 4 million
element array. Attempts at using a substrate of 3000 particle width
were unsuccessful as the SGi workstations would not allow the
initialisation of a 9 million element array). A marker is
then placed at the surface particle on one of the edges of the film
(horizontal co-ordinate of 0) and the marker is advanced along the
surface according to rules dependant on the location of other
particles in the immediate vicinity of the marker. The rules for
advancing the marker can be visualised as being the path a
particle-sized "insect" would take if it were to walk
along the film's surface, it can walk horizontally, vertically and
upside down but it can't cross gaps one or more particles wide. The
marker's position is then output to a data file after each step,
producing a file containing the co-ordinates of particles located on
the film's surface, the film's "coastline". 3.ii. Analysis
of the film surfaces produced - Box counting methods
Equation
1 suggests an inverse power relationship between the number of
boxes needed to cover a line and the size of the boxes used. This
means that if box-counts (N) were made at various box-sizes (r), on
a particular line, the results would show a straight line fit when
plotted on a log(N) - log(r) graph. The slope of the line of best
fit would be equal to -df and hence the fractal dimension of the
line would be able to be determined. In order to
perform this kind of fractal analysis on the film surface data a
method was created which would automate the box-counting. The
program takes a data
file, containing the co-ordinates of a line, or "coast" as
its argument. The program then stores the data file as a two
dimensional array of 1's and 0's. This array is then divided into a
series of squares of a fixed dimension (the "boxes") and
each box is searched to see if it contains any part of the coast
(represented by a 1). If a part of the coast is encountered, the
box-count is updated and the next box is examined. The program
performs this procedure in a loop nested 4-deep, that iterates
vertically and horizontally along, and then within, the boxes shows
a film surface overlaid by the grid that the box-counting program
would use to perform a box-count of the surface length with
box-size, r = 10. Box-counts are
performed at all sizes, r, that are factors of the horizontal range
of the coast. Every box-count result and the scale at which it was
performed is output to a data file. The data files
produced by the box-counting program were then analysed using
Mathematica's graphical and list-processing capabilities. Once a
box-count data file was read into a Mathematica notebook, a list
containing the log values of the box-count, N, and the box-size, r,
was created. This list was then used to obtain a straight line fit
to the log values of N and r, in order to determine the fractal
dimension of the film surface being analysed. The box-count data and
the straight line fit were displayed on the same log-log plot in
order to see how well the box-count data obeyed the relation given
in Equation
1. 3.iii. Films examinedFilms were
grown on three different widths of substrates, 500, 1000 and 2000
particles wide, and consisted of between 30 000 and 720 000
particles. The fractal dimensions for the film surfaces were
calculated using the above described box-counting techniques. The actual lengths for the film surfaces, in terms of number of particles, was able to determined using the box-count results for boxes of size r = 1. This allowed the comparison between film substrate length, film surface length and fractal dimension of the film surface. 4. Results
Figure
1
displays the log-log plot of box-count(N) against box-count(r) for
the surface of a 100 000 particle film grown on a 500 particle wide
substrate. A straight line fit of the points is included. The slope
of the line is -df = -1.32. displays
the fractal dimensions for the surfaces of films containing various
numbers of particles grown on substrates that were 500, 1000 and
2000 particles wide. 5. DiscussionThe results in
Figure 2 display no clear
relationship between the fractal dimension of a film surface and the
number of particles used to produce the film. It does, however show
a distinct difference in fractal dimension for films grown on
different length substrates, with longer substrates producing films
with surfaces of lower fractal dimension. Interpretation
of this results is somewhat difficult. It was expected that each of
the films produced would have much the same fractal dimension as
studies have shown that the fractal dimension of the films produced
in ballistic deposition is around 1.3.[3
] The results (at least for films grown on substrates of 500 and
1000 particle widths) do not seem to reflect a constant
characteristic fractal dimension for the film surfaces, nor do they
reflect any obvious relationship between vertical thickness of the
film (which is proportional to the number of film particles) and the
fractal dimension of the surface of the film. It should be
noted, however, that the variation in the measured surface fractal
dimensions, between films containing different numbers of particles,
decreased with increased substrate length. The measured fractal
dimensions for the surface of films grown on a substrate 500
particles long varied from 1.26 to 1.38, whereas those of films
grown on 2000 particle-long substrates only varied between 1.23 and
1.26 (NB: some of these measure values of df don't appear in Figure
2 due to space restrictions). This
observation suggests that in the shorter width films there are
factors affecting the measurement of the true fractal dimension of
the film surface, while films grown on longer substrates have a
relatively constant fractal dimension of 1.25 +/- 0.02. It is
interesting to note that the famous Koch "snowflake" curve
which, despite being qualitatively quite different to the film
surfaces analysed here, has a similar fractal dimension of log4/log3
1.26.[1]
The variation
in the fractal dimensions of shorter width film surfaces could be
due to the proportionally large size of the film particles compared
to the length of the substrate. Because the particles used in the
simulations have dimensions of 1x1 unit there is an artificial limit
to the scale on which the surface length can be measured. This limit
is proportionately much larger for films grown on a 500
particle-long substrate compared with one grown on 2000
particle-long substrate and hence may interfere with attempts to
produce an accurate value for the fractal dimension for the film
surface. The limit to
the scale of measurement can be quite clearly seen in
Figure
1, which is typical of the log(N)-log(r) plots
which were used to calculate the fractal dimension for the various
film surfaces. The point lying on the vertical axis of the plot
corresponds to the number of boxes of size r = 1 needed to cover the
film surface. This is the actual length, in terms of particle size,
of the film surface. Unlike theoretical fractal shapes which can be
viewed at ever decreasing scales and can, therefore end up with
infinite lengths, the films studied in this project have a finite
length due to the scale limit imposed by the discrete simulation. Figure
1
provides some insight into the operation of the automated
box-counting procedure, when the positions of the data points in
relation to the line of best fit are considered. It is noted that
some points, including that corresponding to the box-count when r =
1, lie below the trendline. These points represent box-sizes at
which the box-count procedure is working close to optimally. By
optimally, it is meant that the grid that is used to measure the
length of the film surface has been superimposed in such a way that
the minimum amount of boxes needed to cover the surface are used. By way of
example, in Figure
1 the point corresponding to r = 25 lies below
the trendline Another
unusual trend emerged from looking at the box-count results for the
films analysed in this project. Without fail, the estimation of
actual surface length, L= N x r, was greatest at r = 2. This was
surprising as Equation
2 suggests that the measured length should increase
with decreased scale and hence it would be expected that
measurements of length, using boxes of size r > 1, would be
always underestimates of the actual length, which is obtained with
boxes of size r = 1. It is not clear why box sizes of r = 2 always
produced overestimates of the actual length. It is likely, however,
that this over-estimation may have contributed to the higher (and
more variable) than expected values for the fractal dimensions of
the films grown on 500 and 1000 particle wide substrates. It is not surprising that there is quite a range in actual lengths of the surfaces of films grown on same size substrates ( 30% in the films grown on the 2000 particle substrate). This is because the invagination of the film surfaces and the random nature of the deposition process will lead to situations where the addition of 1 or 2 particles in the right places will cut off reasonably large sections of the surface and hence decrease the surface length significantly. It is interesting to note that this same process doesn't seem to affect the fractal dimension of the surfaces. The films grown on the 2000 particle substrates display the most variance in surface length and the least variance in measured fractal dimension. 6. ConclusionIt was found
that, for films produced in ballistic deposition simulations, there
was no clear relationship between the length of a film surface, the
number of particles that make up the film, and the fractal dimension
of the film, for films produced in ballistic deposition simulations.
However, there was a suggestion in the analysis of films grown on a
2000 particle substrate, that the fractal dimension of the surface
of these films varies only slightly around a value that is
independent of the number of particles that the film contains.
Further study of more films grown on longer substrates is required
to ascertain whether there is a characteristic fractal dimension for
films grown using ballistic deposition, as suggested in [3]
and by the above results. If such a
characteristic fractal dimension was found for simulated film growth
then it may also apply to real films grown using ballistic
deposition. If this were found to be true then it would be useful in
situations where knowledge of a particular film's surface dimensions
is required to be accurate to a small scale. Knowledge of the film's
fractal dimension, along with measurement of the surface size at a
larger scale, would provide a good estimate of the surface size at
smaller scales, direct measurement of which can require significant
amounts of resources. The length of
film surfaces grown on the same substrates were found to vary by up
to 30%, however it was found that this variance was not reflected in
the fractal dimensions of the film surfaces. The average film
surface length was found to increase linearly with increased
substrate length, as expected. The films studied in this project
were found to have surfaces 6.3 times as long as the substrate they
were grown on. It was hoped
that different random number generators would be used in the
computer simulations of the ballistic deposition, however time
constraints prevented this. It would be instructive if comparisons
were made between the above results, obtained from films generated
using the drand48() random number generator, and the results for
films grown using "better" random number generators. For a
start, it would allow comparisons between films generated on the
same substrates from the same number of particles, but using
different sequences of random numbers in the simulation. This would
allow greater insight into the question of whether all films
generated using ballistic deposition have similar fractal
dimensions, as well as producing more "realistic" results,
as it would be expected that no two real-world films would grow in
the exact same fashion. There are
other areas of this project which require further investigation. A
method to obtain more optimal automated box-count results would
increase the precision of the measurement of a shape's fractal
dimension. While the study of three dimensional films and the
extension of the results obtained in this report to two dimensional
surface areas is another logical step that would have greater
applicability to real thin films. Finally, it
should be noted that the surface lengths studied in this project are
unlikely to reflect the effective surface length of the film when it
is taking part in many reactions. Some of the more highly
invaginated sections of the film surface are unlikely to play any
part in surface reactions. A more realistic scenario may be to use
random walk simulations[4]
to obtain the parts of the film surface that are more likely to take
part in diffusion-type reactions, for example. <>
7. AcknowledgementsI would like to thank A/Prof Bernard Pailtorpe, Dr Nicole Bordes and Daniel Mitchell for their help in this project. The results tabled in this project were generated using the facilities of Sydney Vislab which is supported by the Australian Research Council. <> 8. References
1.
B.B. Mandelbrot, The Fractal Geometry of Nature, (W H Freeman
& Company 1982).
2.
R. Baiod, P. Kessler, P. Ramanlal, L. Sander and R. Savit,
"Dynamical scaling of the surface of finite-density ballistic
deposition", Phys. Rev. A, 38, 3672, (1988).
3.
R. H. Landau and M. J. Paez, Computational Physics : problem
solving with computers, (Wiley 1997).
4.
H. E. Stanley and P. Meakin, "Multifractal phenomena in physics
and chemistry", Nature, 335, 405, (1988). W.
T. Elam, S. A. Wolf, J. Sprague, D. U. Gubser, D. Van Vechten and G.
L. Barz Jr., "Fractal Aggregates in Sputter-Deposited NbGe2
Films", Phys. Rev. Lett., 54, 701, (1985). H.
E. Stanley and A. L. Barabasi, Fractal Concepts in Surface Growth,
(Cambridge University Press 1995). |
