992
Langmuir 2005, 21, 992-1000
Fractal Analysis Methods for Solid Alkane Monolayer
Domains at SiO2/Air Interfaces
Lydia Knüfing,† Hauke Schollmeyer,‡ Hans Riegler,*,‡ and Klaus Mecke†
Australian National University, RSPhysSE, Applied Mathematics, A.C.T. 0200, Australia,
Max-Planck-Institut für Kolloid- und Grenzflächenforschung, Am Mühlenberg,
D-14476 Potsdam/Golm, Germany, Institut für Theoretische Physik, Universität
Erlangen-Nürnberg, Staudtstrasse 7, D-91058 Erlangen, Germany
Received September 17, 2004. In Final Form: November 5, 2004
A systematic evaluation of various fractal analysis methods is essential for studying morphologies of
finite and noisy experimental patterns such as domains of long chain alkanes at SiO2/air interfaces. The
derivation of trustworthy fractal dimensions crucially relies on the definition of confidence intervals for
the assumed scaling range. We demonstrate that the determination of the intervals can be improved
largely by comparing the scaling behavior of different morphological measures (area, boundary, curvature).
We show that the combination of area and boundary data from coarse-grained structures obtained with
the box-counting method reveals clear confidence limits and thus credible morphological data. This also
holds for the Minkowski density method. It also reveals the confidence range. Its main drawback, the
larger swing-in period at the lower cutoff compared to the box-counting method, is compensated by more
details on the scaling behavior of area, boundary, and curvature. The sandbox method is less recommendable.
It essentially delivers the same data as box-counting, but it is more susceptible to finite size effects at the
lower cutoff. It is found that the domain morphology depends on the surface coverage of alkanes. The
individual domains at low surface coverage have a fractal dimension of ≈1.7, whereas at coverages well
above 50% the scaling dimension is 2 with a large margin of uncertainty at ≈50% coverage. This change
in morphology is attributed to a crossover from a growth regime dominated by diffusion-limited aggregation
of individual domains to a regime where the growth is increasingly affected by annealing and the interaction
of solid growth fronts which approach each other and thus compete for the alkane supply.
1. Introduction
Most solid structures in nature are not in thermodynamic equilibrium, and their morphologies are determined
not only by the interaction energies between the molecules
but also by the aggregation kinetics. Often the morphology
of a spatial structure reflects the aggregation process of
the pattern. A typical and widespread example is the
relation between diffusion-limited aggregation and fractal
morphologies.1
The fractal characterization of geometrical shapes have
been studied in mathematical morphology for many
years.2-5 For voxelized6 patterns, the box-counting and
the sandbox method have been well established to
determine the fractal dimension df. Less well-known are
“intrinsic volumes”, i.e., so-called Minkowski functionals,
which provide a whole family of fractal dimensions and
* To whom correspondence should be addressed. E-mail:
[email protected].
† Australian National University and Universität ErlangenNürnberg.
‡ Max-Planck-Institut für Kolloid- und Grenzflächenforschung.
(1) Meakin, P. Fractals, Scaling and Growth far from Equilibrium;
Cambridge University Press: Cambridge, 1998.
(2) Fractals in Science; Bunde, A., Havlin, S., Eds.; Springer-Verlag:
Berlin, 1994.
(3) Falconer, K. Techniques in Fractal Geometry; John Wiley &
Sons: Chichester, 1997.
(4) Falconer, K. Fractal Geometry. Mathematical Foundations and
Applications; John Wiley & Sons: Chichester, 1990.
(5) Stoyan, D.; Stoyan, H. Fraktale, Formen, Punktfelder. Methoden
der Geometrie-Statistik; Akademie-Verlag: Berlin, 1992.
(6) Voxelization is the conversion of a continuous geometric object
into a set of voxels which approximates the object. “Voxel” means “volume
element” or “volume cell”; it is the 3D conceptual analogon of the 2D
pixel.
scaling amplitudes.7-9 Various methods have been developed in integral geometry10-12 to characterize the shape
of particles and domains, in particular the definition of
morphological measures such as the intrinsic volumes of
spatial structures. Typically, the methods are tested on
idealized patterns, e.g., spatial configurations created by
computer simulations. The analysis of real experimental
data poses additional, substantial challenges. Methods
that may be superior theoretically may be inferior for use
on real data because, for instance, they are sensitive to
experimental noise or to limited scaling regimes. Quite
frequently, the size of real systems is only a decade or
even less and the fractal dimension df can only be
estimated with large error margins. Methods based on
integral geometry are especially well suited for small
samples with small scaling regimes because intrinsic
volumes are statistically robust due to their additivity
property.7 Additionally, they provide consistency tests for
fractal parameters such as the effective scaling dimension
(7) Mecke, K. Additivity, Convexity, and Beyond: Applications of
Minkowski Functionals in Statistical Physics. In Statistical Physics
and Spatial Statistics: The Art of Analyzing and Modelling Spatial
Strucutres and Pattern Formation; Mecke, K., Stoyan, D., Eds.; Lecture
Notes in Physics, Vol. 554; Springer-Verlag: Heidelberg, 2000; pp 72184.
(8) Mecke, K. Complete Family of Fractal Dimensions Based on
Integral Geometry. Physica A, to be submitted.
(9) Knüfing, L.; Mecke, K. Determining Morphological Scaling
Amplitudes of Fractals. Physica A, to be submitted.
(10) Mecke, K. Integralgeometrie in der Statistischen Physik; Verlag
Harri Deutsch: Frankfurt, 1994.
(11) Mecke, K. Integral Geometry and Statistical Physics. Int. J.
Mod. Phys. B 1998, 12, 861-899.
(12) Klain, D.; Rota, G.-C. Introduction to Geometric Probability;
Cambridge University Press: Cambridge, 1997.
10.1021/la0476783 CCC: $30.25 © 2005 American Chemical Society
Published on Web 01/04/2005
Fractal Analysis of Alkane Monolayer Domains
Langmuir, Vol. 21, No. 3, 2005 993
Figure 1. Suggested scenario for the nucleation and growth
of solid domains in the case of submonolayer coverage.
and the correlation length. If each of the intrinsic volumes
shows a similar scaling within a certain range, one can
assume that the common exponent equals the fractal
dimension. Deviations of the functional form of the
intrinsic volumes indicate finite size effects or the
breakdown of scaling behavior. Thus, the estimation of
fractal dimensions depends strongly on the definition of
confidence intervals, which may be larger for one analysis
technique than for another.
An example of two-dimensional fractal morphologies is
domains of long chain alkanes at solid/gas interfaces. Such
domains are suitable model systems to test characterization methods because the morphologies can be varied by
the preparation parameters. It is for example possible to
prepare samples with a wide range of (average) alkane
coverages.13 Solid alkane layers can only attain thicknesses
of about molecular length (e.g., ≈41 Å for n-triacontane
(C30H62 ) C30)) or multiples thereof because in their solid
state, the molecules assume their all-trans configuration
and aggregate in a rotator or crystalline phase with the
molecular axis normal to the interface.13-16
Therefore, if the overall alkane surface coverage is not
sufficient for a complete solid monolayer (“submonolayer
coverage”) the surface will be covered only partially by
solid monolayer sections.14-18,23 This leads to morphologies
reaching from singular domains to a closed alkane film
with holes. The suggested scenario for the growth of the
solid monolayer sections, as it occurs upon cooling a liquid
film with submonolayer coverage to below its melting
temperature, is shown in Figure 1. It is assumed that
upon sufficient undercooling below the surface freezing
temperature, Tssv, randomly solid supercritical nuclei form.
These will grow into two-dimensional domains through
the lateral molecular transport and attachment of alkane
molecules. The morphology of the domains depends on
the solidification conditions, i.e., the surface coverage, the
cooling rate, etc. Thus, the morphology reflects the
prevailing two-dimensional nucleation, transport, and
solidification processes. Typically, dendritic or seaweed
shapes will appear due to morphological instabilities of
the growth fronts.19-22 The system is similar to thin film
(13) Merkl, C.; Pfohl, T.; Riegler, H. Phys. Rev. Lett. 1997, 79 (2),
4625.
(14) Holzwarth, A.; Leporatti, S.; Riegler, H. Europhys. Lett. 2000,
52 (6), 653.
(15) Schollmeyer, H.; Ocko, B.; Riegler, H. Langmuir 2002, 18, 4351.
(16) Schollmeyer, H.; Struth, B.; Riegler, H. Langmuir 2003, 19, 5042.
(17) In fact, this is only a first approximation. The topology of the
system is most likely more complicated with additional layers of alkane
molecules lying flat under or above the domains and on the substrate
surface between the domains.
(18) Volkmann, U. G.; Pino, M.; Altamirano, L. A.; Taub, H.; Hansen,
F. Y. J. Chem. Phys. 2002, 116, 2107.
(19) Ihle, T.; Müller-Krumbhaar, H. Phys. Rev. E 1994, 49 (4), 2972.
Figure 2. Selection of morphologies obtained for surface
coverages between 24.9% (sample 249) and 83.3% (sample 833)
after binarization. The pictures show an area of typically 100
µm × 100 µm. The black sections represent the solid alkane
regions.
growth by vacuum deposition.23-26 It differs in that it has
only a limited reservoir of molecules and the transport is
governed by the crossover from three-dimensional (comparatively thick liquid film at the beginning of the domain
growth) to two-dimensional flow (depleted, thin liquid film
toward the end of the domain growth) of individual
molecules.
In the following, we will characterize the morphologies
of alkane domains as a function of the surface coverage
with three frequently used methods, analyze the significance and usefulness of the resulting data, and draw
conclusions from the data on the growth process.
2. Experimental Sample Preparation, Data
Collection, and Data Processing
The samples were prepared by melt growth as described in
detail elsewhere.13-16 A film of toluene solution of C30 is deposited
via spin coating onto a piece of planar, smooth silicon wafer
surface (1 cm × 2 cm) with a natural oxide layer of about 15 Å
thickness. The solution concentration and the spin coating
conditions determine the overall surface coverage of triacontane.
After the evaporation of the toluene, the samples are heated to
about 80 °C for a few minutes. Thus all toluene will evaporate,
and the C30 melts (interfacial melting point of C30 ≈ 70 °C, bulk
melting temperature ≈ 67 °C) and completely wets the substrate
surface. Then the samples are cooled to room temperature at a
rate of about 2 °C/s (cooling rate at 70 °C). At room temperature,
they were investigated by surface force microscopy (SFM;
Nanoscope III, Digital Instruments, Inc., Santa Barbara, CA) in
the noncontact (“tapping”) mode. As mentioned already in the
Introduction, the described preparation conditions lead to
domains with fractal morphology which consist of a monolayer
of alkane molecules in their all-trans configuration, oriented
normal to the interface. The microscopy pictures were binarized
with two states (see Figure 2), the areas of solid alkane (black)
and the area in between (white), respectively. The binarization
threshold was deduced from the gray-scale distributions of the
original microscopy pictures. The gray-scale data show a roughly
(20) Brener, E.; Müller-Krumbhaar, H.; Temkin, D. Phys. Rev. E
1996, 54 (3), 2714.
(21) Jürgens, H.; Peitgen, H.-O.; Saupe, D. Sci. Am. 1990, 263 (2),
60.
(22) Kurz, W.; Fisher, D. J. Fundamentals of Solidification; Trans
Tech Publications Ltd.: Switzerland, 1998.
(23) Mo, H.; Taub, H.; Volkmann, U. G.; Pino, M.; Ehrlich, S. N.;
Hansen, F. Y.; Lu, E.; Miceli, P. Chem. Phys. Lett. 2003, 377, 99.
(24) Brinkmann, M.; Graff, S.; Biscarini, F. Phys. Rev. B 2002, 66
(16), 165430.
(25) Meyer zu Heringdorf, F.-J.; Reuter, M. C.; Tromp, R. M. Nature
2001, 412, 517.
(26) Luo, Y.; Wang, G.; Theobald, J. A.; Beton, P. H. Surf. Sci. 2003,
537, 241.
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Knüfing et al.
Figure 3. Results of the analysis of the morphologies shown in Figure 2 with the box-counting method. Both the behavior of domain
area and that of the boundary length (from coarse-grained structures) are shown. The solid lines represent the best fits to the area
data for data points within a range given in the brackets which results in the indicated fractal dimension df. The fitting range is
also shown by the vertical solid lines. For comparison, the dotted lines present the best fit for the fitting range given in the brackets
assuming a (nonfractal) scaling dimension of 2.
bimodal distribution representing the domain areas and the SiO2
surface in between. The minimum between the two gray-scale
maxima was selected as the binarization threshold. A visual
comparison of the data before and after the binarization and
tests with various threshold values show that the approach leads
to reproducible and consistent results. Data with ambiguous
binarization results, e.g., because of substantial gray-scale
variations within the pictures due to baseline shifts, are easily
identified and were not used for further analysis.
Figure 2 presents a selection of binarized SFM pictures (typical
resolution, 512 × 512 pixels; the pictures represent areas of ≈100
µm × 100 µm) with various surface coverages which are
representative of the data analyzed in the following. The samples
are labeled with numbers between 0 and 1000 indicating their
domain surface coverage in ‰ (0 ) no domains, 1000 ) 100%
domain surface coverage ) closed solid film). The following figures
show only a representative fraction of the pictures and data which
were in fact obtained and analyzed, respectively. We were not
particularly selective in analyzing only the most perfect data
except that samples with crude defects (e.g., dust particles) were
not taken.
3. Fractal Analysis
Fractal patterns such as diffusion-limited aggregation
clusters are usually characterized by a noninteger scaling
dimension which describes the increase of its visible area
when the observation window is increased. The main
problem in determining such fractal dimensions is often
the size of experimental data which do not allow observation of a scaling behavior over more than a decade.
Additionally, data often exhibit substantial noise which
may obscure the scaling behavior of the visible area.
Therefore, a systematic evaluation of the applicability of
fractal analysis techniques is necessary, before conclusions
can be drawn about the fractal dimension of a spatial
structure. In the following, we evaluate three methods to
determine fractal dimensions of the solid domains in
alkane monolayers.
3.1. Box-Counting Method. In the case of the boxcounting method,1-3,5 a grid of increasing box sizes r
(measured in units of voxel size) is used to completely
cover the complete voxelized area. For each r, the number
#(r) of boxes which intersect with a fractal structure, i.e.,
a domain, is counted. The resulting 1/#(r) values are then
plotted vs r in a double logarithmic plot. The slope of the
power-law regression line f(x) ) cxa yields an estimate of
the fractal dimension df ) a. Figure 3 shows the results
of the box-counting method for samples with domain
coverages ranging from 24.9 to 88.3%. Besides the 1/#(r)
as function of r (“area”), Figure 3 also shows boundary
length data (“boundary”, measured in units of box length
r) as a function of the box size. These data show the
boundary length of coarse-grained structures which are
obtained when each box of length r which contains at least
one pixel of a structure is taken as completely filled. Also
shown are regression lines for two different fractal
dimensions df with the corresponding fitting ranges given
in the squared brackets. The dashed lines are fitted to the
area data for lengths r beyond a certain threshold value
(≈the correlation length limit, see below) with a fixed slope
of df ) 2. The solid regression lines present fits of the
experimental area data with the slope as a fitting
parameter; i.e., from these fits the estimated df values
were derived. The solid vertical lines show the range within
which the fits of the solid lines were performed. The lower
bound of the fits is always at r ) 3 (except for sample 833
which is fitted for 1 e r e 3). This is due to finite size
effects for data points for r e 3, as discussed in more detail
below. The upper cutoff is derived from the behavior of
the coarse-grained boundary lengths: Only a range of r
has been used for which the scaling of the boundary length
is approximately the same as that of the area. At large r,
the size of the box has grown past the correlation length
ξ of the fractal structure and the samples show bulk
behavior. This is reflected by the steep increase of the
boundary data. Data beyond the correlation length limit
can be fitted quite nicely with a slope of 2, which is
demonstrated with the dashed regression lines.
Fractal Analysis of Alkane Monolayer Domains
Figure 4. Results from a box-counting examination of random
occupation of a square lattice with different site-occupation
probabilities p ) 0.55, 0.61, 0.74, and 1.0. The deviation from
the slope of 2 and its dependence on p for small r are obvious.
To give an example, for sample 456 the fit for the
estimated dimension df was carried out for 3 e r e 25. The
lower bound is given by the general cutoff line at r ) 3.
The upper bound is derived from the behavior of the
boundary length measure which starts to bend off for r ≈
25. As can be seen from the data of Figure 3, the range
which is useful for a reasonable estimation of the fractal
dimension is decreasing with increasing surface coverage.
Whereas the scaling behavior for sample 249 is quite
obvious and can be quantified reliably over more than 1
decade (from 3 to 56), for samples with more than 60%
coverage a reasonable confidence interval hardly exists,
which implies a nonfractal behavior.
It is not advisable to use box sizes with dimensions
close to the voxel size for the derivation of the fractal
dimension df, as is sometimes found in the literature.27,28
Studies carried out on random occupations of lattice sites
(random voxels) show that for r < 3 the method in itself
affects the data points, resulting in a “virtual” dimension
dv. Figure 4 presents results from a box-counting examination of randomly occupied lattices with different siteoccupation probabilities p ) 0.55, 0.61, 0.74, and 1.0. The
deviation from the slope of 2 and its dependence on p for
small r are obvious. An analysis yields virtual fractal
dimensions dv ) 1.20, 1.33, 1.61, and 1.99, respectively,
for r < 3 (see Figure 5). As expected, for p f 1.0 the effect
of smaller box sizes vanishes. The bias is the result of the
number #(p) of occupied cells of size r inside a system of
size L which depends on the occupation probability p in
the following way:10
2
#(p) ) (1 - (1 - p)r )
(
f y ) y0 + log p + 2 1 +
(Lr)
2
1-p
ln(1 - p) x + O(x2)
p
(1)
)
with x ) -log(r), y ) log(#), and y0 ) 2 log(L). Thus a
virtual fractal dimension can be obtained which does
depend on the occupation probability of the cells of the
system. Figure 5 shows the derivative of the equation
above and the dependence of the fractal dimension dv on
p for small box sizes (see Figure 4).
Concluding, we emphasize that the estimation of the
true fractal dimension df is only possible within the
confidence interval or 3 < r < ξ shown in Figure 3, where
(27) Klemm, A.; Müller, H.-P.; Kimmich, R. Phys. Rev. E 1997, 55,
4413; Physica A 1999, 266, 242-246.
(28) Müller, H.-P.; Weis, J.; Kimmich, R. Phys. Rev. E 1995, 52, 5195;
1996, 54, 5278.
Langmuir, Vol. 21, No. 3, 2005 995
Figure 5. Virtual fractal dimensions dv ) 1.20, 1.33, 1.61, and
1.99, respectively, for r < 3 resulting from the analysis of a
random site occupation of a lattice (see Figure 4).
the upper cutoff ξ decreases with the alkane coverage.
Taking into account data points at r < 3 or r > ξ would
significantly change the value of the estimate for the fractal
dimension. The deviation of the scaling of the boundary
length from that of the area provides an excellent indicator
for the confidence interval.
3.2. Sandbox Method. For an analysis with this
method, observation windows Dr(x) of increasing box sizes
r(x) are centered at pixels x ∈ C of the structure (at any
pixels, i.e., at the boundary of the structures as well as
inside domain arms). Inside the window, the number
#(Dr(x)) of pixels which belong to the structure is counted,
i.e., the observable size or volume of the structure inside
the window. To improve statistics, the mean number of
pixels #(r) is taken and plotted versus the size r of the
window Dr in a double logarithmic graph. As with the
box-counting method, the slope of a regression line (y )
cxa) yields an estimate of df ) a. Figure 6 presents the
analysis from the data of Figure 2 with the fitting ranges
and the resulting fractal dimension, presented in an
analogous way as in Figure 3. Again, the estimation of
the fractal dimension depends drastically on the range of
the fits of the regression lines. As for the box-counting
method, this fit range cannot be determined unambiguously from the area data only. Therefore, again also the
boundary lengths of the cluster sections (but without
coarse graining) inside the observation window have been
measured and plotted as function of r. Compared to the
boundary data derived from the box-counting method, the
sandbox boundary lengths show a much weaker change
of the slope with increasing r (most probably, a f 2 for
area and boundary at large r). This is attributed to the
derivation of the boundary length data from unchanged
structures without coarse graining.
To nevertheless extract confidence limits from the area
and boundary data obtained by the sandbox method as
presented in Figure 6, we tested yet another approach. If
one assumes that both area A and boundary B scale as
fractals with dimensions depending on r, A ∝ rdA(r) and B
∝ rdB(r), respectively, then the ratio ∆ ) B/A of boundary
divided by area scales as ∆ ∝ rdB(r)-dA(r). Thus, if dB(r) )
dA(r), then ∂∆/∂r ) 0, independent from the actual values
of dB(r) and dA(r), and with dB(r) and dA(r) being possibly
still a function of r. If dB(r) * dA(r), then ∂∆/∂r ) f(r).
Figure 6 presents the ratio between boundary and area.
To make small differences in the ratio visible, the y-values
of the data sets were multiplied by 10 and shifted
accordingly. This ratio is indeed changing with r in a more
pronounced way than the boundary data alone. Essentially
all samples show for r < 20 a constant, nonzero slope
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Figure 6. Analysis of the data of Figure 2 with the sandbox method. The fitting ranges and the resulting fractal dimensions are
displayed in a analogous way as for the box-counting method (see Figure 3). Contrary to the box-counting analysis shown in Figure
3, the boundary data are obtained without prior coarse graining. As auxiliary data, also the ratio between boundary and area is
presented.
∂∆/∂r. Hence dB(r) * dA(r). At r ≈ 20, ∆ levels off and at
r > 30, ∂∆/∂r ≈ 0, i.e., dB(r) ≈ dA(r). In terms of our criterion
for a suitable fitting range (dB(r) ≈ dA(r)), this means a
lower cutoff at r < 30. However, the data indicate no upper
cutoff.
We tried yet another approach to determine a confidence
interval and finally the fractal dimension. To find the
limits of a suspected scaling behavior, one may determine
pointwise fractal dimensions within certain, small intervals and analyze how these dimensions change with
changing midpoints. If there is some scaling-like behavior
within a certain range, one can expect no or only small
variations of the pointwise-determined fractal dimensions.
The scaling range limits are the midpoints with significant
variations of the fractal dimensions. Figure 7 presents
the “local” fractal dimensions as they are obtained from
the slope of the area regression line within certain
intervals. The graphs show df as a function of the midpoints
rmid of intervals at rmid ) (r1 + r2)/2 with increasing interval
size I ) r2 - r1 with increasing r (I ) 3-4 (r < 20), I )
6 (r g 20), I ) 10 (r g 100)). From the behavior of the local
fractal dimension, the lower and upper cutoffs for the
overall fractal dimensions were estimated under the
assumption of a minimum variation in df over a maximum
range of r. The cutoffs derived by this method are the ones
shown in Figure 6 (vertical lines). Figure 6 also presents
the regression lines and fractal dimensions derived within
these limits, as well as regression lines fitted to large r
with df ) 2 (dashed lines) for comparison. Figure 7 shows
that the lower limit generally must be assumed at r ≈
10-15, substantially larger than the value for the boxcounting. Partially this may be attributed to using
intervals which “sense” beyond the “real” limit, although
the width of the intervals (3-4) is substantially smaller
than the difference between the lower limits suggested by
the two analyzing methods.
As for the box-counting method, the estimated value of
the fractal dimension df is biased toward 2 for r f 1 as
Figure 7. Local fractal dimensions determined from the slope
of the regression lines within small intervals. The x-axis denotes
the midpoints xmid ) (x1 + x2)/2 of the intervals, and the y-axis
the fractal dimension. There is an increasing interval size I
with increasing r: I ) 3-4 (r < 20), I ) 6 (20 < r < 100), I )
10 (r > 100).
can be seen from Figure 7. The first few window sizes are
therefore not to be taken into account when deriving the
fractal dimension. Even samples with a fairly well-defined
fractal dimension (according to the box-counting approach,
see for instance sample 288) show with the sandbox
method a “swing-in” phase for r < 10. Together with the
upper cutoffs due to the limited correlation length ξ (which
are not obvious from the data of Figure 6 but can be derived
from the data of Figure 7), this severely reduces the useful
fitting range, especially for higher surface coverage. Even
though, for example, a dimension df * 2 can be extracted
for sample 610 from the data, it becomes clear from Figure
Fractal Analysis of Alkane Monolayer Domains
Langmuir, Vol. 21, No. 3, 2005 997
Figure 8. Area and boundary data of the morphologies of Figure 2 (except for 83.3% surface coverage) obtained with the Minkowski
functional method. The solid regression lines were fitted to the area within the cutoff limits as specified in the brackets resulting
in the fractal dimensions indicated (for the cutoff criteria, see the main text).
7 that it is not very meaningful. The rise of df(610) for rmid
already at rmid g 10 indicates a correlation length ξ e 10
which is consistent with a visual inspection of the sample
structure.
In conclusion, we emphasize that the boundary data
from structures which were not coarse grained reveal no
clear confidence limits. The ratio between area and
boundary data shows only the lower cutoff, whereas the
pointwise fractal dimension indicates the lower and the
upper cutoff. Compared to the box-counting method, the
lower cutoff for the sandbox method is substantially higher.
Within their respective confidence limits, both methods
lead to comparable fractal dimensions.
3.3. Minkowski Density Method. Similar to the
sandbox method, for this method the number of pixels
which are part of the structure is counted for an observation window Dr(x) with increasing size r. The pixel in the
bottom left corner of Dr(x) must be an element of the cluster
x ∈ {C}. Again, to obtain the fractal dimension, the mean
number of pixels #(r) inside an observation window Dr is
plotted versus r and the slope of the regression line is
determined. The main difference from the sandbox method
lies in the way the mean number is scaled according to
window size (“area”) and boundary (“boundary”) inside.
Both are plotted as #(r)/r2; i.e., the density is shown
instead of the global measures. Accordingly, the fractal
dimension derived from the slope a of the regression lines
is df ) a + 2 (analogous to the “Minkowski sausage
method”,5 but instead of circles, squares are being used
here). As with the other methods used above, it allows
cross-checking the results for df by analyzing the behavior
of the boundary. In theory, the leading scaling terms for
boundary and area are the same, namely, a ) df - 2.
Figure 8 presents the analysis of the samples of Figure
2 (except for the highest surface coverage) with this
method. As with the other methods, for the significance
of df it is again important to select a meaningful fitting
range (confidence interval) over a range as wide as possible.
Indeed, in agreement with theoretical predictions, area
and boundary show pretty similar slopes for r > 10. The
disparity between area and boundary for small r can be
Figure 9. Minkowski functionals for randomly occupied
lattice sites with various occupation probabilities p. The
deviation from the slope of 2 and its dependence on p for small
r are obvious.
attributed to finite size effects (see below) and indicates
the lower cutoff limit. However, the upper cutoff limit is
not clear from the “standard” yardstick (similar slopes for
area and boundary). The slopes of area and boundary are
very similar up to the maximum r, obviously even beyond
the correlation limit ξ. However, contrary to the data from
the sandbox method, the slopes change at certain r, which
most likely is related to the correlation limit. If one derives
the lower cutoff from the region of disparity of the slopes
of area and boundary and the upper cutoff from the change
in the slopes of area and boundary, respectively, one can
indeed obtain reasonable fractal dimensions (see solid lines
in Figure 8) in accordance with the previous methods.
However, the fitting range is still quite limited and rarely
extends over more than half a decade (as in the case of
sample 355). For high coverage (see sample 611), this
method yields no fractal dimension below d ) 2.
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Knüfing et al.
Figure 10. Curvature data from the morphologies of Figure 2 (except for 83.3% surface coverage).
Since one voxel is per definition always inside the
window, #(r)/r2 is biased toward 1 for r f 1. This is
demonstrated in Figure 9, where random voxels with
various occupation probabilities were analyzed with the
Minkowski density method. The analysis strongly recommends that r < 10 should not be taken into account for
the estimation of the fractal dimension.
Following the idea in integral geometry that the set of
Minkowski functionals is a complete family of morphological additive measures,10,11 one may use the scaling of
each member of this family of measures to analyze the
fractal behavior of spatial structures.7-9 Therefore, besides
the area and boundary length also the integral curvature
χ (which is related directly to the Euler characteristic) as
a third measure has been extracted from the experimental
data (Figure 10). The curvature measure is calculated as
described in detail elsewhere.29 As with the boundary
terms, the sample is divided into boxes of size r and each
box containing at least one pixel of the cluster is taken as
completely filled (coarse graining). Then a negative or a
positive value (+1 or -1) is assigned to the curvature of
each individual vertex (edge) depending on the configuration and type of the three pixels which form the vertex.
The integral curvature value for each sample and box size
r is the negative mean of the sum of the individual
curvature values normalized to the box size. Usually the
leading term of the integral curvature shows a scaling
behavior similar to the area and boundary data. Therefore
its slope and behavior could provide additional information
on the useful range for the fit of the regression lines
(confidence interval) and thus the fractal dimension. The
result of a curvature analysis of the morphologies from
Figure 2 is presented in Figure 10. As can be seen, the
highest coverage, sample 610, obviously yields no useful
fractal dimension, whereas the samples with lower
coverage show ranges with slopes reminiscent of the
expected values. Again, the cutoff limits for the fitting
ranges are not clear from the data. If one assumes a lower
cutoff around r ) 10, as suggested by the analysis
presented in Figure 9, and an upper cutoff, where the
slope of the curvature data is changing compared to its
value at r ≈ 10, one can deduce in a few cases meaningful
(29) Mecke, K. Phys. Rev. E 1996, 53, 4794-4800.
numbers. Whereas the low surface coverage samples
(samples 249 and 288) yield slopes which are much too
low, samples 355 and 456, respectively, indeed suggest
meaningful numbers near 1.5. Further analysis is necessary to decide whether the difference in the scaling of the
curvature to the other measures, i.e., the deviation from
the scaling of area and boundary, is inherent to the
curvature measure or a mere artifact of the finite size and
noise of the experimental data. However, the curvature
measure does allow one to specify the confidence interval,
i.e., upper and lower limits of scaling, and clearly rules
out a fractal behavior for coverages above 50%.
4. Discussion
The morphologies of Figure 2 are all seaweed-like.
Already some time ago a quick analysis of individual
domains, i.e., samples with low surface coverages,14
indicated a fractal dimension of ≈1.7. From this we
concluded a growth scenario with diffusion-limited aggregation (DLA) as presented in Figure 1. However, this
is only a simplified view. There are clearly differences in
the morphologies for high and low surface coverages. For
low coverage, the domains are well separated. At high
coverages, the spaces between the domains are comparable
to the distances between the branches of the same domain.
These observations initiated this study focusing on the
following questions: Which method delivers significant
numbers on the fractal dimension if there is any reasonable
scaling behavior at all, and if so do they reflect the
morphological changes with surface coverage?
For surface coverages below 50%, all three methods yield
for the area data fairly similar fractal dimensions around
1.7 within the confidence limits of the regression lines.
For box-counting and the sandbox method, the regression
lines through all area data points give results which are
similar to those from only the limited confidence range.
The Minkowski density method, on the other hand, is
more discriminating. There are ranges with different
(approximately) linear slopes; i.e., only the selection of
confidence intervals in r leads to reasonable estimates of
the fractal dimension df. For coverages above 50%,
confidence limits are important for all three methods.
Without lower and upper cutoffs, the area data would
suggest df ≈ 2, i.e., no fractal morphology at all. This
Fractal Analysis of Alkane Monolayer Domains
conflicts with the visible morphologies in Figure 2 which
show a fractal behavior on a small interval at small length
scales (see the minimum of the fractal dimensions in
Figure 7 which deviates from df ≈ 2).
It is clear, in the real world, that there are always finite
size limits for a scaling behavior. In the presented case,
the lower limit is approximately the voxel size, and the
upper limit is typically the size of the domains (or the
picture size). Due to the different algorithms of the
different methods for the evaluation of the data points,
these real space limits will translate differently into the
confidence limits for the regression lines.
A test with randomly occupied voxels shows that for
the box-counting method (see Figure 4) the lower cutoff
is very close to the voxel size. Already all data for r larger
than about twice the voxel size show approximately the
correct scaling behavior. If r gets closer to 1, the scaling
behavior depends on the site-occupation probabilities. This
leads to the virtual fractal dimensions shown in Figure
5. To play it safe, for the box-counting analysis a lower
cutoff of 3 was selected. The analysis of the lower cutoffs
for the sandbox and Minkowski density methods reveals
a stronger sensitivity of these methods to the voxel size.
This is obvious from an analysis of randomly occupied
voxels with the Minkowski density method (see Figure 9),
analogous to what is shown for box-counting in Figure 4.
Independent of the site occupancy, a swing-in behavior
reaches up to r ) 10. This allows a lower cutoff for these
methods of r ) 10. It is found that this cutoff is quite
similar to the one obtained by the analysis of the sandbox
method with the local slopes of the regression lines in
small intervals. The swing-in period is r g10 for the
interval midpoints.
Upper cutoffs are not as easily obtained as lower cutoffs.
Compared to the voxel size, they are more a characteristic
of the individual system. As a new concept, this report
uses not only the area behavior but also other morphological measures to determine the (upper) confidence
limits. It is assumed that within the confidence range
area, boundary and, possibly, also curvature reveal a
similar scaling behavior.
With box-counting, we show that this approach is indeed
quite convincing. For surface coverages below ≈50%, the
scaling behavior of area and coarse-grained boundary is
quite similar within a certain range. The area data show
no clear upper cutoff, whereas for the boundary data there
is a pronounced upper cutoff. This upper cutoff leads to
a useable confidence range of about 1 order of magnitude
or more which gives the obtained fractal dimensions some
significance. At higher coverages, the upper boundary
cutoff of the boundary merges with the lower cutoff from
the voxel size. This renounces a (careless) interpretation
of the area data at high coverages which suggest a fractal
dimension df ≈ 2.
With the sandbox method, we also analyzed the area
and the boundary behavior. However, contrary to the
analysis with box-counting, the boundary data were
obtained without coarse graining. In this case, neither
area nor boundary data indicate any upper or lower cutoffs.
Both curves are quite linear and run approximately
parallel due to the dominance of the term ∼r2 which hides
the fractal scaling behavior. Only the ratios between
boundary and area data reveal some dependency on r.
They indicate the lower cutoff, but they also do not supply
any useful hints on the upper bound. Only another
approach, the segment-wise determination of the slope of
the regression line, reveals some indications of lower and
upper cutoffs and thus gives the fractal dimensions which
were derived within these limits a certain credibility.
Langmuir, Vol. 21, No. 3, 2005 999
In the case of the Minkowski density method, all three
measures, area, boundary, and curvature, are quite
sensitive to r and to the coverage. All area data indicate
some swing-in for r < 5, which is consistent with the results
from random nonfractal structures (Figure 9). The swingin for the boundary data seems to reach to r ≈ 10. If one
assumes for the area data a confidence range which is
represented by the linear range extending from the lower
swing-in to some upper limit (indicated by a break in the
area curve toward lower slope), then one obtains reasonable, significant numbers for df for coverages up to 50%.
For higher coverages, the area data alone suggest that
there is no meaningful fractal dimension (essentially df
) 2). Up to moderate coverages, this interpretation of the
area data is nicely supported by the parallel slope of the
boundary data within the suggested confidence range.
Sample 456 is a borderline case where the area confidence
limits barely overlap with the suggested boundary confidence limits.
Last, the curvatures are most sensitive to the morphologies. As expected, the curvature data from coverages
beyond 50% indicate that there is no fractal behavior. At
lower coverages, for samples 249 and 288, an extended
linear curvature range suggests clear upper cutoffs. But
the fractal dimensions of the curvature in this range differ
substantially from those of the other measures. Only
sample 355 supplies a fractal dimension of the curvature
similar to the area and boundary data.
Figure 2 shows that overall density distributions change
from rather heterogeneous (considerable space between
individual domains) to quite homogeneous (the gaps
between neighboring domains equal the gaps between
branches of the same domain). This is correlated to the
fractal dimension. At low coverages, with some significance, it is ≈1.7, whereas at high coverages, the patterns
are homogeneous at large scales with scaling dimensions
d ) 2 of the morphological measures (area, boundary,
curvature). It can be assumed that at low coverage the
domains grow in a fairly “prototype” DLA process with
singular domains supplied from alkanes coming from “far
outside”. Eventually the growth process stops because the
alkane supply runs dry. At higher coverage, the growth
process is modified. The interaction between growing
domains eventually interferes, and the growth/alkane
supply stops locally at different times. The most advanced
growing arms of adjacent domains approach each other
and stop growing due to local alkane depletion, while other
branches lagging behind are still growing. This may induce
some annealing and growth anisotropy (normal to the
radius some alkane supply is still coming in). This may
explain the tendency of increasing fractal dimensions df
upon increasing the coverage in the range up to 50%. A
dedicated study is necessary to reveal how the alkane
mobility, the average domain fill factor, and the fractal
dimension are related.
5. Summary and Conclusion
We analyzed the morphology of experimental data of
alkane domains with three different methods. With the
box-counting and the sandbox method, the scaling behavior (fractal dimension) of domain areas and boundaries
was quantified as a function of domain surface coverage.
With the Minkowski density method, all three Minkowski
measures, i.e., also the curvature, were determined. Our
study focused on the extraction of confidence limits. The
significance of scaling data strongly depends on the
confidence limits.
Quite unexpectedly, the three methods differ notably
in their sensitivity to the real space lower cutoff (voxel
1000
Langmuir, Vol. 21, No. 3, 2005
size). The evaluation of random structures as well as
experimental data show that the lower cutoff for boxcounting is close to the real space limit whereas the
sandbox and Minkowski density methods have substantial
swing-in periods which reduce the useful confidence range
by nearly 1 order of magnitude. We further demonstrate
for the first time that the boundary scaling behavior can
be used successfully to determine the upper cutoffs. To
this end, the boundary data have to be derived from the
coarse-grained structures. Alternatively also the density
may be used (Minkowski density method) instead of the
global measures. In this case, the boundary scaling usually
reveals upper and lower limits. This is important progress
toward the evaluation of the significance of scaling data.
It is found that the area scaling as well as the confidence
range depend on the domain surface coverage. Up to about
50% coverage, all three methods yield similar fractal
dimensions of ≈1.7 (slightly increasing with coverage) with
Knüfing et al.
sufficient credibility (a confidence limit of at least about
1 order of magnitude). This fractal dimension is in
agreement with the suggested DLA growth scenario. At
higher surface coverages, a characterization of the morphologies with a fractal dimension is questionable; the
confidence range is not sufficient. Supposedly the domains
do not grow independent from each other as for prototype
DLA. Annealing and supply shortages of liquid alkane
due to solidification fronts approaching each other influence the domain morphology.
Acknowledgment. L.K. thanks Mark Knackstedt for
the generous support at ANU. K.R.M. acknowledges
financial support from Deutsche Forschungsgemeinschaft
(DFG Grant No. ME 1361/6). H.R. and H.S. appreciate
helpful discussions with Helmuth Möhwald.
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