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Articles Aqueous Sol-Gel Process in the Silica-Metasilicate System. A Microrheological Study Ma´rio Alberto Tenan,*,† David Mendez Soares,‡ and Celso Aparecido Bertran§ Centro de Cieˆ ncias Exatas e de Tecnologia, Universidade de Mogi das Cruzes, Avenida Caˆ ndido Xavier de Almeida Souza, 200 08780-911, Mogi das Cruzes, SP, Brasil, Instituto de Fı´sica Gleb Wataghin and Instituto de Quı´mica, Universidade Estadual de Campinas (UNICAMP) 13083-970, Campinas, SP, Brasil Received January 15, 2000. In Final Form: September 15, 2000 The electrochemical quartz crystal microbalance (EQCM) was used to investigate the sol-gel process from an aqueous sodium metasilicate solution at 25 °C and pH 3. From the EQCM data it was possible to obtain information on the changing rheological properties of the system during the whole process. Besides sensing changes in the rheological properties, the EQCM detected film formation on the sensor surface. This additional information was used to estimate, using a model for film formation kinetics, the sol particle size (the model prediction was confirmed by light scattering measurements). The particle size estimation was of importance in the analysis of the sol viscosity behavior during the earlier stages of the particle aggregation process. The analysis, based on a classical model for the viscosity of a dispersion of charged particles in an electrolyte, provided some insight into the initial aggregation phenomena and microgel formation. Determination of the gelling point was made by examining the evolution of the shear storage modulus of the system. The gelation point was identified as being the time at which the storage modulus rose abruptly from zero. The rheological behavior of the system beyond the gelation point was analyzed in terms of the loss tangent, which value decreased noticeably before becoming constant. The observed decrease in the loss tangent, as a measure of the increasing importance of the elastic energy storage relative to the energy dissipation in the gel, was evidence that Si-O-Si bond formation continued to take place for some time after the sol-gel transition.
1. Introduction Investigations of the sol-gel process are of considerable interest due to the possibility of using this kind of process as a synthetic route for the preparation of outstanding ceramic materials.1-4 The sol-gel process consists of the hydrolysis and condensation of inorganic chains in solution, leading to the formation of a network which after drying results in an amorphous ceramic precursor. The homogeneity and reactivity of the amorphous precursor determine the crystallization temperature and formation of desirable phases during the calcination step in the ceramic material preparation. Thanks to its characteristics, the sol-gel process provides a wet route for the preparation of precursors with the required reactivity and homogeneity as well as low crystallization temperatures.3,4 Since homogeneity depends on the control of the initial steps in the transformation of a sol into a gel, a better understanding of the gelation kinetics is essential in * To whom correspondence should be addressed. E-mail:
[email protected]. Fax: +5511 47987253 Phone: +5511 47987210. † Centro de Cie ˆ ncias Exatas e de Tecnologia. ‡ Instituto de Fı´sica Gleb Wataghin. § Instituto de Quı´mica. (1) Tursiloadi, S.; Imai, H.; Hirashima, H. J. Ceram. Soc. Japan 1995, 103, 1069. (2) Vercauteren, S.; Keizer, K.; Vansant, E. F.; Luyten, J.; Leysen, R. J. Porous Mater. 1998, 5, 241. (3) Senguttuvan, G.; Settu, T.; Kuppusamy, P.; Kamaraj, V. J. Mater. Synth. Process. 1999, 7, 175. (4) Barrera-Solano, C.; Esquivias, L.; Messing, G. L. J. Am. Ceram. Soc. 1999, 82, 1318.
relating the properties of the produced ceramic to the starting sol-gel process.5-7 Microscopic as well as macroscopic methods have been employed in the investigation of the gelation kinetics. The microscopic methods are related to the structural aspects involved in the formation of both sol and gel from the starting substances such as, for example, salts and alkoxides. Typically those methods are optical8 and spectroscopic;5,9 small-angle X-ray scattering is another example of microscopic method used in the investigation of the sol-gel transition.10,11 On the other hand, rheometry, using rheometers in different configurations, constitutes a macroscopic way of investigating gelling kinetics.7,12-14 Piezoelectric sensors in electrochemical quartz crystal (5) MunozAguado, M. J.; Gregorkiewitz, M. J. Colloid Interface Sci. 1997, 185, 459. (6) Meixner, D. L.; Dyer, P. N. J. Sol-Gel Sci. Tech. 1999, 14, 223. (7) Santos, L. R. B.; Santilli, C. V.; Pulcinelli, S. H. J. Non-Cryst. Solids 1999, 247, 153. (8) Hugerth, A.; Nilsson, S.; Sundelof, L. O. Int. J. Biol. Macromol. 1999, 26, 69. (9) Kelts, L. W.; Effinger, N. J.; Melpolder, S. M. J. Non-Cryst. Solids 1986, 83, 353. (10) Yamane, M.; Inoue, S.; Yasumori, A. J. Non-Cryst. Solids 1984, 63, 13. (11) Orcel, G.; Hench, L. L.; Artaki, I.; Jonas, J.; Zerda, T. W. J. Non-Cryst. Solids 1988, 105, 223. (12) Paulsson, M.; Hagerstrom, H.; Edsman, K. Eur. J. Pharm. Sci. 1999, 9, 99. (13) Chiu, H. T.; Wang, J. H. Polym. Eng. Sci. 1999, 39, 1769. (14) Puyol, P.; Cotter, P. F.; Mulvihill, D. M. Int. J. Dairy Tech. 1999, 52, 81.
10.1021/la0000522 CCC: $19.00 © 2000 American Chemical Society Published on Web 11/28/2000
EQCM Study of Si-O-Si Bond Formation in Sol-Gel
microbalances15 can also be used for the same purpose. In contrast with conventional rheometers, those probes have the capability of inducing perturbations with very small amplitudes, i.e., amplitudes comparable to typical sizes of the microstructures in the analyzed system. In this article we discuss the application of the electrochemical quartz crystal microbalance (EQCM) to the study of silica gelation from an aqueous silica sol prepared by the hydrolysis of the metasilicate ion. A brief description of the apparatus and how it can be used to follow the gelation process are presented in Sections 2 and 3. The EQCM data and the related information on the system rheological properties such as the viscosity and the elastic modulus are presented in Section 4. Section 5 is dedicated to the analysis of a film formation on the quartz crystal sensor surface at the beginning of the process. From that analysis it was possible to infer the sol particle size. Other particle characteristics, such as time evolution of the microgel aggregation, gel point determination, and the viscoelastic behavior in the gel phase, are discussed in Section 6. Concluding remarks are presented in Section 7. 2. The Electrochemical Quartz Crystal Microbalance The purpose of this section is to present a brief description of a modified quartz crystal microbalance and show how it can be used to monitor minute mass changes as well as changes in the rheological properties of a contacting fluid medium. For reviews on the quartz crystal microbalance applications we refer to refs 16, 17. The microbalance used in our laboratory consists of a thin circular plate cut from an oriented piezoelectric quartz crystal (specifically an AT-cut plate). Both faces (each 14 mm in diameter) are coated with gold films (electrodes), each having a thickness of ca. 200 nm and an effective area of 0.28 cm2. A driver18 applies appropriately an alternate electric tension to the electrodes to induce the crystal to oscillate in its fundamental mode parallel to its faces. The resonant mechanical oscillations are basically fixed by the crystal thickness (thickness ∼0.28 mm, frequency ∼6 MHz), whereas damping depends on the characteristics of the mounting and the surrounding media. Due to the possibility of using one side of the microbalance as a working electrode in an electrochemical cell while simultaneously sensing the properties mentioned above, the apparatus is also known as the electrochemical quartz crystal microbalance, or EQCM. Near resonance the electrical behavior of the compound oscillator (crystal and contacting media) corresponds to that of a series connection of an inductor (L′), a capacitor (C′), and a resistor (R′), known as the motional arm of the equivalent electrical circuit of the oscillator. Due to piezoelectricity, inductance L′ is related to oscillating masses, capacitance C′ to elastic properties and resistance R′ to dissipation in the system.16,17 At the selected resonance frequency, fres ) 2π/(L′C′)1/2 the electrical impedance of the oscillator is essentially resistive (R′). The special features of the driver used in our laboratory19 permit monitoring changes of both the resonant frequency and the resonant resistance R′ during a process such as that analyzed in Section 4. To understand how the knowledge of these changes is helpful in obtaining (15) Soares, D. M.; Kautek, W.; Frubo¨se, C.; Doblhofer, K. Ber. BunsenGes. Phys. Chem. 1994, 98, 219. (16) Buttry, D. A.; Ward, M. D. Chem. Rev. 1992, 92, 1355. (17) Tenan, M. A.; Soares, D. M. Braz. J. Phys. 1998, 28, 405. (18) Soares, D. M. Meas. Sci. Technol. 1993, 4, 549. (19) Soares, D. M.; Tenan, M. A.; Wasle, S. Electrochim. Acta 1998, 44, 263.
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information on the mechanical properties of the monitored system, one should consider the following points: (i) The change Ze in the electrical impedance of the oscillator caused by loading the crystal with a compound layer is given approximately by ref 19
Ze ) ∆R′ + j(4πL′0∆fres)
(1)
where L′0 is the inductance of the unloaded crystal, ∆R′ is the resistance change and ∆fres is the corresponding frequency shift. (ii) The electrical impedance change Ze and the mechanical impedance Zm at the sensing surface of the crystal are related by
Ze )
Zm κ2
(2)
where κ is the crystal electromechanical constant.19 (iii) In the case of our interest, Zm corresponds to the interaction of the crystal with two layers of different materials: a thin rigid film and a viscous or viscoelastic semi-infinite layer. As discussed in Section 4, the EQCM data analysis indicates that at the beginning of the experiment a very thin film (20 to 30 nm thickness) was rapidly deposited on the surface of the electrode and remained strongly attached to it during the whole solgel process. The mechanical impedance Zm of this compound layer is given approximately by ref 20.
ωmf + xFG A
Zm = j
(3)
The first term on the right-hand side of eq 3 is the contribution of the film; in this term, j is the imaginary unit, (-1)1/2, ω is the angular frequency of the mechanical oscillations, mf is the thin film mass, and A is the electrode area. The second term is the contribution of the viscous or viscoelastic semi-infinite layer with density F and complex shear modulus G. The complex shear modulus G is composed of two terms: G ) G′ + jG′′. The real term G′, known as the shear storage modulus, represents the elastic response to stresses developed in the medium, whereas the imaginary component G′′, known as the shear loss modulus, is related to dissipation. The latter is frequently expressed in terms of the viscosity η, namely G′′ ) ωη.21 (iv) Equations 1-3 can be used to calculate the unknown values of the mechanical properties from the measured EQCM frequency and resistance data. More specifically, they are helpful in evaluating the mechanical properties of the monitored system in our experiment because the time scale for the kinetics of film formation on the electrode was quite small compared to that for the system’s evolution from the sol state (viscous fluid) to the gel state (viscoelastic fluid). Another way to infer the rheological properties of the contacting layer from the EQCM data is to consider the physical model for the compound oscillator.17,21 This approach is based on the physics of wave propagation through the contacting media, namely, air (viscous fluid of low viscosity), quartz crystal (elastic medium), rigid film of negligible thickness, and semi-infinite viscous (sol) or viscoelastic (gel) layer. Thanks to the coupling between the elastic and electrical properties of quartz, it is possible to evaluate the electrical impedance of the compound oscillator, as a function of the parameters characterizing (20) Granstaff, V. E.; Martin, S. J. J. Appl. Phys. 1994, 75, 1319. (21) Kanazawa, K. K. Faraday Discuss. 1997, 107, 77.
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the film (deposited mass, mf) and the fluid layer (complex shear modulus, G). By adjusting the values of mf and G, it is possible to reproduce the EQCM frequency and resistance data, thus obtaining the desired information. 3. Experimental Section A 10% w/w metasilicate solution was prepared by dissolving Na2SiO3‚5H2O salt in water immediately before use. To initiate the gelation process at pH 3, 9 mol/L sulfuric acid was added to the solution and the desired pH was detected with the help of a pH indicator paper (Merck). Double distilled water and analytical grade chemicals (Merck) were used throughout. As a result of the pH lowering, a silica sol was rapidly formed and the kinetics of the gelation process at pH 3 and 25 °C was then monitored by the piezoelectric sensor of the EQCM adapted to the chemical cell22 containing ca. 50 mL of solution. The EQCM sensor consisted of a 6 MHz polished quartz crystal (KVG; Neckarbiscofsheim) coated with 200 nm thick gold layers with a chromium underlayer. Monitoring was accomplished by measuring the variations of (1) the quartz crystal resonance frequency and (2) the resistance related to the impedance of the crystal-driver circuit. In an independent experiment, we used the dynamic light scattering technique (equipment: Malvern Instruments, Model4700) to determine the particle size distribution in the silica sol obtained by solution pH lowering as described above.
Figure 1. EQCM resonant frequency shift relative to air, as a function of time.
4. Silica Gelation Monitoring by the EQCM As is well-known23,24 the silica gelation process takes place in three stages: (i) polymerization of monosilicic acid (Si(OH)4) to form amorphous SiO2 spherical particles, (ii) growth of particles, and (iii) particle aggregation. Sol formation (stages (i) and (ii)) starts from the moment of the solution preparation and, depending on the experimental conditions, ends before the beginning of stage (iii). Growing aggregates in stage (iii) start as branched chains that form progressively three-dimensional network regions in the sol. These microgel regions grow at the expense of the sol regions until they link together to form the complete gel network. In our experiment, sol formation preceding stage (iii) was achieved by a convenient choice of pH and SiO2 concentration of the metasilicate solution. Stage (iii) was then monitored by measuring changes in the EQCM resonant frequency and resistance. Time evolution of those measured quantities is shown in Figures 1 and 2. A film formation on the electrode at the first moments of the process (duration ∼11 min) is clearly identified in these figures. This process is characterized by frequency changes without any change in the EQCM resistance. The rheological properties of the fluid system and the mass of the film deposited initially on the electrode surface were evaluated with the help of the physical model for the EQCM,17,21 as described in Section 2. In the calculation we considered the values for the electromechanical parameters of quartz,25 those for the crystal geometry, and the experimental value for the solution density, F ) 1.06 × 103 kg/m3. Figures 3 and 4 are plots of the viscosity η and the shear storage modulus G′, as functions of time, respectively. Figure 5 shows the mass mf of the film deposited on the electrode surface, as a function of time. (22) Lima, P. T.; Bertran, C. A.; Soares, D. M. Extended Abstracts of the 49th Annual Meeting of the International Society of Electrochemistry; ISE: Kytakyushu, Japan, 1998; p 181. (23) Iler, R. K. The Chemistry of Silica; Wiley: New York, 1979; p 173. (24) Brinker; C. J.; Scherer, G. W. Sol-Gel SciencesThe Physics and Chemistry of Sol-Gel Processing; Academic Press: Boston, 1990; p 100. (25) Eggers, F.; Funck, Th. J. Phys. E 1987, 20, 523.
Figure 2. EQCM resonant resistance shift relative to air, as a function of time.
Figure 3. Fluid viscosity as a function of time.
As can be seen in Figure 3 the fluid viscosity did not change appreciably during the whole process when compared with results from usual experiments (near the gelation point, sudden viscosity changes from ∼1 mPa s to ∼102 mPa s or more have been reported; see ref 24, pp 304-311). To understand this result it should be noted that in our experiment the amplitude of the shear motion induced by the vibrating crystal was very small (using the physical model of ref 17 we can estimate that the amplitude was on the nm scale, see also ref 21). Thus, by preserving the structure of the three-dimensional gel network, the shear motion with small amplitude dissipated energy essentially in the liquid retained by the elastic structure. In contrast, large-amplitude motions induced in other experiments irreversibly affect the structure, with the ultimate result of increasing dissipation in the system.
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in Figure 5. Film mass deposition obeyed the exponential law -k(t-t0) mf ) m(∞) ] f [1 - e
Figure 4. Fluid shear storage modulus as a function of time.
(4)
as described by a sigmoidal function fitted to the experimental data set. The corresponding fitting parameters -1 for eq 4 have the values m(∞) f ) 1.077 µg, k ) 0.01 207 s and t0 ) -5.531 s, with a χ2 value of 0.00 196. To model the film formation process we consider the grown charged silica particles in random motion near the electrode surface. Some of those particles eventually impinge on the surface and interact on contact, in which case they adhere to the electrode. We assume that the energy of attraction between the surface and the adhered particles is large compared with the thermal energy of the latter, so that although lateral Brownian motion is allowed on the surface the rate of particle detachment can be considered negligible. This assumption turned out to be justified, as its ultimate consequence, namely the predicted particle size, was confirmed experimentally by light scattering measurements. Let θ(t) be the fraction of the electrode surface covered with the adherent particles at time t. The rate of change of the coverage factor, dθ/dt, due to adhesion is, in principle, proportional to (a) the fraction (1 - θ) of the surface still uncovered at time t and (b) the number density of particles in the dispersion contacting the electrode. However, as shown below, changes in the number density are not significant for the conditions of our experiment; thus we propose that the kinetics of film formation is governed by the equation
Figure 5. Mass deposited on the electrode surface, as a function of time.
dθ ) k(1 - θ) dt
For a discussion on the difficulty of identifying the gelation point from viscosity data we refer to ref 24, p 304. The informative way to look at gelation is to analyze the results in Figure 4. The abrupt rise of the magnitude of the storage modulus observed at about 1600 s can be identified as the gel point. This interpretation conforms with the picture of gelation described at the beginning of this section, i.e., a significant elastic response to stress appears when the last links are formed among large clusters to create a continuous 3D network extending throughout the system. Before closing this section, it should be mentioned that even though stable colloidal suspensions can exhibit a finite shear storage modulus in the limit of infinite frequency, our interpretation of the 6-MHz EQCM results is germane. In fact, (i) the measured storage modulus in the sol phase (G′ ) 0) is not in contradiction with the model prediction for G′ of hard sphere colloids.26 For the conditions of our experiment (volume fraction φ ≈ 0.01, particle diameter d ) 32 nm and temperature T ) 298 K), it follows from Figure 1 of ref 26 that G′ < 10-2kBT/d3 ≈ 1 Pa, in the infinite frequency limit. Second, (ii) since the particles were formed prior to the beginning of the measurements and (iii) the gel state was observed visually at the end of them, the abrupt change in G′ that occurred at ∼1600 s can be attributed to the formation of the 3D aggregate network structure.
where the constant k depends on the temperature and the free energy change due to the binding process. Integration of eq 5 gives
5. Film Formation The time evolution of the mass mf deposited on the electrode surface at the beginning of stage (iii) is shown (26) Lionberger, R. A.; Russel, W. B. J. Rheol. 1994, 38, 1885.
θ ) 1 - e-kt
(5)
(6)
The coverage factor θ(t) can be written as the ratio between the number N(t) of adherent particles at time t and the total number Ntot of those particles forming a close-packed array that completely covers the electrode surface at the end of the process. Accordingly, it follows from eq 6 that
N ) Ntot(1 - e-kt)
(7)
Assuming that particles in the sol are of uniform size, we can write the mass of the film as the product of the number of particles on the electrode, N, and the mass of a silica particle, mparticle. Using eq 7 we write for mf
mf(model) ) Ntotmparticle(1 - e-kt) or (∞) (1 - e-kt) mf(model) ) mf(model)
(8)
(∞) ) Ntotmparticle. For an estimate of Ntot we where mf(model) consider a hexagonal close packing array of spherical particles of radius a covering the electrode surface of area A. In this case,27
(27) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Plenum Press: New York, 1970; Vol. 2, p 766.
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Ntot )
Tenan et al.
1 A 2 2x3 a
Table 1. Particle Size Distribution Obtained from Light Scattering Experiment Performed in a Silica Sol Immediately after Preparation
Expressing the particle mass in terms of its density, Fparticle, and considering the above relation between Ntot and thegeometric parameters A and a, the limiting value of (∞) , can be rewritten as the film mass, mf(model) (∞) mf(model) )
2π Fparticle Aa 3x3
(9)
Particle size can be inferred from eq 9 after equating eq 8 with eq 4. For the numerical estimate we consider Fparticle ) 2 × 103 kg/m3. This value seems to be a quite reasonable choice for the average density of a particle constituted of an amorphous SiO2 core surrounded by silanol groups (ref 23, p 217). The other values to be considered in the calculation are those for the fitting parameters in eq 4 and the experimental effective electrode area A ) 0.28 cm2. From these considerations we get a particle radius a ≈ 16 nm. Table 1 shows the results for the particle size distribution obtained from the light scattering experiment performed in a silica sol immediately after preparation. Those results show that the silica sol was polydisperse. The dispersion was characterized by an average particle radius of 21 nm, with most of the particles having radii between 10 and 23 nm. This independent determination of the particle radius corroborates our assumption that a compact monolayer was formed on the electrode surface. The discrepancy found between the values obtained by the light scattering technique and those determined by the EQCM measurements can be attributed mainly to two factors: (i) modeling the film as a close packing array of monosized spheres and (ii) small differences in particle size distributions resulting from independent preparations of the silica sols. Let us now discuss the diffusion process that the “particle-sink” electrode induces in the contacting fluid dispersion. The particle distribution in the fluid is expected to change in the direction perpendicular to the electrode surface whereas fluid flow as induced by the small-amplitude shear oscillations of the crystal takes place in a direction parallel to the surface. Thus, one may consider that diffusion and fluid motion are uncoupled, and the transport equation for the particles in thedispersion near the electrode reduces to the diffusion equation
∂2n ∂n )D 2 ∂t ∂y
(10)
In eq 10 n stands for the number density of particles, D ≈ 2.9 × 10-12 m2/s is the diffusion coefficient for silica particles (ref 28, pp 740, 779, and 787), and y is the Cartesian coordinate in the direction orthogonal to the electrode. As the initial condition we set
n(y,0) ) n0
(11)
and as the boundary conditions we consider
n(∞,t) ) n0
size (nm)
number
4.2 5.1 6.2 7.6 9.3 11.3 13.9 16.9 20.7 25.3 30.9 37.7 46.1 56.3 68.8 84 102.7 125.4 153.2 187.2 228.7 279.4 341.3 417
0 0.1 0.3 1.1 2.4 4.3 6.4 8.5 10.3 11.4 11.7 11.2 10 8.2 6.1 4.1 2.3 1.1 0.3 0.1 0 0 0 0
J0 ) -
Ntotk -kt 1 dN )e A dt A
(13)
Equation 13 is the balance equation for particles at the boundary y ) 0, i.e., the flux density of particles
J0 ) -D
∂n ∂y
|
y)0
through the electrode-dispersion interface should be equal to minus the rate of change in the number of particles adherent to the electrode surface, per unit area. Time evolution of the particle distribution in the sol near the electrode, as determined by eqs 10-13, is shown in Figure 6. For the numerical evaluation of the curves in Figure 6 the particle number density in the bulk was taken as n0 ≈ 7.9 × 1020 particles/m3. This value was obtained by taking into account the experimental value for the solution density, the Na2SiO3 concentration in the initial solution, the estimated particle radius, and the assumed density for a particle. Since diffusion is a slow mass transfer process, the particle number density near the sink decreases quite rapidly at the first moments of the electrode coating process. However, as the surface becomes covered with the adherent silica particles, the flux density J0 decreases to zero and diffusion still proceeds in order to replenish all the particles used up during the film deposition process. It should be stressed that even at the maximum depletion rate the particle density near the electrode does not change appreciably (maximum change ≈ 6%, see Figure 6). This result justifies our assumption leading to eq 5. 6. Rheology of the Sol-Gel System
(12)
and (28) Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2nd ed.; Marcel Dekker: New York, 1986.
Sol Phase. At the very beginning of stage (iii), i.e., before aggregation becomes significant, the sol in contact with the oscillating crystal can be modeled as an isotropic Newtonian dispersion of slightly charged spheres in a 0.5 M Na2SO4 aqueous solution (all the Si atoms from the Na2SiO3 are assumed to react and form the silica particles). At pH 3 and a temperature of 25 °C the particles are
EQCM Study of Si-O-Si Bond Formation in Sol-Gel
Figure 6. Particle distribution in the dispersion near the electrode, at different times. (a) t ) 1 s; (b) t ) 10 s; (c) t ) 120 s; (d) t ) 240 s; (e) t ) 360 s; (f) t ) 665.5 s (end of film formation); (g) t ) 960 s; (h) t ) 1320 s.
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at high concentrations, we will use it to get an estimate for κ. This gives κ ≈ 4.02 × 109 m-1. To estimate ζ we consider its relation with the experimental mobility u of a particle, ζ ) ηu/C, where C is a constant whose numerical value depends on the magnitude of the product κa (ref 28, p 751). In our case κa ≈ 64.3, which corresponds to C ≈ 0.857 (ref 28, p 756). Taking as a typical value for the mobility of the SiO2 particles u ≈ -1.4 × 10-8 m2/(V s) (see ref 28, p 787), we get ζ ≈ -28 mV, which gives a charge Q ≈ -1.9 × 10-17 C. This value corresponds to a surface charge density expressed as OH- ions of ca. 4 × 10-2 ions/nm2. This is a quite reasonable estimate considering the pH and salt concentration of the solution in our experiment (cf. ref 23, p 356). In studying the rheology of the suspension, some variables should be considered. One of them is the volume fraction, φ ) (4πa3/3)n0, occupied by the dispersed particles. In our case, the volume fraction amounts to φ ) 0.0149. Another variable is the dimensionless viscosity, η/η0, or the ratio of the viscosity of the dispersion, η, to that of the dispersing liquid, η0. An empirical relation between the relative viscosity and the volume fraction for a suspension of uniform spherical particles is given by (see refs 23, p 361, and 29)
η η0
Figure 7. Shear loss modulus versus EQCM resonant resistance change. The difference in the straight lines shown in the figure is statistically significant as the two points defining the first straight line correspond to the set of the 14 experimental data points obtained prior to the sol-gel transition.
expected to bear a negative surface charge (ref 24, p 104). For a rough estimate for the charge Q on a particle of radius a ) 16 nm we consider the relation (ref 28, p 779) Q ) aζ(1 + κa), where is the permittivity of the medium (which is taken as that for water (ref 28, p 691)), ζ is the zeta potential, and κ is the reciprocal of the diffuse double layer thickness for the particles. Although the DebyeHu¨ckel approximation (ref 28, p 691) should not be valid
(14)
Considering eq 14 as being appropriate to describe the initial behavior of the sol in our experiment, we can get an estimate for the viscosity η0 of the dispersing medium. For φ ) 0.0149 and η ) 1.163 mPa s, eq 14 gives η0 ) 1.119 mPa s, a value which differs by less than 1% from the viscosity of a 0.5 M Na2SO4 solution at 25 °C.30 A more recent theoretical analysis of the rheology of suspensions of rigid spheres31 predicts the following expression for the relative viscosity
η η0
Figure 8. Time evolution of the loss tangent (G′′/G′) in the gel phase. The pronounced decrease in the loss tangent value indicates that Si-O-Si bonds continued to form until t ≈ 5000 s.
) 1 + 2.5φ + 10.05φ2
) 1 + 2.5βφ + 2.5(βφ)2 +
3 L5 2 φ 40 a
()
(15)
The model takes into account the interaction of two charged particles under the action of Brownian motion, electrostatic repulsion, and viscous forces. In eq 15 the coefficient β J 1 accounts for the primary electroviscous effect related to the deformation of the electrical double layer around an isolated sphere. The first O(φ2) term is due to far-field hydrodynamic interactions and the second one is the dominant nonequilibrium contribution from the electrostatic particle-particle interactions. This second O(φ2) term is characterized by a separation parameter L such that in a thin transition layer at a distance r ≈ L from the center of a particle the repulsive electrostatic interactions become comparable to those arising from Brownian motions and viscous effects. A comparison between eq 15 with β ≈ 1 and eq 14 leads to L ≈ 2.5a. It is worth comparing this result for L with the mean separation s between the centers of two particles in a uniform distribution of density n0. For the conditions of our experiment, s ) n01/3 ≈ 6.9a. Thus we get L < s, as should be expected for a low rate aggregation process. (29) Thomas, D. G. J. Colloid Sci. 1965, 20, 267. (30) Stokes, R. H.; Mills, R. Viscosity of Electrolytes and Related Properties, The International Encyclopedia of Physical Chemistry and Chemical Physics; Pergamon Press: Oxford, 1965; p 123. (31) Russel, W. B. Effects of Interactions between Particles on the Rheology of Dispersions, in Theory of Dispersed Multiphase Flow; Academic Press: New York, 1983; p 17.
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As the particles collide and form aggregates, the model of hard sphere monodispersion loses its validity. Parameters such as porosity and ultimately elasticity of the aggregates should be taken into account to better describe the rheology of the system. However, if we assume that eq 14 still holds for the data marked as point (a) in Figure 3, that equation predicts a volume fraction φ ) 0.0268. Since the average density of the system was observed to be constant over the whole process, this almost 2-fold increase in the volume fraction can be interpreted as evidence of spongy clusters holding liquid in their structures. As discussed above the results for viscosity below t ≈ 1600 s provided some understanding of the initial aggregation phenomena leading the sol to microgels. On the other hand, as is discussed in the following paragraphs, the results for G′ and G′′ obtained from the EQCM data above t ≈ 1600 s allow us to characterize clearly the solgel transition and the behavior of the system in the gel phase. Sol-Gel Transition. Another way to identify gelation from the EQCM data is by plotting the shear loss modulus, G′′ ) ωη, against the resonant resistance change, ∆R′, see Figure 7. As shown in the figure, a discontinuity of the straight lines occurs between points (a) and (b) which correspond to those marked as (a) and (b) in Figure 4, respectively. Since the resonant resistance R′ and the moduli G′ and G′′ are interrelated (see Section 2, in particular eqs 2 and 3), the difference in the straight lines in Figure 7 reflects the discontinuity in the G′ value at ∼1600 s, thus corroborating our previous interpretation. Gel Phase. Finally, it is instructive to analyze Figure 8, which shows the evolution of the rheology of the gel in
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terms of the loss tangent. The loss tangent, defined as G′′/G′, is a dimensionless measure of the ratio of the energy loss to the energy stored in each cycle of the acoustic wave penetrating the viscoelastic medium. As shown in Figure 8, the relative importance of dissipation decreased until t ≈ 5000 s. That decrease observed in the loss tangent suggests that bond formation did not stop at the gel point. This would occur not only because the incipient 3D network was compliant enough to allow segments of it to move and make new bonds but also because particles in the remaining sol within the gel skeleton continued to attach themselves to it. No further changes in the loss tangent were observed after t ≈ 5000 s, this being an indication that the 3D network formation came to completion. 7. Conclusion The EQCM used in our laboratory is a wide-range detector for Newtonian and non-Newtonian fluids, producing minimal perturbations in the analyzed systems and permitting quantitative measurements of viscous and/ or viscoelastic properties of them. Thanks to this feature it was possible to monitor the sol-gel process from the metasilicate solution and detect the gelation point. That point was identified by examination to be the time at which the system developed elasticity. Furthermore, detection of a film formed on the electrode surface enabled us to estimate the particle size. Acknowledgment. Financial Support: FAPESP, CNPq and FAEP/OMEC. LA0000522
