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Perikinetic Aggregation of Alkoxylated Silica Particles in Two Dimensions
Journal of Colloid and Interface Science 218, 77– 87 (1999) Article ID jcis.1999.6369, available online at http://www.idealibrary.com on
Perikinetic Aggregation of Alkoxylated Silica Particles in Two Dimensions Peter H. F. Hansen and Lennart Bergstro¨m 1 Institute for Surface Chemistry, P.O. Box 5607, SE-114 86 Stockholm, Sweden Received June 17, 1998; accepted June 22, 1999
effect of possible cluster rearrangement due to thermal fluctuations, shear, convection, and settling has been analyzed (10 – 13). Studies with such more complex models have shown that the fractal dimension is not as unique and universal a structure parameter as was originally envisioned. Complemented with the power law of prefactors, fractal scaling has been limited to a certain region in the aggregation process (8, 14). However, the analysis of the fractal dimension is still a good discriminator to distinguish between different aggregation processes. It is a measure of scaling in a self-organizing system and is sensitive to the dependence of driving force, particle bond strength, and time scales of the aggregation processes. Light scattering in three dimensions is commonly used for studying aggregation; however, it has a major complication. The scattered light is a convolution between the structural parameter of the individual clusters and the spatial configuration of clusters. The latter is a combination of the cluster size distribution and the spatial distribution of clusters (15, 16). For DLCA, deconvolution is possible using simulation of the aggregation to obtain a cluster size distribution as input data (15), a procedure that results in a dependence of the accuracy of the extracted kinetic information from the simulation model applied. For RLCA, however, the polydispersity of the cluster size distribution inhibits the separation of the structure parameter and the spatial configuration of clusters; this is a consequence of a correlation with the spatial distribution of aggregates (16). In contrast, using optical inspection of a twodimensional system, the presence of the configuration of aggregates becomes an asset (17, 18), providing that the statistical problem of direct imaging can be solved by automated image sampling and processing of a large number of images. The experimental studies applying imaging techniques in two dimensions started to emerge in the mid-1980s (19 –21). Those observations inspired more thorough investigations of colloidal aggregation at an air–liquid interface (22–26). The majority of this work has involved electrostatic-stabilized particles spread at the air–water interface where aggregation was induced by addition of electrolyte. Although these studies have provided important insight regarding the aggregation process, this system suffers from an ill-defined interaction potential (27), which might explain the large span of the measured
Aggregation and cluster formation of a two-dimensional colloidal system consisting of 2-mm alkoxylated silica particles trapped at the air–toluene interface have been studied. This novel system allows particle interactions to be controlled by varying the length of the grafted alkyl chains; simple estimates suggest particle bond strengths of 15 and 30 kT for the octadecyl- and octyl-coated system, respectively. Video-enhanced microscopy and image analysis enabled a simultaneously study of kinetics and structure of the ensemble of clusters in situ. The octyl system displayed a DLCAlike structure, D f ' 1.45, whereas the octadecyl system resulted in a more dense structure, D f ' 1.55. The temporal evolution of the cluster-mass distribution displayed a transition point between regimes of slower and faster aggregation for both systems, which was interpreted as a transition from DLCA to convection-limited cluster aggregation (CLCA). © 1999 Academic Press Key Words: perikinetic aggregation; particle interactions; fractal dimension; dynamic scaling; image analysis.
1. INTRODUCTION
Aggregation is a fundamental process of substantial academic interest and technological relevance (1–5). Early studies identified two limiting types of cluster– cluster aggregation behavior: diffusion-limited cluster aggregation (DLCA), characterized by a sticking probability of one, which makes aggregation limited by particle diffusion only; and reaction-limited cluster aggregation (RLCA), characterized by a sticking probability significantly lower than one, which makes the initial aggregation rate very slow. The two regimes not only display different fractal dimensions [D f is 1.44 6 0.04 for DLCA and 1.55 6 0.03 for RLCA (6)] but show differences in the aggregation kinetics as well, e.g., expressed in the decay and broadness of the cluster size distribution (5). Computer simulations have provided much of the fundamental knowledge in this field. During the past decade, computer simulations have evolved from simple models using random trajectories and fixed sticking probabilities toward more complex simulations using Brownian dynamics algorithms and pair-interaction potentials between particles (7–9). Also, the 1
To whom correspondence should be addressed. E-mail: lennart. [email protected]. 77
0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
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fractal dimensions, varying from 1.2 to 1.6 (19 –23, 26). These problems have hampered analysis of the results and limited the comparison between experiments and computer simulations to mainly qualitative features. Hence, there is a need for twodimensional systems where both the strength of interparticle attraction and the external forces, e.g., convection or shear, are well controlled. In this paper a novel system for investigating two-dimensional aggregation will be presented. It is based on the use of alkoxylated silica particles trapped at the air–liquid interface of nonaqueous liquids, thus minimizing the influence of electrostatic phenomena. This system permits the short-range particle interactions, the interaction strength, to be controlled in two ways: first, by varying the range of the steric repulsion by means of the thickness of the coating, and second, by controlling the magnitude of the van der Waals attraction between the particles through the contact angle between alkoxylated particle and liquid (28, 29). We will describe the properties of this system and present results on cluster structure and the kinetics in relation to the mechanisms of cluster transport for two systems: octyl- and octadecyl-coated silica particles trapped at the air–toluene interface. 2. MATERIALS AND METHODS
2.1. Preparation and Spreading of Alkoxylated Silica Particles Silica particles were alkoxylated by grafting of two different alcohols, octanol and octadecanol, to the particle surface. The silica (Tokuyama Soda Co., Japan) particles were spherical and highly monodisperse with a mean particle diameter of 1.97 mm and a standard deviation of 0.07 mm, as determined by scanning electron microscopy (SEM). Grafting of alcohols on the silica particles followed the work by Van Helden et al. (30) and Ballard et al. (31). The grafting process simply involves a condensation reaction between the silanol and the alcohol group. The grafting reaction is degenerated by the presence of water; hence, the first step was to dry the silica powder under vacuum at 60°C for 24 h. Afterward, the dried silica powder was suspended in an alcohol with a selected chain length, previously purified by distillation. The weight of added alcohol was three times the weight of silica powder to obtain a suitable concentration of the resulting dispersion. The homogeneity was ensured by sonication of the suspension for several hours. The grafting proceeded by an esterification (condensation) reaction (31) by simply heating the dispersion, while stirring, to a temperature of 195°C for 4 h. When esterification was concluded, the grafted particles were rinsed from the remaining alcohol by centrifugation of the dispersion, followed by redispersing of the sediment in fresh liquid. This rinsing procedure was repeated three times, with a mixture of cyclohexane and chloroform (40%, 60%) the first time and pure cyclohexane the
second and third times. The particles were found to be hydrophobic. They were not dispersible in water but were easy to disperse in cyclohexane. Finally, the particles were stored in cyclohexane, in which they can remain for several years without degrading (30). Before use, a small quantity of coated particles was simply dried by evaporation of cyclohexane and redispersed in ethanol to a weight fraction of 0.2%. Spreading of the alkoxylated silica particles at the air– toluene interface proved to be a crucial process, which demanded much skill and practice. The thoroughly cleaned cell was filled with toluene up to a height of approximately 1.5 mm, before a droplet of 5 ml ethanol suspension was gently put in contact with the air–liquid interface by slowly lowering a microsyringe. When the droplet reached the organic liquid, the low interfacial tension between ethanol and toluene caused the droplet to immediately spread. Gentle application of the droplet was necessary to minimize aggregation and even immersion of the alkoxylated silica particle into the toluene phase during spreading. For optimum spreading of the particles, it was found necessary to ultrasonicate the ethanol dispersion for an extended time, approximately 90 min, and allow the dispersion to heat up to around 40°C. At this temperature a very long-range interparticle repulsion was attained at the air–toluene interface which prevented clustering of particles during spreading. This will be discussed in more detail later. Successful spreading was characterized by a very low concentration of dimers (less than 1%) and the formation of a two-dimensional colloidal film with long-range order. This colloidal film lost its long-range order after an induction time of less than 1 h, allowing single particles to approach each other and start to form dimers and larger aggregates. This aggregation process was observed for several hours until most of the particles had combined into a single network. During this observation a series of images were collected every 15 min. 2.2. Instrumentation and Data Acquisition A POC Bachofer cell served as the container for the organic liquid. It consisted of a cylindrical Teflon wall (diameter of 42 mm) and a circular glass slice (0.17 mm thick) as bottom. The parts were tightened together in a holder and covered by a glass lid. In this cell, aggregation of the particles floating at the air–liquid interface was observed using an inverted microscope (Zeiss Axiovert 100), equipped with long-working-distance phase-contrast optics placed on a vibration-reducing stone table. This setup provided a good optical path with minimal disturbance from convection or interference by condensed liquid on the optics; see Fig. 1 for more details. The aggregation process was captured by a 310 CCD camera (Hamamatsu C540501) mounted to the microscope. The video signal was contrast enhanced by a preprocessor (Hamamatsu Argus-20) before the images were digitized using an SLIC frame grabber (MultiMedia Access Corp.) and stored on a Sun Ultra UNIX work station. As a reference, the images were also video recorded
2D PERIKINETIC AGGREGATION
FIG. 1. Schematic picture of the setup.
(Sony SVO 9500 MDP). The demand of resolving single particles was realized through the choice of a phase-contrast objective with magnification 203, resulting in a maximum field-of-view of 260 3 200 mm. Under these circumstances, each particle (2-mm diameter) occupied the equivalent area of 5 3 5 pixels, which was found to be the limit for obtaining the positions of single particles within a cluster with subpixel precision. 2.3. Image Processing Images were processed and analyzed to produce data files where the position of the particles was listed cluster by cluster. For this analysis, we used the software microGOP 2000/s, developed by Contextvision, a software designed to characterize images by their texture or by ordinary gray-scale image operations. The first step in the image processing was to separate particles within clusters to enable the determination of particle positions. We developed a procedure using ordinary image operations as well as texture operations to separate particles from each other. This procedure involved the combination of two methods. The first method started by smoothing the grayscale image with a filter to remove high-frequency noise. A texture method was then applied whereby the most probable structure of circles and lines was identified and mapped into a pseudocolor image. Next, thresholds were set to extract the most probable location of single particles. This operation resulted in a binary image containing separated particles. However, the binary images also contained artifacts; spots corresponding to nonexisting particles were found within pores, and noise outside the cluster edge was also frequently included in the image. These artifacts were removed using a second method of image processing. This method began with thresholding of the unprocessed gray-scale image, thereby displaying the areas containing clusters of particles as single whole objects. Furthermore, a second binary image containing borders between particles (areas of bright pixels) in the gray-scale image was
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created by a watershed algorithm. Next, the two images were combined producing a binary image containing groups of separated spots corresponding to particles. The low magnification of the gray image inhibits separation of all particles; however, pores were empty and no extra particles or noise was present in this binary image. Thus, by using the image from the second method as a mask, extra particles and noise from the image obtained by the first method were removed, thus producing a binary image with only separated particles. This corrected binary image was used for all further analysis. Particles within a certain distance from each other were identified as belonging to the same cluster and the position of each particle was measured by the center of gravity. A measure of the precision of the image processing and the image analysis can be estimated from the distribution of radial separation between particles connected in a dimer, illustrated in Fig. 2. The histogram displays a Gaussian distribution centered around a separation distance of 2.5 mm. The deviation of this measured average separation distance from the expected distance of 2 mm is related to the use of phase-contrast optics. Phase-contrast microscopy enhances the border between two particles, which creates nonspherical images of the particles constituting the pair. Hence, the position, measured as the center of gravity of the particle images, is shifted. The magnification 203 results in a pixel size of 0.4 3 0.4 mm, which corresponds to the width of the separation distance distribution (Fig. 2). We found it necessary to use this relatively high magnification (5 pixels per particle diameter) to minimize the occurrence of “false” clusters, i.e., two particles being interpreted as a dimer or cluster of higher order without actually being in contact. This allows us to determine the number and size of small clusters with high accuracy. The price for this improved resolution lies in the limited field of view; we are limited to observing clusters of R g , 100 mm.
FIG. 2. Distribution of separation distances between particles in objects defined as dimers in the image analysis sequence.
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FIG. 3. Schematic illustration of the effect of the degree of immersion and the range of steric repulsion on the total interparticle energy at the air–liquid interface.
3. RESULTS AND DISCUSSION
3.1. Characteristics of Two-Dimensional Colloidal System We have used a model colloid system consisting of alkoxylated silica particles trapped at the air–liquid interface. Due to the many complications in aqueous systems, we have chosen to use organic liquids of low polarity which minimize the effects of electrostatic interactions. The present system builds on the work of Kondo et al. (28, 29), which, in turn, was a development of Onoda’s (19) study of trapped particles at the air– liquid interface. The main advantage of the 2D alkoxylated silica system is that the magnitude of the attractive interactions can be controlled. The total interaction energy is essentially controlled by two parameters: (i) the degree of immersion (which is related to the contact angle) and (ii) the range of the steric repulsion. The effect of these two parameters on the magnitude of the interparticle attraction is illustrated in Fig. 3. This results in a model system that is very flexible since it allows independent control of both the attractive and repulsive interaction through the choice of organic liquid and length of the grafted alkyl chains. We have estimated interparticle energies between the partly immersed alkoxylated silica particles at the air–toluene interface using the Williams and Berg model (22). The model is based on an attractive van der Waals energy and a repulsive double-layer interaction. We have simply modified the model by the introduction of a repulsive steric interaction instead of the double-layer repulsion. The magnitude of the van der Waals interaction depends on the degree of immersion since the Hamaker constant in air always is higher than the effective Hamaker constant in a medium (32). The degree of immersion is related to the contact angle between the particle and the liquid; a high contact angle results in a low degree of immersion— hence, the van der Waals attraction becomes strong. The
range and magnitude of the steric repulsion depend on the thickness of the grafted alkyl layer. The capillary interaction between separated particles was found to be negligible for the investigated particle size (33). Also, for particles with a contact angle less than 80°, there is enough distance between the respective three phase lines of two particles in contact to safely ignore capillary interactions. Hence, the degree of immersion and the range of the steric repulsion control the total interaction energy. In the present study, the range of the steric repulsion is varied by grafting of octyl and octadecyl chains for which the layer thickness can be estimated from the molecular dimensions (1.4 and 2.5 nm, respectively). We have estimated the Hamaker constants for silica across air (vacuum) and toluene. The Hamaker constant across air, A air 5 15.8 kT, was taken from Bergstro¨m (34). The Hamaker constant across toluene was calculated using Lifshitz theory (34, 35). The necessary spectral parameters for toluene are not readily available; however, if we assume that toluene and benzene have identical spectral characteristics, an assumption that is supported by the very small difference in refractive index, n 5 1.496 and 1.501 for toluene and benzene, respectively, we obtain A toluene 5 0.8 kT. The contact angle for alkoxylated silica against benzene was previously determined to u 5 36° (29). Using this value also for the toluene system, we obtain an effective Hamaker constant of 1.21 kT for alkoxylated silica at the air–toluene interface, which results in an estimated magnitude of the attractive energy at contact of 2V M(C8) ' 30 kT for the thinner octyl coating and 2V M(C18) ' 15 kT for the thicker octadecyl coating for silica particles of radius 1 mm. 3.2. Observations during Spreading and Aggregation In this section, we will discuss the general qualitative features of the spreading and aggregation process. The process proceeds through four stages: spreading and initiation, formation of an ordered 2D colloidal film, cluster– cluster aggregation, and finally formation of a continuous particle gel. All stages except for the initiation are illustrated in Fig. 4. Below, some aspects of the different stages will be discussed in more detail. The alkoxylated silica particles are spread from an ethanol dispersion onto the toluene surface. During the spreading stage, particles spread rapidly over the air–toluene interface and depending on pretreatment of the ethanol dispersion, we could observe irreversible cluster formation as well as a more homogeneous spreading of single particles contained in stripes. Regions essentially empty of particles separated these stripes from each other. We found that successful spreading of single particles followed by the evolution of long-range order was considerably improved if the ethanol suspension had a temperature of approximately 40°C prior to spreading. It appears that the particles obtained a long-range repulsive interaction at this temperature, which prevented formation of clusters during the
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spreading stage. If no ultrasonication and heating of the ethanol suspension were performed, excessive clustering was observed and we did not observe any formation of a dilute colloidal film as discussed below. This temperature-induced repulsive interaction also assisted in leveling out the particle-rich stripes at the air–toluene interface. In fact, after a relatively short time, on the order of 10 –20 min, the particles formed a two-dimensional film with substantial long-range order. The mean distance between the particle was of the order of 15–20 mm, which corresponds to an area fraction of 1–2%. It should be mentioned that this colloid film did not cover the entire toluene surface; the film essentially consisted of a homogeneous region in the middle of the cell covering approximately 10 –20% of the entire available area. Figure 4a shows a small section of such a region. The film displays a long-range order with a tendency to hexagonal packing. Some dimers and even trimers can be observed; for the spreading to be considered successful, no more than 1% of dimers and even lower concentrations of clusters of higher order were accepted. At present, the origin of the temperature-induced repulsion is unknown. Possible causes may relate to dissociation and charge formation of free silanol groups that still remain after alkoxylation. One running speculation is that such a reaction at the particle surface may produce a slowly relaxing electrostatic interaction. This suggestion is supported by Ballard (31), who found an intriguing reaction between a hydrophobized surface and methanol, which was explained as a dissociation process between the methanol and free silanol groups. The effect of temperature on such a reaction remains to be elucidated and we are currently investigating this phenomenon in detail. The long-range repulsion only persists for a certain time. After 30 – 60 min, the particles start to diffuse over larger distances, the long-range order is lost, and clusters start to form. Figure 4b is a typical representation of the beginning of the clustering stage where clusters consisting of several tens of particles coexist together with monomers. As clustering proceeds, the clusters grow in size (Fig. 4c) and eventually a large particle network is formed (Fig. 4d). The time spent to reach this stage is called t max and was 4 – 6 h for the octyl system and 6 – 8 h for the octadecyl system. Once the particle network formed, we were unable to detect any rearrangement or further
FIG. 4. Images from the different stages of the aggregation process represented by the octyl-coated silica system: (a) initial state, t 5 4 min, where the ordered colloidal film has formed; (b) early clustering stage, t 5 90 min, where smaller clusters coexist with a substantial amount of singular particles; (c) later clustering stage, t 5 270 min, where the cluster sizes have grown but the size distribution is very wide since singular particles still exist; (d) final stage, t 5 400 min, where almost all clusters have joined into one large cluster, a particle network. This event defines the maximum aggregation time t max.
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FIG. 5. Area fraction, f (t), of particles as a function of aggregation time for (h) octyl-coated and (E) octadecyl-coated system. The lines correspond to a power law fit with the exponent a 5 0.42 and 0.58 for the octyl and octadecyl systems, respectively.
evolution of the clusters or network structure and the experiment was terminated. In addition to random Brownian motion, we also observed a drift, due to convection, of particles and clusters. This drift was observed to vary in magnitude and direction but seemed not to affect the local relative motion between monomers. Convection currents may be due to remnant temperature gradients in the liquid cell or stem from the movement of the cell. Although these manipulations were performed with great care, it is possible that the cluster trajectories could be affected. All observations were performed in the center of the area containing particles to avoid any possible edge effects. The area fraction of particles varied slightly within and between the particle-rich regions. However, we observed that with time, i.e., while the cluster was growing, the area fraction of particles within the particle-rich regions increased. Figure 5 shows how the area fraction increases from a value f ' 0.02 to a final value of f ' 0.05, which corresponds to the final density of the particle network where most clusters have become connected to one object spanning a large area. It should be pointed out that this network is not cell spanning in the classical sense; the network is simply a large-scale structure that consists of a homogeneous collection of clusters.
where R i is the position of particle i and ^R& is the mean position, equal to the center of mass of the cluster. In Fig. 6, we find the usual linear interval suggesting that the clusters are self-similar with a fractal dimension D f. The fractal dimension was obtained from a fit to the data ranging from s $ 10 and resulted in 1.57 6 0.04 for the octyl-coated particles and 1.63 6 0.04 for the octadecyl-coated particles. This plot contains data collected from all times during the aggregation process for the two systems. Included in the data is the measure of approximately 7500 clusters for each system having a size range from 2 to 1100 particles. The plot is linear all the way down to very small clusters (s . 10) but has a more scattered look for the large clusters because of poorer statistics. From the data, we have also performed an analysis of the scaling of R g for different aggregation times. Figure 7 shows that D f tends to increase with time (or cluster size) for the octadecyl-coated system while it stays approximately constant for the octylcoated system. At early times, t , 150 min, both systems display fractal dimensions in the range of D f ' 1.5. At later times, however, there is an increase in the fractal dimension for the octadecyl system toward a plateau value around D f(C18) 5 1.65 6 0.05 whereas the octyl system stays approximately constant at D f(C8) 5 1.53 6 0.05. The structural analysis was complemented using a boxcounting algorithm to deduce the fractal dimension, D f, within the clusters (36). The fractal dimension of clusters of a certain mass, s, was obtained from a fit to the linear part of a log–log plot of the number of boxes containing particles, n b, as a function of generalized length, d, defined from the total number 1/2 of boxes, d [ n tot , where the values of n b were averages over all clusters of the actual size. Additionally, each cluster was analyzed in six directions to remove the bias of the square grid.
3.3. Cluster Structures Cluster structures were characterized by determining the fractal dimension from the scaling law between the mass of a cluster, expressed as the number of particles in the cluster s, and the size of the cluster, here measured by the radius of gyration, R g. s } R Dg f,
Rg 5
1 s
Î
[1]
O ~R 2 ^R&! , s
2
i
i
[2]
FIG. 6. Double-logarithmic plot of the radius of gyration, R g, vs the number of particles in the cluster, s, for the octyl system (h) and (E) the octadecyl system, which has been offset by a factor 10 for clarity. The lines are least-squares fits to the data and yield D f(C8) 5 1.57 6 0.04 and D f(C18) 5 1.63 6 0.04.
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2D PERIKINETIC AGGREGATION
The number-average cluster mass, N(t), is defined as
O sn~s,t! N~t! 5 O n~s,t! , s
[4]
s
and the weight-average cluster mass, S(t), is obtained by
O s n~s,t! S~t! 5 O sn~s,t! . 2
s
[5]
s
FIG. 7. Time dependence of the fractal dimension, D f, measured from the scaling of R g for the octyl system (h) and the octadecyl system (E). The error bars show the 95% confidence interval.
Box-counting analysis shows that the fractal dimension is higher for the octadecyl system than for the octyl system for all cluster sizes (Fig. 8). The fractal dimension increases with cluster size for both systems, ranging from D f ' 1.3 at s ' 40 up to a plateau value of 1.45 and 1.55 at cluster mass around s ' 250 particles for the octyl- and octadecyl-coated system, respectively. The two methods of structural analysis, R g analysis and box counting, both show that the fractal dimension of the clusters increases with aggregation time (or cluster size). However, the box-counting method yields a significantly lower fractal dimension compared to the R g method. Such a difference is not unexpected since the two methods implicitly weight parts of the structural features differently. Measuring fractal dimensions via R g implicitly weights the peripheral parts of the cluster more, a method that becomes sensitive to for example, cluster anisotropy. This study focuses on the structure of smaller clusters (R g , 100a 0 ), which are inherently more sensitive to cluster anisotropy; hence, the box-counting method probably presents a better measure of the structural features of the clusters.
Hence, N(t) reflects the aggregation rate whereas S(t) reflects the population change. In analogy to polymer molecular weight distributions, the ratio of the first and second moments of the mass distribution, P(t), frequently referred to as the polydispersity (37), P~t! 5 S~t!/N~t!,
[6]
may be used as a measurement of the width of the cluster size distribution. The time and mass dependence of the cluster-mass distribution and its moments, N(t) and S(t), constitute the core of the kinetic information. Dynamic scaling for the cluster-mass distribution can be written as n~s,t! } t 2w s 2 t f~s/t z !,
[7]
where w, t , and z are the scaling exponents and f( x) is the cutoff function such that f( x) ; x d for x ! 1 and f( x) ! 1 for x @ 1. Alternatively, the dynamic scaling can be written in the form
3.4. Aggregation Kinetics The cluster-mass distribution, n(s,t), may be used to study the kinetics of the aggregation process (3–5). The definition of n(s,t) varies somewhat; in this study it denotes the number of clusters of mass s at time t divided by the total area of inspection in the measurement, i.e., the number of images times the image size. There is a direct relation between n(s,t) and the surface fraction of particles, f (t), given by
f ~t! 5
O sn~s,t!. s
[3]
FIG. 8. Cluster-mass dependence of the fractal dimension, D f, measured with the box-counting method for the octyl system (h) and the octadecyl system (E). The error bars show the 95% confidence interval.
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[8]
where F( x) can take one of two different universal forms, depending on whether or not the combined cluster transport and sticking probability between particles exceed a critical value (5). Above the critical point, F( x) falls algebraically with x for x ! 1 and F( x) ! 1 for x @ 1. Below the critical point, F( x) ! 1 for both x ! 1 and x @ 1. Inserting Eqs. [7] in [3] and substituting the summation with an integral result in the following relation (3):
f ~t! } t z~22 t !2w .
[9]
Furthermore, inserting Eq. [7] into Eqs. [5] and [4] yields scaling laws for S(t) and N(t) according to S~t! } t z ,
[10]
H tt ,, z
N~t! }
w
t , 1, t . 1.
[11]
The time evolution of the cluster-mass distribution is shown in Fig. 9. The data are divided into groups of s described by l such that l a [ s [ {2 a21 1 1; 2 a } for a 5 1, 2, . . . and l 0 [ s 5 1, i.e., the group containing the monomers. The sum of the cluster-mass distribution in each group is normalized with the area fraction of particles, f(t), and plotted against the mean size of the clusters in respective group, ^s& l . We observe qualitative differences when comparing the octyl system (Fig. 9a) and the octadecyl system (Fig. 9b). In the octyl system a second peak in the distribution develops early whereas in the octadecyl system a broader second peak develops first at later stages. This difference is related to the aggregation kinetics; a rapid occurrence of a pronounced peak in the cluster-mass distribution shows that the formation of medium clusters from monomers and dimers is faster than the formation of large clusters from medium clusters. In both systems, at long aggregation times, the second peak in the cluster-mass distribution becomes very broad, which indicates the formation of a particle network. The kinetics of the cluster-mass distribution defined as in Eq. [8] may further be represented in a reduced form, scaling according to n~s,t!s 2 } F~s/S~t!!.
[12]
Figure 10 shows the reduced cluster-mass distributions collapsed into a master curve. The appearance of F( x) is that of F( x) 5 x 2 g( x) with the cutoff function g( x) ! 1 when x ! 1 and x @ 1 and is characteristic of a system dominated by the kinetics of large clusters (5). The data from different times are somewhat scattered but follow a master curve characteristic for each system. The peak of the master curve for the octyl system
FIG. 9. Normalized cluster-mass distribution, n(s,t)/ f (t), for the (a) the octyl system and (b) the octadecyl system. For clarity, the data are collected in groups, l, defined as l a [ s [ {2 a21 1 1; 2 a } for a 5 1, 2, . . . and l 0 [ s 5 1.
is more pronounced than for the octadecyl system. This indicates that the octyl system is characterized by a stronger cutoff for the reduced mass s/S(t) , 1 as a result of the more pronounced aggregation of small clusters. The time dependence of the number-average cluster mass, N(t), is shown in Fig. 11. The data may be divided into two regimes, an initial, slow aggregation regime and a later, fast aggregation regime, with a sudden change between the two regimes at a critical point. The data from both regimes are fitted to the simple power law in Eq. [11]. We note that the difference in aggregation rate in the initial, slow aggregation regime for the two systems reflects the ratio between the estimated particle bond strength at 30 and 15 kT of the octyl and octadecyl system, respectively. However, there is no significant difference in aggregation rate between the two systems during the later, fast aggregation regime. The weight-average cluster mass, S(t), displays a transition
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FIG. 11. Number-average cluster mass, N(t), plotted against time for the octyl system (h) and the octadecyl system (E). A power law fit to the data at the two regimes yields the scaling exponent, w, for N(t).
FIG. 10. Reduced cluster-mass distribution, n(s,t)s 2 , as a function of the reduced cluster mass, s/S(t) at different times: (a) the octyl system; (b) the octadecyl system. The scaling shows clusters in the size range 1 # s # 30 particles.
between two regimes as well (Fig. 12). We fitted the data to a power law in the first, scattered regime as well as in the less scattered second regime. The resulting z exponents of Eq. [10] (see Table 1) suggest that the time evolution of cluster masses is similar for the octyl and octadecyl systems in the first regime. This situation is changed at the transition to the second regime, where the octadecyl system develops large clusters at a higher rate than the octyl system. We note that the scaling exponents, w and z, exhibit typical values of DLCA or RLCA (37) in the slow aggregation regime but increase with by least a factor of 10 for the fast aggregation regime (Table 1). The evolution of cluster masses is further analyzed by plotting the time dependence of the polydispersity, P(t) (Fig. 13). The plot suffers from the scattered nature of the S(t) data; however, we find that the octyl system is characterized by a continuous weak increase in polydispersity whereas the octa-
decyl system displays a strong increase in polydispersity at later aggregation times. This difference in polydispersity is related to the characteristics of the cluster growth process. For a system where the dominating cluster growth process is attachment of small clusters or single particles to large clusters (i.e., heterogeneous aggregation), the two moments, S(t) and N(t), will increase with similar rates; hence, the polydispersity does not change with time. For homogeneous aggregation, i.e., when cluster growth mainly proceeds through the aggregation of clusters of similar size, P(t) will increase with time (37). The data in Fig. 13 suggest that the evolution of cluster masses in the octyl system is dominated by aggregation of the smallest clusters, leaving the polydispersity nearly constant. The substantial increase in polydispersity of the octadecyl system suggests that homogeneous aggregation between clusters is important. Hence, when a lower sticking probability is introduced, the probability of bond formation between clusters increases with points of possible cluster connection, i.e., cluster
FIG. 12. Number-average cluster mass, S(t), plotted against time for the octyl system (h) and the octadecyl system (E). A power law fit to the data at the two regimes yields the scaling exponent, z, for S(t).
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TABLE 1 Values of Exponents Obtained from the Dynamic Scaling of the Cluster-Mass Distribution a System
w(slow)
w(fast)
z(slow)
z(fast)
Octyl Octadecyl
0.25 0.10
3.00 3.05
0.31 0.31
4.34 5.65
a
The scaling exponent w is related to N(t), while z is related to S(t). Slow and fast refer to the two aggregation regimes (Figs. 11 and 12).
mass, and the domination of the more mobile small clusters becomes gradually suppressed (4, 5). Both N(t) and S(t) display a sudden change, which does not agree with the theory of perikinetic aggregation. Suspecting that convection might be of importance, we performed a simple analysis of dimensionality based on the Pe´clet number. The Pe´clet number, Pe, identifies the dominance between transport due to diffusion and convection. Pe 5
3 pm d 2 U , kT
[13]
where m is the kinetic viscosity, d is the radius of a solid sphere, U is the effective speed due to the convection, and kT is the thermal energy. For a monomer with d 5 a 0 (1 mm) and an average speed of U ' 1 m m/s, the Pe´clet number is Pe ' 1.6 3 10 23 . (1 mm/s is an order of magnitude estimate of the mean convection speed based on the drift of large clusters.) A rough estimate of Pe for different clusters is obtained from fractal scaling, d ' R g } (a 0 s) 1/D f. Using Eq. [13], the values above, and D f 5 1.33 (box-counting estimate for small clusters, Fig. 8) results in Pe < Cs 3/ 2 ,
[14]
where C is a numerical factor, essentially on the order of unity, and s is the cluster mass. Convection is dominating when Pe @ 1, which corresponds to clusters with a mass on the order of s c ' 100 2/3 ' 20 particles. Indeed, the transition from slow to fast aggregation occurs for both the octyl and octadecyl systems when the cluster-mass distributions show a peak in the size range 16 , s # 32 (Fig. 9). This simple analysis suggests that the second, fast aggregation regime is dominated by convection-limited aggregation (CLCA) of large clusters. Convection induces mobility of large clusters and the collision frequency increases. However, the success of bond formation at a cluster collision is still size dependent for the octadecyl-coated system, which results in a stronger increase of S(t) and a gradual increase of P(t) compared to the octyl-coated system. This explanation of the drastic change in aggregation rate differs from previous suggestions of similar observations. Robinson and Earnshaw (23–25) explained the change in aggrega-
tion rate with a transition from DLCA to RLCA. It is interesting to note that many of the features characterizing a transition to CLCA or from DLCA to RLCA are similar, making it difficult to distinguish between these alternative explanations. 4. SUMMARY AND CONCLUSIONS
We have developed a two-dimensional colloidal model system that enables us to control the particle interaction and study the cluster– cluster aggregation in situ. The system consists of silica particles coated by grafted alkyl chains of different chain lengths and trapped at the air–liquid interface of an organic liquid. The total interaction energy is essentially controlled by two parameters: (i) the degree of immersion (which is related to the contact angle) and (ii) the range of the steric repulsion. This results in a model system that is very flexible since it allows the independent control of both the attractive and repulsive interaction through the choice of organic liquid and length of the grafted alkyl chains. In this study, we have only varied the range of the steric repulsion by grafting of octyl and octadecyl chains, which results in an estimated magnitude of the attractive energy at contact of 30 and 15 kT, respectively. Video-enhanced microscopy and image analysis were used to determine the cluster structure as a function of time. The use of a relatively high magnification, corresponding to 5 pixels per particle diameter, allowed us to determine the number and mass of clusters with high accuracy. Analysis of the obtained images showed that both systems produced self-similar clusters. The fractal dimension was measured with two methods: scaling of radius of gyration with cluster mass, and a boxcounting algorithm. The octyl system displayed a DLCA-like structure, D f ' 1.45, whereas the octadecyl system resulted in a more dense structure, D f ' 1.55. Applying dynamic scaling on the cluster-mass distribution identified a transition point between regimes of slower and faster aggregation. The scaling exponents displayed typical values of DLCA and RLCA at the regime of slow aggregation
FIG. 13. Temporal evolution of the polydispersity, P(t), for the octyl system (h) and the octadecyl system (E).
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but were increased tenfold at the regime of fast aggregation. The analysis suggested an increased collision frequency of the large clusters as a source for this behavior. Simple analysis of dimensionality, based on the Pe´clet number, indicated that convection affected the kinetics, as this made large clusters more mobile. We noted similarities in kinetics between nonrearranging systems that display a transition from DLCA to convection-limited cluster aggregation (CLCA) or from DLCA to RLCA. ACKNOWLEDGMENTS The authors acknowledge the late Prof. John Earnshaw for his helpful introduction to the field of colloidal aggregation in two dimensions. We also thank Prof. Brian Vincent for fruitful discussions. The Swedish Research Council for Engineering Sciences, TFR, supported this research.
REFERENCES 1. Meakin, P., Adv. Colloid Interface Sci. 28, 249 (1988). 2. Family, F., and Landau, D. P., “Kinetics of Aggregation and Gelation.” North-Holland, Amsterdam, 1984. 3. Vicsek, T., and Family, F., Phys. Rev. Lett. 52, 1669 (1984). 4. Meakin, P., Vicsek, T., and Family, F., Phys. Rev. B 31, 564 (1985). 5. Family, F., Meakin, P., and Vicsek, T., J. Chem. Phys. 83, 4144 (1985). 6. Jullien, R., and Botet, R., “Aggregation and Fractal Aggregates.” World Scientific, Singapore, 1987. 7. Dickinson, E., in “The Structure, Dynamics and Equilibrium of Colloidal Systems” (D. M. Bloor and E. Wyn-Jones, Eds.), p. 707. Kluwer Academic, Dordrecht/Norwell, MA, 1990. 8. Bos, M. T. A., and van Opheusden, J. H. J., Phys. Rev. E 53, 5044 (1996). 9. Heyes, D. M., J. Phys.: Condens. Matter 7, 8857 (1995). 10. Kantor, Y., and Witten, T. A., J. Phys. Lett. 45, L675 (1984). 11. Allain, C., and Cloitre, M., Adv. Colloid Interface Sci. 46, 129 (1993). 12. Wessel, R., and Ball, R. C., Phys. Rev. A 46, R3008 (1992).
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13. Torres, F. E., Russel, W. B., and Schowalter, W. R., J. Colloid Interface Sci. 145, 51 (1991). 14. Sorensen, C. M., and Roberts, G. C., J. Colloid Interface Sci. 186, 447 (1997). 15. Lin, M. Y., Klein, R., Lindsay, H. M., Weitz, D. A., Ball, R. C., and Meakin, P., J. Colloid Interface Sci. 137, 263 (1990). 16. Haw, M. D., Poon, C. K., and Pusey, P. N., Phys. Rev. E 56, 1918 (1997). 17. Robinson, D. J., and Earnshaw, J. C., Phys. Rev. Lett. 71, 715 (1993). 18. Earnshaw, J. C., and Robinson, D. J., Nuovo Cimento Soc. Ital. Fis. 16, 1141 (1994). 19. Onoda, G. Y., Phys. Rev. Lett. 55, 226 (1985). 20. Hurd, A. J., and Schaefer, D. W., Phys. Rev. Lett. 54, 1043 (1985). 21. Skjeltorp, A. T., Phys. Rev. Lett. 58, 1444 (1987). 22. Williams, D. F., and Berg, J. C., J. Colloid Interface Sci. 152, 218 (1992). 23. Robinson, D. J., and Earnshaw, J. C., Phys. Rev. A 46, 2045 (1992). 24. Robinson, D. J., and Earnshaw, J. C., Phys. Rev. A 46, 2055 (1992). 25. Robinson, D. J., and Earnshaw, J. C., Phys. Rev. A 46, 2065 (1992). 26. Stankiewicz, J., Vı´lchez, M. A. C., and Alvarez, R. H., Phys. Rev. E 47, 2663 (1993). 27. Robinson, D. J., and Earnshaw, J. C., Langmuir 9, 1436 (1993). 28. Kondo, M., Shinozaki, K., Bergstro¨m, L., and Mizutani, N., Langmuir 11, 394 (1995). 29. Kondo, M., Shinozaki, K., and Mizutani, N., “The 3rd International Union of Materials Research Societies International Conference on Advanced Materials Proceedings,” Elsevier, Amsterdam/New York, 1995. 30. Van Helden, A. K., Jansen, J. W., and Vrij, A., J. Colloid Interface Sci. 81, 354 (1980). 31. Ballard, C. C., Broge, E. C., Iler, R. K., John, D. S. St., and McWhorter, J. R., J. Phys. Chem. 65, 20 (1961). 32. Israelachvili, J. N., “Intermolecular and Surface Forces,” 2nd ed. Academic Press, London, 1991. 33. Chan, D. Y. C., Henry, J. D., Jr., and White, L. R., J. Colloid Interface Sci. 79, 410 (1981). 34. Bergstro¨m, L., Adv. Colloid Interface Sci. 70, 125 (1997). 35. Hough, D. B., and White, L. R., Adv. Colloid Interface Sci. 14, 3 (1980). 36. Bunde, A., and Havlin, S. (Eds.), “Fractals and Disordered Systems.” Springer-Verlag, Berlin/New York, 1991. 37. Elaissari, A., and Pefferkorn, E., J. Colloid Interface Sci. 143, 343 (1991).
Perikinetic Aggregation of Alkoxylated Silica Particles in Two Dimensions Peter H. F. Hansen and Lennart Bergstro¨m 1 Institute for Surface Chemistry, P.O. Box 5607, SE-114 86 Stockholm, Sweden Received June 17, 1998; accepted June 22, 1999
effect of possible cluster rearrangement due to thermal fluctuations, shear, convection, and settling has been analyzed (10 – 13). Studies with such more complex models have shown that the fractal dimension is not as unique and universal a structure parameter as was originally envisioned. Complemented with the power law of prefactors, fractal scaling has been limited to a certain region in the aggregation process (8, 14). However, the analysis of the fractal dimension is still a good discriminator to distinguish between different aggregation processes. It is a measure of scaling in a self-organizing system and is sensitive to the dependence of driving force, particle bond strength, and time scales of the aggregation processes. Light scattering in three dimensions is commonly used for studying aggregation; however, it has a major complication. The scattered light is a convolution between the structural parameter of the individual clusters and the spatial configuration of clusters. The latter is a combination of the cluster size distribution and the spatial distribution of clusters (15, 16). For DLCA, deconvolution is possible using simulation of the aggregation to obtain a cluster size distribution as input data (15), a procedure that results in a dependence of the accuracy of the extracted kinetic information from the simulation model applied. For RLCA, however, the polydispersity of the cluster size distribution inhibits the separation of the structure parameter and the spatial configuration of clusters; this is a consequence of a correlation with the spatial distribution of aggregates (16). In contrast, using optical inspection of a twodimensional system, the presence of the configuration of aggregates becomes an asset (17, 18), providing that the statistical problem of direct imaging can be solved by automated image sampling and processing of a large number of images. The experimental studies applying imaging techniques in two dimensions started to emerge in the mid-1980s (19 –21). Those observations inspired more thorough investigations of colloidal aggregation at an air–liquid interface (22–26). The majority of this work has involved electrostatic-stabilized particles spread at the air–water interface where aggregation was induced by addition of electrolyte. Although these studies have provided important insight regarding the aggregation process, this system suffers from an ill-defined interaction potential (27), which might explain the large span of the measured
Aggregation and cluster formation of a two-dimensional colloidal system consisting of 2-mm alkoxylated silica particles trapped at the air–toluene interface have been studied. This novel system allows particle interactions to be controlled by varying the length of the grafted alkyl chains; simple estimates suggest particle bond strengths of 15 and 30 kT for the octadecyl- and octyl-coated system, respectively. Video-enhanced microscopy and image analysis enabled a simultaneously study of kinetics and structure of the ensemble of clusters in situ. The octyl system displayed a DLCAlike structure, D f ' 1.45, whereas the octadecyl system resulted in a more dense structure, D f ' 1.55. The temporal evolution of the cluster-mass distribution displayed a transition point between regimes of slower and faster aggregation for both systems, which was interpreted as a transition from DLCA to convection-limited cluster aggregation (CLCA). © 1999 Academic Press Key Words: perikinetic aggregation; particle interactions; fractal dimension; dynamic scaling; image analysis.
1. INTRODUCTION
Aggregation is a fundamental process of substantial academic interest and technological relevance (1–5). Early studies identified two limiting types of cluster– cluster aggregation behavior: diffusion-limited cluster aggregation (DLCA), characterized by a sticking probability of one, which makes aggregation limited by particle diffusion only; and reaction-limited cluster aggregation (RLCA), characterized by a sticking probability significantly lower than one, which makes the initial aggregation rate very slow. The two regimes not only display different fractal dimensions [D f is 1.44 6 0.04 for DLCA and 1.55 6 0.03 for RLCA (6)] but show differences in the aggregation kinetics as well, e.g., expressed in the decay and broadness of the cluster size distribution (5). Computer simulations have provided much of the fundamental knowledge in this field. During the past decade, computer simulations have evolved from simple models using random trajectories and fixed sticking probabilities toward more complex simulations using Brownian dynamics algorithms and pair-interaction potentials between particles (7–9). Also, the 1
To whom correspondence should be addressed. E-mail: lennart. [email protected]. 77
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fractal dimensions, varying from 1.2 to 1.6 (19 –23, 26). These problems have hampered analysis of the results and limited the comparison between experiments and computer simulations to mainly qualitative features. Hence, there is a need for twodimensional systems where both the strength of interparticle attraction and the external forces, e.g., convection or shear, are well controlled. In this paper a novel system for investigating two-dimensional aggregation will be presented. It is based on the use of alkoxylated silica particles trapped at the air–liquid interface of nonaqueous liquids, thus minimizing the influence of electrostatic phenomena. This system permits the short-range particle interactions, the interaction strength, to be controlled in two ways: first, by varying the range of the steric repulsion by means of the thickness of the coating, and second, by controlling the magnitude of the van der Waals attraction between the particles through the contact angle between alkoxylated particle and liquid (28, 29). We will describe the properties of this system and present results on cluster structure and the kinetics in relation to the mechanisms of cluster transport for two systems: octyl- and octadecyl-coated silica particles trapped at the air–toluene interface. 2. MATERIALS AND METHODS
2.1. Preparation and Spreading of Alkoxylated Silica Particles Silica particles were alkoxylated by grafting of two different alcohols, octanol and octadecanol, to the particle surface. The silica (Tokuyama Soda Co., Japan) particles were spherical and highly monodisperse with a mean particle diameter of 1.97 mm and a standard deviation of 0.07 mm, as determined by scanning electron microscopy (SEM). Grafting of alcohols on the silica particles followed the work by Van Helden et al. (30) and Ballard et al. (31). The grafting process simply involves a condensation reaction between the silanol and the alcohol group. The grafting reaction is degenerated by the presence of water; hence, the first step was to dry the silica powder under vacuum at 60°C for 24 h. Afterward, the dried silica powder was suspended in an alcohol with a selected chain length, previously purified by distillation. The weight of added alcohol was three times the weight of silica powder to obtain a suitable concentration of the resulting dispersion. The homogeneity was ensured by sonication of the suspension for several hours. The grafting proceeded by an esterification (condensation) reaction (31) by simply heating the dispersion, while stirring, to a temperature of 195°C for 4 h. When esterification was concluded, the grafted particles were rinsed from the remaining alcohol by centrifugation of the dispersion, followed by redispersing of the sediment in fresh liquid. This rinsing procedure was repeated three times, with a mixture of cyclohexane and chloroform (40%, 60%) the first time and pure cyclohexane the
second and third times. The particles were found to be hydrophobic. They were not dispersible in water but were easy to disperse in cyclohexane. Finally, the particles were stored in cyclohexane, in which they can remain for several years without degrading (30). Before use, a small quantity of coated particles was simply dried by evaporation of cyclohexane and redispersed in ethanol to a weight fraction of 0.2%. Spreading of the alkoxylated silica particles at the air– toluene interface proved to be a crucial process, which demanded much skill and practice. The thoroughly cleaned cell was filled with toluene up to a height of approximately 1.5 mm, before a droplet of 5 ml ethanol suspension was gently put in contact with the air–liquid interface by slowly lowering a microsyringe. When the droplet reached the organic liquid, the low interfacial tension between ethanol and toluene caused the droplet to immediately spread. Gentle application of the droplet was necessary to minimize aggregation and even immersion of the alkoxylated silica particle into the toluene phase during spreading. For optimum spreading of the particles, it was found necessary to ultrasonicate the ethanol dispersion for an extended time, approximately 90 min, and allow the dispersion to heat up to around 40°C. At this temperature a very long-range interparticle repulsion was attained at the air–toluene interface which prevented clustering of particles during spreading. This will be discussed in more detail later. Successful spreading was characterized by a very low concentration of dimers (less than 1%) and the formation of a two-dimensional colloidal film with long-range order. This colloidal film lost its long-range order after an induction time of less than 1 h, allowing single particles to approach each other and start to form dimers and larger aggregates. This aggregation process was observed for several hours until most of the particles had combined into a single network. During this observation a series of images were collected every 15 min. 2.2. Instrumentation and Data Acquisition A POC Bachofer cell served as the container for the organic liquid. It consisted of a cylindrical Teflon wall (diameter of 42 mm) and a circular glass slice (0.17 mm thick) as bottom. The parts were tightened together in a holder and covered by a glass lid. In this cell, aggregation of the particles floating at the air–liquid interface was observed using an inverted microscope (Zeiss Axiovert 100), equipped with long-working-distance phase-contrast optics placed on a vibration-reducing stone table. This setup provided a good optical path with minimal disturbance from convection or interference by condensed liquid on the optics; see Fig. 1 for more details. The aggregation process was captured by a 310 CCD camera (Hamamatsu C540501) mounted to the microscope. The video signal was contrast enhanced by a preprocessor (Hamamatsu Argus-20) before the images were digitized using an SLIC frame grabber (MultiMedia Access Corp.) and stored on a Sun Ultra UNIX work station. As a reference, the images were also video recorded
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FIG. 1. Schematic picture of the setup.
(Sony SVO 9500 MDP). The demand of resolving single particles was realized through the choice of a phase-contrast objective with magnification 203, resulting in a maximum field-of-view of 260 3 200 mm. Under these circumstances, each particle (2-mm diameter) occupied the equivalent area of 5 3 5 pixels, which was found to be the limit for obtaining the positions of single particles within a cluster with subpixel precision. 2.3. Image Processing Images were processed and analyzed to produce data files where the position of the particles was listed cluster by cluster. For this analysis, we used the software microGOP 2000/s, developed by Contextvision, a software designed to characterize images by their texture or by ordinary gray-scale image operations. The first step in the image processing was to separate particles within clusters to enable the determination of particle positions. We developed a procedure using ordinary image operations as well as texture operations to separate particles from each other. This procedure involved the combination of two methods. The first method started by smoothing the grayscale image with a filter to remove high-frequency noise. A texture method was then applied whereby the most probable structure of circles and lines was identified and mapped into a pseudocolor image. Next, thresholds were set to extract the most probable location of single particles. This operation resulted in a binary image containing separated particles. However, the binary images also contained artifacts; spots corresponding to nonexisting particles were found within pores, and noise outside the cluster edge was also frequently included in the image. These artifacts were removed using a second method of image processing. This method began with thresholding of the unprocessed gray-scale image, thereby displaying the areas containing clusters of particles as single whole objects. Furthermore, a second binary image containing borders between particles (areas of bright pixels) in the gray-scale image was
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created by a watershed algorithm. Next, the two images were combined producing a binary image containing groups of separated spots corresponding to particles. The low magnification of the gray image inhibits separation of all particles; however, pores were empty and no extra particles or noise was present in this binary image. Thus, by using the image from the second method as a mask, extra particles and noise from the image obtained by the first method were removed, thus producing a binary image with only separated particles. This corrected binary image was used for all further analysis. Particles within a certain distance from each other were identified as belonging to the same cluster and the position of each particle was measured by the center of gravity. A measure of the precision of the image processing and the image analysis can be estimated from the distribution of radial separation between particles connected in a dimer, illustrated in Fig. 2. The histogram displays a Gaussian distribution centered around a separation distance of 2.5 mm. The deviation of this measured average separation distance from the expected distance of 2 mm is related to the use of phase-contrast optics. Phase-contrast microscopy enhances the border between two particles, which creates nonspherical images of the particles constituting the pair. Hence, the position, measured as the center of gravity of the particle images, is shifted. The magnification 203 results in a pixel size of 0.4 3 0.4 mm, which corresponds to the width of the separation distance distribution (Fig. 2). We found it necessary to use this relatively high magnification (5 pixels per particle diameter) to minimize the occurrence of “false” clusters, i.e., two particles being interpreted as a dimer or cluster of higher order without actually being in contact. This allows us to determine the number and size of small clusters with high accuracy. The price for this improved resolution lies in the limited field of view; we are limited to observing clusters of R g , 100 mm.
FIG. 2. Distribution of separation distances between particles in objects defined as dimers in the image analysis sequence.
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FIG. 3. Schematic illustration of the effect of the degree of immersion and the range of steric repulsion on the total interparticle energy at the air–liquid interface.
3. RESULTS AND DISCUSSION
3.1. Characteristics of Two-Dimensional Colloidal System We have used a model colloid system consisting of alkoxylated silica particles trapped at the air–liquid interface. Due to the many complications in aqueous systems, we have chosen to use organic liquids of low polarity which minimize the effects of electrostatic interactions. The present system builds on the work of Kondo et al. (28, 29), which, in turn, was a development of Onoda’s (19) study of trapped particles at the air– liquid interface. The main advantage of the 2D alkoxylated silica system is that the magnitude of the attractive interactions can be controlled. The total interaction energy is essentially controlled by two parameters: (i) the degree of immersion (which is related to the contact angle) and (ii) the range of the steric repulsion. The effect of these two parameters on the magnitude of the interparticle attraction is illustrated in Fig. 3. This results in a model system that is very flexible since it allows independent control of both the attractive and repulsive interaction through the choice of organic liquid and length of the grafted alkyl chains. We have estimated interparticle energies between the partly immersed alkoxylated silica particles at the air–toluene interface using the Williams and Berg model (22). The model is based on an attractive van der Waals energy and a repulsive double-layer interaction. We have simply modified the model by the introduction of a repulsive steric interaction instead of the double-layer repulsion. The magnitude of the van der Waals interaction depends on the degree of immersion since the Hamaker constant in air always is higher than the effective Hamaker constant in a medium (32). The degree of immersion is related to the contact angle between the particle and the liquid; a high contact angle results in a low degree of immersion— hence, the van der Waals attraction becomes strong. The
range and magnitude of the steric repulsion depend on the thickness of the grafted alkyl layer. The capillary interaction between separated particles was found to be negligible for the investigated particle size (33). Also, for particles with a contact angle less than 80°, there is enough distance between the respective three phase lines of two particles in contact to safely ignore capillary interactions. Hence, the degree of immersion and the range of the steric repulsion control the total interaction energy. In the present study, the range of the steric repulsion is varied by grafting of octyl and octadecyl chains for which the layer thickness can be estimated from the molecular dimensions (1.4 and 2.5 nm, respectively). We have estimated the Hamaker constants for silica across air (vacuum) and toluene. The Hamaker constant across air, A air 5 15.8 kT, was taken from Bergstro¨m (34). The Hamaker constant across toluene was calculated using Lifshitz theory (34, 35). The necessary spectral parameters for toluene are not readily available; however, if we assume that toluene and benzene have identical spectral characteristics, an assumption that is supported by the very small difference in refractive index, n 5 1.496 and 1.501 for toluene and benzene, respectively, we obtain A toluene 5 0.8 kT. The contact angle for alkoxylated silica against benzene was previously determined to u 5 36° (29). Using this value also for the toluene system, we obtain an effective Hamaker constant of 1.21 kT for alkoxylated silica at the air–toluene interface, which results in an estimated magnitude of the attractive energy at contact of 2V M(C8) ' 30 kT for the thinner octyl coating and 2V M(C18) ' 15 kT for the thicker octadecyl coating for silica particles of radius 1 mm. 3.2. Observations during Spreading and Aggregation In this section, we will discuss the general qualitative features of the spreading and aggregation process. The process proceeds through four stages: spreading and initiation, formation of an ordered 2D colloidal film, cluster– cluster aggregation, and finally formation of a continuous particle gel. All stages except for the initiation are illustrated in Fig. 4. Below, some aspects of the different stages will be discussed in more detail. The alkoxylated silica particles are spread from an ethanol dispersion onto the toluene surface. During the spreading stage, particles spread rapidly over the air–toluene interface and depending on pretreatment of the ethanol dispersion, we could observe irreversible cluster formation as well as a more homogeneous spreading of single particles contained in stripes. Regions essentially empty of particles separated these stripes from each other. We found that successful spreading of single particles followed by the evolution of long-range order was considerably improved if the ethanol suspension had a temperature of approximately 40°C prior to spreading. It appears that the particles obtained a long-range repulsive interaction at this temperature, which prevented formation of clusters during the
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spreading stage. If no ultrasonication and heating of the ethanol suspension were performed, excessive clustering was observed and we did not observe any formation of a dilute colloidal film as discussed below. This temperature-induced repulsive interaction also assisted in leveling out the particle-rich stripes at the air–toluene interface. In fact, after a relatively short time, on the order of 10 –20 min, the particles formed a two-dimensional film with substantial long-range order. The mean distance between the particle was of the order of 15–20 mm, which corresponds to an area fraction of 1–2%. It should be mentioned that this colloid film did not cover the entire toluene surface; the film essentially consisted of a homogeneous region in the middle of the cell covering approximately 10 –20% of the entire available area. Figure 4a shows a small section of such a region. The film displays a long-range order with a tendency to hexagonal packing. Some dimers and even trimers can be observed; for the spreading to be considered successful, no more than 1% of dimers and even lower concentrations of clusters of higher order were accepted. At present, the origin of the temperature-induced repulsion is unknown. Possible causes may relate to dissociation and charge formation of free silanol groups that still remain after alkoxylation. One running speculation is that such a reaction at the particle surface may produce a slowly relaxing electrostatic interaction. This suggestion is supported by Ballard (31), who found an intriguing reaction between a hydrophobized surface and methanol, which was explained as a dissociation process between the methanol and free silanol groups. The effect of temperature on such a reaction remains to be elucidated and we are currently investigating this phenomenon in detail. The long-range repulsion only persists for a certain time. After 30 – 60 min, the particles start to diffuse over larger distances, the long-range order is lost, and clusters start to form. Figure 4b is a typical representation of the beginning of the clustering stage where clusters consisting of several tens of particles coexist together with monomers. As clustering proceeds, the clusters grow in size (Fig. 4c) and eventually a large particle network is formed (Fig. 4d). The time spent to reach this stage is called t max and was 4 – 6 h for the octyl system and 6 – 8 h for the octadecyl system. Once the particle network formed, we were unable to detect any rearrangement or further
FIG. 4. Images from the different stages of the aggregation process represented by the octyl-coated silica system: (a) initial state, t 5 4 min, where the ordered colloidal film has formed; (b) early clustering stage, t 5 90 min, where smaller clusters coexist with a substantial amount of singular particles; (c) later clustering stage, t 5 270 min, where the cluster sizes have grown but the size distribution is very wide since singular particles still exist; (d) final stage, t 5 400 min, where almost all clusters have joined into one large cluster, a particle network. This event defines the maximum aggregation time t max.
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FIG. 5. Area fraction, f (t), of particles as a function of aggregation time for (h) octyl-coated and (E) octadecyl-coated system. The lines correspond to a power law fit with the exponent a 5 0.42 and 0.58 for the octyl and octadecyl systems, respectively.
evolution of the clusters or network structure and the experiment was terminated. In addition to random Brownian motion, we also observed a drift, due to convection, of particles and clusters. This drift was observed to vary in magnitude and direction but seemed not to affect the local relative motion between monomers. Convection currents may be due to remnant temperature gradients in the liquid cell or stem from the movement of the cell. Although these manipulations were performed with great care, it is possible that the cluster trajectories could be affected. All observations were performed in the center of the area containing particles to avoid any possible edge effects. The area fraction of particles varied slightly within and between the particle-rich regions. However, we observed that with time, i.e., while the cluster was growing, the area fraction of particles within the particle-rich regions increased. Figure 5 shows how the area fraction increases from a value f ' 0.02 to a final value of f ' 0.05, which corresponds to the final density of the particle network where most clusters have become connected to one object spanning a large area. It should be pointed out that this network is not cell spanning in the classical sense; the network is simply a large-scale structure that consists of a homogeneous collection of clusters.
where R i is the position of particle i and ^R& is the mean position, equal to the center of mass of the cluster. In Fig. 6, we find the usual linear interval suggesting that the clusters are self-similar with a fractal dimension D f. The fractal dimension was obtained from a fit to the data ranging from s $ 10 and resulted in 1.57 6 0.04 for the octyl-coated particles and 1.63 6 0.04 for the octadecyl-coated particles. This plot contains data collected from all times during the aggregation process for the two systems. Included in the data is the measure of approximately 7500 clusters for each system having a size range from 2 to 1100 particles. The plot is linear all the way down to very small clusters (s . 10) but has a more scattered look for the large clusters because of poorer statistics. From the data, we have also performed an analysis of the scaling of R g for different aggregation times. Figure 7 shows that D f tends to increase with time (or cluster size) for the octadecyl-coated system while it stays approximately constant for the octylcoated system. At early times, t , 150 min, both systems display fractal dimensions in the range of D f ' 1.5. At later times, however, there is an increase in the fractal dimension for the octadecyl system toward a plateau value around D f(C18) 5 1.65 6 0.05 whereas the octyl system stays approximately constant at D f(C8) 5 1.53 6 0.05. The structural analysis was complemented using a boxcounting algorithm to deduce the fractal dimension, D f, within the clusters (36). The fractal dimension of clusters of a certain mass, s, was obtained from a fit to the linear part of a log–log plot of the number of boxes containing particles, n b, as a function of generalized length, d, defined from the total number 1/2 of boxes, d [ n tot , where the values of n b were averages over all clusters of the actual size. Additionally, each cluster was analyzed in six directions to remove the bias of the square grid.
3.3. Cluster Structures Cluster structures were characterized by determining the fractal dimension from the scaling law between the mass of a cluster, expressed as the number of particles in the cluster s, and the size of the cluster, here measured by the radius of gyration, R g. s } R Dg f,
Rg 5
1 s
Î
[1]
O ~R 2 ^R&! , s
2
i
i
[2]
FIG. 6. Double-logarithmic plot of the radius of gyration, R g, vs the number of particles in the cluster, s, for the octyl system (h) and (E) the octadecyl system, which has been offset by a factor 10 for clarity. The lines are least-squares fits to the data and yield D f(C8) 5 1.57 6 0.04 and D f(C18) 5 1.63 6 0.04.
83
2D PERIKINETIC AGGREGATION
The number-average cluster mass, N(t), is defined as
O sn~s,t! N~t! 5 O n~s,t! , s
[4]
s
and the weight-average cluster mass, S(t), is obtained by
O s n~s,t! S~t! 5 O sn~s,t! . 2
s
[5]
s
FIG. 7. Time dependence of the fractal dimension, D f, measured from the scaling of R g for the octyl system (h) and the octadecyl system (E). The error bars show the 95% confidence interval.
Box-counting analysis shows that the fractal dimension is higher for the octadecyl system than for the octyl system for all cluster sizes (Fig. 8). The fractal dimension increases with cluster size for both systems, ranging from D f ' 1.3 at s ' 40 up to a plateau value of 1.45 and 1.55 at cluster mass around s ' 250 particles for the octyl- and octadecyl-coated system, respectively. The two methods of structural analysis, R g analysis and box counting, both show that the fractal dimension of the clusters increases with aggregation time (or cluster size). However, the box-counting method yields a significantly lower fractal dimension compared to the R g method. Such a difference is not unexpected since the two methods implicitly weight parts of the structural features differently. Measuring fractal dimensions via R g implicitly weights the peripheral parts of the cluster more, a method that becomes sensitive to for example, cluster anisotropy. This study focuses on the structure of smaller clusters (R g , 100a 0 ), which are inherently more sensitive to cluster anisotropy; hence, the box-counting method probably presents a better measure of the structural features of the clusters.
Hence, N(t) reflects the aggregation rate whereas S(t) reflects the population change. In analogy to polymer molecular weight distributions, the ratio of the first and second moments of the mass distribution, P(t), frequently referred to as the polydispersity (37), P~t! 5 S~t!/N~t!,
[6]
may be used as a measurement of the width of the cluster size distribution. The time and mass dependence of the cluster-mass distribution and its moments, N(t) and S(t), constitute the core of the kinetic information. Dynamic scaling for the cluster-mass distribution can be written as n~s,t! } t 2w s 2 t f~s/t z !,
[7]
where w, t , and z are the scaling exponents and f( x) is the cutoff function such that f( x) ; x d for x ! 1 and f( x) ! 1 for x @ 1. Alternatively, the dynamic scaling can be written in the form
3.4. Aggregation Kinetics The cluster-mass distribution, n(s,t), may be used to study the kinetics of the aggregation process (3–5). The definition of n(s,t) varies somewhat; in this study it denotes the number of clusters of mass s at time t divided by the total area of inspection in the measurement, i.e., the number of images times the image size. There is a direct relation between n(s,t) and the surface fraction of particles, f (t), given by
f ~t! 5
O sn~s,t!. s
[3]
FIG. 8. Cluster-mass dependence of the fractal dimension, D f, measured with the box-counting method for the octyl system (h) and the octadecyl system (E). The error bars show the 95% confidence interval.
¨M HANSEN AND BERGSTRO
84 n~s,t! } s 22 F~s/t z !,
[8]
where F( x) can take one of two different universal forms, depending on whether or not the combined cluster transport and sticking probability between particles exceed a critical value (5). Above the critical point, F( x) falls algebraically with x for x ! 1 and F( x) ! 1 for x @ 1. Below the critical point, F( x) ! 1 for both x ! 1 and x @ 1. Inserting Eqs. [7] in [3] and substituting the summation with an integral result in the following relation (3):
f ~t! } t z~22 t !2w .
[9]
Furthermore, inserting Eq. [7] into Eqs. [5] and [4] yields scaling laws for S(t) and N(t) according to S~t! } t z ,
[10]
H tt ,, z
N~t! }
w
t , 1, t . 1.
[11]
The time evolution of the cluster-mass distribution is shown in Fig. 9. The data are divided into groups of s described by l such that l a [ s [ {2 a21 1 1; 2 a } for a 5 1, 2, . . . and l 0 [ s 5 1, i.e., the group containing the monomers. The sum of the cluster-mass distribution in each group is normalized with the area fraction of particles, f(t), and plotted against the mean size of the clusters in respective group, ^s& l . We observe qualitative differences when comparing the octyl system (Fig. 9a) and the octadecyl system (Fig. 9b). In the octyl system a second peak in the distribution develops early whereas in the octadecyl system a broader second peak develops first at later stages. This difference is related to the aggregation kinetics; a rapid occurrence of a pronounced peak in the cluster-mass distribution shows that the formation of medium clusters from monomers and dimers is faster than the formation of large clusters from medium clusters. In both systems, at long aggregation times, the second peak in the cluster-mass distribution becomes very broad, which indicates the formation of a particle network. The kinetics of the cluster-mass distribution defined as in Eq. [8] may further be represented in a reduced form, scaling according to n~s,t!s 2 } F~s/S~t!!.
[12]
Figure 10 shows the reduced cluster-mass distributions collapsed into a master curve. The appearance of F( x) is that of F( x) 5 x 2 g( x) with the cutoff function g( x) ! 1 when x ! 1 and x @ 1 and is characteristic of a system dominated by the kinetics of large clusters (5). The data from different times are somewhat scattered but follow a master curve characteristic for each system. The peak of the master curve for the octyl system
FIG. 9. Normalized cluster-mass distribution, n(s,t)/ f (t), for the (a) the octyl system and (b) the octadecyl system. For clarity, the data are collected in groups, l, defined as l a [ s [ {2 a21 1 1; 2 a } for a 5 1, 2, . . . and l 0 [ s 5 1.
is more pronounced than for the octadecyl system. This indicates that the octyl system is characterized by a stronger cutoff for the reduced mass s/S(t) , 1 as a result of the more pronounced aggregation of small clusters. The time dependence of the number-average cluster mass, N(t), is shown in Fig. 11. The data may be divided into two regimes, an initial, slow aggregation regime and a later, fast aggregation regime, with a sudden change between the two regimes at a critical point. The data from both regimes are fitted to the simple power law in Eq. [11]. We note that the difference in aggregation rate in the initial, slow aggregation regime for the two systems reflects the ratio between the estimated particle bond strength at 30 and 15 kT of the octyl and octadecyl system, respectively. However, there is no significant difference in aggregation rate between the two systems during the later, fast aggregation regime. The weight-average cluster mass, S(t), displays a transition
2D PERIKINETIC AGGREGATION
85
FIG. 11. Number-average cluster mass, N(t), plotted against time for the octyl system (h) and the octadecyl system (E). A power law fit to the data at the two regimes yields the scaling exponent, w, for N(t).
FIG. 10. Reduced cluster-mass distribution, n(s,t)s 2 , as a function of the reduced cluster mass, s/S(t) at different times: (a) the octyl system; (b) the octadecyl system. The scaling shows clusters in the size range 1 # s # 30 particles.
between two regimes as well (Fig. 12). We fitted the data to a power law in the first, scattered regime as well as in the less scattered second regime. The resulting z exponents of Eq. [10] (see Table 1) suggest that the time evolution of cluster masses is similar for the octyl and octadecyl systems in the first regime. This situation is changed at the transition to the second regime, where the octadecyl system develops large clusters at a higher rate than the octyl system. We note that the scaling exponents, w and z, exhibit typical values of DLCA or RLCA (37) in the slow aggregation regime but increase with by least a factor of 10 for the fast aggregation regime (Table 1). The evolution of cluster masses is further analyzed by plotting the time dependence of the polydispersity, P(t) (Fig. 13). The plot suffers from the scattered nature of the S(t) data; however, we find that the octyl system is characterized by a continuous weak increase in polydispersity whereas the octa-
decyl system displays a strong increase in polydispersity at later aggregation times. This difference in polydispersity is related to the characteristics of the cluster growth process. For a system where the dominating cluster growth process is attachment of small clusters or single particles to large clusters (i.e., heterogeneous aggregation), the two moments, S(t) and N(t), will increase with similar rates; hence, the polydispersity does not change with time. For homogeneous aggregation, i.e., when cluster growth mainly proceeds through the aggregation of clusters of similar size, P(t) will increase with time (37). The data in Fig. 13 suggest that the evolution of cluster masses in the octyl system is dominated by aggregation of the smallest clusters, leaving the polydispersity nearly constant. The substantial increase in polydispersity of the octadecyl system suggests that homogeneous aggregation between clusters is important. Hence, when a lower sticking probability is introduced, the probability of bond formation between clusters increases with points of possible cluster connection, i.e., cluster
FIG. 12. Number-average cluster mass, S(t), plotted against time for the octyl system (h) and the octadecyl system (E). A power law fit to the data at the two regimes yields the scaling exponent, z, for S(t).
¨M HANSEN AND BERGSTRO
86
TABLE 1 Values of Exponents Obtained from the Dynamic Scaling of the Cluster-Mass Distribution a System
w(slow)
w(fast)
z(slow)
z(fast)
Octyl Octadecyl
0.25 0.10
3.00 3.05
0.31 0.31
4.34 5.65
a
The scaling exponent w is related to N(t), while z is related to S(t). Slow and fast refer to the two aggregation regimes (Figs. 11 and 12).
mass, and the domination of the more mobile small clusters becomes gradually suppressed (4, 5). Both N(t) and S(t) display a sudden change, which does not agree with the theory of perikinetic aggregation. Suspecting that convection might be of importance, we performed a simple analysis of dimensionality based on the Pe´clet number. The Pe´clet number, Pe, identifies the dominance between transport due to diffusion and convection. Pe 5
3 pm d 2 U , kT
[13]
where m is the kinetic viscosity, d is the radius of a solid sphere, U is the effective speed due to the convection, and kT is the thermal energy. For a monomer with d 5 a 0 (1 mm) and an average speed of U ' 1 m m/s, the Pe´clet number is Pe ' 1.6 3 10 23 . (1 mm/s is an order of magnitude estimate of the mean convection speed based on the drift of large clusters.) A rough estimate of Pe for different clusters is obtained from fractal scaling, d ' R g } (a 0 s) 1/D f. Using Eq. [13], the values above, and D f 5 1.33 (box-counting estimate for small clusters, Fig. 8) results in Pe < Cs 3/ 2 ,
[14]
where C is a numerical factor, essentially on the order of unity, and s is the cluster mass. Convection is dominating when Pe @ 1, which corresponds to clusters with a mass on the order of s c ' 100 2/3 ' 20 particles. Indeed, the transition from slow to fast aggregation occurs for both the octyl and octadecyl systems when the cluster-mass distributions show a peak in the size range 16 , s # 32 (Fig. 9). This simple analysis suggests that the second, fast aggregation regime is dominated by convection-limited aggregation (CLCA) of large clusters. Convection induces mobility of large clusters and the collision frequency increases. However, the success of bond formation at a cluster collision is still size dependent for the octadecyl-coated system, which results in a stronger increase of S(t) and a gradual increase of P(t) compared to the octyl-coated system. This explanation of the drastic change in aggregation rate differs from previous suggestions of similar observations. Robinson and Earnshaw (23–25) explained the change in aggrega-
tion rate with a transition from DLCA to RLCA. It is interesting to note that many of the features characterizing a transition to CLCA or from DLCA to RLCA are similar, making it difficult to distinguish between these alternative explanations. 4. SUMMARY AND CONCLUSIONS
We have developed a two-dimensional colloidal model system that enables us to control the particle interaction and study the cluster– cluster aggregation in situ. The system consists of silica particles coated by grafted alkyl chains of different chain lengths and trapped at the air–liquid interface of an organic liquid. The total interaction energy is essentially controlled by two parameters: (i) the degree of immersion (which is related to the contact angle) and (ii) the range of the steric repulsion. This results in a model system that is very flexible since it allows the independent control of both the attractive and repulsive interaction through the choice of organic liquid and length of the grafted alkyl chains. In this study, we have only varied the range of the steric repulsion by grafting of octyl and octadecyl chains, which results in an estimated magnitude of the attractive energy at contact of 30 and 15 kT, respectively. Video-enhanced microscopy and image analysis were used to determine the cluster structure as a function of time. The use of a relatively high magnification, corresponding to 5 pixels per particle diameter, allowed us to determine the number and mass of clusters with high accuracy. Analysis of the obtained images showed that both systems produced self-similar clusters. The fractal dimension was measured with two methods: scaling of radius of gyration with cluster mass, and a boxcounting algorithm. The octyl system displayed a DLCA-like structure, D f ' 1.45, whereas the octadecyl system resulted in a more dense structure, D f ' 1.55. Applying dynamic scaling on the cluster-mass distribution identified a transition point between regimes of slower and faster aggregation. The scaling exponents displayed typical values of DLCA and RLCA at the regime of slow aggregation
FIG. 13. Temporal evolution of the polydispersity, P(t), for the octyl system (h) and the octadecyl system (E).
2D PERIKINETIC AGGREGATION
but were increased tenfold at the regime of fast aggregation. The analysis suggested an increased collision frequency of the large clusters as a source for this behavior. Simple analysis of dimensionality, based on the Pe´clet number, indicated that convection affected the kinetics, as this made large clusters more mobile. We noted similarities in kinetics between nonrearranging systems that display a transition from DLCA to convection-limited cluster aggregation (CLCA) or from DLCA to RLCA. ACKNOWLEDGMENTS The authors acknowledge the late Prof. John Earnshaw for his helpful introduction to the field of colloidal aggregation in two dimensions. We also thank Prof. Brian Vincent for fruitful discussions. The Swedish Research Council for Engineering Sciences, TFR, supported this research.
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