Contribution to the Determination of Kinetic Parameters of the Sol–Gel Transformation by Rheological Measurements

Journal of Colloid and Interface Science 230, 268–271 (2000) doi:10.1006/jcis.2000.7095, available online at http://www.idealibrary.com on

Contribution to the Determination of Kinetic Parameters of the Sol–Gel Transformation by Rheological Measurements Wolfram Vogelsberger,1 Andreas Seidel, and Rolf Fuchs Institute of Physical Chemistry, Faculty of Chemistry and Earth Science, Friedrich-Schiller-University Jena, Lessingstraße 10, D-07743 Jena, Germany Received October 5, 1999; accepted July 17, 2000

The system tetraethoxysilane(TEOS)–water–ethanol has been studied by rheological measurements. Different molar ratios of TEOS : water (1 : 4, 1 : 10, and 1 : 20) are studied at different temperatures (30, 40, and 50◦ C). The dynamic viscosity (rotating mode) at a constant shear rate (100 s−1 ) and the elastic and viscous moduli (oscillating mode) at a constant frequency (1 Hz) are determined. The viscosity–time curves are evaluated by application of a nucleation and particle growth model. Good agreement between experiments and theory is observed. The model allows the determination of the complex rate constant of silica precipitation. The temperature-dependent measurements gave the possibility to determine the apparent energy of activation by common methods. The results are in agreement with data from the literature. The gel time defined as intersection point of elastic and plastic moduli and its dependence on temperature are evaluated by the Smoluchowski model. The energy of activation for the coagulation was determined and found to be in the correct order of magnitude. ° C 2000

Academic Press

Key Words: sol–gel; gelation mechanism; kinetics; rheology; TEOS–wather–ethanol system.

INTRODUCTION

Synthesis of different materials via the sol–gel route is advantageous in many cases. It is widely used in the preparation of porous bodies and nanoparticles as well. The sol formation and the sol–gel transformation are accompanied by drastic changes of the rheological properties of the reacting system. A summarizing discussion of the situation can be found in (1, p. 303). Results can also be found by several authors, e.g., (1–9). Two physically different processes are responsible for the changes of the viscosity. At the beginning of the synthesis, the increase in viscosity is due to nucleation and growth processes of sol particles. Later in time, the coagulation of the sol particles forming larger aggregates and finally the gel network dominates the rheological behavior of the system. It is the aim of this article to present simple methods to discriminate between these two mechanisms. A way will be shown to calculate kinetic parameters of sol particle formation and coagulation. In this connection it is indispensable

1

To whom correspondence should be addressed.

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to measure both viscous and elastic properties of the reaction mixture, respectively. The simple system of tetraethoxysilane– water–ethanol is chosen as an example. Investigations are made for different compositions and different reaction temperatures in an acidic medium using hydrochloric acid as a catalyst. EXPERIMENTAL

A detailed description of the experiments can be found in Ref. (10). The products used are tetraethoxysilane (TEOS > 98%, Merck, Darmstadt, Germany), hydrochloric acid (p.a., Merck), and ethanol (96%, Bundesmonopolverwaltung f¨ur Branntwein, Germany). The water used was double purified by deionization. The molar ratio TEOS : ethanol = 1 : 0.36 has always been used to obtain moderate reaction times whereas the molar ratio TEOS : water is varied (T/H = 1 : 4, 1 : 10, and 1 : 20). Mixtures of TEOS–ethanol and water–hydrochloric acid were prepared in different vessels. The reaction begins with the mixing of these solutions. In the first time this reaction mixture was stirred (500 min−1 ) for homogenization. The temperature was kept constant in a thermostat. After the homogenization of the system, the pH was measured by use of a glass electrode (Schott, Mainz, Germany). All reaction conditions and the time after that app when the system to all appearances is gelled, tg , are collected in Table 1. The sol–gel transformation was followed by rheological measurements with the universal dynamic spectrometer UDS 200 (Paar Physica, Stuttgart, Germany) using a single gap cylinder system. It was necessary to use self-made measuring equipment (body from Teflon, beaker from special steel) to avoid corrosion. The first period of the process was measured under rotating conditions with a constant shear rate γ˙ = 100 s−1 to determine the dynamic viscosity, η. η versus γ˙ measurements show that no shear thinning occurs up to η = 30 mPa s (11). Only these data are used for the theoretical evaluation. A short time before gelation takes place, measurements were started in the oscillating mode with a constant frequency of f = 1 Hz and a maximum amplitude of the deformation angle ϕ = 2 mrad. Characteristic parameters in this phase of the sol–gel transformation are the elastic (storage) modulus, G 0 , the viscous (loss) modulus, G 00 , and the complex shear modulus, G ∗ = G 0 + i G 00 (12).

268

269

KINETIC PARAMETERS OF THE SOL–GEL TRANSFORMATION

TABLE 1 Reaction Conditions for the Gel Synthesis, Results of the Rheological Measurements, and Parameters and Correlation Coefficients of the Regression Analyses after Eqs. [3], [5], [6], and [8] T/H ratio ϑ/◦ C

1:4

1 : 10

1 : 20

30

40

40

50

30

40

50

50

30

40

50

Stirring time/min pH app tg /min

60 2.16 4020

30 2.12 1771

30 2.19 1860

30 2.23 840

80 2.68 2640

80 2.79 900

80 2.80 420

80 2.80 419

110 2.82 3180

80 2.85 1260

80 2.91 558

−log10 (ηS,0 ) s[log10 (ηS,0 )] −log10 (A0 /s−1 ) s[log10 (A0 /s−1 )] RC

1.15 0.03 8.72 0.01 0.998

1.22 0.13 8.34 0.03 0.992

1.14 0.06 8.36 0.02 0.998

1.35 0.13 7.91 0.03 0.991

1.78 0.09 8.20 0.02 0.996

2.04 0.19 7.67 0.03 0.992

2.78 0.38 7.20 0.04 0.994

1.89 0.23 7.16 0.04 0.997

2.53 0.19 8.03 0.03 0.994

3.02 0.31 7.50 0.03 0.989

3.97 0.72 7.02 0.06 0.988

1H 6= /(kJ mol−1 ) S0 RC

70.8 ± 4.8 −3.5 ± 1.9 −0.9953

90.8 ± 2.0 5.7 ± 0.8 −0.9995

90.3 ± 1.1 5.9 ± 0.4 −0.9999

E A /(kJ mol−1 ) ln(kf,0 A0 ) RC

73.4 ± 4.9 4.8 ± 2.5 −0.9956

93.4 ± 2.0 12.5 ± 0.8 −0.9995

92.9 ± 1.0 12.7 ± 0.4 −0.9999

tg /min G g /Pa E A /(kJ mol−1 ) ln(8kB /3a) RC coag

4058 3.98

1773 3.31

1828 3.40

814 2.83

2397 3.30

45.6 ± 1.3 1.7 ± 0.5 −0.999

RESULTS

Determination of the Kinetic Parameters of Sol Formation Typical viscosity versus reaction time curves are shown in Fig. 1 in terms of the specific viscosity ηS = (η − η0 )η0−1 where η0 is the viscosity of the solvent. Such a behavior of the viscosity can be observed if the volume fraction of the solid in the suspension is increased (13). The volume fraction increases if sol particles are formed and grow. It has been shown earlier (5) that this behavior of the viscosity can be interpreted in terms of

897 2.66

425 1.72

419 2.60

3060 2.74

1145 2.08

51.0 ± 1.7 4.1 ± 0.6 −0.999

528 1.85

50.1 ± 2.0 3.3 ± 0.8 −0.999

a model that includes nucleation and particle growth in a system where the volume may be assumed to be constant and the supersaturation reduces during nucleation (6). The model bases on the general accepted overall reaction equation kf

≡SiOH + Si(OH)4 * ) ≡Si–O–Si(OH)3 + H2 O.

[1]

kd

Here, kf and kd are the rate constants of polycondensation and dissolution, respectively. Besides the well-known temperature dependence, the rate constants kf and kd are complex functions of pH and background electrolyte concentration (14). For the constant kf the following expression can be obtained:

kf =

kneu +

[H+ ] 0 K S,1 exp(y0 )

n

[M]S,tot · 1 +

· kpos + [H+ ] 0 K S,1 exp(y0 )

0 K S,2 exp(y0 ) [H+ ]

+

0 K S,2

· kneg o

exp(y0 ) [H+ ]

with µ

[M]S,tot

FIG. 1. Time dependence of the specific viscosity for the gels prepared with a molar ratio of TEOS : water = 1 : 10 (r, 30◦ C; m, 40◦ C; d, 50◦ C). The solid lines are the results of nonlinear regression calculations after Eq. [3].

m ¶ K S,2 [H+ ] = [M]S · 1 + m + + . K S,1 [H ]

[2]

[M]S,tot is the total concentration of all monomeric silica species under saturation conditions. [M]S is the concentration of unm charged monomeric silicic acid under saturation conditions. K S,1 m and K S,2 are the equilibrium constants of the deprotonation of positively charged and uncharged monomeric silicic acid. 0 0 and K S,2 are the intrinsic equilibrium constants of the K S,1

270

VOGELSBERGER, SEIDEL, AND FUCHS

deprotonation of ≡SiOH+ 2 and ≡SiOH groups, respectively, and y0 is the dimensionless surface potential that depends strongly on the background electrolyte concentration. kneu , kpos , and kneg are the rate constants of cleavage of siloxane bonds of uncharged, positively charged, and negatively charged surface species, respectively. See also (15). Using Einsteins formula ηS = η · η0−1 − 1 = L8 (16), the corresponding rate equation can be written for the overall reaction (5) ¶¾ ½ ¾ ½ µ B0 ηS 1 B0 ηS dηS = A0 yηS ln y · 1 − · 1− − , [3] dt y y y with the abbreviations A0 = δOH

kB [M]S,tot T kf = A0 T kf and B0 = (L VS [M]S,tot )−1. σ [4]

L is a shape factor, 8 is the volume fraction of the sol particles, and t is the reaction time. A0 is the complex rate constant of the overall sol formation process containing kf (A0 = 2.76 × 10−6 mol L−1 K−1 · T · kf ). The value of the constant B0 is determined experimentally (17) to be 13.16. y is the supersaturation, δOH is the number of surface hydroxyl groups per unit of surface area, kB is the Boltzmann constant, T is the temperature, and σ is the surface tension of silica. Nonlinear regressions corresponding to Eq. [3] were carried out (18) to determine the complex rate constants A0 . The numerical integration of Eq. [3] occurred by the Runge–Kutta method of fifth degree. This method needs an additional parameter ηS,0 the regression value for the specific viscosity of the first measuring point. The logarithms of A0 and ηS,0 as well as their errors and the correlation coefficient, RC , are given in Table 1. The solid lines in Fig. 1 are calculated by means of these parameters. Sufficient agreement is observed between experiment and theory. The parameter ηS,0 is small and its influence on the sol formation is not significant. As expected, its value is smaller if the amount of water in the system is increased. In accordance with the general behavior of rate constants, A0 increases with increasing temperature. A0 increases also if the water content of the reaction mixture becomes larger. As can be seen from Table 1, the pH increases with an increasing amount of water. Thus, the observed increase of A0 is probably a pH effect due to Eq. [2]. Global parameters describing the temperature dependence of A0 may be obtained in the common manner. With the Eyring formula for kf Eq. [4] results in the following linearized form: ln

entropy, respectively. R is the gas constant, and h is the Planck constant. Another possibility is to substitute kf in Eq. [4] by the Arrhenius ansatz, which leads to the following linearized equation: ln

A0 EA 1 = ln(kf,0 A0 ) − · . T R T

[6]

Here are E A the energy of activation and kf,0 the pre-exponetial factor of the rate constant. The results of the linear regression calculations for Eqs. [5] and [6] are shown in Table 1. It is easy to make certain that the relation E A = 1H 6= + R · T is fulfilled by the measured quantities. The values of the quantities in Table 1 are in agreement with data found in the literature, which are collected in Ref. (6). Values of 38 ≤ E A ≤ 80 kJ mol−1 are reported. Determination of Kinetic Parameters of Coagulation Typical curves for the time dependence of storage and loss moduli observed in the systems investigated are shown in Fig. 2. The appropriate definition of the gel time is a matter of numerous discussions, e.g., (1, pp. 304, 9, 19–21). From these papers it becomes clear that measurements of the storage and loss moduli should be used. The scientific best founded definition is to determine the point of gelation as the point of frequency-independent loss tangent (19). Also, often used is the definition tan δ = 1, i.e., the crossover of storage and loss moduli (e.g., (21)), which is sometimes frequency dependent. It is demonstrated in (9) that the differences between these definitions and even the visually determined gel time are small in a TEOS system. Therefore, the equality of G 0 and G 00 is used to determine tg in these investigations. The data of the intersection points (G g = G 0 = G 00 ) are collected in Table 1. It becomes clear from Table 1 that the app visually determined tg and the rheological-determined tg do not differ significantly from each other.

µ 0 ¶ A kB L 1S 6= 1H 6= 1 A0 = ln · + − · 2 T h mol R R T = S0 −

1H 6= 1 · . R T

[5]

1H 6= and 1S 6= are the activation enthalpy and the activation

FIG. 2. Time dependence of the storage and loss moduli for the gels prepared with a molar ratio of TEOS : water = 1 : 10 (30◦ C: r, G 0 ; +, G 00 . 40◦ C: m, G 0 ; ∗, G 00 . 50◦ C: d, G 0 ; ×, G 00 ).

KINETIC PARAMETERS OF THE SOL–GEL TRANSFORMATION

The tg values can be used to give a quantitative description of the particle coagulation kinetics by the von Smoluchowski model (22, 1, p. 331). After this model the concentration of particles at time t, cP , is given by a second-order rate equation: P

1 1 − = kc · t = a. cP (t) cP (t = 0)

[7]

P

cP (t) is the concentration of all particle aggregates inclusive of single particles at time t. cP (t = 0) is the concentration of single particles at the beginning of the coagulation process. The rate constant of coagulation, kc , can be expressed by an Arrhenius ansatz. Von Smoluchowki calculated the preexponential factor for the diffusional encounters of spherical particles to be kc0 = 4πD · rH (22) where rH is the hydrodynamic radius of the sol particles, the effective kinetic radius. The diffusion coefficient, D, can be replaced by the Stokes– Einstein relation D = kB T /6πηr (r is the sol particle radius). With these three relations Eq. [7] results for the gel time, tg , in the following linearized form: µ ln

η tg T

¶

µ = ln

2kB rH · 3a r

¶

coag

−

EA . RT

[8]

coag

Von Smoluchowki stated rH /r ≥ 2. E A is the activation energy of coagulation. The results of the evaluation of Eq. [8] are shown coag in Table 1. E A is found to be in the range of 40–80 kJ/mol given in Ref. (1, p. 309). The fact that the gel time has a minimum at a T/H ratio of 1 : 10, i.e., the coagulation rate has a maximum, coag coag is not caused by a minimum of E A . In contrast, E A seems to have a maximum. Thus, the behavior of the kc is reflected by the pre-exponential factor. This quantity is dependent on the particle concentrations Pat the beginning of the reaction, cP (t = 0), and at the gel time, cP (t = tg ) (Eq. [7]). According to our previous considerations about an increasing rate of particle formation with increasing pH, i.e., increasing amount of water, it is possible to assume that P cP (t = 0) increases in the same direction. On the other hand, cP (t = tg ) may decrease if the reaction mixture is deluded. The existence of a maximal coagulation rate therefore could be explained qualitatively. SUMMARY

It has been shown that rheological measurements are suitable for studying the kinetics of the sol–gel transformation. Measurement of the dynamic viscosity in the first period of the process provides the possibility to determine sol particle formation and growth in dependence on time. An overall rate constant, A0 , is obtained. From the temperature dependence of A0 it is possible to estimate global activation parameters. The activation energies are in agreement with literature data. However, these measurements give no information about the mechanism of the reaction. Their meaning is only to have a possibility to calculate A0 dependent on the reaction temperature.

271

If the sol–gel process comes in an advanced stage, it is suitable to measure the storage and the loss moduli. The gel time is defined as the time when these moduli become equal. Kinetic evaluation of the gel time provides information on the coagulation of the sol particles to form the gel structure. The rate constant of coagulation can also be expressed by an Arrhenius ansatz. Analysis of the kinetic parameters show that an observed minimum of the gel time, i.e., a maximum of the rate constant of coagulation, is mainly caused by two effects. An increasing amount of water in the reaction mixture increases the pH. Increasing pH increases the rate of particle formation and their growth. The result is a decrease of the gel time. On the other hand, the increasing amount of water reduces the particle concentration in the system that is accompanied by an increase in gel time of the system. The main processes involved in the sol–gel process are the following ones: particle formation by nucleation, particle growth, reduction of the particle concentration by Ostwald ripening, and formation of the gel network by particle coagulation. The first three processes may be studied by measurement of the dynamic viscosity whereas the last process can be quantified by the measurement of the rheological moduli. Thus, it has been shown that rheological measurements give the possibility of quantifying the kinetics of the whole sol–gel process. ACKNOWLEDGMENT The authors would like to acknowledge the “Deutsche Forschungsgemeinschaft” for financial support of this work.

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