Formation Mechanism of Amorphous TiO2 Spheres in Organic

The curves A and B in Figure 1 represent the yields of precipitate for ñ = 0 and 3, respectively, as functions of t, and all curves with ñ from 0 to 3...
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J. Phys. Chem. C 2008, 112, 18437–18444

18437

Formation Mechanism of Amorphous TiO2 Spheres in Organic Solvents 2. Kinetics of Precipitation Tadao Sugimoto*,† and Takashi Kojima‡ Institute of Multidisciplinary Research for AdVanced Materials, Tohoku UniVersity Katahira 2-1-1, Aoba-ku, Sendai 980-8577, Japan ReceiVed: April 5, 2008; ReVised Manuscript ReceiVed: September 30, 2008

This paper (Part 2) is a report on the kinetics of precipitation in a sol-gel system developed in Part 1 of this series for the formation of monodispersed amorphous TiO2 spheres through hydrolysis of titanium butoxide (TBO) in its homogeneous solution of a mixed solvent of butanol/acetonitrile (vol. ratio 1:1) with ammonia. The precipitation was performed by adding 5 cm3 of a composite solution of 1.0 M H2O and 0.20 M NH3 to 5 cm3 of 0.10 M TBO in 1 s and the following aging, under constant agitation at 25 °C. From Nielsen’s chronomal analysis on the change of the electric conductivity of the system, 83% of the precipitation of the hydrolysis product was found to be completed by 2.5 s from the start of the admixing. The preceding hydrolysis of TBO was likely to have finished earlier than the start of the nucleation of the product particles, as revealed from the kinetic analysis on the hydrolysis and precipitation processes. The particle growth proceeded through a two-step process consisting of a very fast initial step finished at latest by 2.5 s and the following much slower one. The cause of the two-step process was explained in terms of the high initial reactivity of the particle surfaces prior to the adsorption of ammonia, as well as the possibility of precipitation of higher clusters under the extremely high supersaturation, and the following deactivation of the particle surfaces by retarded adsorption of ammonia. The second step was found to proceed in a third-order reaction-controlled kinetics supposedly by direct deposition of the trimer of hydrolysis product in equilibrium with the monomer and dimer. Finally, mechanisms of particle formation in relevant sol-gel systems of TiO2 and ZrO2 are discussed in the light of the present results. Introduction

Experimental Section

In Part 1 of this series,1 it was found that the precipitation of titanium hydroxide particles reached 90% by 10 s, and a much slower precipitation process followed thereafter in a homogeneous system for the synthesis of the monodispersed spherical particles of titania by the hydrolysis of titanium butoxide (TBO) in a butanol/acetonitrile mixed solvent in the presence of ammonia. The condensation of the hydroxide with dehydration was found to proceed virtually within the individual particles after their precipitation, independently of the precipitation event. The objective of Part 2 of this series is to make clear the kinetic mechanism of the entire precipitation process as much as possible, consisting of the hydrolysis of TBO and the deposition of titanium hydroxide as the hydrolysis product of TBO for the nucleation and the growth of the titania particles in the above system, using data on the change of yield of the precipitate with time and the evolution of electric conductivity with the progress of the precipitation. For the data analysis, we used a kinetic equation for the hydrolysis of metal alkoxide in general, derived specifically for the analysis on the hydrolysis process of TBO in the present system, and Nielsen’s chronomal method2 for the analysis on the growth mechanism of the titania particles by the deposition of the hydrolysis product.

The kinetic study was performed with the standard synthetic system in Part 1 of this series.1 The procedure is as follows. A 5 cm3 sample of Solution B, consisting of 0.20 M NH3 and 1.0 M H2O in a mixed solvent of BuOH/AN (volume ratio 1:1), was injected into the same volume of Solution A, containing 0.10 M TBO in the same mixed solvent of BuOH/AN, in ca. 1 s using a micropipette while stirring with a magnetic stirrer at 25 °C to complete the uniform mixing at the same time, followed by aging under constant agitation at the same temperature. Thus, the initial concentrations of the solutes were nominally 0.050 M TBO, 0.50 M H2O, and 0.10 M NH3. For tracing the yield changing with time, 3 cm3 of a sample was withdrawn at prescribed times after mixing Solutions A and B, and immediately percolated. The contents of titanium ion in the filtrate were determined by UV spectroscopy after mixing a part of the filtrate with triethanolamine to form a stable complex with titanium ion, according to the procedure in the experimental section of Part 1 of this series.1 Automatic in situ measurement of the electric conductivity changing with time was also carried out as described in Part 1.

* To whom correspondence should be addressed. E-mail: tdosugimoto@ pop06.odn.ne.jp. † Current address: Manazuru Institute for Superfine Particle Science, Manazuru 1912-4, Manazuru-machi, Kanagawa 259-0201, Japan. ‡ Current address: Department of Applied Chemistry and Biotechnology, Faculty of Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan.

Results and Discussion 1. Overview on the Precipitation Process. The heavy solid line in Figure 1 shows the changing yield of the precipitate within 60 s in the standard system, as calculated from the concentration of all solute remaining in the supernatant. The yield of the precipitate sharply increased to ca. 90% within 10 s, followed by much slower increase up to 100% by 10 min. From this characteristic behavior of the precipitation, it was expected

10.1021/jp802954u CCC: $40.75  2008 American Chemical Society Published on Web 11/04/2008

18438 J. Phys. Chem. C, Vol. 112, No. 47, 2008

Sugimoto and Kojima mass balances for titanium species and water are given from eq 2 as -1 hm(hm-1 + h-1 m+1 + · · · + h3 )[Xm] + [X4] ) [X0]0

(3)

and -1 hm(mhm-1 + (m + 1)h-1 m+1 + · · · + 3h3 )[Xm] +

4[X4] + [Y] ) [Y]0 (4) where [X0]0 and [Y]0 are the initial concentrations of TBO and water, respectively. Thus, if the degree of hydrolysis y is defined by the total yield of Ti(OH)4 including particles and solute (y ≡ [X4]/[X0]0), one obtains the following differential equation for the degree of hydrolysis, y, from eqs 2-4 Figure 1. Experimental yield of precipitate in the standard system plotted as a function of time (heavy solid line), and theoretical yields calculated as functions of time in cases A (∼n ) 0) and B (∼n ) 3) on assumption that the precipitation rate is limited by the hydrolysis rate of TBO with the yield ) 0.89 at 10 s.

in Part 1 of this series1 that there are at least two distinct steps in the precipitation. That is, the first step is an exceedingly rapid precipitation process associated with fast hydrolysis of TBO or immediately after the hydrolysis, while the succeeding slow step seems to be only for the deposition of the remaining hydroxide solute after the hydrolysis of TBO. In order to confirm this expectation and find its detail, let us analyze the behavior of the supernatant solute. Since the precipitation process in the present system consists of the hydrolysis of alkoxide and the deposition of the hydrolysis product onto growing particles after nucleation, what we have to do first is the analysis on the rate-determining steps of the entire process. If the hydrolysis step is much faster than the deposition step of the hydrolysis product, we will observe only the deposition process after an instantaneous production of the hydroxide solute. On the other hand, if the hydrolysis of alkoxide is much slower than the deposition of the hydroxide, the rate of precipitation is determined by the hydrolysis rate, and a kind of steady state in a balance of the generation of the hydroxide monomer and its consumption for deposition is established. Prior to the distinction between these different cases, we may need to make clear the kinetic nature of the hydrolysis of alkoxide. The stepwise hydrolysis process of TBO may generally be written as

Ti(OBu)4-n(OH)n+H2O f Ti(OBu)3-n(OH)n+1 + BuOH

(1) for n ) 0, 1, 2, and 3. Insofar as the amount of water is enough to totally hydrolyze a metal alkoxide, one may assume that each step of this sequential process is essentially irreversible. Thus, if the overall hydrolysis rate is limited by the reaction rate for the slowest hydrolysis of a certain species Ti(OBu)4-m(OH)m (0 e m e 3) denoted by Xm, the concentration of species Xn for 0 e n e m-1 is zero, and each species of Xn for m + 1 e n e 3 is in a steady state of generation and consumption. In this case,

-d[Xm] ⁄ dt ) hm[Xm][Y] ) hm+1[Xm+1][Y] ) · · · ) h3[X3][Y] ) d[X4] ⁄ dt (2) where [Xn] is the concentration of Xn, Y is H2O, and hn is the hydrolysis rate constant of Xn. [X4] means the total concentration of Ti(OH)4 in the liquid phase and in the precipitate. Here, the

dy ⁄ dt ) h[X0]0(1 - y){([Y]0 ⁄ [X0]0 - 4) + (4 - n˜)(1 - y)}

(5) where -1 -1 h ≡ (hm-1 + h-1 m+1 + · · · + h3 )

(6)

and

n˜ ≡

-1 mhm-1 + (m + 1)h-1 m+1 + · · · + 3h3 -1 hm-1 + h-1 m+1 + · · · + h3

(7)

Here, h corresponds to the overall hydrolysis rate constant for all species Xn from n ) m to 3 in the steady state: viz., hm[Xm][Y] ) h([Xm] + [Xm+1] + · · · + [X3])[Y]. Also, n˜ corresponds to the average n value of a Xn species from n ) m to 3 in the steady state, and thus 0 e n˜ e 3. For example, if the hydrolysis of Ti(OBu)(OH)3 (m ) 3) or X3 is the rate-determining step of -1 the whole hydrolysis process, then h ) h3 and h-1 0 ) h1 ) h-1 ˜ ) 3 in eq 7. The solution of eq 5 is 2 ) 0 in eq 6, and n generally given by

y)

exp(Rt) - 1 exp(Rt) - β

(8)

where

R ≡ h([Y]0 - 4[X0]0); β ≡

4 - n˜ [Y]0 ⁄ [X0]0 - n˜

(9)

Since [Y]0/[X0]0 ) 10 with 0 e n˜ e 3 in the present system, the variable range of β is within 1/7 e β e 2/5. If the rate constant of the particle growth is assumed to be so large that the rate of the precipitation is entirely limited by the rate of the hydrolysis, the degree of hydrolysis, y, is equivalent to the yield of the precipitate. In this case, since [X0]0 ) 0.05 M, [Y]0 ) 0.5 M, and y ) 0.89 at t ) 10 s in the standard system in Part 1 of this series,1 we obtain β ) 2/5, R ) 0.177, and h ) 0.589 M-1 s-1 when n˜ ) 0, while β ) 1/7, R ) 0.207, and h ) 0.690 M-1 s-1 when n˜ ) 3, from eqs 8 and 9. The curves A and B in Figure 1 represent the yields of precipitate for n˜ ) 0 and 3, respectively, as functions of t, and all curves with n˜ from 0 to 3 are located in the very narrow range between them. The significant deviation of these theoretical curves from the experimental curve except at 10 s clearly tells us that the actual precipitation process is not simply limited by the hydrolysis process of TBO, but at least a two-step process, consisting of the very rapid first step for precipitation much earlier than 10 s, whose rate is limited by the hydrolysis rate or independent of the foregoing hydrolysis process, and the following much slower second step only for the precipitation of the residual hydrolysis product. Thus, at least, the hydrolysis

Formation Mechanism of Amorphous TiO2 Spheres

J. Phys. Chem. C, Vol. 112, No. 47, 2008 18439

dr DVm ) (C - C∞) dt r

(10)

where D is the diffusion coefficient of the monomer, Vm is the molar volume of the solid of the particles, C is the concentration of the monomer at t, and C∞ is the solubility of the particles. In the present system, C is the concentration of X4 remaining in the solution phase. If the initial concentration is denoted by C0, the degree of precipitation, ξ, may be defined as

ξ≡

Figure 2. Change of the conductivity with aging time in the standard system.

process must have finished much earlier than 10 s. If we provisionally simulate the change of concentration of the supernatant solute for the slower step after 10 s with a firstorder kinetics [C ) C0 exp(-kt)], the rate constants k () -ln(C/ C0)/t), obtained from the slope of the straight line of -ln(C/C0) vs t, is calculated as 0.0125 s-1. On the other hand, Figure 2 exhibits the change of electric conductivity with time in the standard system with ammonia within 60 s, revealing a sharp increase of conductivity to its maximum at 2.5 s, followed by a slower attenuation. If we simulate the decreasing conductivity after 10 s based on the first-order kinetics and assume proportionality of the conductivity above the baseline ()1.65 µS cm-1 reached after aging for ca. 60 min) to the concentration of the hydrolysis product above the solubility in a given system, we obtain 0.0121 s-1, in agreement with 0.0125 s-1 obtained from the increasing yield after 10 s in Figure 1 within the experimental error. Although there is no theoretical reason for the simulation by the first-order reaction for the slow process after 10 s, the agreement in both the changing pattern and the apparent rate constant between the two different approaches may suggest that the electric conductometry detected the same precipitation process as that observed from the changing yield. Since the electric conductivity is basically responsible only for the change of the concentration of the electroconductive hydrolysis product, while the total concentration of the supernatant solute for the calculation of the changing yield includes not only the concentration of the hydrolysis product but also of the remaining TBO, if any, the agreement between these different measurements strongly supports the earlier conclusion that the hydrolysis of TBO is not involved in the precipitation process after 10 s. 2. Chronomal Analysis on the Growth Mechanism. It is now evident from the aforementioned analysis that the hydrolysis of TBO has been finished at latest by 10 s, followed by a simple deposition process of residual hydroxide monomer onto the growing particles independent of the hydrolysis process. Thus, one may safely apply Nielsen’s chronomal method to the analysis of the growth mechanism of the titania particles after 10 s.2,3 But, at first, let us try the chronomal analysis for the precipitation process later than 2.5 s, regardless of some possibility of hydrolysis still remaining even after 2.5 s, using the electric conductivity change in Figure 2 on assumption that the electric conductivity is reduced in proportion to the reduction of the concentration of the hydroxide monomer with precipitation. If the growth of a particle of radius, r, is in the diffusioncontrolled mode, the linear growth rate, dr/dt, is given by

C0 - C C0 - C∞

(11)

In addition, if the initial particle size is negligibly small, ξ is given by ξ ) (r/r∞)3. In this case, one obtains the following relationship:

∫0ξ x1⁄3(1dx- x) )

3DVm(C0 - C∞) r∞2

t

(12)

on the assumptions that the particles are grown at a constant particle number in a simple supersaturated solution of only hydroxide monomer, and that the initial ξ is zero, where r∞ is the final particle radius. If the integral on the left-hand side and the coefficient of t on the right-hand side are designated by ID and KD, respectively, as

ID ≡

∫0ξ x1⁄3(1dx- x) , KD ≡

3DVm(C0 - C∞) r∞2

(13)

ID plotted against t will give a straight line with a slope of KD. If the growth mechanism is kept unchanged from t ) 0, the straight line will pass through the origin. On the other hand, if a particle is grown in a reactioncontrolled mode, dr/dt may be given by

dr ) kpVm(C - C∞)p dt

(14)

where kp is the rate constant of the surface reaction, and p is a dimensionless positive constant. In this case, the following relationship holds

∫0ξ x2⁄3(1dx- x)p )

3kpVm(C0 - C∞)p t r∞

(15)

If the integral on the left-hand side and the coefficient of t on the right-hand side are designated by Ip and Kp, respectively, as

Ip ≡

∫0

ξ

3kpVm(C0 - C∞)p dx K ≡ p r∞ x2⁄3(1 - x)p

(16)

Ip plotted against t will give a straight line with a slope of Kp. If the growth mechanism is unchanged from t ) 0, the straight line must pass through the origin. Figure 3 demonstrates curves obtained by plotting ID and Ip of different p values as functions of t, using the data of electric conductivity from 2.5 to 60 s for the standard system in Figure 2 on the assumptions that (C - C∞) is proportional to the corresponding conductivity above its baseline and that ξ ) 0.89 at t ) 10 s from Figure 1. All these curves do not pass the origin, showing that the growth mechanism at least before 2.5 s is different from thereafter for all cases. The plot of ID against t on assumption of the diffusion-controlled growth after 2.5 s is not only far from a straight line, but also the diffusion coefficient D calculated from the initial slope (∼0.050 s-1) near 2.5 s is 4.4 × 10-13 m2 s-1, about 4 orders of magnitude lower

18440 J. Phys. Chem. C, Vol. 112, No. 47, 2008

Figure 3. Chronomal functions on assumption of reaction-controlled growth (Ip: p ) 1-4) or diffusion-controlled growth (ID) for the standard system. A linear chronomal function is obtained only when a reactioncontrolled growth at p ) 3 is assumed.

than an ordinary diffusion coefficient on the order of 10-9 m2 s-1. For the calculation of D, we used eq 13 with KD ) 0.05 s-1, r∞ ) 0.225 µm, Vm ) 38.6 cm3 mol-1 (d ) 3 g cm-3 from ref 4, Ti(OH)4 ) 115.9 g mol-1), and C0 - C∞ ) 0.05 mol dm-3. Thus, the growth kinetics after 2.5 s must be a reactioncontrolled one. For the plotting against time after 2.5 s, a linear relationship is obtained only when we assume reaction-controlled growth with p ) 3. Here, it is noteworthy that the linear relationship in the range from 10 to 60 s, obtained from both the data of yield in Figure 1 and the electric conductivity in Figure 2, can directly be extended to 2.5 s from 10 s, based only on the data of the electric conductivity in Figure 2. This fact strongly supports that the precipitation process in the time range from 2.5 to 10 s is the same as that in the range after 10 s, i.e., simple deposition of the hydroxide monomer onto growing particles free from the hydrolysis process of TBO. Since ξ ) 0.83 at t ) 2.5 s from the electric conductometry, the actual peak of the conductivity must be located much earlier than 2.5 s, and its height must be far higher than observed in Figure 2. The rapid generation of the hydrolysis product, corresponding to the sharp increase of conductivity, seems to be too fast to follow with the electric conductometer used. The growth kinetics after 2.5 s can be described by the reaction-controlled growth formula in eq 14 with p ) 3. The k3 value is calculated as k3 ) 5.18 × 10-8 m7 mol-2 s-1 from eq 16 with the slope of the straight line K3 ) 3.33 s-1. However, if one extrapolates this straight line to t ) 0, one obtains I3 ) 13.8 at t ) 0, equivalent to ξ ) 0.77. Thus, at least up to ξ ) 0.77, the precipitation must be governed by a different preceding process of a much higher Kp. If we assume p ) 3 in the early stage as well, and if the high initial overall K3 and the corresponding k3 are denoted by K3′ and k3′, we obtain that K3′ ) 8.80 s-1 and k3′ ) 1.37 × 10-7 m7 mol-2 s-1 from the slope of the dotted straight line drawn from the origin to the point on the original straight line with the slope ) 3.33 s-1 at t ) 2.5 s in Figure 3. This overall K3′ ()8.80 s-1) is the minimum value of K3′ corresponding to a hypothetical case that the rapid

Sugimoto and Kojima precipitation starts from ξ ) 0 at t ) 0 and lasts until ξ ) 0.83 at t ) 2.5 s without changing the rate constant. As has been shown in Part 1 of this series,1 ammonia has no significant effect on the initial rapid precipitation within its tested concentration range up to 1.0 M. In this initial stage, the solubility of the hydroxide monomer is not sufficiently lowered by ammonia, and thus the precipitation is not fully accelerated by ammonia yet. Thus, the initial fast precipitation with K3′ g 8.80 s-1 seems to be mainly due to the rapid reaction of the hydrolysis product on the highly solvated rough surfaces of the nascent particles before ammonia becomes completely effective. In this context, the distinct two-step growth process is also observed even in the absence of ammonia as shown in Figure 2 in Part 1. But, in this case, the slow progress of precipitation in the second step is likely to be due to the gradually decreasing solubility of the precipitate itself initially much higher than in the presence of ammonia, as referred to in Part 1. It is now evident that the hydrolysis process has been finished by 2.5 s in the standard system, and that 83% of the hydrolysis product has been precipitated by then. Since the important processes, such as the hydrolysis of TBO, the nucleation of titania particles, and their initial growth process, are all involved in this short time, we need to extract as much information as possible from this stage for understanding the entire particle formation process. Although the available experimental data in the standard system are limited only to the facts that the hydrolysis process is completed by 2.5 s and that the precipitation yield ξ ) 0.83 at 2.5 s, it is possible to specify the upper and lower limits of the growth rate constant of the generated titania particles under the constraints of the these experimental facts in two typical cases: (a) The hydrolysis of TBO is so fast as to finish before the start of the precipitation of the hydrolysis product, and thus the growth rate is limited by the deposition rate of the hydroxide monomer onto the growing particles (deposition-limited growth); (b) the hydrolysis rate is so slow that the growth rate is limited by the hydrolysis rate of TBO (hydrolysis-limited growth). If we obtain the upper and lower limits of the growth rate constant in the two extreme cases, we will be able to predict which is more likely, the depositionlimited growth or the hydrolysis-limited growth. 3. Ranges of the Growth Rate Constant in the Initial Stage. Even if the hydrolysis of TBO is so fast as to finish instantly, it lasts, at least, for 0.4 s necessary for the added water to reach the stoichiometric quantity ()the quadruple moles of TBO) for the complete hydrolysis of 5 cm3 of 0.10 M TBO during the injection of a 5 cm3 BuOH/AN solution of 1.0 M H2O and 0.20 M NH3 (Solution B) for 1 s. On the other hand, the hydrolysis of TBO has been proved to finish at latest by 2.5 s from the chronomal analysis on the data of electric conductometry. Regardless of the hydrolysis rate, 83% of the hydrolysis product must be precipitated by 2.5 s. Under these conditions, let us give insights into the dynamic processes and estimate the ranges of the growth rate constant in the initial stage in the two extreme cases on assumption of the third-order growth rate. a. Deposition-Limited Growth. Let us first consider the fastest hydrolysis of TBO, which proceeds simultaneously with the addition of the Solution B and virtually finishes at the same time with the addition of the stoichiometric amount of water at 0.4 s during the addition of water totally for 1 s. Since it is known that the hydrolysis of X3 is the rate-determining step of the entire hydrolysis process, 5,6 we further assume that n˜ ) 3 in eq 5, so that X0 ()Ti(OBu)4) is totally hydrolyzed into X3 ()Ti(OBu)(OH)3) by 0.3 s, and the X3 is finally changed into

Formation Mechanism of Amorphous TiO2 Spheres X4 ()Ti(OH)4) within the time range from 0.3 to 0.4 s, followed by the remaining addition of excess water until 1 s. In this case, the generation rate of X4 is 5.0 × 10-3 mol s-1. If, in addition, the precipitation of the hydroxide monomer is sufficiently fast, the concentration of the monomer may reach a peak soon after 0.3 s in the steady state in the mass balance of the generation of the monomer and its consumption for the nucleation and the growth of the nuclei. Then, the nucleation will be terminated and the generated nuclei will continue to grow with monomer steadily furnished. The mean diameter of the nuclei at the end of the nucleation can be estimated as 0.072 µm ()2σ), since the standard deviation of the size distribution (σ) of the final product of mean diameter 0.45 µm was 0.036 µm, which is kept constant during the particle growth of monodisperse particles if they are grown by the reactioncontrolled kinetics.3 At this moment, the precipitation is limited only to 0.4% of the total (ξ ) (0.072/0.45)3), and the linear growth rate of a nucleus, dr/dt, at the end of nucleation is calculated as 29.3 µm s-1 from the generation rate of hydroxide monomer ()5.0 × 10-3 mol s-1), the mean nucleus radius ()0.036 µm) and the final particle number in the system ()4.05 × 1011). If we calculate the supersaturation (C - C∞) for the growth of the same sized nuclei on assumption of the diffusioncontrolled growth with an ordinary diffusion coefficient of 10-9 m2 s-1, the supersaturation is found to be 2.7 × 10-2 M from eq 10 with dr/dt ) 29.3 µm s-1. This means that when the supersaturation reaches 2.7 × 10-2 M, the system goes into a steady state of the generation of X4 and its consumption for the diffusion-controlled particle growth. If a reaction-controlled growth is assumed at the same supersaturation of 2.7 × 10-2 M, the corresponding k3′ value with p ) 3 is calculated as 3.7 × 10-5 m7 mol-2 s-1 from eq 14. In other words, if the growth rate constant k3′ is extremely large as above 3.7 × 10-5 m7 mol-2 s-1, the particles are grown by the diffusion-controlled kinetics, but its growth rate is basically limited by the hydrolysis rate in the steady state. Hence, if the growth rate constant k3′ is smaller than 3.7 × 10-5 m7 mol-2 s-1, the maximum supersaturation for the growth of the nuclei must become higher than 2.7 × 10-2 M, and the growth kinetics is switched to the reaction-control. Since the supersaturation cannot exceed 0.05 M, there must be the minimum value of k3′ for the growth of the nuclei in balance with the fastest hydrolysis process, that is, 6.06 × 10-6 m7 mol-2 s-1, as given from eq 14 with dr/dt ) 29.3 µm s-1 and C - C∞ ) 0.05 M. Therefore, if k3′ > 6.06 × 10-6 m7 mol-2 s-1, the nucleation occurs during the fastest hydrolysis of X3 from 0.3 to 0.4 s, and the generated nuclei are grown in a steady balance with the hydrolysis of X3 until the end of the hydrolysis at 0.4 s, followed by brief growth at most for additional 0.05 s by rapid deposition of the residual hydroxide monomer. In this case, at least 80% of the total hydrolysis product must be precipitated by 0.45 s, as calculated from eq 15, and the much slower second growth stage with k3 ) 5.18 × 10-8 m7 mol-2 s-1 follows thereafter. If the nucleation occurs after the hydrolysis reaction, the initial supersaturation must be equal to 0.05 M, and the upper limit of the k3′ value for the growth of the nuclei in this case is 6.06 × 10-6 m7 mol-2 s-1. Meanwhile, the lower limit of the k3′ value in this case corresponds to the rate constant of the particle growth starting from t ) 0.4 s with ξ ) 0 until t ) 2.5 s with ξ ) 0.83 through simple deposition of monomer under the initial supersaturation of 0.05 M. This lower limit of k3′ is 1.63 × 10-7 m7 mol-2 s-1, as calculated on the basis of the corresponding K3′ value ()10.5 s-1) given from eq 15 or graphically

J. Phys. Chem. C, Vol. 112, No. 47, 2008 18441 from Figure 3 on assumption of ξ ) 0 at t ) 0.4 s and ξ ) 0.83 at t ) 2.5 s. In summary of deposition-limited growth, if 1.63 × 10-7 m7 mol-2 s-1 e k3′ e 6.06 × 10-6 m7 mol-2 s-1, the nucleation occurs after the hydrolysis process, and the following rapid growth stage lasts at longest till 2.5 s. In this case, the initially rapid and the following slow growth processes proceed independently of the hydrolysis process. b. Hydrolysis-Limited Growth. In this case, the initial particle growth is assumed to be limited by the slowest hydrolysis of TBO lasting until 2.5 s, in which the hydrolysis rate is a function of water concentration changing not only with its consumption by reaction, but also with its constant supply from outside for the initial 1 s. Since our primary purpose is the estimation of the lowest k3′ value to achieve the hydrolysis-limited growth in the case of the slowest hydrolysis, we may use a much more simplified model free from this complex situation. Namely, in addition to the former assumption of n˜ ) 3, we assume a hydrolysis process of X3 starting from 0.4 s, by which the mixing of Solutions A and B is completed, and finishing its 99% by 2.5 s, i.e., y ) 0, [X3] ) 0.05 M, and [Y] ) 0.35 M at t ) 0.4 s; y ) 0.99, [X3] ) 5.0 × 10-4 M, and [Y] ) 0.30 M at t ) 2.5 s. In this case, β ) 1/7, R ) 2.12 s-1 and h ) 7.07 M-1 s-1 from eqs 8 and 9. If the formation rate of Ti(OH)4 per unit volume, d[X4]/dt, is denoted by J, there is the following relationship in the steady mass balance in this model:

J ) h[X3]([Y]0 - 4[X0]0 + [X3]) ) 4πr2n′ k3′ C3

(17)

where n′ is the number concentration of the particles kept constant, C is the concentration of X4 in the solution phase, and the solubility of the particles, C∞, in the supersaturation (C - C∞) is neglected. In the steady state, the derivatives of J as to time for the generation and the consumption of the hydroxide monomer must also be kept equal as

dJ dJ d[X3] dJ dC ) ) dt d[X3] dt dC dt

(18)

If the dJ/d[X3] and dJ/dC are denoted by fh and fg, they are given from eq 17 as

fh ≡ h([Y]0 - 4[X0]0 + 2[X3]); fg ≡ 12πr2n′ k3′ C2

(19)

Here,

fg ⁄ fh g 1

(20)

must be fulfilled at the same time to keep the steady state, since |dC/dt| e |d[X3]/dt| must be satisfied in eq 18 for the prompt change of C in response to the reduction of [X3] for the generation of X4. From eqs 17, 19, and 20,

fg 3[X3]([Y]0 - 4[X0]0 + [X3]) ) g1 fh C([Y]0 - 4[X0]0 + 2[X3])

(21)

As we have assumed that the steady state is terminated at t ) 2.5 s, it must hold that fg/fh ) 1 at t ) 2.5 s (y ) 0.99). In this case, C ) 1.5 × 10-3 M at t ) 2.5 s as calculated from eq 21 with fg/fh ) 1 and [X3] ) 5.0 × 10-4 M. We also obtain k3′ ) 1.26 × 10-5 m7 mol-2 s-1 from eq 17 with 4πr2n′ ) (36πVm2n′)1/3([X0]0y - C)2/3. Hence, if this high k3′ value were retained until t ) 2.5 s, we would have C + [X3] ) 2.0 × 10-3 M at t ) 2.5 s. But, its actual value is much higher such as 8.5 × 10-3 M from ξ ) 0.83 at t ) 2.5 s, and thus the initially high k3′ value must have turned much earlier than 2.5 s to the low k3 value ()5.18 × 10-8 m7 mol-2 s-1) for the final precipitation stage. The turning point is estimated to be t )

18442 J. Phys. Chem. C, Vol. 112, No. 47, 2008

Sugimoto and Kojima

1.4 s (y ) 0.90) from the assumption of C + [X3] ) 8.5 × 10-3 M in eq 17. The degree of hydrolysis y at the start of the steady state, or the end of the nucleation stage, is given by y ) 0.41 at t ) 0.62 s (0.22 s after 0.4 s) from eq 17 with k3′ ) 1.26 × 10-5 m7 mol-2 s-1 and r ) 0.036 µm. The high k3′ value proper to hydrolysis-limited growth falls well within the range of k3′ for the rapid particle growth starting during the fastest hydrolysis in deposition-limited growth (k3′ > 6.06 × 10-6 m7 mol-2 s-1). In summary of hydrolysis-limited growth, when the hydrolysis lasts until 2.5 s, h must be 7.07 M-1 s-1 and k3′ must be k3′ g 1.26 × 10-5 m7 mol-2 s-1 to achieve the hydrolysis-limited growth. As a consequence of deposition- and hydrolysis-limited growth, the hydrolysis rate constant h must be no less than 7.07 M-1 s-1, so that the hydrolysis reaction is completed by 2.5 s. For the complete achievement of the hydrolysis-limited growth regardless of the hydrolysis rate, the growth rate constant k3′ must be at least greater than 1.26 × 10-5 m7 mol-2 s-1. If the precipitation occurs immediately after the preceding hydrolysis process so that the particles grow in the deposition-limited growth mode, the k3′ value must be in the range of 1.63 × 10-7 m7 mol-2 s-1 e k3′ e 6.06 × 10-6 m7 mol-2 s-1. If we consider the continuous transition from k3′ to the low growth rate constant k3 ()5.18 × 10-8 m7 mol-2 s-1), the deposition-limited growth in which k3′ is in the moderate range within 1.63 × 10-7 m7 mol-2 s-1 e k3′ e 6.06 × 10-6 m7 mol-2 s-1 is more likely, compared to the hydrolysis-limited growth with the discretely high k3′ g 1.26 × 10-5 m7 mol-2 s-1. In the case of the deposition-limited growth with the moderate k3′, the h value of the hydrolysis process must be much greater than 7.07 M-1 s-1 so as to be finished earlier than the start of the precipitation of the hydrolysis product, e.g., almost at the same time with the addition of water sufficient for the complete hydrolysis of TBO. This prediction will be verified in Part 3 of this series, in which the precipitation rate is found to be independent of the preceding hydrolysis rate. 4. Possibility of Clustering of the Hydroxide Monomer. Ogihara et al.7 found a first-order reaction-controlled mechanism for the growth of ZrO2 particles by deposition of zirconium hydroxide monomer in ethanol in the absence of ammonia, in which the deposition of the hydrolysis product was much slower than in our system. If the affinity of the solvent to hydroxide monomer is reduced by partly replacing the alcoholic medium with acetonitrile like in our system, the clustering of the hydroxide monomer will be enhanced, so that the precipitation will dramatically be accelerated. If we postulate an equilibrium among monomer and clusters such as dimer, trimer, tetramer, etc.,

κ2 )

C2 2

C

, κ3 )

Cn C3 , · · · κn ) , ··· C2C Cn-1C

(22)

and

Cn )

n

κiCn ) βnCn ∏ i)2

(23)

where κ2, κ3, and κn are the consecutive stability constants of dimer, trimer, and n-mer, respectively; C, C2, C3, and Cn are the concentrations of monomer, dimer, trimer, and n-mer, respectively; and βn is the cumulative stability constant of n-mer defined by

βn ≡

n

κi ∏ i)2

(24)

Since the particles are amorphous, the solute molecules deposited on the surfaces of the growing particles may be incorporated

into the particles without the surface process for forming twodimensional nuclei on the surfaces necessary for the formation of crystalline particles. In such a case, the state of the solute molecules in the solution phase may directly be reflected to the growth mode. If the particle growth proceeds via deposition of some specific n-mer, and if the solubility of the hydroxide, C∞, is negligibly small compared to the supersaturated concentration of monomer C, the linear growth rate, dr/dt, may be given by

dr ) k0nVmCn ) k0nVmβnCn ) knVmCn dt

(25)

where k0n is the first-order rate constant of the n-mer, and kn is the corresponding nth-order growth rate constant for the monomer defined as

kn ≡ k0nβn

(26)

but kn ) k0n and βn ) 1 at n ) 1. Hence, the third-order reactioncontrolled growth in the second precipitation stage in our system may suggest direct deposition of trimer and indirect deposition of monomer and dimer in equilibrium with the trimer onto the growing particles. Incidentally, the rapid precipitation in the first stage has been ascribed to the high k3′ as a measure of surface reactivity in the initial stage. However, if we assume a higher value of exponent p in Ip of eq 16 as an index of the order of the surface reaction or the extent of the clustering, Ip at a given ξ(t) is apparently lifted, so that Kp as the slope of Ip vs t is also raised, regardless of the magnitude of kp. Hence, there seems to be a possibility of additional contribution of a higher p value to the fast precipitation in the first stage, when the concentration of monomer and thus the probability of its clustering are high, although this does not affect the prediction of the depositioncontrolled growth. 5. Discussion on the Interpretation of the Growth Mechanism. Jean and Ring8 studied the kinetics of the precipitation of uniform TiO2 particles by hydrolysis of titanium ethoxide in ethanol in the absence of ammonia, and concluded that the particles were grown by the diffusion-controlled mechanism, based on the decreasing coefficient of variation ()standard deviation of particle radius divided by the mean radius) of the size distribution with time and on their chronomal analysis. However, the coefficient of variation always decreases with time, regardless of the growth mechanism, unless the particles are grown with some additional processes, such as extensive renucleation and/or coagulation. In addition, if the particles are grown by diffusion-controlled kinetics, not only the coefficient of variation, but also the standard deviation itself must significantly decrease with time.3,9 Notwithstanding, their data show that the standard deviation rather increased with time for all growth conditions. Moreover, the diffusion coefficients calculated from the chronomal data ranging from 2 × 10-14 to 9 × 10-14 m2 s-1 are extremely smaller than those of ordinary molecular diffusion on the order of 10-9 m2 s-1 by a factor of 4-5 orders of magnitude. The enormous discrepancy may be inexplicable on the basis of the diffusion-controlled growth mechanism, even if some clustering or oligomerization of the hydroxide monomer is assumed. As a consequence, it seems that the particles in their systems were basically grown by a reaction-controlled kinetics. The increase of the standard deviation with time may suggest more or less involvement of renucleation and/or coagulation, and it may be one of the reasons why the chronomal analysis failed to indicate the reactioncontrolled kinetics. On the other hand, Ogihara et al.7 studied the kinetics of precipitation of ZrO2 particles by the hydrolysis

Formation Mechanism of Amorphous TiO2 Spheres of zirconium butoxide similarly in ethanol without ammonia. They found that the ZrO2 particles were grown by the firstorder reaction-controlled growth kinetics from the chronomal analysis, in consistence with the standard deviation kept constant with the elapse of time. The first-order reaction control in their system seems reasonable if one considers the very slow precipitation process, probably free of the clustering of the hydroxide monomer. In this sense, the criticism of Dirksen and Ring,10 based on the same criterion of Jean and Ring,8 to this conclusion of Ogihara et al. may not be pertinent. However, there is a more fundamental problem common to the arguments in these two studies on titania and zirconia in ethanol. First, in the TiO2 system, a kind of overall equilibrium between the reactants of hydrolysis, such as Ti(OC2H5)4 and H2O, and the hydrolysis product TiO2 · xH2O was assumed, because the final yield of the precipitate was apparently proportional to [Ti(OC2H5)4]0 and the square of [H2O]0. In other words, the hydrolysis of titanium ethoxide was regarded as a reversible reaction, stopped halfway by an equilibrium finally reached, and thus the precipitation rate of the particles was assumed to be determined by the hydrolysis rate. On the other hand, in the ZrO2 system, the precipitation was assumed to be caused by the polycondensation of partly or fully hydrolyzed alkoxide monomer in the solution phase. Thus, in this case, the precipitation rate of the ZrO2 · xH2O particles must be determined by the condensation rate in the solution phase. In both arguments, the growth rate of the particles is predetermined by a process preceding to the deposition of the hydroxide monomer onto the growing particles. In such cases, the chronomal analysis to specify the rate-determining step in the deposition process of the hydroxide monomer, either the diffusion step or the surface reaction step, is meaningless.3 However, it is now obvious from the study in Part 1 of this series1 that the titanium butoxide is hydrolyzed instantaneously on mixing with ample water in a mixed solvent of butanol/acetonitrile even without ammonia. In addition, the hydrolysis of TBO is likely to finish earlier than the start of the precipitation of the hydrolysis product, while the condensation of the hydroxide monomer, if any, occurs much later in the solid precipitate, independently of the precipitation event. If the results in our system are applicable to the sol-gel systems of titania and zirconia by hydrolysis of titanium ethoxide and zirconium butoxide in ethanol, the chronomal analysis may safely apply to these systems as well. Presumably, the idea of equilibrium between the reactants of the hydrolysis and the hydroxide product might have come from the strong dependence of the precipitation rate and the product yield on the initial concentrations of alkoxide and water. However, in view of the extremely fast reaction of metal alkoxide on contact with only a small amount of water in the presence of abundant alcohol as the medium, the hydrolysis of metal alkoxide must basically be an irreversible reaction. Hence, it seems reasonable to consider that the hydrolysis of alkoxides in ethanol is also irreversible and finished almost instantly on mixing with ample water, and that the following process is only for the nucleation and growth of particles by deposition of the hydroxide monomer fully produced in the beginning. The strong dependence of the precipitation rate and the final yield of the precipitate on the concentrations of alkoxide and water in ethanol may readily be understood if one considers the increase in the concentration of the hydrolysis product with increasing concentration of alkoxide and the dramatic reduction of the solubility with increasing concentration of excess water as shown in Part 1 of this series.1 In this context, Barringer and Bowen11 postulated an equilibrium among the

J. Phys. Chem. C, Vol. 112, No. 47, 2008 18443 species of Ti(OR)4-n(OH)n with n ) 0-3 in the hydrolysis of Ti(OR)4 and the final irreversible step from Ti(OR)(OH)3 to Ti(OH)4 as the rate-determining step of the precipitation, based on the strong dependence of the induction time for the nucleation of product particles on the initial concentrations of water and titanium ethoxide in ethanol. However, if we consider the irreversible nature of the hydrolysis of metal alkoxides, it seems that every step of Ti(OR)4-n(OH)n + H2O f Ti(OR)3-n(OH)n+1 + ROH for n ) 0-3 is in a steady state, instead of the equilibrium of reversible elemental reactions during the hydrolysis reaction, as described in eq 1. Also, although it is reasonable to assume that the final step from Ti(OR)(OH)3 to Ti(OH)4 is the rate-determining step of the hydrolysis process, it may not be the rate-determining step of precipitation, as expected from the extremely fast hydrolysis and the much slower precipitation of the hydrolysis product in ethanol. As to the possibility of condensation of Ti(OH)4 in ethanol, Barringer et al.4,11 reported only 10-16% weight loss by evaporation of water in TG-DTA analysis with water-washed dry powder of titania prepared in ethanol without ammonia, where the theoretical weight loss from Ti(OH)4 to TiO2 is 31.1%. Thus, if we also take into account the high affinity of ethanol to water, a possibility of condensation or dehydration in the solid phase and/or liquid phase during precipitation in ethanol cannot be excluded. But, in view of the dramatically accelerated precipitation of the hydroxide monomer with increasing concentration of excess water, it is unlikely that the condensation as a kind of dehydration process to be promoted by lack of water is the incentive to the precipitation. Eiden-Assmann et al. prepared hydrous titania12,13 or zirconia13 spheres by hydrolysis of titanium ethoxide in ethanol or zirconium alkoxides in different alcoholic solvents, in the presence of polyethyleneoxides or a salt of alkali metal salts. They found that titania particles prepared in the presence of some specific polyethyleneoxides were particularly porous, unlike other titania particles synthesized in the absence of polymers and zirconia particles even in the presence of the same kind of polymers. The authors explained the formation mechanism of the porous titania in terms of the aggregation of primary particles sterically stabilized by the polymers, because they basically interpreted the growth mechanism of these particles in terms of the aggregation of preformed primary particles. It is true that the aggregative growth model is often used for explaining the formation of many kinds of amorphous or polycrystalline monodisperse particles.14-16 But, if this explanation is based mainly on the final particle structure, such as a mosaic assembly, or only on theoretical derivation starting from an aggregative growth model, one may need more definite evidence, since there are numerous examples of amorphous or polycrystalline monodisperse particles grown by direct deposition of solute species.3,17-22 In this case, it is not rare to find that coexisting inorganic or organic matters and/or even the solvent are incorporated into the particles at the same time with the integration of the solute. Specifically, when the primary particles are not actually observed, and/or when there is no definite reason for the formation of the monodispersed particles by aggregation of primary particles, careful analysis on the formation process is needed for concluding the growth mechanism. The particle growth in sol-gel systems may correspond to this case. In particular, the highly uniform size distribution in our present system implies clear-cut separation between the nucleation and the growth stages, as well as little possibility of coagulation, at least, after the particles are grown above ca. 70 nm in mean diameter. In such a system, the particle growth

18444 J. Phys. Chem. C, Vol. 112, No. 47, 2008 may not be explained by aggregation of preformed primary particles in which there is no special reason for the production of uniform particles. In addition, the chronomal analysis on the present system for titania particles in butanol/acetonitrile with ammonia or on the system of Ogihara et al.7 for zirconia particles in ethanol without ammonia seems to suggest direct deposition of solutes such as the trimer or the monomer of each hydrolysis product, and thus some quantitative scrupulous analyses may be required for the elucidation of the growth mechanism of the titania particles by hydrolysis of titanium ethoxide in ethanol as well. Conclusions The hydrolysis of TBO is expected to finish with addition of the stoichiometric molarity of water ()the quadruple of TBO) by ca. 0.4 s in the fastest case, in which the hydrolysis rate is determined by the addition rate of water, and at latest by 2.5 s in the slowest case, as revealed from Nielsen’s chronomal analysis based on in situ electric conductometry. About 83% of the subsequent precipitation process is finished by 2.5 s after the start of water addition. The precipitation process for the particle growth consists of a very fast first step for at least 77% of the total precipitation, but finished earlier than 2.5 s, and the following much slower second step completed by 10 min. The cause of the two-step growth was explained in terms of the high reactivity of the hydroxide monomer on the particle surfaces in the early stage before ammonia becomes effective, and the deactivation of the hydroxide monomer by adsorption of ammonia in the later stage. The hydrolysis process seems to finish earlier than the start of precipitation, followed by the two-step precipitation of the hydroxide monomer. The growth process of the product particles in the second stage was found to proceed in the third-order reaction-controlled

Sugimoto and Kojima kinetics, suggesting direct deposition of the trimer of the hydroxide monomer unit in equilibrium with the monomer and dimer. References and Notes (1) Sugimoto, T.; Kojima, T. J. Phys. Chem. C, in press. (2) Nielsen, A. E. Kinetics of Precipitation; Pergamon: Oxford, 1964. (3) Sugimoto, T. Monodispersed Particles; Elsevier: Amsterdam, 2001. (4) Fegley, Jr. B.; Barringer, E. A. In Better Ceramics Through Chemistry; Brinker, C. J., Clark, D. E., Ulrich, D. R., Eds.; Elsevier: New York, 1984; p 187. (5) Boyd, T. J. Polym. Sci. 1951, 7, 591. (6) Winter, G. J. Oil Colour Chem. Assoc 1953, 36, 689. 1953, 36, 695. (7) Ogihara, T.; Mizutani, N.; Kato, M. J. Am. Ceram. Soc. 1989, 72, 421. (8) Jean, J. H.; Ring, T. A. Langmuir 1986, 2, 251. (9) Sugimoto, T. AdV. Colloid Interface Sci. 1987, 28, 65. (10) Dirksen, J. A.; Ring, T. A. J. Am. Ceram. Soc. 1990, 73, 131. (11) Barringer, E. A.; Bowen, H. K. J. Am. Ceram. Soc. 1982, 65, C-199. (12) Eiden-Assmann, S.; Widoniak, J.; Maret, G. Chem. Mater. 2004, 16, 6. (13) Widoniak, J.; Eiden-Assmann, S.; Maret, G. Colloids Surf., A 2005, 270, 329. (14) Ocan˜a, M.; Rodoriguez-Clemente, R.; Serna, C. J. AdV. Mater. 1995, 7, 212. (15) Privman, V.; Goia, D. V.; Park, J.; Matijevic´, E. J. Colloid Interface Sci. 1999, 213, 36. (16) Niederberger, M.; Co¨lfen, H. Phys. Chem. Chem. Phys. 2006, 8, 3271. (17) Sugimoto, T. Chem. Eng. Technol. 2003, 26 (3), 313. (18) Sugimoto, T.; Sakata, K.; Muramatsu, A. J. Colloid Interface Sci. 1993, 159, 372. (19) Shindo, D.; Park, G.; Waseda, Y.; Sugimoto, T. J. Colloid Interface Sci. 1994, 168, 478. (20) Sugimoto, T.; Dirige, G. E.; Muramatsu, A. J. Colloid Interface Sci. 1996, 182, 444. (21) Sugimoto, T.; Muramatsu, A. J. Colloid Interface Sci. 1996, 184, 626. (22) Sugimoto, T.; Itoh, H.; Mochida, T. J. Colloid Interface Sci. 1998, 205, 42.

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