Growth Kinetics of Nanosize Silica in a Nonionic Water-in-Oil Microemulsion: A Reverse Micellar Pseudophase Reaction Model

Journal of Colloid and Interface Science 218, 68 –76 (1999) Article ID jcis.1999.6232, available online at http://www.idealibrary.com on

Growth Kinetics of Nanosize Silica in a Nonionic Water-in-Oil Microemulsion: A Reverse Micellar Pseudophase Reaction Model K. Osseo-Asare 1 and F. J. Arriagada Department of Materials Science & Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802 Received December 19, 1997; accepted March 26, 1999

particles in the NP-5 microemulsion system are reported. Emphasis is given to the effect of the reverse micellar population on the growth process, so that the water-to-surfactant molar ratio (R) is the variable of interest. The experimental results are analyzed quantitatively in terms of a reverse micellar pseudophase model (8 –10), according to which the microemulsion reaction medium may be viewed as a two-phase system consisting of a reverse micellar pseudophase and a bulk oil phase.

The growth kinetics of silica nanoparticles synthesized by the microemulsion-mediated alkoxide hydrolysis method was investigated with tetraethoxysilane (TEOS) as the silica precursor and polyoxyethylene (5) nonylphenylether (NP-5)/cyclohexane/ammonium hydroxide as the water-in-oil microemulsion system. The time evolution of the mean diameter of the silica particles was determined for different values of the water-to-surfactant molar ratio (R). Particle growth was found to be a slow process, where under typical experimental conditions at room temperature ([TEOS] 5 0.024 M, 29.6 wt% NH 3, water-to-TEOS molar ratio (h) 5 7.8) the particles achieved their terminal size after several days. During the early stages of the reaction, particle growth followed first-order kinetics, and the observed first-order growth rate constants decreased with increase in R. A reverse micellar pseudophase model (which considered the partition of reactants between the reverse micellar pseudophase and the bulk oil phase) was developed to analyze the growth kinetics under the hypothesis that TEOS hydrolysis was rate controlling. The pseudophase model predicted an inverse relationship between the observed growth rate and R, in agreement with experiment. The roles of steric effects and the bound state of water molecules, in retarding the hydrolysis rate, were highlighted by examining the effect of R on the TEOS hydrolysis rate constant in the reverse micellar pseudophase. © 1999 Academic Press Key Words: silica; w/o microemulsion; nanoparticles; tetraethoxysilane; TEOS; polyoxyethylene nonylphenyl ether; growth kinetics; pseudophase model.

BACKGROUND

Kinetics of TEOS Hydrolysis and Particle Growth in Homogeneous Alcoholic Solutions The formation of silica by the alkoxide hydrolysis process may be described by the overall reaction Si~OR! 4 1 2H2O 5 SiO2 1 4ROH.

The detailed mechanism involves both hydrolysis and polycondensation reactions. It is recognized that, in the presence of excess water, the base (ammonia)-catalyzed hydrolysis reaction of tetraethoxysilane (TEOS) in short-chain alcohol solvents is first order with respect to both the ammonia concentration and the TEOS concentration (11–14). Kay and Assink (15) suggested that in both acid- and base-catalyzed systems the rate is also first order with respect to the water concentration, while Byers et al. (11) indicated that the base-catalyzed hydrolysis rate has a stronger dependence on the water concentration (order of 1.5). The rate of TEOS hydrolysis is also affected by the nature of the solvent, and this has been attributed to both steric effects and hydrogen bonding (11, 12, 16 –18). In general, the hydrolysis rate decreases for longer alkyl chains of the solvent (steric effects) and for solvents which hydrogen bond strongly with water molecules (11, 12, 17). The detailed mechanisms of nucleation and growth leading to the formation of monodisperse silica particles are still a matter of controversy (11, 12, 18 –29). A monomer-addition growth model has been presented by Matsoukas and Gulari

INTRODUCTION

The synthesis of nanosize silica particles by the ammoniacatalyzed hydrolysis of tetraethoxysilane (TEOS) in the polyoxyethylene (5) nonylphenyl ether (NP-5)/cyclohexane/ ammonium hydroxide water-in-oil microemulsion system was discussed in previous papers from this laboratory (1–7). The particle size was found to be a strong function of the waterto-surfactant molar ratio (R) and of the ammonia concentration in the aqueous phase. In the present paper, kinetic studies aimed at providing further insight into the mechanisms of formation of silica 1

To whom correspondence should be addressed.

0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.

[1]

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GROWTH KINETICS OF NANOSIZE SILICA

69

(21–23), while Zukoski and co-workers (24 –27) have offered a nuclei-aggregation model. A compromise between these two models has been proposed by van Blaaderen et al. (28, 29). According to these investigators, particle growth involves monomer addition, with the growth rate controlled by the rate of the alkoxide hydrolysis. On the other hand, particle nucleation proceeds via the aggregation of siloxane subparticles. Growth Kinetics in Reverse Micellar Media In the light of the above considerations, two specific factors which seem of relevance to the present work can be identified: (a) the importance of silica solubility and supersaturation on the growth mechanism, and (b) the sensitivity of the TEOS hydrolysis rate to steric or hydrogen bond effects. The water-in-oil microemulsions used in the present experiments are initially ethanol free; this is considered to be an important difference when compared with the case of homogeneous synthesis in alcoholic solutions. It is known that the solubility of amorphous silica is much lower in alcoholic media than in alcohol-free solutions (30). Thus, for a given content of silicic acid and pH, the level of supersaturation will be considerably lower in the aqueous pool of the ethanol-free reverse micelles than in a homogeneous alcoholic solution under similar conditions. The second factor of importance to this work is the observation that the hydrolysis rate of TEOS is significantly retarded by steric effects or by conditions that promote binding of water molecules to the reaction medium (e.g., by hydrogen bonding with the alcohol solvent). For example, Byers and Harris (12) reported that replacing n-butanol by tert-butanol as solvent induced about a sevenfold decrease in the hydrolysis rate. Since TEOS hydrolysis in the reverse micellar system is expected to be affected by both steric effects from the surfactant layer and by the state of solubilized water (i.e., “bound” to the surfactant polar groups or “free” in the water pool (1, 7, 31)), it is likely that hydrolysis would play an even more important role in the overall growth kinetics in the microemulsion system than in the case of homogeneous alcoholic media. The oxyethylene groups of the surfactant molecules can interact with water molecules via hydrogen bonding. When the amount of water in the organic phase is very low, most of the water molecules are hydrogen bonded to the surfactant polar groups and they are said to be bound. With increase in the water content, a stage is eventually reached where unbound or free water molecules become available. Spectroscopic experiments conducted in our laboratory (1) with a fluorescence probe (ruthenium tris(bipyridyl), Ru(Bpy) 3) indicate that in the NP5/cyclohexane/water system, free water molecules become available when the water-to-surfactant molar ratio (R) exceeds ;1. Using a variety of spectroscopic techniques (NMR, ESR, near-IR), Kawai et al. (31) identified three states of the solubilized water in polyoxyethylene (6 and 10) nonylphenyl ether/ cyclohexane/water microemulsion systems, i.e., water bound

FIG. 1. Time evolution of the mean diameter of SiO 2 particles produced in the NP-5/cyclohexane/NH 4OH reverse micellar system at different R values; [TEOS] 5 0.024 M, 29.6 wt% NH 3, h 5 7.8, T 5 22°C.

directly to the oxyethylene groups, water bound to the hydrated polar groups, and bulk-like water in the inner polar core. EXPERIMENTAL PROCEDURES

Materials and Methods The materials and procedures for the preparation of particles in the NP-5/cyclohexane/NH 4OH reverse micellar system have been described previously (1–7). Reverse microemulsions with water-to-surfactant molar ratios (R) in the range of 0.7 to 5.4 and with concentrated ammonium hydroxide (29.6 wt% NH 3) as the aqueous solubilizate were used. The total microemulsion volume was typically fixed at 5.07 mL and NP-5 concentration was varied in the range 0.056 – 0.274 M; a constant amount (0.04 mL) of concentrated ammonium hydroxide was added. The overall TEOS concentration and the water-to-TEOS molar ratio (h) were maintained constant at 0.025 M and 7.8, respectively. Dispersion samples were extracted at different reaction times and deposited on TEM grids. Number-average particle diameters and size distributions were determined on enlarged TEM micrographs with a ZIDAS image analysis system (Zeiss). Under the experimental conditions investigated, final particle sizes were obtained after about 160 h reaction time. RESULTS AND DISCUSSION

Effect of R on the Time Evolution of Particle Size The number-average particle diameters (d n) obtained as a function of reaction time at selected R values are shown in Fig.

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OSSEO-ASARE AND ARRIAGADA

2d@TEOS#/dt 5 d@SiO2#/dt 5 k@TEOS#.

FIG. 2. Time evolution of the normalized standard deviation of SiO 2 particles produced in the NP-5/cyclohexane/NH 4OH reverse micellar system at different R values; [TEOS] 5 0.024 M, 29.6 wt% NH 3, h 5 7.8, T 5 22°C.

1. As can be seen in this figure, particle growth proceeds rapidly at short times, especially in the low R range. For example, particles produced at an R value of 0.68 have achieved ca. 81% of their final size after 21 h, while the equivalent figure decreases to about 77, 70, and 65% for R values of 1.56, 3.17, and 5.37, respectively. The time dependence of the size distribution of the particles, expressed as the standard deviation over the number-average diameter (normalized standard deviation, in %), is shown in Fig. 2. As seen in this figure, the size distribution of the particles decreases rapidly at the beginning of the reaction (ca. the first 30 h), and then it decreases slowly, becoming almost time-independent at longer reaction times. In general, the kinetics of particle growth in the reverse micellar system was slower than that observed for homogeneous media. In a homogeneous alcoholic solution, terminal particle sizes are achieved within a few hours under typical reaction conditions (e.g., 0.17 M TEOS, 1.3 M NH 4OH, and a water-to-TEOS ratio of 20 (26)). In contrast, growth to the final size takes several days in the reverse micellar system ([TEOS] 5 0.02 M, 15.3 M NH 3 in aqueous phase, water-to-TEOS ratio of 8). The slow growth rates observed in the reverse microemulsion may be associated with the compartmentalization of reagents in this microheterogeneous reverse micellar system.

[2]

The fractional conversion, X SiO2, was evaluated as (d t /d f) 3 , where d t and d f are, respectively, the particle diameter at time t and the final diameter. The results obtained for the particle size data for up to 60 h of reaction are presented in Fig. 3. As seen in this figure, particle growth is well represented by first-order kinetics for all R values investigated, but only up to about 21 (R 5 0.68) to 30 h (R 5 5.37) of reaction. At these times, the conversion of TEOS to silica ranges from 56.4 (R 5 0.68) to 37.1% (R 5 5.37). The deviation from first-order kinetics suggests a change in reaction mechanism; this behavior is currently under further investigation. The first-order growth rate constants (k g) obtained from the initial slopes in Fig. 3 are presented in Fig. 4. Also shown in this figure are the rate constants for two additional sets of synthesis experiments conducted under similar conditions ([TEOS] 5 ca. 0.025 M and concentrated ammonium hydroxide (wt% NH 3 5 29.6, i.e., [NH 3] 5 15.3 M)). The value of the rate constant k g in these additional experiments was evaluated from the particle size attained after 21 h and from the corresponding final size. As seen in Fig. 4, there is reasonable reproducibility among all three sets of data. The results, indicate that the growth rate constant k g decreases as R increases. The maximum value for the rate constant (observed at the

Growth Rate Law The particle size versus time data (Fig. 1) were analyzed by assuming first-order kinetics with respect to TEOS concentration; i.e.,

FIG. 3. First-order growth kinetics of SiO 2 particles in the NP-5/cyclohexane/NH 4OH reverse micellar system at different R values; [TEOS] 5 0.024 M, 29.6 wt% NH 3, h 5 7.8, T 5 22°C.

GROWTH KINETICS OF NANOSIZE SILICA

FIG. 4. Effect of the water-to-surfactant molar ratio (R) on the first-order growth rate constant (k g) of SiO 2 particles in the NP-5/cyclohexane/NH 4OH reverse micellar system; [TEOS] 5 ca. 0.025 M, 29.6 wt% NH 3, T 5 22°C.

lowest R value) is about 0.00073 min 21 for the present conditions, i.e., [TEOS] 5 0.024 M (referred to total microemulsion volume; [TEOS] 5 ca. 4.8 M, referred to the water pool if TEOS were completely solubilized in the reverse micellar pseudophase), water to TEOS molar ratio (h) 5 7.8, and [NH 3] 5 15.3 M (referred to the water pool). This value for the rate constant can be compared with a typical growth rate constant of 0.009 min 21 obtained in a homogeneous ethanolic solution ([TEOS] 5 0.17 M, [NH 3] 5 1 M, h 5 20 (24)).

71

viewed as having a thickness of a few atomic layers and as being highly structured. For example, from studies on the rheological behavior of silica suspensions, Greenberg et al. (32) suggested that ca. 5 layers of water molecules were strongly adsorbed on a given moving silica particle. Sasaki and Maeda (34) concluded from TEM and dynamic light scattering investigations that silica particles of ;6-nm radius are surrounded by a 7-nm-thick hydration layer. The presence of a hydration layer on silica plates and colloidal silica has also been demonstrated by direct force measurements (35–38). In addition, water molecules may be bound not only by interaction with hydroxylated groups at the silica surface (30) but also by the oxyethylene groups of the surfactant molecules. Under these conditions TEOS hydrolysis may be inhibited. Accordingly, the hydrolyzed TEOS-containing reverse micelles may be considered as the main source of monomer for particle growth. Within the general framework described in Fig. 5, the observed growth kinetics may be controlled by: (a) slow release of monomer via TEOS hydrolysis in the empty reverse micelles, (b) slow surface reaction of monomer addition to the growing particle, and (c) slow transport processes as determined by the dynamics of intermicellar mass transfer. The possibility that the rate of growth is controlled by the rate of hydrolysis of TEOS is explored below. As noted in the background section, the hydrolysis rate of TEOS molecules may have an important role on the overall growth kinetics in the reverse micellar media. Steric and hydrogen bonding effects can retard hydrolysis significantly, and both inhibiting effects may be enhanced in the reverse micelles where the reaction locale involves the oil/water interface. Hydrolysis-Controlled Growth Kinetics The observed growth kinetics can be analyzed in terms of the overall hydrolysis and condensation reactions

Particle Growth in the Reverse Micellar System The situation in the microemulsion phase wherein the silica particles are growing is illustrated schematically in Fig. 5. The fluid phase may be viewed as consisting of particle-filled reverse micelles and “empty” reverse micelles (i.e., free of silica particles, but containing hydrolyzed TEOS molecules). In addition, a significant fraction of the TEOS molecules remains in the bulk oil phase during the reaction. Growth may be the result of (a) transfer of hydrolyzed TEOS species from a reverse micelle to a particle-filled reverse micelle, and/or (b) direct interaction of unhydrolyzed TEOS molecules with a particle-filled reverse micelle. In the latter case, prior to growth, hydrolysis of TEOS has to proceed in the thin “water-shell” or hydration layer (30, 32–38) surrounding the particles. It is believed that hydrolysis of TEOS in a particle-filled reverse micelle may be less significant than hydrolysis in an empty reverse micelle. The water shell is

FIG. 5. Growth mechanisms in the SiO 2/NP-5/cyclohexane/NH 4OH reverse micellar system.

72

OSSEO-ASARE AND ARRIAGADA OH 2

Si~OR! 4 1 4 H2O O ¡ Si~OH! 4 1 4 ROH

1 dm p 5 k9H@TEOS#. V 0 dt

[3]

kH k C OH 2

- 0 SiO2~s! 1 2 H2O, Si~OH! 4 |

[4]

OH 2k D

where k H, k C, and k D are, respectively, the rate constants for hydrolysis, condensation, and dissolution. In homogeneous alcoholic media, the hydrolysis rate (r 5 d[TEOS]/dt) is first order with respect to TEOS, water, and catalyst (OH 2) concentrations (14, 15), as represented in Eq. [5],

[10]

The rate expression of Eq. [10] corresponds to a first-order growth rate. Therefore, the rate constant for growth, k g (Fig. 4), can be identified with the overall first-order hydrolysis rate constant, k9H. Thus, we obtain 1 dm p d@TEOS# 52 5 k9H@TEOS# 5 k g@TEOS# V 0 dt dt

[11]

@TEOS# 5 @TEOS# iexp~2k gt!,

[12]

or 2

r 5 k H@TEOS#@H2O#@OH #,

[5]

2

where [TEOS], [H 2O], and [OH ] are the molar concentrations of the indicated species. Under conditions of excess water and constant catalyst concentration, the hydrolysis rate (r) can be expressed as r 5 k9H@TEOS#,

[6]

where k9H 5 k H[H 2O][OH 2] is the pseudo-first-order hydrolysis rate constant. Assuming that condensation involves only the participation of silicic acid (the monomer), under conditions of constant water and catalyst concentrations, the resulting rate expressions are d@TEOS# 5 2k9H@TEOS# dt

[7]

d@Si~OH! 4 # 5 k9H@TEOS# dt 1 k9CA~@Si~OH! 4 # eq 2 @Si~OH! 4 #! 1 dm p 5 2k9C A~@Si~OH! 4 # eq 2 @Si~OH! 4#! V 0 dt

[8] [9]

where k9C is the overall condensation rate constant, A the total surface area of the solid particles, [Si(OH) 4] and [Si(OH) 4] eq the molar concentrations of silicic acid at time t and at equilibrium, respectively, m p the total moles of silica in the product particles at time t, and V 0 the total volume of the microemulsion phase. Under conditions where the hydrolysis process is slow as compared with condensation, the steady-state approximation can be made. That is, it can be assumed that the concentration of the intermediate (Si(OH) 4) is small and constant during most of the reaction. If d[Si(OH) 4 ]/dt 5 0, then it follows from Eqs. [8] and [9] that

where [TEOS] i is the initial molar concentration of TEOS. The Pseudophase Model The results presented above suggest that the growth kinetics of silica particles is controlled by the rate of TEOS hydrolysis. The observed decrease in the growth rate constant (k g) as R increases (Fig. 4), however, is apparently in contradiction with what would be expected. In principle, hydrolysis should be favored at large R, since under these conditions water is mostly free and a well-defined hydrophilic domain has been formed within the reverse micelles (1–7). This apparent contradiction may be resolved by considering the microheterogeneity of the microemulsion fluid system. The observed kinetics can be interpreted in terms of a pseudophase model (8 –10), which assumes that the reaction of interest (hydrolysis in this case) may take place in two separate pseudophase domains. In this model, the microemulsion medium (of total volume V 0 ) is viewed as a two-phase system consisting of a reverse micellar pseudophase (of volume V m) and a bulk oil phase (of volume V b). Furthermore, each reactant is distributed between both phases according to a partition constant P i . A schematic representation of the system is shown in Fig. 6. In the pseudophase model, the observed rate (r) of the reaction is considered to be an averaged value over the whole volume of the system, and it can be expressed in terms of the respective reaction rates in the reverse micellar and bulk oil phases (r m and r b) and the volume fraction of each pseudophase (f m and f b), r 5 r mfm 1 r b~1 2 f m! 5 k obs@TEOS# 0 @H2O# 0 @OH 2# 0 ,

[13]

where k obs is the experimentally observed rate constant and [TEOS] 0, [H 2O] 0, and [OH 2] 0 refer to the concentrations of the indicated species in the total microemulsion volume (V 0 ). In principle, hydrolysis can occur independently in the reverse micellar pseudophase and in the bulk oil phase as

73

GROWTH KINETICS OF NANOSIZE SILICA

It is thus assumed implicitly in Eq. [18] that the distribution of reactants between both pseudophases is at equilibrium and not affected by the ongoing chemical reactions. This assumption can be justified by recognizing that the exchange of the reactant species (TEOS, H 2O, OH 2) through droplet collision and the intermicellar matter exchange occur very rapidly (e.g., on the microsecond time scale (39, 40)). By inserting Eq. [18] (for each of the species of interest) in Eq. [16], the expression relating k obs with k m is obtained as k obs 5

FIG. 6. model.

Schematic representation of the reverse micellar pseudophase

r m 5 k m@TEOS# m@H2O# m@OH 2# m

[14]

r b 5 k b@TEOS# b@H2O# b@OH 2# b.

[15]

In Eqs. [14] and [15], the subscripts m and b indicate quantities referred, respectively, to the reverse micellar pseudophase and the bulk oil phase. Thus, k m and k b are, respectively, the hydrolysis rate constants in the reverse micellar pseudophase and the bulk oil phase. The reasonable assumption can be made here that hydrolysis in the bulk oil phase is negligible (i.e., r b 5 0), since water and OH 2 ions reside mainly in the reverse micellar pseudophase. Therefore, combining Eqs. [13] and [14], we obtain

k obs 5 k mP TP WP OHfm,

The volume fraction of the reverse micellar pseudophase (f m) can be expressed as

S

[17]

D

1 1 f m~P i 2 1! . Pi

f m 5 n s@NP-5# 0 1 n H2O@H2O# 0 1 n NH3@NH3# 0

[21]

f m 5 n s@NP-5# 0 1 n aq@H2O# 0,

[22]

or

From the mass balance for species i averaged over the microemulsion volume, the total concentration of species i is given by @i# 0 5 @i# m

[20]

[16]

The concentration of the ith-species referred to the total microemulsion volume (i.e., [i] 0 ) can be related to its local concentration in the reverse micellar pseudophase (i.e., [i] m) through the corresponding partition constant, P i . This constant is given by P i 5 @i# m/@i# b.

[19]

where P T 5 [TEOS] m/[TEOS] b, P w 5 [H 2O] m/[H 2O] b, and P OH 5 [OH 2] m/[OH 2] b. Recall that f m is the volume fraction of the reverse micellar pseudophase, which includes both the micellized surfactant and the solubilized aqueous phase. In a dilute microemulsion system (as in this work), the volume (V m) of the reverse micellar pseudophase is much smaller than that of the bulk organic phase volume (V b), and therefore f m is much smaller than unity. The molar volume of the surfactant (NP-5) is 0.440 M 21. Therefore, given the experimental conditions ([NP-5] 0 5 0.056 – 0.274 M, 5.07 L total microemulsion volume, and 0.04 mL aqueous ammonium hydroxide solution), the volume fraction f m is not expected to exceed 0.13. Thus, for a microemulsion system of low concentrations of surfactant and aqueous phase, Eq. [19] reduces to

r 5 k m@TEOS# m@H2O# m@OH 2# mfm 5 k obs@TEOS# 0 @H2O# 0 @OH 2# 0 .

k mP TP WP OHfm , @1 1 f m~P T 2 1!#@1 1 f m~P W 2 1!# 3 @1 1 f m~P OH 2 1!#

[18]

where n s, n H2O, and n NH3 are, respectively, the molar volumes of surfactant, water, and ammonia, n aq the equivalent molar volume of the ammonia solution, and [NP-5] 0 the micellized surfactant concentration with respect to the total microemulsion volume (V 0 ). By using the fact that the water-to-surfactant molar ratio (R) is given by R 5 [H 2O] 0 /[NP-5] 0 , the volume fraction of the reverse micellar pseudophase can be expressed as

f m 5 @H2O# 0$ n s/R 1 n aq%. Inserting Eq. [23] in Eq. [20], we obtain

[23]

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OSSEO-ASARE AND ARRIAGADA

k obs 5 k mP TP WP OH@H2O# 0 $ n s/R 1 n aq%.

[24]

From Eqs. [5], [11], and [13], the growth rate constant k g is given by k g 5 k obs@H2O# 0 @OH 2# 0 .

[25]

Thus, k g can be expressed as k g 5 k mP TP WP OH@H2O# 20 @OH 2# 0 $ n s/R 1 n aq%.

[26]

Under conditions of constant water and hydroxyl ion concentrations, we obtain finally k g 5 K$ n s/R 1 n aq%,

[27] FIG. 7. Test of the pseudophase model (Eq. [27]) of growth kinetics of SiO 2 particles in the NP-5/cyclohexane/NH 4OH reverse micellar system.

where the constant K is given by K 5 k mP TP WP OH@H2O# 20 @OH 2# 0 .

[28]

According to Eq. [27], the growth rate constant k g should be independent of R (k g ' K n aq) at large R values (i.e., at low surfactant concentrations; as noted in the experimental section, R was changed by varying the surfactant concentration), while k g should be inversely proportional to R (k g ' K n s/R) at low R values (i.e., at high surfactant concentrations). The trend predicted by Eq. [27] correlates very well with the effect of R on the rate constant k g shown in Fig. 4. As seen in this figure, at low R values k g decreases with increase in R, but it tends to a plateau at large R values. In terms of the above pseudophase model, these trends may be further rationalized as follows. An increase in the surfactant concentration translates into an increase in the volume fraction of the reverse micellar pseudophase (Eq. [22]). But this pseudophase is the locale of the rate-determining hydrolysis reaction. Thus, a high surfactant concentration corresponds to an increase in the volume fraction of reaction sites (f m) and should therefore lead to enhanced kinetics (Eq. [20]). Assessment of the Pseudophase Model As indicated by Eq. [27], the growth rate constant k g should be linearly dependent on 1/R. The k g data presented in Fig. 4 are plotted versus 1/R in Fig. 7. As seen in this figure, the experimental data conform to the expected linear dependency only at small 1/R values, i.e., in the high R range. A possible explanation for this behavior is discussed below. The proposed pseudophase model can be further tested by recognizing that the ratio of the molar volumes of the surfactant and aqueous phase (n s/n aq) can be determined by the slope (K n s) and intercept (K n aq) in Eq. [27]. The values of n s and n aq are known (0.440 M 21 for NP-5 and 0.031 M 21 calculated for the concentrated ammonia solution), so that the expected n s/n aq

value is 14. The calculated n s/n aq value from the slope and intercept taken from Fig. 7 is about 5.7, which is considered to be in reasonable agreement with the expected value in view of the assumptions made in the model. The pseudophase model applied here accounts only for the effective concentration of the reactants in the reverse micellar pseudophase, as governed by their respective partition constants. Since the reverse micellar pseudophase is viewed as a continuous separate phase, the model does not explicitly consider other effects arising from the discrete nature of the reverse micelles, which may affect the reaction kinetics. Such effects (e.g., electrostatic interaction, steric effects) which influence the reactivity of the reactants in the microenvironment of the reverse micelles are expected to be reflected in the rate constant associated with the reverse micellar pseudophase, k m (8). According to these ideas, for constant water and hydroxyl ion concentrations, the parameter K in Eqs. [27] and [28] is not necessarily a constant, but it may be a function of R if the intrinsic value of k m (the hydrolysis rate constant in the reverse micellar pseudophase) depends on R. In the present case, the hydrolysis rate of TEOS is expected to depend on the value of R. Thus, for example, it is likely that the higher the proportion of water molecules bound directly to the surfactant polar groups in the reverse micelles (low R (1, 31)), the slower will be the hydrolysis rate. This view is consistent with the retarding effect of strong water–solvent (alcohol) interactions observed in silica synthesis by the alkoxide sol– gel process in homogeneous alcoholic media (11, 12, 16). Therefore, the observed effect of R on the rate constant k g (Figs. 4 and 7) may actually include two different contributions, i.e., that associated with the concentration of reactants in the reverse micellar pseudophase and that arising from the discrete nature of the reverse micelles. Separation of the relative contributions of each of these effects on the rate constant k m cannot be achieved with the current experimental data.

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GROWTH KINETICS OF NANOSIZE SILICA

have been much more pronounced in the lower R range had k m been independent of R. These observations are also consistent with the trend shown in Fig. 7; at large 1/R values (i.e., low R), k m decreases, and therefore the observed rate constant k g tends to level off rather than being proportional to 1/R. In summary, the above analysis, based on a pseudophase reaction model, supports the view that the growth rate of silica particles in the reverse micellar system is controlled by TEOS hydrolysis (i.e., the release of the “active” monomer), at least during the initial stages of growth (below ;30 h). It appears from the observed deviation (at longer times, above ;30 h; see Fig. 3) from first-order kinetics that a mechanism different from hydrolysis is becoming rate controlling in the later stages of growth. It is possible that the rate of the surface reaction (condensation), rather than hydrolysis, is becoming important in the overall growth rate at the later stages of growth. REFERENCES

FIG. 8. Effect of the water-to-surfactant molar ratio (R) on the parameter K (Eq. [27]) in the NP-5/cyclohexane/NH4OH reverse micellar system.

Inspection of Eq. [27] indicates that if the parameter K is constant throughout the R range, then the quotient k g/( n s/R 1 n aq) must be independent of R. Figure 8 shows that this is not the case, since K increases with R. This result is in line with the previous ideas, suggesting that k m is a strong function of R. As indicated by Fig. 8, the parameter K starts to level off at R values around 1.8, which is in reasonable agreement with the R region where there is a transition from bound to free water within the reverse micelles (1, 31). If furthermore the variation in K is attributed only to the variation of the rate constant k m, then the results of Fig. 8 indicate that k m may increase about four- to fivefold when R increases from 0.5 to values larger than 2. Such an increase in the hydrolysis rate constant k m due to diminishing steric hindrance and to the more free nature of the water molecules is very reasonable; for example, in homogeneous media the hydrolysis rate constant may increase sevenfold by reducing steric hindrance, as obtained by replacing tert-butyl alcohol by n-butanol (12). The observed rate constant k g increases as the volume fraction (f m) of the reverse micellar pseudophase increases (i.e., as R decreases), as indicated by Eq. [27]. On the other hand, it seems that k m decreases as f m increases (i.e., as R decreases), as suggested by the results of Fig. 8. These two competitive effects are reflected on the observed rate constant k g, which, as shown in Fig. 4, decreases as R increases. It can be easily seen that the effect of R on the growth rate shown in Fig. 4 should

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