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Applied Surface Science 353 (2015) 1143–1149

Contents lists available at ScienceDirect

Applied Surface Science

jou rn al h om ep age: www.elsev ier .com/ locate /apsusc

bservation of a crossover in kinetic aggregation of Palladium colloids

. Ghafari a, M. Ranjbarb, S. Rouhania,∗

Department of Physics, Sharif University of Technology, Tehran PO Box 11135-9161, IranDepartment of Physics, Isfahan University of Technology, Isfahan PO Box 84156-83111, Iran

r t i c l e i n f o

rticle history:eceived 7 February 2015eceived in revised form 19 May 2015ccepted 2 July 2015vailable online 14 July 2015

eywords:alladium colloidsydrophobicityractals

a b s t r a c t

We use field emission scanning electron microscope (FE-SEM) to investigate the growth of palladiumcolloids over the surface of thin films of WO3/glass. The film is prepared by Pulsed Laser Deposition(PLD) at different temperatures. A PdCl2 (aq) droplet is injected on the surface and in the presence ofsteam hydrogen the droplet is dried through a reduction reaction process. Two distinct aggregationregimes of palladium colloids are observed over the substrates. We argue that the change in aggregationdynamics emerges when the measured water drop Contact Angel (CA) for the WO3/glass thin films passesa certain threshold value, namely CA ≈ 46◦, where a crossover in kinetic aggregation of palladium colloidsoccurs. Our results suggest that the mass fractal dimension of palladium aggregates follows a power-lawbehavior. The fractal dimension (Df) in the fast aggregation regime, where the measured CA values vary

◦ ◦

from 27 up to 46 according to different substrate deposition temperatures, is Df = 1.75(± 0.02) – thevalue of Df is in excellent agreement with kinetic aggregation of other colloidal systems in fast aggregationregime. Whereas for the slow aggregation regime, with CA = 58◦, the fractal dimension changes abruptlyto Df = 1.92(± 0.03). We have also used a modified Box-Counting method to calculate fractal dimensionof gray-level images and observe that the crossover at around CA ≈ 46◦ remains unchanged.

. Introduction

In the 1970s Benoit Mandelbrot (1) introduced the idea of fractaleometry to bring a number of earlier studies of irregular shapesnd natural processes together. Mandelbrot called attention tohe particular geometrical properties of objects such as the sur-ace of clouds, the shore of continents, and the branches of treeshich possess a rather special kind of geometrical complexity.e coined the name fractal for these complex shapes to express

he fact that they are characterized by a non-integer (fractal)imensionality. Fractal dimensions are established using geometricower-law scaling relationships between each dimensional geom-try and characteristic length scales of the object. The descriptiveoncepts provided by fractals have proven to have a deep connec-ion with many aspects of the natural world including structuresrom microscopic aggregates to the cluster of galaxies [8].

The aggregation or flocculation of small particles to form largertructures and clusters have attracted serious attention since

broad spectrum of aggregation processes including gelationnd polymerization processes in polymer science [1], coagula-ion processes in aerosol and colloidal physics, percolation and

∗ Corresponding author.E-mail addresses: rouhani@ipm.ir, srouhani@sharif.ir (S. Rouhani).

ttp://dx.doi.org/10.1016/j.apsusc.2015.07.010169-4332/© 2015 Published by Elsevier B.V.

© 2015 Published by Elsevier B.V.

nucleation in phase transitions and critical phenomena, rouleauxformation by red blood cell adhesion in hematology and crystal-lization, and dendritic growth processes in material science canbe well-described by scaling concepts [2,3]. In particular, sev-eral experimental investigations have been carried out for theirreversible kinetic aggregations in aqueous colloidal systems [4–7]suggesting that the resulting clusters are fractals. These investi-gations were followed by intense theoretical and computationalactivities to elucidate their statistical properties [8–10].

In a recent study [11], WO3/glass thin films were prepared bypulsed laser deposition (PLD) at 100 mTorr oxygen pressure. Inthe presence of H2, a wet-gasochromic switching with an edge-to-center coloring nature was observed when aqueous PdCl2 wasused as hydrogen catalyst. Hydrogen gas acts as a common reduc-ing agent with no residual chemical impact on the system; hence,the resulting catalyst layer has high purity. Palladium nanoparti-cles were formed over the substrate through a reduction reactioninside the droplet and a peculiar aggregation of Pd colloids on acontinuous gray layer of palladium was reported.

Colloidal nanoparticles mostly enter structured materials asone-, two-, or three-dimensional networks, grown at a solid–solid,

fluid–fluid, or solid–fluid interface. Colloidal particles are solid orliquid matters with various sizes ranging from a few nanome-ters up to many micrometers which are suspended inside a gas,liquid, or solid. These particles demonstrate essential properties

dx.doi.org/10.1016/j.apsusc.2015.07.010http://www.sciencedirect.com/science/journal/01694332http://www.elsevier.com/locate/apsuschttp://crossmark.crossref.org/dialog/?doi=10.1016/j.apsusc.2015.07.010&domain=pdfmailto:rouhani@ipm.irmailto:srouhani@sharif.irdx.doi.org/10.1016/j.apsusc.2015.07.010

1 face Science 353 (2015) 1143–1149

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144 M. Ghafari et al. / Applied Sur

ompared to small molecules, because of their mesoscopic sizeith the resulting large surface-to-volume ratio and intermedi-

te dynamics [12]. Unlike highly concentrated suspensions thatre close to their gelation point at drop deposition, the drying ofow to semi-concentrated nanocolloids form open aggregates ofower surface coverage � s and fractal dimension Df ≤ dspace. Suchrowth processes are critically determined by interpaticle inter-ctions, particle–substrate interactions, and drying kinetics [13].ne of the forms of these colloids are solid particles suspended in

liquid medium, such as noble metal catalysts (Pd, Pt) inside anqueous, which is what we are referring to here.

We have found that the resulting clusters possess different frac-al dimensions in accordance with different contact angle values inwo regimes, namely the fast aggregation regime with CA � 46◦,nd the slow aggregation regime with CA � 46◦, which governs theinetics of aggregation in each regime. This work is the first directttempt to relate the hydrophobicity of the substrate to the fractalroperties of the colloidal clusters formed over it.

This paper is organized as follows. In the first section the detailsf the experimental setup is provided. In the second section, aethod to interpret the fractal properties of aggregates using their

wo-dimensional projected image is introduced, the validity of theeasurements is argued, and the possibility of having three dimen-

ional aggregates is offered. Section 3 mainly elaborates this issuey implementing a modified Box-Counting method to calculate theractal dimension of gray-level images and indicates that the resultschieved in section two remains unchanged. Finally, we discuss themplications of our work.

. Experimental setup

Thin WO3 films were fabricated by pulsed laser deposition (PLD)ethod from tungsten oxide pressed powder onto circular glass

ubstrates of 11 mm diameter, which prior to deposition, wereleaned ultrasonically by methanol and DI water. The employedeposition system was a stainless steel chamber with a base pres-ure of 1 × 10−5 Torr. In order to deposit stoichiometric tungstenxide films, the PLD process was carried out in 100 mTorr oxy-en (purity 99.9%) pressure. To obtain films with different surfaceydrophilicities, the substrates were held at different temperaturesuring deposition process by means of an electric heater directlylaced on the back of substrates. The processes of deposition wereone under different substrate temperatures (STs) including roomemperature, 100, 200, 300 and 400 ◦C. To ablate the tungsten oxideargets, 5000 pulses of a KrF laser (� = 248 nm, � = 10 ns, laser energyf 200 mJ and RR = 10 Hz) was delivered to the surface of the rotat-ng target at a 45◦ angle. The substrate to target distance was keptonstant at 7 cm. In order to make the palladium precursor, a PdCl2olution were prepared by solving 0.02 g PdCl2 powder (5 N) into9.9 cm3 DI water and 0.1 cm3 HCl. Then drops with constant vol-mes of 0.07 cm3 of the 0.2 g/l PdCl2 solution were put on theurface of WO3 layers. Finally, the palladium nano-particles wereynthesized inside the drops by hydrogen-reduction method. To dohis, a constant flow (2 l/min) of 10% H2/Ar mixture gas was deliv-red into a sealed chamber contacting PdCl2/WO3/glass samples.

The surface morphology of obtained PdCl2/WO3/glass samplesas observed on a FE-SEM (Hitachi model S4460) instrument.ydrophilicity of WO3/glass samples were investigated by the con-

act angle measurements which were performed in atmospheric airt room temperature using a commercial contact angle meter (Datahysics OCA 15plus) with ±1◦ accuracy (see Fig. 1). A droplet was

njected on the surface using a 2 �l micro-injector. Fig. 2 shows set of FE-SEM images (scale bars = 5 �m) from aggregation ofd nanoparticles on WO3/glass substrates with different surfaceydrophobicity in the drop-drying process. Because of the metallic

Fig. 1. Contact angle (CA) of water for the WO3/glass thin films as a function ofsubstrate temperature.

nature of palladium nano-particles, their brightness is much higherthan the non-metallic WO3 substrates in electron microscopy andcould be easily recognized in our FE-SEM images from the flatsubstrates. The aggregates resemble to fractal likes structures ofdifferent shapes and branching depending on the hydrophobicityof WO3 substrate.

3. Interpretation of the fractal properties fromtwo-dimensional projected image: results and discussion

There are a number of techniques that have been appearedin the literature to determine the mass fractal dimension of anaggregate including the light scattering, settling, and image analy-sis method. Each method has its own limitations and benefits (2).One of the major advantages of image analysis is that it providesinvaluable information about particle morphology compared to theother two methods. On the other hand, the disadvantage lies onthe measurement statistics for an ensemble of particles. Here, weuse field emission electron scanning microscope (FE-SEM) which ispreferred over other optical and electron microscopes since it pre-vents the images from the danger of modifying the floc structureor biasing the floc orientation in sampling (2).

Although the aggregates are three-dimensional inside the solu-tion, we assume that, upon drying through reduction reaction,they collapse to form nearly flat two-dimensional structures andwe essentially tried to extract their 2-D mass fractal dimensionto see whether it shows scale invariance properties or not. Thereare several methods to find the mass fractal dimension of a struc-ture from its two dimensional projected image, namely the NestedSquare Method (NSM), Perimeter Grid Method (PGM), and Ensem-ble Method (EM) [14]. Here, we apply the Nester Square Methodmainly because we have a few ensemble of clusters–five clustersat each substrate deposition temperature. For calculating the frac-tal dimension of clusters using NSM, we first turn the images intobinary sequences, then partition them into squares of increasingsize from 1 × 1 up to 40 × 40 square pixels – note that the size ofeach cluster is roughly 160 × 160 square pixels – and finally countthe number (N) of squares needed to cover the entire cluster. Thismethod is basically similar to the ordinary Minkowski–Bouliganddimension in which we cover the object with tiles having the samedimensionality as the containing space. Since our binary images aretwo dimensional, the image is covered with squares of a certain sizeusing the minimum number of squares to fully cover the image ofthe object:

dimbox(S) = lim∈→0ln(N)

ln(1/ ∈) (1)

M. Ghafari et al. / Applied Surface Science 353 (2015) 1143–1149 1145

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ig. 2. Shows a set of FE-SEM images of Pd/WO3 (scale bars = 5 �m). The contact eposition temperature from (a) to (e).

The 2-D mass fractal dimension (Df) is estimated by the linearegression slope of the ln–ln plot of the 1/boundary against pixelumbers (N) (Fig. 3).

The plots indicate that the fractal dimension follows a power-aw behavior for each of the clusters and they can be indeed treateds two-dimensional fractals. It can also be inferred that there is arossover in the fractal dimension of aggregates. For aggregatesith CA � 46◦ the average fractal dimension is Df = 1.75(± 0.02),

hereas for the dense aggregates with CA = 58◦ the fractal dimen-

ion jumps to Df = 1.92(± 0.03). We calculated the average fractalimension for the fast aggregation regime, with Df = 1.75(± 0.02),sing the weighted average method (see Equation (2)). Dfi = xi ±

of water (CA measurements) as hydrophobicity factor is shown at each substrate

�i (for i = 1, 2, 3, ∧4) represents the estimate fractal dimension ofan ensemble of aggregates for CA values 46◦, 36◦, 33◦, and 27◦,respectively. To calculate the weighted average we have:

Dfaverage =∑i=1

4 xi/�i2∑i=1

4 1/�i2(

4∑ 1 )−1/2 (2)

�average =

i=1�i2

According to the experiment in ref [11], the rate of gasochromicswitching and consequently the formation palladium aggregates on

1146 M. Ghafari et al. / Applied Surface Science 353 (2015) 1143–1149

Fig. 3. Plot of ln(N) vs. ln(1/∈), where the dashed lines are the least-square fit to the data of each cluster. Approximate fractal dimension is the average over an ensemble offi n be s1 interceo

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ve clusters with CAs equal to (a) 58◦ , and (b) 46◦ . Similar scale-invariance trend ca.74 (±0.03), 1.71 (±0.05), and 1.77 (±0.03), respectively. A slight variation in the y-f the images and therefore caused by the resolution of each image.

he surface decreases considerably when CA = 58◦–coloration pro-ess gradually happens in about 17–30 min for the fast aggregationegime (at 300 ◦C, 200 ◦C, 100 ◦C, and 25 ◦C substrate depositionemperatures) and in 51 min for the slow aggregation regimeat 400 ◦C). This, in fact, suggests that the hydrophobicity of theubstrate is a major factor responsible for controlling the ratef deposition as well as the aggregation of the particles. Thesearticles originate from hydrogen-induced nucleation inside theolution and are inherently under random Brownian motion. Weelieve that as the hydrophilicity decreases, the droplet has lessendency to wet the surface and as a results the concentration ofhe Pd colloids inside the aqueous will increase. Since it takes muchonger for the nanoparticles to aggregate at lower hydrophilicity,he particles have more time to diffuse inside the solution andttract each other to form a more dense branching fractal-like shapeith a fractal dimension value closer to d = 2.

We then attempted to see if the internal structure of clustersxhibits scale invariance behavior and to further investigate if anyhase transition occurs near CA ≈ 46◦ by finding the two-point cor-elation density function c(r) using the equation below:

(r) = (d/2)2�d/2�r

〈M(r + �r) − M(r)

rd−1

〉(3)

where d is the dimension of space, r is the location of a par-icle and �r is the distance between two particles in the pixelo-ordinate system, 〈. . .〉 represents the averaging over the posi-ion of particles, and M(r) is equal to 1 at the location of a particle

nd 0 elsewhere. For a two-dimensional projection of these fractals,f we take l as the monomer diameter in the pixel co-ordinates, wexpect c(r) ∝ r−˛ for l ≤ r � Rg where ̨ = 2 − Df and Rg is the radiusf gyration (Rg varies from 40 to 60 pixels for relatively different

ig. 4. Plot of ln(c(r)) vs. ln(r), where the dashed lines are the expected slope of the regress the average two-point density correlation function of five clusters. (a) Approximate cos 40 pixels, so we expect to see the power-law behavior in the range 1.61 ≤ ln(r) < 3.69. (adius of gyration is 53 pixels, so we expect to see the power-law behavior in the range 1.

een for clusters with CA values 36◦ , 33◦ , and 27◦ with measured fractal dimensionpt of each regression line is believed to happen due to different size magnifications

cluster sizes) and can be estimated in the pixel co-ordinates usingthe following equation [15]:

Rg2 = 1

Npixels

Npixels∑i=1

(ri − rmean)2 (4)

where rmean is the location of centre of mass and Npixels is thenumber of counted pixels of the projected image of each cluster.

However, due to the finite size effects of the clusters we are notable to precisely show that c(r) exhibits a long range power-lawbehavior. It can be clearly seen from Fig. 2 that the clusters havemany regions of overlapping particles – especially at 400 ◦C depo-sition temperature where clusters with giant two-dimensionalspherical cores in the middle of the aggregates exist – and thereforethe curves are noisy (see Fig. 4). Yet, an extended linear regime ofcorrelation function can still be recognized in the range of r ≈ 5 tor ≈ 40 in the pixel coordinate system.

The limiting slopes, shown by dashed lines in Fig. 4, suggestthat the set of data points for two-point correlation function areconsistent with the measured Hausdorff dimension Df = 2 − ̨ fromthe NSM method (Fig. 5).

Similar measurements on the two-point correlation functionsand fractal dimension of gold and silica colloidal aggregates withDf ≈ 1.75 for fast aggregation and Df ≈ 2.05 for the slow aggre-gation regime were also reported elsewhere (see ref [4,5,16,17])which reasonably coincide with the results of our experimentaldata. Therefore, we tried to investigate the possibility of having a

phase transition by plotting the scaling exponent ̨ as a function ofcontact angle (CA) and observe a drop in the value of ̨ for CA = 58◦:

Universality in colloid aggregation has been checked for threechemically different colloidal systems under diffusion-limited and

sion line according to the measured mass fractal dimension ( ̨ = 2 − Df). Each graphrrelation density function for clusters with CA = 58◦; the average radius of gyrationb) Approximate correlation density function for clusters with CA = 46◦; the average61 ≤ ln(r) < 3.97; same trends are seen for clusters with CA values 36◦ , 33◦ , and 27◦ .

M. Ghafari et al. / Applied Surface Science 353 (2015) 1143–1149 1147

Fig. 5. Plot of ̨ vs. water contact angle (CA). The dashed line is the predicted valueoac

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Fig. 6. Determination of Ns using DBC method, where the dashed cubes are thenumber of boxes needed to cover the grid (1, 1) and the gray bars represent the

f ̨ in the fast aggregation regime. Each point represents the value of ̨ exponentveraged over an ensemble of five clusters which changes with respect to differentontact angle measures (CAs vary from 27◦ to 58◦).

eaction-limited aggregation conditions [18]. Here, we offer whatight be another candidate in colloidal systems which exhibits uni-

ersal behavior in the limiting cases of fast and slow aggregation.owever, more precise measurements on the aggregation process

s needed to reach this conclusion – especially near CA ≈ 46◦.Strictly speaking, we do not offer any proof of assuming that

he collapse of the aggregates is geometrical under projection fromhree to two dimensions. But we know that if the object is mass frac-al with fractal dimension less than two (geometric transparencyondition), then the fractal dimension is preserved upon projec-ion. In addition, if the fractal dimension approaches to Df ≈ d, aituation known as the geometric opacity appears, where the frac-al dimension may not be preserved upon projection (2)–this mighte the case for the aggregates with CA = 56◦. We also believe thathe relative offset in the magnitude of c(r) for aggregates also hap-ens because of the projection of the aggregates. Thus, we seek fornother method which enables a better estimate of the number ofarticles in a cluster when the image might suffers from geometricpacity conditions.

In the next section, an efficient method in image processing isntroduced to show that although the 3-D fractal dimension is dif-erent from the projected two-dimensional images, the crossoveroint can still be recognized.

. A modified box-counting method to calculate fractalimension of gray-level images

As we discussed in the previous section, the image analysisethod is mostly based on the conversion of the projected aggre-

ates into binary images. A software then analyzes these processedmages to measure the two-dimensional fractal dimension of theggregate by determining the power-law relation between the areaf the aggregate and the characteristic length scale. Therefore, it isifficult to extract their three-dimensional properties such as theolume or exact number of particles using binary images. Thereave been some experimental and computational efforts to calcu-

ate the three-dimensional fractal dimension of aggregates basedn their projected images (3) (4) (5) (6). Each of these works offers

solution to a limited scope of experimental conditions. Here,e introduce an image processing technique that can reduce the

dverse effects of projecting a structure in three-dimensional spacento a plane. By looking at the FE-SEM images, one can spot the vari-tions in height of each cluster based on the changes in the incidentntensity of light – The closer a section of a cluster gets to the scan-

ing microscope, the brighter it appears in the image. As a result,e can imply that by binarizing the images, as we did in Section 2,e are actually throwing away the information about the height of

ach cluster. Consequently, we cannot have a correct estimate of

image intensity surface–each bar represents the gray level value of one pixel in theimage plane.

the volume and number of particles in each cluster – especially forclusters with regions of many overlapping particles. However, FE-SEM provides us with a gray-level image, enabling us to divide theheight of each cluster into several unit sections where each sectiondenotes a set of particles. Following this approach, we can makea better approximation for the total number of particles inside acluster and negate the problem of projecting the image.

Fractal geometry has gradually established its significance inthe study of image characteristics when Pentland [19,20] providedthe first theory about the human perception of smoothness androughness of surfaces, with fractal dimension of 2 corresponding tosmooth surfaces and fractal dimension of 3 corresponding to a max-imum rough surface and considered the image intensity surface asfractal Brownian function (fBf) and measured the fractal dimen-sion from Fourier power spectrum of fBf. Since then, many othertheories such as reticular cell counting [21] and variations of box-counting method [22–24], which were applicable to a wider classof fractals, were introduced. Amongst these different approacheswas a successful method introduced by Sarkar and Chaudhuri,also known as the differential box-counting (DBC) method, whichproves to possess a relatively fast algorithm and yield more accurateresults compared to the previous methods [25]. The DBC methodis introduced as follows: First, consider an image of size M × M asa three-dimensional spatial surface with (x,y) denoting 2-D posi-tion and z denoting the gray level of the projected image – the graylevel increases in value as the image surface intensity increases.Then, the (x,y) space is partitioned into non-overlapping grids ofsize s × s, such that M/2 ≥ s > 1 where s is an integer. Similar to theordinary box-counting method (see Equation (1)), in order to countthe total number of pixels (Ns) in the scale s using differential box-

counting method, we use the following procedure: On each gridthere is a column of boxes of size s × s × s′, where s′ indicates theheight of each box and is defined such that

⌊G/s′⌋

=⌊

M/s⌋

, where

1148 M. Ghafari et al. / Applied Surface Science 353 (2015) 1143–1149

F the nu

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ig. 7. If the boxes are shifted in the z direction (see the arrow in the right image),

. . .� denotes the floor function and G is the total number of grayevels. For instance, you can see the measured number of boxes forn arbitrary image intensity in Fig. 6 (where s = 5, and s′ = 4). Let theinimum and maximum gray level in the (i, j) grid fall into kth and

th boxes, respectively. The boxes covering this grid are counted inumber as:

s (i, j) = l − k + 1 (5)

where ns(i, j) is the contribution of Ns at (i, j) point in the gridnd the subscript ‘s’ denotes the scale factor. In Fig. 6, for example,e have ns (i, j) = 4 − 1 + 1 = 4.

Considering contributions from all grids, then Ns can be mea-ured:

s =∑

i,j

ns(i, j) (6)

Finally, the fractal dimension can be estimated by measuring thelope of the straight line best fitting the points in ln(Ns) vs. ln(1/s)

lot [25]. However, this classical DBC method has some drawbacks

extensively discussed in [26,27] – such as over-counting or under-ounting the number of boxes. Here, we briefly address one ofhe major drawbacks and then take a modified DBC method [26],

ig. 8. The plot of ln(Ns) vs. ln(1/∈) using the SDBC method, where ∈ = s/M and the dasimension is the average over ensemble of five clusters with CAs equal to (a) 58◦ , and (b) nd 27◦ with measured fractal dimension 2.65(±0.03), 2.54(±0.01), and 2.64(±0.03), respappen due to different size magnifications of the images and therefore caused by the res

mber of boxes needed to count the intensity surface will be reduced from 4 to 3.

known as the Shifting Differential Box-counting (SDBC) algorithm,so as to improve the accuracy of estimating the fractal dimension.

The DBC method can potentially over-count the number ofboxes covering the image intensity surface. For an instance, as itcan be seen from Fig. 7, if the boxes are appropriately shifted alongthe z direction, no more than three boxes are needed to cover thegray level variation of the intensity surface on a specific block ofthe image plane.

As mentioned earlier in this section, the Differential Box-Counting method subdivides the gray level variation into someintervals of size s and measures the number of boxes requiredto cover the image intensity in the z direction by computing⌈

zmax/s⌉

−⌊

zmin/s⌋

, where �. . .� denotes the ceiling function. Inthe Shifting Differential Box-counting algorithm, we measure thenumber of boxes by computing

⌈(zmax − zmin + 1)/s

⌉. In this way,

we resolve the problem of quantization effect, discussed above, andcount the least number of boxes required for covering the imageintensity surface more effectively. It has been proven that the SDBCalgorithm obtains the estimate fractal dimension closer to the exact

value compared to the DBC method [26]. Hence, by applying theSDBC algorithm to our FE-SEM images, we have found that a suddenchange in fractal dimension from D3 = 2.56(± 0.01) in the fast aggre-gation regime to D3 = 2.47(± 0.06) in the slow aggregation regime

hed lines are the least-square fit to the data of each cluster. Approximate fractal46◦ . Similar scale-invariance trend can be seen for clusters with CA values 36◦ , 33◦ ,ectively. A slight variation in the y-intercept of each regression line is believed toolution of each image.

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ccurs. The reason for emphasizing that there is a sudden change inractal dimension is mainly because the fractal dimension does notary noticeably in the fast aggregation regime; the fractal dimen-ion is D3 = 2.64(± 0.03) for CA = 27◦, D3 = 2.65(± 0.03) for CA = 36◦,nd D3 = 2.60(± 0.05) for CA = 46◦ where it drops to D3 = 2.47(± 0.06)or CA = 58◦ (see Fig. 8).

It is also worth pointing out that if we take a closer look at thelusters at different substrate temperatures (see Fig. 2), it becomesvident that the clusters at 400 ◦C are fairly smoother (lower vari-tion in gray intensity surface) than any other cluster at differentubstrate temperatures.

This result indicates that even by calculating the three-imensional fractal dimension of the aggregates and making aetter estimate for the total number of particles inside a cluster, therossover can still be observed. This is a clear signal that geometricature of the objects is changing at around CA ≈ 46◦.

. Summary and discussion

In this paper we have discussed the observation of an abrupthange in fractal dimension of DLA-like clusters formed whenepositing Palladium on WO3/glass substrate. We argue that therossover happens when the CA value passes a certain thresh-ld, namely CA ≈ 46◦. The two-dimensional fractal dimension ofhe aggregates changes from Df = 1.75(± 0.03) to Df = 1.92(± 0.03).ince the measured fractal dimension of palladium colloid aggre-ates happens to be similar to earlier studies on gold, silica,nd polystyrene colloid aggregates, the possibility of having twoimiting regimes of colloidal aggregation is also discussed. Inddition, the probability of having geometric opacity has beenddressed and a modified box-counting method has been sug-ested so as to obtain a better estimate for the total number ofarticles inside each cluster and to minimize the unfavorable effectsf 2-D image projection. Abrupt changes in the geometry of aggre-ates have been observed under various conditions (for exampleee [28–32]). Most of these changes happen when temperature ishe agent. Here we believe that the direct agent is hydrophobic-ty. To our knowledge this is the first time that hydrophobicity haseen found to be the agent of crossover. Also in most of the reportedeometrical changes, in fact self-similarity disappears altogether,hereas here we observe that the scaling behavior remains intact.

eferences

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[

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Observation of a crossover in kinetic aggregation of Palladium colloids1 Introduction2 Experimental setup3 Interpretation of the fractal properties from two-dimensional projected image: results and discussion4 A modified box-counting method to calculate fractal dimension of gray-level images5 Summary and discussionReferences