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On the measurement of the fractal dimension of aggregated particles by electron microscopy :
experimental method, corrections and comparison with numerical models
M. Tence, J.P. Chevalier, R. Jullien
To cite this version:
M. Tence, J.P. Chevalier, R. Jullien. On the measurement of the fractal dimension of aggregated particles by electron microscopy : experimental method, corrections and comparison with numerical models. Journal de Physique, 1986, 47 (11), pp.1989-1998. �10.1051/jphys:0198600470110198900�.
�jpa-00210394�
1989
On the measurement of the fractal dimension of aggregated particles by
electron microscopy : experimental method, corrections and comparison
with numerical models
M. Tence (*), J. P. Chevalier (+ ) and R. Jullien (*)
(*) Laboratoire de Physique des Solides, Bât. 510, Université Paris-Sud, 91405 Orsay Cedex, France (+) C.E.C.M.-C.N.R.S., 15 rue G. Urbain, 94407 Vitry-sur-Seine Cedex, France
Résumé. - Des images digitalisées en fond noir annulaire d’agrégats polydispersés de billes de fer sont obtenues avec un microscope électronique piloté par ordinateur. On discute des différentes méthodes capables
d’extraire la dimension fractale des agrégats et on analyse les différentes sources d’erreur. Les résultats sont
comparés avec des simulations à l’ordinateur utilisant une version polydisperse du modèle d’ agrégation par
collage d’amas. Les simulations montrent que la polydispersité n’affecte pas la valeur de la dimension fractale.
Le résultat expérimental (D =1,9 ± 0,1 ) est consistant avec un modèle d’agrégation par collage d’amas avec trajectoires linéaires.
Abstract. - Digital annular dark-field images of aggregated iron polydisperse particles are obtained using a computer-controlled STEM. Different methods are discussed on how to extract the fractal dimension of the aggregates, and the sources of error are analysed. The results are compared with computer simulations on a
polydisperse version of the cluster-cluster aggregation model. Simulations show that polydispersity does not
affect the fractal dimension. The experimental result for the fractal dimension ( D =1.9 ± 0.1) is consistent with the cluster-cluster model with linear trajectories.
J. Physique 47 (1986) 1989-1998 NOVEMBRE 1986,
Classification
Physics Abstracts
61.14 - 05.40
1. Introduction.
The work of Forrest and Witten [1] opened a new
area of study in electron microscopy, for small complicated objects, such as aggregated particles.
Through measurements of micrographs of smoke- particle aggregates, they showed that long-range
correlations exist and that these obey a power law.
The exponent in this power law was identified as the fractal (or Hausdorff) dimension [2] of the aggregate
and hence characteristic of a specific aggregation
mechanism. Since this study, considerable interest has focussed on aggregation phenomena, both through computer modelling [e.g. 3-6] and through
further experimental studies using imaging in the
electron microscope or various scattering experi-
ments [e.g. 7-11]. Owing to this combined (theoreti-
cal and experimental) approach it is now possible to
compare measured fractal dimensions in real aggre-
gates with those from computer models involving
different aggregation mechanisms and hence gain insight on the mechanism involved for specific preparation conditions. However for such a compari-
son to be valid, it is essential to examine critically
how measurements are made and what are the
sources of systematic error. The aim of this paper is
to test such a measurement on more general aggrega- tes than considered previously (particles having
different sizes and overlapping frequently) and using
the most suitable electron microscopy techniques (Digital Annular Dark-field imaging in a Scanning
Transmission Electron Microscope). We will then
discuss the possible systematic errors, their effects and how these can be corrected and finally compare the results with those obtained through computer
modelling of a polydispersed system in a cluster- cluster aggregation process [5]. We have published preliminary results of this work [12], emphasizing
the techniques of electron microscopy which have
been used.
2. Specimens and specimen preparation.
The iron powder used in this study was prepared by
a novel cryogenic technique [13]. A solid iron ingot
is levitation melted by radio frequency induction, in
a cryogenic environment (typically liquid argon). At
the interface between the liquid argon and the molten iron a turbulent reaction occurs. Here we
believe that either vapourized iron or molten iron beads are quenched by the cryogenic liquid. The
« boiled off » argon then flows through several
decantation tanks containing hexane. From the mud which settles at the bottom of the tanks, a black powder can be obtained by drying.
Specimens for electron microscopy are prepared
either by mixing the powder with ethanol in a watch
glass, or by ultrasonically dispersing the powder in
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470110198900
ethanol (using a standard ultrasonic cleaner). A
carbon on formvar electron microscope grid is then dipped in the ethanol suspension and allowed to dry.
Although we found no difference in aggregate appearance between these two slightly different
methods of preparation, the aggregates appear to be
more homogeneously distributed on the grid in the
latter case. We infer that this method does not
substantially break up the aggregates or allow the further combination of aggregates or particles. This
fact was confirmed by the observations. The measu- rements presented here were obtained by ultrasoni- cally dispersing the powder.
Bright-Field imaging in a conventional transmis- sion electron microscope (Fig. 1) shows that the iron
powder consists of aggregates of spherical particles.
The particles have sizes ranging from about 100 A to 2 000 A in diameter, with a mean size of about 500 A. We also note that particles overlap frequen- tly. Furthermore, some of the beads are not spheri- cal, but appear to be welded together, suggesting
that aggregation occurred at high temperature when the metal is molten or just solidifying. These points (range of size, overlap and different shapes) show
that methods of particle counting are likely to be
difficult in this case.
Diffraction patterns from such particles show that they have a bcc structure as expected. The patterns also exhibit extra rings corresponding to Fe3o4.
Dark-Field imaging demonstrates that this oxide is in the form of a thin coating ( - 15 A or so ) on the particles. This coating is not expected to affect the
measurements.
3. Experimental procedure and choice of signal for imaging.
Previous work on the measurement of the fractal dimension of aggregates from electron micrographs
first used bright-field imaging techniques and then employed either counting the particles (if these are
Fig. 1 - - Transmission electron bright-field image of a single aggregate. It is important to note that particles of
the same size need not have the same contrast and that extinctions (thickness fringes) occur in the larger particles.
all of the same size) [8-10] or digitizing the image
into a binary distribution [1]. Other authors have also suggested elaborate image treatments to pro- duce binary distributions from bright field images [14]. These treatments are all essential if one uses
bright field imaging, since the bright-field signal is
not simply proportional to the amount of matter present. This effect is well known (see e.g. [15]) and
is due to the strong interaction between the electrons and the sample (in this case the most severe limita-
tion is due to the very strong dependence of the scattering on the Bragg orientation of the crystallyne particles). Such effects can be clearly seen in the bright field image (Fig. 1) and give rise to thickness fringes and to variable intensity for particles of the
same size. The same effects occur in conventional dark-field images, where one tilts the incident beam and forms the image with part of the polycrystalline Debye ring. Here only the particles which are in the
correct Bragg orientation for this particular selected angular range (both radial and azimuthal) will be imaged. These difficulties are further compounded
when the particles overlap.
In our case, the nature of the specimens used (particles of varying size, together with frequent overlap) precluded the use of any such treatment based on conventional bright or dark field images.
Furthermore we wished to try the most appropriate
of techniques recently developed. We therefore chose to use the signal from an Annular Dark-Field detector using a 100 kV scanning transmission elec-
tron microscope equipped with a field emission gun
(FEG-STEM). Furthermore since the images are to
be processed, they were acquired using a digital
scan. The instrument and the Annular Dark-Field
(ADF) detector have been fully described previously [16-18]. The digital acquisition and processing unit,
which is linked to the microscope has also been
described elsewhere [19].
The signal from the ADF detector has considera- ble advantages for this kind of work over both the
bright- and dark-field conventional images, whose
contrast is produced essentially by Bragg scattered (elastic) electrons. The ADF signal is from high angle elastically scattered electrons, inelastically
scattered electrons subsequently elastically scattered
to high angles, and high-angle elastically scattered
electrons which are then inelastically scattered. (The
inelastic scattering is largely small angle). Crystalline
contrast effects are strongly attenuated since we now
average the contrast over a large angular range (0 to
2 1T in azimuth and from one or two to about 10
Bragg angles). This means that the signal can be
considered to vary linearly with the amount of iron present up to thicknesses of about 700 A [20]. In any
case it varies monotonically to much greater thicknesses, although the linear law then becomes a
much poorer approximation.
To record images with a sufficiently large dynamic
range, due to both low intensity from the support film and high values from large particles, the output from the ADF detector is encoded into numerical values using a voltage-frequency convertor. This
1991
averages the signal over the dwell time per pixel and
hence allows high signal intensities (i.e. more than 108 counts s-1) to be recorded, when individual pulse counting is no longer feasible.
Figure 2 is an example of a digitally recorded
ADF image from a typical aggregate. We note that the overlapping particles are clearly taken into account, that particles of the same size have the
same intensity and that large particles show
thickness countours corresponding to spherical shape
with no drastic plateau effect due to failure of the linear law for large thicknesses. Thus this signal is
most appropriate to measure mass scaling in aggrega- tes.
For adequate comparison with the underlying theory and with regards to self-similarity, we have attempted to measure as large a range of aggregate size as possible, and hence we have recorded images
at 4 fixed electron optical magnifications (x 20 000 ;
x 50 000 ; x 100 000 ; x 200 000). These have been calibrated and have been found to be within 3 %,
with distortion (i.e. difference between horizontal and vertical magnification) of less than 2.5 %. To enable good resolution of the smallest particles at
the lowest magnification, 512 x 512 digital images proved necessary and this was maintained for the other magnifications, as well. To compare data from
images taken at different magnifications (i.e. because
an image pixel does not correspond to the same
value of area, and hence the intensity is not equiva-
lent to the same specimen mass) the intensity of each pixel p is normalized by y a factor M1 where M is the
magnification [21].
4. Image treatment and calculation of the fractal dimension.
The 512 x 512 images, once recorded and stored on
tape, are treated in the following manner. After
normalization for different magnifications, the computer program written for this study finds the
Fig. 2. - Scanning transmission annular dark field image.
The contrast is now approximately linearly related to the
mass.
outer limit of the aggregate (imaged one by one, i.e.
one per frame) and draws a close fitting rectangle
around it. If there are several aggregates in the frame (i.e. one large one and a small one close to it)
this routine can distinguish them as long as they do
not interpenetrate. The average intensity of the image outside this rectangle is then measured and this value is then used for the value of the back-
ground subtraction, i.e. to account for the scattering
from the carbon/formvar support film. Since the support film is the same for all aggregates analysed
on the same grid, this value is also used to normalize the intensity for each aggregate for fluctuations in the incident electron beam, and hence enable all the aggregates to be compared. This is, in general,
necessary for all electron micrographs, but especially
so here, since the intensity from the Field Emission Gun can decrease by a factor of 2 over several minutes and the tip requires frequent re-generation.
The « stripped and normalized » images of aggre- gates are analysed to yield fractal dimensions by two
different methods, both based on a scaling relation
between mass and size. In the first method used,
which is similar to the method proposed by Forrest
and Witten [1], the scaling relation is probed inter- nally for each aggregate. That is, for a given origin
chosen to be on a particle in the central portion of
the aggregate, concentric squares are drawn, and
within each square, the intensity (assumed now to be linearly proportional to the mass) is summed. To minimize the effects of the arbitrary choice of origin,
three origins are in fact used and for each size of square the arithmetic mean of the mass, for the three
origins, is taken. This tends to reduce the effects
owing to the breakdown of scaling and to the finite
size of the particles for small squares. From In/In
plots of mass versus size of square, a curve is
obtained, the slop of which gives a value of D, the fractal dimension. We will call this the « nesting
squares » method.
The second method used simply seeks to verify the scaling relation between mass and size for all the aggregates. Thus we simply integrate the intensity
over the whole aggregate and relate this to a measure of the size of the aggregate. We have chosen to use the value of J a2 + b2 for the size,
where a and b are the sides of the close fitting rectangle used previously. From a log-log plot of
mass versus size for all the aggregates measured, we
obtain a set of points, which can be fitted to a
straight line, again with slope D. We have also tried
using other measures of aggregate size (e. g.
but we have found that this makes only a very slight difference on the value of the slope D (about 10 % of the standard deviation on
the value).
5. Results of the measurements.
Altogether 36 aggregates were measured with an approximately normal distribution in size (or rather operational magnification). From this set, 3 aggrega-
tes were excluded. One because they were too many other small aggregates, another because the image
was saturated (streaking in the background) and a
third because the aggregate was very small (about
50 particles) and essentially linear and so judged non-representative. Of the 33 remaining, 3 consisted of a large and a small aggregate on the same frame.
The small aggregates in these 3 frames were also eliminated since we think that the magnification
used did not lead to appropriate sampling. This then
leaves 33 aggregates in the analysis, with two orders
of magnitude in « mass » between the smallest and
the largest.
Concerning what we call the « nesting squares » method, a number of problems were encountered.
The log-log plots produced curves which could be considered to have a linear central point. This was
not always clear, since fluctuations occurred even in the central portion. Some examples are given in our preliminary paper [12]. In that work [12] we had
estimated the slope of these linear portions using a
ruler. Here we have carried out a least-squares fit,
but without any weighting (cf. Forrest and Witten [1]
who weighted according to the intensity in the square). However, if we decide to discard part of the
curve, either the top, bottom or both since it is
expected that these will not be linear and that this
departure from linearity can be accounted for by
discrete finite size effects and by truncation effects,
we find that we obtain a continuous variation in the values of D depending on how much of the curve is discarded. For example if we discard only the top end (eliminating truncation effects) and assuming
that the averaging of different origins has reduced
particle size effects at the lower end, then we obtain
the following values from a least squares fit :
The percentages given in the bracket correspond to
the percentage of the total size of the curve that has
Fig. 3. - Log-log plot of mass/size for 33 aggregates measured. A least-squares fit on the slope yields
D =1.89 ± 0.09.
been fitted. It is thus clear that it is difficult to decide which value can be. considered as « correct » and
why. This will be discussed fully in the next section,
but it should be noted that the values obtained here
by this method are all systematically smaller than that estimated ( D = 1.8 ± 0.2) in our preliminary
report [12]. This is due to two factors, one which was
a systematic overestimate of the slope estimated
with a ruler (!) and the other is that we now use a correct background subtraction technique. We now
subtract a measured average background, which is necessarily larger than the minimum pixel value used
before. This will necessarily lead to a smaller value
of the fractal dimension.
The result for the total mass/size scaling method
for all the aggregates is more satisfactory. Figure 3 is
a log-log plot for the 33 aggregates studied. The scatter of point is quite considerable, but a least squares fit is physically reasonable and this gives :
The physical meaning of this value will be discussed later in terms of the various models for aggregation.
However several remarks about the measures can now be made. Firstly the last value is significantly
different from the value obtained by the « nesting
squares » method, and secondly it is slightly less than
the value we had previously reported [12] of
D = 2.0 ± 0.1. This decrease can be explained by
the improved method of background subtraction discussed above. The problem of background sub-
traction has not been of relevance to other authors up to now since they have either counted particles or
binarised their images.
6. Projection and finite size effects in the K nesting
squares w method.
Many authors [e.g. 2] state that the projection of a 3-
dimensional aggregate to 2-dimensions will not affect the value of the fractal dimension D calculated by
the « nesting squares » method for values of D less than 2. For D > 2, this method cannot give more
than an effective fractal dimension equal to 2, whilst the mass/size method would yield the correct value
in principle. In fact this reasoning is correct only for
infinite aggregates, and for D less than but close to 2, finite size effects will affect the projection, and
hence the results obtained using the « nesting squa-
res » method.
This idea can be demonstrated by considering
some kind of « ideal » continuous representation of
a finite-sized 3-d fractal of maximum radius R and with spherical symmetry, defined by the following
mass density :
The coefficient in front of the term rD -3 is calculated
so as to normalize the density to the total mass M of
the aggregate. The consideration of an abrupt cut off
1993
at r = R is not necessarily very physical, but we
believe that this will only be a secondary effect. The
essential point is that this cut-off exists, thus taking
the finite size into account explicitly.
This can be easily projected by calculating the
mass m contained in a cylinder of radius ), whose
axis goes through the origin. Integrating over the variable r, one has:
with
After some straightfordward changes of variables,
one finds
This expression can be used to calculate m ( c ) numerically. To see how this quantity behave, when
C -+ 0, one can rewrite it as :
and then, when ( --+ 0
When ( --+ 0 one can expand the diverging integral :
Thus when § - 0, m ( ( ) contains two competing
terms in §2 and ( D. One can write in general:
and
This can be seen on the log-log curves shown in figure 4, when, for D > 2, the slope of the curves, as
In i tends to - oo, becomes independent of D. A
R
more quantitative picture can be obtained by defi- ning a size dependent dimension Deff ( f ) ; this is given by the slope of the log-log curve m ( § ) for a
given f. R Thus:
Fig. 4. - Results for the mass m ( § ) contained in a
cylinder of radius in the case of a fractal « sphere » with finite cut-off R. The figure shows a plot of In m ( § ) (where M is the total mass of the sphere), against In
’ for various values of D.
R
Figure 5 is a plot of D eff ( § ) versus D for different
This curves gives an idea of the error made in the
« nesting square » method, since from the curves obtained by this method, values of De ff can be obtained ; i.e. when the effective fractal dimension is evaluated at a point corresponding to a square
going to half the edge distance, a quarter the edge distance, etc. For example one sees that for a
reasonable value of eIR, of order 1/8 or 1/16, the
difference between the true fractal dimension and the effective (or apparent) one is of the order of 0.2
