Temperature effect on fractal structure of silica aggregates

Temperature Effect on Fractal Structure of Silica Aggregates PEI T A N G , * D A V I D E. COLFLESH,$ AND B E N J A M I N C H U * , i -,l Departments of *Chemistry, t Materials Science and Engineering, and SAnatomical Sciences HSC, State University of New York at Stony Brook, Long Island, New York 11794-3400 Received October 5, 1987; accepted J a n u a r y l 1, 1988

Light scattering and transmission electron microscopymeasurements are reported for aggregation of silica particles in suspension at high temperatures (50 ° and 80°C). Fractal dimensions df of 1.79-1.85 were determined using a variety of experimental procedures and data treatment methods. The values are smaller than df ~ 2 for silica clusters formed at room temperatures. An aggregation mechanism for the silica aggregateswith more open structures is presented. © 1988AcademicPress.Inc. I. I N T R O D U C T I O N

In recent years, considerable interest has developed in the formation of r a n d o m structures under nonequilibrium conditions. Meanwhile, the fractal concept has become an important approach in describing the geometry of r a n d o m structures produced by aggregation phenomena. In order to answer the question why these aggregates are fractal objects and how the fractal dimension is related to aggregation phenomena, theorists have devoted considerable efforts to build various models (1-3), such as cluster-cluster aggregation and diffusion-limited aggregation, to simulate aggregation processes. In the clustercluster aggregation model, both particles and clusters move about randomly until most of them aggregate, thus forming larger and larger self-similar objects having power law densitydensity correlations. One of the main results from this model is a fractal dimensionality of 1.75-1.8 for three-dimensional systems undergoing diffusion-limited aggregation. If some parameters in the model are modified, the fractal dimension will be changed, For example, when the sticking probability of clusters approaches zero, a fractal dimension of 2.1 in Author to whom correspondence should be addressed (Department of Chemistry).

the three-dimensional systems was obtained (4). This kind of aggregation process is referred to as reaction-limited aggregation, with df = 2.1 and 2.0 from realistic (polydisperse) and monodisperse (hierarchical) models, respectively. Experimentally, m a n y factors m a y influence the aggregation process. For a colloidal system, the colloids are aggregated by reduction of the charge on the surface of the particles causing them to stick upon diffusion-induced collision. Therefore, chemical reactivity, kinetic energy, mobility, mass, etc., of the clusters a r e notable factors governing the aggregation process. Over a certain range o f p H values and salt concentrations, extremely rapid aggregation can be induced, yielding clusters with a fractal dimension df of 1.75 (5, 6). Silica aggregates, with df = 1.75, were observed to restructure so as to yield d f ~ 2.1 (7). Very slow aggregations in an environment of low salt concentrations usually give rise to fractal dimensions of around two ( 8 - 1 0 ) . All the reported experiments were performed only at r o o m temperatures. It is the purpose of the present work to report the results of a study of the aggregation process for colloidal silica spheres at different temperatures. We found that silica aggregates formed between 50 ° and 80°C are still statistically scale-invariant objects. In comparing

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Journalof ColloidandInterfaceScience,Vol. 126, No. 1,November 1988

FRACTAL STRUCTURE OF SILICA

X 1 0 4 tO 3.2 X 10 s c m - 1 . The light scattering measurements followed the cluster growth until the static structure factor S ( q ) exhibited simple power-law dependence on q (S(q) q-dr) over the entire q range accessible by our light scattering spectrometer (15 ° ~ 0 ~< 140°). The final aggregated particle suspensions which had been dialyzed in order to remove all NaC1 were observed by using transmission electron microscopy ( H I T A C H I H U 12 ). The images obtained in this way can show very clear aggregated structures without interference from NaC1.

with the results measured at r o o m temperatures (df ~ 2.1), the high-temperature products have a m o r e open structure. Both light scattering and electron microscope image measurements showed that silica aggregates formed at higher temperatures ( 5 0 ° - 8 0 ° C ) have smaller fractal dimensions (1.79-1.85 ). II. EXPERIMENTAL All samples were prepared from very uniform-sized colloidal silica spheres (Ludox SM). Different concentrations of Ludox SM in water were measured by using small-angle X-ray scattering. F r o m the initial slope of a In I(0) vs 02 plot (desmeared data), a z-average radius of gyration of 58 A was obtained which corresponded to a z-average hard sphere diameter of 150 A as shown in Fig. 1. This value is in good agreement with the 145-A z-average value measured from our transmission electron micrographs. The aggregation was induced by suspending the spheres in 0.5 M N a C 1 solution at different controlled temperatures. The stock silica suspension (30 wt% silica) was diluted to 0.1 wt%. The p H value was adjusted to be 6 so that all data sets were obtained under the same conditions of p H and concentration except for temperature. An argon-ion laser operating at 488 n m was used for light scattering measurements over a range of scattering wave vectors, q, from 4.5

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III. RESULTS

1. Light Scattering Aggregation processes of colloidal silica suspensions at p H = 6 and 0.03 wt% concentration at three different temperatures of 22.0 + 0.8 °, 50.0 + 0.5 °, and 80.0 _ 0.1°C were followed by light scattering intensity measurements. Figure 2 depicts the time evolution of the static structure factor S ( q ) versus the scattering wave vector q for aggregation at 50°C. Aggregations at 22 ° and 80°C have time evolutions similar to that in Fig. 2. A notable feature in Fig. 2 is that as the particles become large, a log-log plot of S(q) versus q exhibits a power-law relationship over the entire q range which can be taken as an indicator for

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slow reaction-limited aggregation, which gives rise to a fractal dimension of 2.03 + 0.02 using linear least-squares fitting by means of the LLSQF routine in the IMSL library. Curves b and c, obtained only 4 to 5 days after the initial aggregation at higher temperatures (50 ° and 80°C, respectively), have slopes of 1.79 __+0.02 and 1.82 +~ 0.01, implying a more open (or looser) fractal structure. On the other hand, all the time evolution results can be very well fitted by Eq. [ 1], as shown by the solid lines in Fig. 2, according to the Fisher-Burford approximation (1 I):

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2. Evolution of the static structure factor S(q) with time during the aggregation process at 50°C. The diamonds, triangles, squares, and circles were measured at 46.5, 57, 74, and 91 h after aggregation, respectively. The solid lines are fits of the data over the indicated range by Eq. [1].

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fractal formation. Figure 3 plots the powerlaw relationships for the three temperatures. Curve a (22°C) was measured around 40 days after the initial aggregation and shows a typical

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FRACTAL

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not as rapid as typical diffusion-limited aggregations observed by Weitz et al. in the colloidal gold system (5, 6), and that there was no rapid restructuring as discussed in Ref. (7) for the silica system since our fractal dimension was kept almost constant until large clusters precipitated. In Ref. (7), the extremely rapid aggregation p r o d u c e d clusters with df = 1.75 ___0.05, but their fractal dimension changed to 2.08 within a period o f minutes to hours due to restructure. Besides, as shown in Fig. 5, m e a s u r e m e n t s o f the h y d r o d y n a m i c radius, Rh, as a function o f time did not satisfy the power-law kinetics with Rh ~ t (~/af), which was used as another criterion o f a m o r e restrictive diffusion-limited aggregation process. O u r Rh values were determined from translational diffusion coefficients D using the Stokes-Einstein relation. Figure 6 shows h o w D could be obtained f r o m an extrapolation to q = 0. In view o f the above two observations, i.e., that the aggregation process t o o k days instead o f minutes and a log-log plot o f Rh versus time did not obey a power law, the aggregations at high temperatures m a y have some characteristics which are different f r o m those o f diffusion-limited aggregation. Another interesting p h e n o m e n o n is that the 10 4

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h y d r o d y n a m i c radii Rh obtained from translational diffusion coefficients o f the aggregates formed at high temperatures (50 ° a n d 8 0 ° C ) are proportional to their radii o f gyration Rg in the range 850 fi < Rg < 8000 fit with a ratio Rh/R~ = 1.5 + 0.09. The value o f the ratio is strongly at variance with Rh/Rg = 0.72 + 0.02 in the range 500 fi < R e < 7000 fi obtained (12) for colloidal silica aggregation at r o o m temperatures. For simulated aggregates, the value Of Rh/Rg is 0.875 (13).

2. Transmission Electron Microscopy ( TEM)

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Fractal structures o f the clusters f o r m e d at different temperatures were examined by using transmission electron micrographs. Typical pictures of the aggregates at 22 ° , 50 ° , and 80°C are illustrated in Fig. 7. The structure in Fig. 7a ( 2 2 ° C ) is quite compact, whereas those in Figs. 7b and 7c (50 ° and 80°C, respectively) show a m u c h m o r e t e n u o u s and ramified appearance. Journal of Colloid and Interface Science, Vol. t26, No. 1, November 1988

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FIG. 7. Electron micrographs of colloidal silica clusters formed at different temperatures: (a) 22°C; (b) 50°C; (c) 80°C. We used the same analysis made by Weitz and Olivera (5), who pointed out that the point-to-point correlation function [ C ( r ) ] approach is valid only for the two-dimensional images of the three-dimensional clusters in solution in the absence of serious overlap in projection. In our case, the compact structure in Fig. 7a clearly showed overlapping of silica particles in the two-dimensional electron micrograph images. Thus, we did not try to determine a fractal dimension from such structures. In Figs. 7b and 7c, the more ramified appearance of the silica aggregates suggests some justification in following the analysis of Weitz and Olivera. We should, however, be aware of the distortion in C ( r ) produced by the overlap and plan to use this approach only as supporting evidence to our light scattering measurements. For an N-particle aggregate, the point-to-point correlation function can be expressed as 1

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Journal of Colloid and Interface Science, Vol. 126,No. 1, November1988

Take a circle with radius r centered at r' which represents the position of one particle. If the circle passes through a particle center, then this particle is counted, i.e., P(r + r') = 1; otherwise P(r + r') = 0. By moving the circle center r' to the center of different particles and by changing the radius r, C(r) of the three-dimensional fractal object projected onto a two-dimensional space can be calculated. Two representative correlation functions calculated for a cluster formed at 80 ° and 50°C are shown in F i g . 8. It is clear that for r ~ L (L is the whole cluster size), C(r) relates to r as

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where a = 2 - dr. The limiting slopes for a typical cluster at 80°C and another at 50°C yield a (80°C) = 0.21 _+ 0.06 and a (50°C) = 0.15 + 0.02, with the corresponding fractal dimensions of 1.79 + 0.06 and 1.85 _+ 0.02, respectively. The clusters, each with more than one thousand single silica particles, were used

FRACTAL STRUCTURE OF SILICA

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to calculate the correlation function C(r) at each temperature. Variations of a from different clusters were smaller than 8%. The average values are listed in Table I. Results obtained in this way are consistent

with those generated by the following analysis method. A particle near the center of gravity of the aggregate on the electron micrograph was picked as the center Of a series of concentric circles of different radii. The n u m b e r of Journal of Colloid and Interface Science, Vol. 126, No. 1, November 1988

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FIG. 7--Continued.

particles n ( r ) within each circle o f radius r is counted. By averaging the results using particles near the center o f gravity o f the aggregate as centers for the concentric circles, we can Journal of Colloid and Interface Science, Vol. 126, No. 1, November 1988

establish a relationship for n ( r ) and r. Figure 9 shows a log-log plot o f n versus r. The results show n ,~ rar with d f = 1.79 ___0.04 at 50°C and dr = 1.82 ___0.02 at 80°C, respectively.

FRACTAL STRUCTURE OF SILICA I

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second one is the sticking of the colliding clusters or particles. The kinetic process can be described as

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If the rate constant k2 ,> kl, the growth rate is controlled by A --* B, a case related to the r (era) so-called diffusion-limited aggregation. The FIG. 8. Point-to-point correlation functions of TEM process from A to B is mainly determined by images of silica aggregatesformed at 80°C (squares) and 50°C (circles). The slope of the linear portion of circles thermal Brownian motions; an increase in is -0.15, yieldingde = 1.85. The slope of the linear portion temperature will cause more frequent colliof squares is -0.21, yielding df = 1.79. sions so as to increase the rate of aggregation. This is one of the reasons why the aggregation is m u c h faster in high-temperature experiments. IV. DISCUSSION At a given frequency of collision, the stickI f we look only at a static property, silica ing probability is another important factor aggregates formed at high temperatures seem which is determined primarily by the nature to be the products of diffusion-limited aggre- of the short-range interaction between two apgation because their fractal dimensions (1.79proaching particles or clusters as illustrated 1.85 as summarized in Table I) are very close schematically in Fig. 10, in which the positive to the results from computer simulation of dif- U in curve A represents a repulsion barrier fusion-limited cluster-cluster aggregation. and is responsible for reaction-limited aggreHowever, the violation of power-law kinetics gation, while curve B represents diffusion( R h ~ t 1~dr) and an aggregation period of sevlimited aggregation. eral days (instead of minutes) reflects that silAccording to the Derjaguin-Landau-Verica aggregation at high temperature does have wey-Overbeek model (14 ), the screened couits own special variations. tombic repulsive barrier between two apIn brief, colloid aggregation can be assumed proaching particles UR is a function of the surto occur in two steps. The first step is the col- face charge and screening length. It is not lision between clusters a n d / o r particles. The surprising that by increasing the ionic strength 10 -5

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Journal o f Colloid and Interface Science, Vol. 126, No. 1, November 1988

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much but, as to be expected, it does increase the total number of higher-energy clusters, resuiting in an increase in the stick probability. On the other hand, higher-temperature operation also increases the diffusion rate, since decreases exponentially with increasing temperature. Both of these effects could promote aggregation so that higher rates of aggregation Occur.

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or by decreasing the pH value, the screening length or the surface charge density on the particles or clusters is reduced so as to lower the repulsive barrier. A diffusion-limited aggregation occurs. If, as in most experimental cases, the repulsive barrier does not vanish totally, the formation of the aggregates will be determined by the number of "high-energy" particles or clusters, which are able to overcome the barrier, and by the frequency with which the high-energy particles or clusters meet each other in terms of effective collisions. For a given height of barrier, the frequencies of effective collisions are different for different sizes of clusters: the larger the size, the slower the motion in an environment of given viscosity, and the less the chance to meet each other. In summary, for a process as described in [3] and with consideration of the viscous force acting upon the clusters by the solvent, the rate constant for aggregate formation can be written as k ~ kB___Texp -

Figure 11 shows a schematic representation of the effects of temperature on the structure of the silica aggregates. At room temperatures, the growth of fractal objects is dominated by aggregation of large colloidal clusters with single particles or smaller-sized clusters. The reason for this mechanism is that the larger clusters are less mobile. Under such a condition, even with a very low repulsive barrier, the aggregation process is mainly performed by the small particles or clusters running into the larger, less mobile clusters. As a result, relatively dense fractal structures are formed, as shown in Fig. 7a for aggregation at 22°C. However, by increasing the temperature to 50 ° and 80°C, the solvent viscosity is decreased exponentially. Then, the translational motions of clusters increase. If the temperature is high enough to enable considerable movements of larger clusters, the contribution from effective collisions between those larger clusters to form even larger aggregates increases as the sticking probabilities of aggregates could become higher. An obvious result of this mechanism

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FiG. 10. Schematic representation of the potential between charged colloidal particles (or aggregates). Curve A represents a reaction-limited aggregation system whereas curve B leads to rapid aggregation.

FRACTAL STRUCTURE OF SILICA room temperature (-20"C)

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reflects that the fractal growth is exponential with respect to time, as normally observed in a reaction-limited aggregation. These two aspects seem to imply that there is a transition f r o m an initial reaction-limited aggregation o f smaller particles to a consequent diffusionlimited aggregation o f larger clusters with high probability o f sticking. The larger the clusters, the larger is the range o f van der Waals attractive interaction between them, and the easier it is for the colliding clusters to overcome the shorter-range repulsion u p o n collision. ACKNOWLEDGMENTS

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is that the speed o f aggregation will n o w be faster than that o f large clusters capturing single small particles. This is why the T E M results at 50 ° and 80°C (see Figs. 7b and 7c) show looser scale-invariant structures. v. CONCLUSION Diffusion-limited and reaction-limited aggregations are two theoretical limits for aggregation processes if the collision frequency and sticking probability are considered as the two d o m i n a n t factors during fractal formation. O u r experimental results show, for the first time, that stable silica aggregates with a fractal dimension o f 1.79-1.85 can be obtained at high temperatures. Although a dimensionality o f 1.79-1.85 is typical for a diffusion-limited aggregation process, the d y n a m i c information

We gratefully acknowledge support of this work by the Department of Energy under Grant DEFG0286ER45237A001. B.C. wishesto thank M. M. Dewey,Health Sciences Center, Stony Brook, for permission to use the electron microscope. REFERENCES 1. Witten, Jr., T. A., and Sander, L. M., Phys. Rev. Lett. 47, 1400 (1981). 2. Meakin, P., Phys. Rev. Lett. 51, 1119 (1983). 3. Kolb, M., Botet, R., and Jullien, R., Phys. Rev. Lett. 51, 1123 (1983). 4. Jullien, R., and Kolb, M., J. Phys. A 17, 639 (1984). 5. Weitz, D. A., and Olivera, M., Phys. Rev. Lett. 52, 1433 (1984). 6. Weitz, D. A., Huang, J. S., Lin, M. Y., and Sung, J., Phys. Rev. Lett. 54, 1416 (1985). 7. Aubert, C., and Cannell, D. S., Phys. Rev. Left. 56, 738 (1986). 8. Schaefer, D. W., and Martin, J. E., Phys. Rev. Lett. 52, 2371 (1984). 9. Schaefer, D. W., and Keefer, K. D., Phys. Rev. Lett. 53, 1383 (1984). 10. Schaefer, D. W., Martin, J. E., and Keefer, K. D., J. Phys. C46, 3 (1985). 11. Fisher, M. E., and Burford, R. J., Phys. Rev. B 10, 2818 (1974). 12. Wiltzius, P., Phys. Rev. Lett. 58, 710 (1987). 13. Chen, Z.-Y., Meakin, P., and Deutch, J. M., Phys. Rev. Lett. 59, 2121 (1987). See also subsequent comments and reply by Pusey, P. N., Rarity, J. G., Klein, R., and Weitz, D. A., Phys. Rev. Lett. 59, 2122 (1987); Wiltzius, P., and van Saarloos, W., Phys. Rev. Lett. 59, 2123 (1987). 14. Verwey, E. J. W., "Theory of Stability of Lyophobie Colloids." Elsevier, Amsterdam, 1948. 15. Chu, B., Ford, J. R., and Dhadwal, H. S., in "Methods of Enzymology"(S. P. Colowiekand N. O. Kaplan, Eds.), Vol. 117, p. 256. Academic Press, Orlando, FL, 1985. Journal of Colloid and Interface Science, Vol. 126, No. I, November 1988