Reversible Agglomeration: A Kinetic Model for the Peptization of Titania Nanocolloids

Journal of Colloid and Interface Science 214, 283–291 (1999) Article ID jcis.1999.6218, available online at http://www.idealibrary.com on

Reversible Agglomeration: A Kinetic Model for the Peptization of Titania Nanocolloids Danijela Vorkapic and Themis Matsoukas 1 Department of Chemical Engineering, Pennsylvania State University, University Park, Pennsylvania 16802 Received October 27, 1998; accepted March 16, 1999

The final result is a clear suspension containing particles whose size ranges from 15 to 100 nm. Since the final size is still larger than the size of the primary particles, the peptized colloid consists of agglomerated structures containing several primary particles. The marginal stability of titania colloids has been demonstrated in a number of studies. The peptized colloid is often prone to reagglomeration, as manifested by a slow increase of size over the period of several days (5). Further evidence is found in the effect of shearing on colloidal stability. Look and Zukoski report that shear rates on the order 1 to 10 2 s 21 can induce aggregation in an otherwise stable suspension of titania (6). The insufficient stabilization of nanosize titania has led to the hypothesis that the slow peptization is indeed the result of the simultaneous action of two opposing mechanisms, the fragmentation of agglomerates, which is mediated by the peptizing agent, and the reagglomeration of the fragments, which is driven by insufficient stabilization of the suspension (5, 1). This hypothesis provided a satisfactory qualitative basis for the interpretation of various experimental aspects of these systems (1). In the context of this model, the highest degree of redispersion will be achieved under conditions that promote fragmentation of agglomerates (deagglomeration) while inhibiting the reagglomeration of the fragments. In order to interpret the kinetics of peptization and assign physical meaning to the kinetic parameters one must begin with a detailed accounting for the simultaneous action of deagglomeration and reagglomeration, and for this one must consider the population balance. The problem of simultaneous fragmentation/reagglomeration, or reversible agglomeration, has received considerable attention due to its relevance to a broad range of problems in colloidal science, polymerization reactions, and multiphase dispersions (7–11). Much of the theoretical work in the area of reversible agglomeration has focused on the solution of the population balance and on scaling relationships for the time evolution of the mean size for various size dependencies of the fragmentation and agglomeration rate constants (12–14). From an experimental point of view, the available data are usually limited to size measurements, while size distributions are rarely known, especially for colloids in the nanometer range.

We formulate a model for the peptization of titania nanocolloids. The model assumes simultaneous agglomeration/deagglomeration with size-independent rate constants. It predicts that the evolution of the particle size exhibits first-order kinetics and that the final particle size scales as R ; C 1/df, where C is the mass concentration of the colloid and d f is the fractal dimension. These predictions are tested experimentally. We find that the model provides a quantitative and consistent description of the peptization process over a wide range of experimental variables and that it allows the determination of the aggregation and deaggregation rate constants from the time evolution of the average size. © 1999 Academic Press

Key Words: titania; nanoparticles; agglomeration; deagglomeration; peptization.

I. INTRODUCTION

The agglomeration of colloids is often a problem, especially when small, monodisperse particles are desired. In some cases it is possible to peptize the agglomerated colloid and redisperse it in the form of unagglomerated particles. The deagglomeration process, also known as peptization, is achieved through mechanical and chemical means and aims at breaking the bonds that hold the primary particles together. The goal is a fully dispersed colloid, but in practice this is not always achieved. This is particularly true when the primary particles are in the low nanometer range because such sizes are difficult to stabilize electrostatically. Nanocolloidal titania formed from the hydrolysis and condensation of titanium alkoxides in water is a good example of this process. In the presence of a large excess of water (100 to 200 water-to-titanium by mole) titanium alkoxides hydrolyze and condense to produce ultrafine primary particles 3 to 10 nm in diameter (1– 4). These particles agglomerate very rapidly, and within seconds of mixing they produce large precipitates of macroscopic dimensions (1). Redispersion is achieved through the action of nitric acid, often aided by stirring or sonication. This is a slow process which has been observed to exhibit first-order kinetics (5) with time constants on the order of days. 1

To whom correspondence should be addressed. 283

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Our approach has been motivated by the need to provide a quantitative interpretation of the kinetics of deagglomeration. First we formulate the mathematical model for this process and develop predictions for the time evolution of the average size. These predictions are then put to test against experimental results obtained from the peptization of alkoxide-derived titania. II. KINETICS OF REVERSIBLE AGGLOMERATION

The premise of this kinetic model is that fragmentation 2 and reagglomeration occur simultaneously during peptization. Agglomeration is a second-order process in the concentration of particles, and in the special case of fast (diffusion-limited) agglomeration its rate constant can be calculated from the pair interaction in the suspension (15). The deagglomeration process is much less understood, especially since it may be brought about by a combination of mechanisms. It is generally believed that the deagglomeration of titania involves electrostatic repulsion among primary particles in an agglomerate and chemical cleavage of the bonds that hold these primaries together (1, 5, 16 –18). High shear and collisions between agglomerated particles can also contribute to deagglomeration. If the solids concentration is not too high the effect of collisions in the breakup process can be neglected and the rate of deagglomeration may be taken to be first order in the number concentration of particles. The equation that governs the total number of particles N can be written as

of size i. We define the average agglomerate particle size, n, as the number of primary particles per cluster; namely,

SD

R N0 5 n5 N r0

df

,

[4]

where r 0 is the size (radius) of the primary particles, R is the size of the mean agglomerate particle, and d f is the fractal dimension. The total number concentration of primary particles, N 0 , remains constant throughout the process and is given by N0 5

3C , 4 pr r 30

[5]

where C is the mass concentration of titania and r is the density of the primary particle. Inserting these definitions into Eq. [1] we obtain the corresponding equation for n: dn N 0 K# a 5 2 K# dn. dt 2

[6]

The relative strength of fragmentation and agglomeration is determined by the dimensionless ratio 2K# dn/K# aN 0 : the size decreases if this ratio is larger than unity, otherwise it decreases. A. Size at Steady State

dN 1 5 K# aN 2 2 K# dN, dt 2

[1]

where N is the total number concentration in the suspension at time t. The average rate constants for agglomeration, K# a, and deagglomeration, K# d, are defined as K# a 5 K# d 5

1 N2 1 N

O O K NN ij

i

i

[2]

j

j

O ~ f 2 1! K N , i

i

i

[3]

i

where N i is the number concentration of particles containing i primary particles, K ij is the agglomeration rate for the pair i and j, K i is the breakage rate of an agglomerate particle of size i, and f i is the mean number of fragments generated by its fragmentation. Notice that f i 2 1 is simply the net change in the total number concentration upon fragmentation of a particle 2 In this discussion the terms “deagglomeration,” “fragmentation,” and “breakup” are treated as synonymous and refer to the breakage (redispersion) of the agglomerate.

The particle size at steady state is obtained from Eq. [6] with dn/dt 5 0: K# a N 0 . n` 5 # Kd 2

[7]

The corresponding mass-equivalent sphere radius of the peptized particle is found from Eq. [7] in combination with Eqs. [4] and [5] to be

F

3C K# a R` 5 r0 8 pr r 30 K# d

G

1/d f

.

[8]

According to this result the final size (linear dimension) scales as C 1/d f provided that K# a and K# d are kept constant. Equation [8] is general and involves no a priori assumptions regarding the dependence of the rate constants on particle size, or any assumptions with respect to the number and distribution of fragments produced with each deagglomeration event. Indeed, the only condition is that agglomeration is second order in N while deagglomeration is first order. Implicitly it is also assumed that n @ 1 such that that fractal arguments apply.

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TABLE 1 Conditions for Titania Precipitation

B. Kinetics In general both K# a and K# d are functions of size and thus vary with time. However, a particularly simple situation arises if the rate constants are assumed to be size independent. In this case the solution to Eq. [6] is easily obtained and takes the form n n 0 2 n ` 2K# d~t2t 0! 511 e , n` n`

[9]

where n 0 is the number concentration of agglomerates at time t 0 and n ` is the steady-state mean size which is obtained from Eq. [7]. Experimentally it is the radius rather than the number of primary particles per agglomerate that is known. To obtain an equation for the size we combine Eqs. [4] and [9] to find

[10]

n0 2 n` exp~K# dt 0 ! 5 const. n`

[11]

Close to steady state, Eq. [10] can be expanded in a Taylor series as R a 2 ~d f 2 1! 22K# dt # 5 1 1 ae 2K dt 2 e 1... , R` 2

[12]

where a 5 b/d f . At sufficiently long times (a exp(2K# dt) ! 1) the higher order terms can be neglected to obtain R # 5 1 1 ae 2K dt , R`

~t @ ln a/K# d!.

TiO 2 Conc. a (mole/liter)

H 2O/Ti a (mole/mole)

pH

A B

0.916 0.454

38.6 92.6

1.10 6 0.05 1.10 6 0.05

a At precipitation; concentrations in the peptizing solution vary with dilution.

not depend on size. The latter assumption is highly simplified and its applicability must be assessed by comparing the model predictions with experimental data. III. EXPERIMENTAL

R # 5 ~1 1 be 2K dt ! 1/d f R` with b5

Sample

[13]

Therefore, the radius approaches the steady state value as a first-order process whose time constant is equal to the deagglomeration rate constant. The agglomeration rate appears implicitly in this result in the value of R ` , as seen from Eq. [8]. The model suggests a simple data reduction scheme from measurements of the particle size as a function of time: the deagglomeration rate can be obtained from a three-parameter fit (R ` , a, and K# d) based on Eq. [13] (a two-parameter fit, if the the experimental value is used for the final size). The agglomeration rate K# a is then obtained from K# d and the known final size, using Eq. [8]. The parameter a may be used to check the validity of the condition t @ ln a/K# d but is of no further significance. We reiterate that Eq. [13] is valid only sufficiently close to the steady-state value and that it is based on the assumption that the rate constants for fragmentation and agglomeration do

Titania nanoparticles were synthesized by reaction between titanium-isopropoxide and water in the presence of nitric acid. Titanium-isopropoxide was supplied by Aldrich and was used without further purification. Distilled water was used to initiate and complete the hydrolysis reaction. Nitric acid, supplied by J. T. Baker, was used as peptization agent and was present during precipitation as well. No alcohol was added. However, since the hydrolysis reaction produces isopropanol, the peptizing mixtures contained 1 to 4 M isopropanol (estimated assuming complete hydrolysis). Two recipes were used to synthesize the agglomerated particles in this study. The difference is in the water-to-titanium molar ratio used during precipitation (see Table 1). In general, high water-to-titanium molar ratios promote the formation of smaller sizes of the peptized colloid (1). Thus, by precipitating samples at various water-to-titanium ratios we can probe a wider range of particle sizes. The procedure for particle synthesis was as follows: A specified amount of nitric acid was mixed with water in a glass bottle and placed in a temperature controlled bath at 50°C. Titanium isopropoxide was added dropwise to the heated solution under constant magnetic stirring at 150 RPM. Precipitation occurred within seconds of the mixing of the reactants and was clearly visible in the form of a highly turbid solution. Peptization was initiated immediately by the presence of nitric acid in the reaction mixture. The temperature was held at 50°C for a period of 12 h, and then it was lowered to 25°C, where it was held constant for the rest of the peptization. The deagglomeration process was followed for several weeks until the clear blue-white solution was observed, indicating the disappearance of large agglomerates. Particle sizes were measured by dynamic light scattering in a 2030AT Brookhaven model with a He–Ne laser (l 5 632.8 nm). Size measurements were performed at 90° scattering angle and 25°C from samples diluted in water in order to avoid multiple scattering and particle interactions. The reported sizes represent the average of three measurements.

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FIG. 1. Peptization of titania agglomerates in nitric acid (pH 1.10). The line is a fit based on Eq. [13].

IV. RESULTS

The results of a typical peptization experiment are shown in Fig. 1. Following precipitation the system is a turbid solution containing large, visible precipitates. Size measurements by dynamic light scattering give reproducible results once the size becomes smaller than about 0.5 mm. Initially the decrease in particle size is rapid. It is then followed by a much slower decrease, which takes place over a period of several weeks, as the final size is approached. Only the slow decrease was fitted using Eq. [13]. The approach to steady state is described by a single decaying exponential, as shown by the inset in Fig. 1 (D ` 5 16.5 nm for this experiment). This fit is very good for peptization times larger than about 1 day but fails at earlier times. Recall, however, that the experimental conditions were not uniform throughout the entire process: during the first 12 h of peptization the temperature was kept at 50°C while the suspension was vigorously stirred; during the remaining 59 days, which constitutes the bulk of our measurements, the temperature was kept at 25°C and stirring was reduced to occasional gentle mixing. This procedure was found in our earlier work to yield the most reproducible results (1) and was adopted in the present work as well. Thus, the disagreement at the early part of the transient can be attributed to the change of experimental conditions. Moreover, Eq. [13] should only be applied sufficiently close to steady state. For the data set shown in Fig. 1 the fitted parameters from Eq. [8] are a 5 1.006, K# d 5 0.159 days 21. Therefore, the condition t @ (ln a)/K# d is satisfied over the time period covered by the exponential fit.

In the first set of experiments to be discussed we test the predicted scaling between the final particle and the concentration of solids. Titania was precipitated as described in the experimental section and the precipitate was allowed to peptize for approximately 6 h. It was then divided into four parts, and each was diluted in different amounts of nitric acid in water so as to maintain a constant pH of 1.10 in all samples. In this manner we obtained four different concentrations of titania, 0.0296, 0.0423, 0.0535, and 0.0727 g/cm 3, in otherwise identical solvent environments. These concentrations were calculated from the degree of dilution and the initial alkoxide concentration assuming stoichiometric conversion to TiO 2. The peptization of the four samples was followed over a period of three weeks and the results are shown in Fig. 2. All samples showed a decrease in size with time. However, the more dilute samples produced smaller final sizes. This is in agreement with Eq. [8], which predicts an inverse relationship between R ` and C. To directly test Eq. [8] we plot R ` against C as shown in Fig. 3. Two different sets of data are shown in this figure, each corresponding to precipitation conditions A and B in Table 1. The data corresponding to precipitation conditions A can be fitted to a power-law relation with exponent 0.582. The corresponding fractal dimension is d f 5 1/0.582 5 1.72. The second set of data corresponding to precipitation conditions B has a somewhat different behavior. At the high concentration end, dilution results in decrease of the particle size. At higher dilutions (lower concentration of TiO 2) the size becomes in-

FIG. 2. Peptization at different solids concentration. The concentration of titania is varied by diluting the sample under constant pH.

REVERSIBLE AGGLOMERATION

FIG. 3. Final particle size as a function of titania concentration. The solid line is a least-squares fit to the data points for sample A. The dotted lines are drawn qualitatively through the data points for sample B. The precipitation conditions for the two samples are different, resulting in larger particles overall for A (see Table 1 for precipitation conditions).

sensitive to dilution and a plateau is reached at a size of about 15 nm. Further dilutions have no effect on the size of the peptized colloid. Indeed, 15 nm is the smallest hydrodynamic diameter we have been able to observe. The agglomeration and deagglomeration rate constants derived from this analysis are shown in Fig. 4. The deagglomeration rate, K# d, is obtained directly from the time constant of the exponential fit in Eq. [13]. The agglomeration rate constant, K# a, is then calculated from Eq. [8], which can also be written as

S

K# a 5 K# d

8 pr r 30 3C

DS D R` r0

287

remain constant with the concentration of titania. Thus the differences in the final sizes of the peptized colloid are entirely due to the titania concentration and can be fully accounted for by Eq. [8]. This test is important in that it shows the rate constants obtained from this analysis to be consistent. Indeed, the rate constants should not depend on the concentration of the colloid but only on the chemistry of the solution (pH and ionic strength). In all of the above experiments the conditions are such that the rate of deagglomeration drives the system to smaller sizes. The same model, however, applies under conditions such that agglomeration overtakes deagglomeration. In this case the system will again reach steady state, but the size increases as it approaches its final value. To test the model under these conditions we performed salt-induced flocculation of the peptized colloid. First, a stable suspension was produced by peptizing the sol over a period of 2.5 months to a final hydrodynamic diameter of 16 6 1 nm. To this stable colloid we added varied amounts of a NaCl solution to produce salt concentrations ranging from 0.05 to 0.44 M. The resulting sols are characterized by constant pH (pH 1.10) but varying ionic strength. The size of the destabilized sols was followed for a period of one week, and the results are plotted in Fig. 5. Upon addition of NaCl all samples showed an increase in size over time. All samples approached a steady state value which was larger at the higher salt concentrations. The attainment of steady state is clear indication that deagglomeration takes place even in the presence of electrolyte. We note that the first measurement, taken within a minute of the addition of the salt,

df

.

[14]

For the primary particle we use 2r 0 5 5 nm based on TEM results reported earlier (1). This value is in the range reported for titania derived from alkoxides, and is also in agreement with X-ray diffraction measurements (19). The density is taken to be r 5 3.84 g/cm 3, corresponding to that of anatase (20). This choice is based on X-ray diffraction analysis of the samples which shows no rutile and only small amounts of brookite. The fractal exponent d f and the deagglomeration rate constant K# d are obtained from the previous analysis, while the final particle size R ` is measured directly. Thus all the necessary parameters are known. As we see in Fig. 4, K# a and K# d

FIG. 4. Agglomeration and deagglomeration rate constants obtained from the data in Fig. 2.

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VORKAPIC AND MATSOUKAS

FIG. 5. Destabilization of the peptized colloid by NaCl. Solid lines are model fits based on Eq. [13].

(K# d by our nomenclature) scales with the concentration of titania as C 0.5 , and on the basis of the fractional exponent they suggested that peptization involves sequential deagglomeration and reagglomeration steps. From the results presented here it is clear that K# d does not depend on the concentration of titania if pH and ionic strength are kept constant. In the experiments of Bartlett and Woolfrey the titania concentration was varied at fixed molar ratio [HNO 3]:[Ti]; thus neither the pH nor the ionic strength were kept constant. Therefore, their result reflects variations in the electrostatic environment of the colloid rather than a true dependence of the peptization rate on the concentration of titania. The value d f 5 1.72, which we obtained from the concentration dependence of the final size (Fig. 3), compares favorably with the d f 5 1.5 obtained by Bartlett and Woolfrey (5), who measured it directly using static light scattering. We did not attempt to measure the fractal dimension by light scattering because the wavelength of our light source (632.8 nm) is too large to probe the internal structure of particles ;100 nm or smaller. The fractal dimension offers suggestions as to the mechanism of agglomeration. Diffusion-limited cluster– cluster aggregation produces d f 5 1.75–1.8 while reaction-limited growth produces higher dimensionalities (d f ; 2) (21). Given the very rapid precipitation of titania during hydrolysis (1) and the marginal stability of small titania particles (5, 6),

always results in a size that is larger than the known initial size of the colloid (16 nm). This initial increase was found to be more pronounced at higher salt concentrations and ranged from less than 0.5 nm at 0.1 M NaCl to 9 nm at 0.44 M NaCl. For this reason we have left the initial size as a parameter to be fitted instead of using the known size before the addition of salt. The fits obtained in this manner are shown in Fig. 5 and show that Eq. [13] provides a very good description of the size evolution. The corresponding rate constants are shown in Fig. 6 and are also tabulated in Table 2. The agglomeration rate is strongly dependent on salt and increases with increasing concentration of NaCl. By contrast, the deagglomeration rate is independent of the salt concentration. This suggests that the redispersion of the agglomerated colloid is not controlled by electrostatic repulsion among charged primary particles because such a mechanism would exhibit a dependence on the ionic strength. V. DISCUSSION

Two specific predictions of the model have been experimentally tested, first, that the size approaches steady state as a first-order process, and second, that the final size has a powerlaw dependence on the concentration of the colloid under constant pH and ionic strength. Both predictions were quantitatively confirmed. Our observation of first-order kinetics is in agreement with the results of Bartlett and Woolfrey (5). These authors also reported that the corresponding kinetic constant

FIG. 6. Agglomeration and deagglomeration rates as a function of salt concentration.

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REVERSIBLE AGGLOMERATION

TABLE 2 Summary of Kinetic Experiments TiO 2 (g/cm 3)

NaCl (M)

D (nm)

K# d (1/day)

K# a (cm 3/s)

W

t 1/2 (days)

N (1/cm 3)

1.39E 1 10 1.25E 1 10 1.54E 1 10 1.58E 1 10

2.52 2.14 2.55 2.91

1.04E 1 16 1.10E 1 16 1.13E 1 16 1.02E 1 16

1.17E 1 09 8.49E 1 08 7.28E 1 08 3.18E 1 08 1.16E 1 08

0.632 0.498 0.799 0.610 0.563

3.46E 1 15 3.20E 1 15 1.71E 1 15 9.79E 1 14 3.85E 1 14

Peptization a 2.96E 2 02 4.23E 2 02 5.35E 2 02 7.27E 2 02

0 0 0 0

20.5 24.5 27.5 35.0

8.86E 2 22 9.87E 2 22 8.01E 2 22 7.81E 2 22

0.398 0.467 0.393 0.344

Salt-induced agglomeration b 7.04E 2 03 6.70E 2 03 5.90E 2 03 5.20E 2 03 4.10E 2 03 a b

0.05 0.10 0.20 0.30 0.44

16.9 17.1 22.9 29.5 44.1

1.06E 2 20 1.45E 2 20 1.69E 2 20 3.88E 2 20 1.07E 2 19

1.581 2.008 1.252 1.639 1.775

Sol preparation conditions according to sample A in Table 1. Sol preparation conditions according to sample B in Table 1.

the agglomeration rate is expected to be dominated by diffusion rather than by slow reaction between clusters. The fractal dimension obtained here supports this notion. From Fig. 3 we also note that the power-law dependence breaks down below about 15 nm diameter. Given that the size of the primary particles is estimated to be 3 to 10 nm, this size represents a cluster of 2 to 25 primary particles (assuming d f ' 1.6–3). At present, 15 nm is the smallest stable size, as measured by light scattering, that we have been able to produce. Figure 3 suggests that this represents a limit that cannot be further reduced. It is possible that the continuous deagglomeration–reagglomeration process results in a structural rearrangement of the cluster that produces a tightly held particle that is not susceptible to further peptization. Interestingly, even though the predicted scaling is not observed in the lower concentration end in Fig. 3, the approach to steady state is still represented quite accurately by a first-order decay, as predicted by Eq. [13]. We caution, however, that the parameters of this fit may not be readily identified with the rate constants of the process as such identification presumes the validity of Eq. [8]. A practical aspect of our model is that it allows us to experimentally obtain the agglomeration and deagglomeration rate constants. The agglomeration rate can be characterized in terms of the stability factor, W (15), K0 W5 # , Ka

[15]

where K 0 5 8kT/3 h is the Smoluchowski coagulation rate for hard spheres of the same size, k is the Boltzmann constant, T 5 298 K is the peptization temperature, and h 5 0.89 cp is the viscosity of the solvent (water). The corresponding stability ratios, shown in Table 2, are on the order of 10 10 for peptization

in the absence of salts. Since the agglomeration rate constant is taken to be independent of size, these stability ratios represent the average over the duration of peptization (21 days). The large magnitude should not be interpreted to mean that the suspension is stable against agglomeration. It is more instructive to compute the half-life of the suspension, calculated as follows (15): t 1/ 2 5

3h W. 4kTN

[16]

Here N is the number concentration of particles and is estimated from Eqs. [4] and [5]. The values given in Table 2 are based on the number concentration at steady state. This halflife is on the order of days. It is interesting to note that t 1/ 2 can be obtained in a simpler way as the inverse of the deagglomeration constant. To show this, we begin with Eq. [6] and set the derivative equal to zero. Using Eqs. [4] and [15] the result becomes K# d 5

K# aN 4kTN 1 5 5 . 2 3hW t 1/ 2

[17]

Due to the change in N, t 1/ 2 changes during peptization, until at steady state it becomes equal to the inverse deagglomeration rate. Thus, the relationship K# dt 1/ 2 5 1 is an alternative statement of the steady state condition and is entirely equivalent to Eq. [8]. Some insight into the peptization process can be gained by studying the deagglomeration rate constant. The fact that K# d does not depend on the ionic strength indicates that the role of electrostatic repulsion among primary particles is unimportant

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VORKAPIC AND MATSOUKAS

in bringing about the disintegration of agglomerates. It is more likely that peptization involves the breakdown of chemical bonds that hold primary particles together, as has been suggested for zirconia colloids (18). This hypothesis is further supported by the solubility of titania in the presence of acids. At pH 1, approximately 1% of titania is in the form of soluble species (19). High-curvature structures, such as small necks connecting nanometer-size crystallites together are most susceptible to acid attack and dissolution, thereby slowly releasing primary particles in solution. In this picture, fragmentation is most probably binary, with one primary particle being released at a time. Since nanometer-size particles are colloidally unstable, they tend to re-attach to larger agglomerates, thus fueling the reverse reaction. Some comments are due with respect to the model assumptions. In general, as long as the rates of agglomeration and deagglomeration are of different order in N, steady state is possible in a batch system. To quantify the steady state value and obtain Eq. [8] we introduced the assumption that the rates for agglomeration and deagglomeration are second and first order, respectively, with respect to the particle concentration. With regard to agglomeration, this is the standard assumption for agglomeration in dilute systems. With respect to deagglomeration, first-order kinetics are justified if the predominant mechanism is due to chemical attack of the bonds that hold primary particles together. In this case the release of fragments from clusters occurs independently. This picture presumes that interparticle collisions and shear, which may also promote fragmentation, do not play a role. Thus our analysis applies to dilute suspensions in the absence of stirring. Perhaps more severe is the assumption that K# a and K# d are constant during peptization. This assumption builds a rather simplified picture of the process, yet it results in very good agreement with the experiment. In diffusion-limited agglomeration without particle interactions the agglomeration rate is K# a 5 8kT a /3 h with a 5 1.075 (22). As a result of the homogeneity of the Smoluchowski coagulation rate, in this case K# a is indeed independent of size. With electrostatic interactions present, the homogeneity of the coagulation kernel is not preserved and variation of K# a with size is expected. Notice, however, that sizes in our experiments vary by a factor of 2 or less. Moreover, it is often experimentally observed that the stability factor is rather insensitive to the size of the colloid (15). It is not unrealistic then to assume that K# a remains relatively constant over the relatively narrow range of sizes encountered here. With respect to the deagglomeration rate constant, the literature offers little guidance as to its expected dependence on size because the kinetics of fragmentation are highly dependent on the underlying mechanism. In size reduction processes the breakup rate is usually expected to decrease with size until a size is reached which does not reduce any further (11, 13, 23–25). The data in Fig. 3 seem to support the notion that an irreducible cluster size exists at around 15 nm. However, there

is no sufficient information in these data to determine the size dependence of K# d. As shown in Fig. 4, the deagglomeration rate is the same for particles whose final size ranges from 20 to 35 nm (see Table 2). The quantitative fit provided by Eq. [13] is further, though indirect, evidence that K# d is indeed constant in our experiments. Most likely the range of sizes encountered here is rather narrow to allow the manifestation of size-dependent effects. As a final comment, it is interesting to look at a closely related system. First-order kinetics are predicted and have been observed in systems involving simultaneous agglomeration and breakup under shear. The kinetic equation used to describe this process is (26, 27) dN 1 5 K# aN 2 K# dN 2 , dt 2

[18]

where K# a, K# d are the rate constants for agglomeration and breakup, respectively, and are functions of the shear rate. This equation has a remarkable similarity to Eq. [1] but an important difference as well: In Eq. [18] the agglomeration rate is taken to be first order and the breakup rate second order in N, the opposite from our model. The scaling for K# a is based on the well-known Smoluchowski result for shear-induced aggregation in the absence of hydrodynamic interactions which predicts K# a ; R 3 ; 1/N (28), hence K# aN 2 ; N. The breakup scaling arises from the assumed mechanism for fragmentation under turbulent shear (26). Equation [18] is mathematically identical to Eq. [1] with the roles of K# a and K# d reversed. Its solution is the same as that for Eq. [13], but the decay constant is now identified as the agglomeration rate, K# a. While both solutions predict exponential transients, there is an important difference: According to Eq. [18] dilution should result in larger size as the second-order process (in this case deagglomeration) is inhibited relative to the first-order process (agglomeration). This is contrary to the observed behavior in the experiments presented here. Thus we can discount the applicability of the shear model to the titania system. VI. CONCLUSIONS

The kinetic model for simultaneous agglomeration– deagglomeration provides quantitative description of the size evolution during the acid peptization of titania nanocolloids. The model predicts first-order decrease of size and power-law scaling of the form R ` ; C 1/d f between the particle concentration and size. Both predictions were verified experimentally. In addition, the agglomeration and deagglomeration can be obtained from the time dependence of the size of the peptizing colloid. The experimentally obtained value for d f , the invariance of the agglomeration and deagglomeration constants with dilution, and the increase of the experimentally obtained agglomeration rate on the concentration of electrolyte demon-

REVERSIBLE AGGLOMERATION

strate the consistency in the parameters of the kinetic model. Finally, from the lack of dependence of the deagglomeration rate constant on the salt concentration we also conclude that deagglomeration is dominated by the chemical breakdown of bonds between primary particles and that electrostatic repulsion helps to stabilize the colloid but has no measurable effect in the deagglomeration step. This analysis is done with no a priori assumptions regarding these processes other than that the rate constants be independent of size. This apparently restrictive condition can be met by limiting the attention to sizes close to the steady-state value and was not found to limit in any way the analysis of the experimental data presented here. Thus process parameters can be studied quantitatively as to their effect in the redispersion of agglomerated colloids. ACKNOWLEDGMENT This work was supported in part by the National Science Foundation under grant CTS 9702653.

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