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Physical Basis for the Formation and Stability of Silica Nanoparticles in Basic Solutions of Monovalent Cations Jeffrey D. Rimer, Raul F. Lobo,* and Dionisios G. Vlachos* Center for Catalytic Science and Technology, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received April 27, 2005. In Final Form: July 1, 2005 The colloidal stability, phase behavior, and solubility of silica nanoparticles (3-10 nm) that are formed in basic solutions of monovalent cations (primarily tetrapropylammonium) are investigated using a combination of chemical equilibria and electrostatic models. The free-energy gain associated with the formation of an electric double layer surrounding the nanoparticle was obtained by solving the PoissonBoltzmann equation. This free energy is an important contribution to the total free energy of the particle and is second only to the formation of Si-O-Si bonds. The free energy of formation of the nanoparticles becomes increasingly negative with an increase in particle size and density, which explains the lower solubility of nanoparticles compared to that of amorphous silica. There is a minimum in the free energy of condensation as a function of size that qualitatively explains why the formation of small particles with a uniform size ( 100 silicon atoms (∆Gc ) -16 kJ/mol of Si).29 The condensation model consists of a set of equilibrium equations (eqs 3, 5, and 6), along with the dissociation of water (pKw ) 14 at 25 °C): Kw
H2O 798 OH- + H+
(8)
(7)
In the model, we consider the total silica in solution, [SiO2], (29) The free energy of oligomerization is reported on the basis of 1 mol of Si(OH)4.
The electroneutrality of the solution is maintained using the following equation:
0 ) [TPA+] + [H+] - [OH-] - [Si(OH)3O-] [tSiO-] (12) which, when combined with eq 3 and eqs 6-11, results in a set of three equations, three unknowns ([Si(OH)3O-], [SiO2]np, and [OH-]), and one fitting parameter, pKa,np. The 9 TPAOH/9500 H2O curve in Figure 1b was fitted with an optimum pKa,np value of 11.2, which was obtained by minimizing the residual error, E, between the experimental and the model pH values:
E)
∑i (pHexp - pHmod )2
(13)
in which the index i refers to each data point. The pH was then predicted for the remaining curves in the phase diagram, as shown in Figure 3a, for nanoparticles with n ) 356 silicon atoms. The model slightly overestimates the pH in region I but captures the transition from monomers to nanoparticles very well. Figure 3b shows the predicted concentrations of monomeric and nanoparticle species. The chemistry in region I is dominated by the deprotonation of monomeric silica with insignificant nanoparticle formation, but at the CAC, an appreciable concentration of neutral silicic
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acid starts to build, leading to condensation and the formation of nanoparticles in solution. The predicted CAC occurs close to a 1:1 ratio of [SiO2] and [TPAOH]initial, which is in excellent agreement with the experimental data.4 After the CAC is reached, the model predicts a decrease in the total concentration of monomeric silica, which is consistent with the reduction of conductivity observed in region II. The physics leading to a well-defined CAC is analogous to that of surfactants that experience a critical micelle concentration (CMC), with the constraint that, in the case of silica, aggregation is coupled to acid-base chemistry. The condition for thermodynamic equilibrium requires that the chemical potential of each species, µoi , is equal for solutions containing monomers (µo1) and various sized aggregates (µon). The CMC is often expressed with respect to the free energy of micelle formation:30,31
Gomic ) µon - µo1 ) RT ln CMC
(14)
and thus, the concentration of micelles, Cn, is given by
{
(
Cn ) n C1 exp
)}
-∆Gomic RT
n
(15)
This equation is identical to eq 11, which is used in the condensation model; therefore, the CAC and CMC are conceptually similar and show that aggregate formation occurs when µon < µo1. It is well-known for inorganic oxides that as the degree of silica polymerization increases, the pKa decreases. On the basis of the relatively large size of the nanoparticle (n > 100), we would expect that pKa,np e 9.0.26 However, we find that pKa,np ) 11.2, which is much higher than that of a silica monomer (9.45) and even a dimer (9.0). The reason is that pKa,np represents an effective constant that incorporates not only formation and breakage of chemical bonds (SiO-H) but also the effect of the charged nanoparticle surface. The concentration of protons near the particle surface is given by the Boltzmann factor:
[H+]o ) [H+]∞ exp -(eψo/kBT)
(16)
in which ψo is the surface potential, [H+]∞ is the bulk proton concentration, e is the charge of an electron, T is the temperature, and kB is the Boltzmann constant. Substituting this into the equilibrium expression represented by eq 6 yields
Ka,np ) Koa,np exp(eψo/kBT) )
[tSiO-][H+]∞ [tSiOH]
(17)
in which the effective constant Ka,np is a product of the true nanoparticle deprotonation constant, Koa,np, and the term exp(eψo/kBT), which accounts for the surface potential of the nanoparticle. To decouple these contributions, the surface potential, ψo, must be known. This is accomplished using a complexation model (see section 2). The results in Figure 3 were determined using particles with 356 silicon atoms, which is specific to nanoparticles that are formed in 9 TPAOH/9500 H2O solutions; however, we know that particle size varies with solution pH. The (30) Evans, D. F.; Wennerstrom, H. The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology meet, 2nd ed.; Advances in Interfacial Engineering Series; Wiley-VCH: New York, 1999. (31) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed; Academic Press: London; Orlando, FL, 1992.
Figure 4. Effect of the particle size on predictions of the condensation model for several x TPAOH/9500 H2O solutions using pKa,np ) 11.2 and keeping ∆Gc constant.
sensitivity of the model to nanoparticle size (values of n) is illustrated in Figure 4. The model predictions are nearly independent of the polymer size for n > 100. As the size of the nanoparticle becomes smaller than 30, the transition at the CAC is smoother, whereas the pH values in region II are not significantly affected. In the 40 TPAOH/9500 H2O solution, the case in which n ) 12 better captures the experimental trends near the CAC, which may indicate the presence of smaller oligomers prior to nanoparticle formation. Similar sensitivity analyses for values of pKa,np show that nanoparticle deprotonation controls the pH in region II but does not affect the CAC or the pH values in region I (see Supporting Information, Section III.B). In other words, region I is controlled by the chemistry of monomeric silica only, at least to a first-order approximation. Uncertainty of Condensation Model Predictions. Here we assess the inherent uncertainties of the condensation model with respect to the following: (1) the degree of internal silanol deprotonation within the nanoparticle, (2) the allowance of the multiple deprotonation of silica species in solution along with the possible formation of dimers prior to the CAC, and (3) the incorporation of TPA+ association with surface silanol groups to determine the effect of the adsorbed cation. Internal Versus External Silanol Groups. The results presented in Figure 3a were obtained assuming that all silica in the nanoparticle can equally undergo deprotonation (one OH per silicon). A diagram illustrating the model is shown in Figure 5a, in which the physical details of the nanoparticle (i.e., size and shape) are missing. The silanol groups are treated as individual entities, shown for simplicity as a linear chain, and neglecting effects that may arise because of the internal porosity of the nanoparticles. To determine the effect of internal deprotonation, the [SiO2]np in eq 8 is divided into external and internal species:
[SiO2] ) [SiO2]monomer + [SiO2]np,surface + [SiO2]np,internal (18) [SiO2]np,surface ) δ[SiO2]np
(19)
Here, δ represents the fraction of the silanol sites of a nanoparticle that are on the surface, which is given by the following expression:
δ)
SA,npNSiOH FVnpNAmSiO2/Mw,SiO2
(20)
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Figure 5. Schematic diagrams illustrating the treatment of the nanoparticle and solution phases in the condensation model for (a) internal and external deprotonation (δ ) 1) and (b) only external deprotonation (δ ) 0.76).
in which F is the nanoparticle density, mSiO2 is the mass fraction of silica in the nanoparticle (see Table 1), Mw,SiO2 is the molecular weight of silica (60 g/mole), SA,np and Vnp are the nanoparticle surface area and volume, respectively, and the surface silanol density, NSiOH, is taken as the value reported for colloidal silica (4.0 sites/nm2).32 Equation 10 is modified such that only the deprotonation of surface sites is considered:
with multiple deprotonation sites in this region, which are not included in the above model. Here, we first consider multiple deprotonations for a solution of monomers in region I, and then we consider a combined monomer/dimer model to explore the effect of oligomers. Because Si(OH)4 is a polyprotic weak acid, it can undergo multiple deprotonations at high pH (e.g., pH > 12):26 Kim
-
δ[SiO2]np ) [tSiOH]s + [tSiO ]s
(21)
in which the subscript s denotes surface species, and all concentrations are in units of molarity. Equation 20 requires empirical data; therefore, we use the SAXS characterization data of the 9 TPAOH/9500 H2O solution.6 The fraction of surface silanol groups in the nanoparticle, δ, is 0.76, when a density of 1.75 g/cm3 and a silica mass fraction of 0.63 are used. Because the nanoparticle composition and size are independent of silica concentration, δ was held constant. The 9 TPAOH/9500 H2O curve was again fit to obtain a recalculated optimum pKa,np of 11.0, which is slightly lower than the previous value (pKa,np ) 11.2)33 because the total number of tSiOgroups must remain the same for charge neutrality. The incorporation of δ brings some physical properties of the nanoparticles into the condensation model (Figure 5b). We can consider two extremes: (1) the case in which all internal and external silica can deprotonate (δ ) 1) or (2) the case in which only surface silanol groups are allowed to deprotonate (δ ) 0.76). The true behavior is between these two extremes, and thus we would expect the pKa,np value obtained from the condensation model to vary by ∼5% depending on the degree of internal deprotonation. In addition, the value of δ depends on the size, shape, and composition of the nanoparticle; therefore, δ values will change for solutions of varying TPAOH composition. However, the large fraction of surface sites available in nanoparticles renders the effect of internal deprotonation sites on phase behavior of secondary importance. Chemistry in Region I. The condensation model systematically overestimates the pH in region I. Because nanoparticle formation of large n does not take place in region I, this difference from experimental data is, in part, due to the errors of the pH measurements at high pH, the solution nonideality,34 and the presence of small oligomers (32) Rutland, M. W.; Pashley, R. M. J. Colloid Interface Sci. 1989, 130, 448. (33) The condensation model was analyzed with varying δ (0.50.76) on the basis of the potential range of nanoparticle densities (1.752.0 g/cm3), and it was shown that the change in pKa, np is negligible (see Supporting Information, Section III. D).
(i-1)i798 SiOi(OH)4-i + H+, i ) 1-4 (22) SiOi-1(OH)5-i
Including only the first deprotonation, pK1m, results in a slight overestimation of the pH, as shown in Figure 6a. The inclusion of the second deprotonation, pK2m, causes a reduction in the predicted pH for x TPAOH/9500 H2O solutions (x ) 18 and 40), whereas the change is relatively insignificant for x ) 4 and 9. The results in Figure 6a show the limitations of simplifying the chemistry in region I to monomers with only one dissociation constant. The most likely reason for the discrepancies between the predicted and experimental pH using pK2m is that we do not account for small oligomers in region I. It has been shown by 29Si NMR that oligomeric species (e.g., dimers, octameric cubes, etc.) are present in solutions of high pH,35-42 but NMR measurements have been restricted to higher silica concentrations. From our results, we cannot say with certainty what fraction of oligomers is present. Because the identity, concentration, and free energy of condensation for each oligomer are unknown, a more explicit analysis of region I is not possible at this time. However, we can gain insight into the effects of oligomers by considering a solution of both a monomer and a dimer, [Si2O(OH)6]. This is accomplished by setting n to 2 in the condensation model and accounting for the first and second (34) The effect of the activity coefficients on the equilibrium equations in the condensation model was analyzed. It was found that the inclusion of activity coefficients does not alter the predicted pH. When second monomer/oligomer deprotonations are included, small changes are observed at high pH (i.e., 40 TPAOH/9500 H2O); however, these changes are insignificant and result in a small increase in the predicted pH. Details of these calculations are provided in Supporting Information (Section V). (35) Sjoberg, S.; Ohman, L. O.; Ingri, N. Acta Chem. Scand., Ser. A 1985, 39, 93. (36) Harris, R. K.; Knight, C. T. G.; Hull, W. E. ACS Symp. Ser. 1982, 194, 79. (37) Harris, R. K.; Newman, R. H. J. Chem. Soc., Faraday Trans. 1977, 73, 1204. (38) Harris, R. K.; Robins, M. L. Polymer 1978, 19, 1123. (39) Engelhardt, G.; Zeigan, D.; Jancke, H.; Hoebbel, D.; Wieker, W. Z. Anorg. Allg. Chem. 1975, 418, 17. (40) Marsmann, H. C.; Lower, R. Chem.-Ztg. 1973, 97, 660. (41) Knight, C. T. G. Zeolites 1989, 9, 448. (42) Kinrade, S. D.; Swaddle, T. W. Inorg. Chem. 1988, 27, 4253.
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Figure 6. (a) Comparison of the monomer model (n ) 0) in region I, both with (solid lines) and without (dashed lines) the second deprotonation. (b) Monomer and monomer/dimer model predictions for a 40 TPAOH/9500 H2O solution. The equilibrium constants used are pKc (-1.3), pK1m (9.5), pK2m (12.6), pK1d (9.0), and pK2d (10.7), which were taken from ref 26.
deprotonation of both the monomer and the dimer, using the following equation for the dimer: Kid
(i-1)iSi2Oi(OH)7-i 798 Si2Oi+1(OH)6-i + H+, i ) 1, 2 (23)
The monomer/dimer model was chosen because the equilibrium constants are known, and thus the model contains no fitted parameters. The results are shown in Figure 6b for the 40 TPAOH/9500 H2O solution. The inclusion of dimers increases the predicted pH. Although this increase is small, developing a model that accounts for larger oligomers will most likely lead to predictions that are closer to the experimental data. Nevertheless, the model consisting only of monomers with a single deprotonation, although not physically accurate at high pH, provides a good description of the data and captures most of the phenomenology of region I in terms of pH and conductivity. TPA Adsorption onto the Nanoparticle Surface. The interactions of TPA+ with the nanoparticle surfaces namely, the TPA+ coverage and site(s) of adsorptionsare unknown; however, we can draw from studies that were performed on other silicate materials to infer how TPA+ association can potentially affect the condensation model.15,32,43 Nikolakis et al. used complexation models to fit the ζ potential of colloidal silicalite-1 particles as a function of pH. From their analysis, it was concluded that there are two sites for TPA+ adsorption on the zeolite surface: deprotonated silanol sites and zeolite pores. It was found that TPA+ predominantly adsorbs to the latter, but, because nanoparticles lack an ordered pore structure,44 we only consider the TPA+ association with deprotonated surface silanol groups: KTPA
(tSiO-)s + TPA+ 798 (tSiO-TPA)s
(24)
The reported pKTPA value is 6.45,15 and the net concentration of TPA+ is
[TPA+]total ) [TPA+] + [tSiO-TPA]s
(25)
Here, we only consider surface silanol deprotonation (δ ) 0.76) to exclude TPA+ association with internal silanol (43) Claesson, P.; Horn, R. G.; Pashley, R. M. J. Colloid Interface Sci. 1984, 100, 250. (44) Kragten, D. D.; Fedeyko, J. M.; Sawant, K. R.; Rimer, J. D.; Vlachos, D. G.; Lobo, R. F.; Tsapatsis, M. J. Phys. Chem. B 2003, 107, 10006.
groups. Thus, eq 21 is modified to include the adsorbed TPA+:
δ[SiO2]np ) [tSiOH]s + [tSiO-]s + [tSiO-TPA]s (26) For pKTPA ) 6.45, there is minimal association of TPA+ with surface silanol groups, and the model predictions are unaffected. The equilibrium constant, KTPA, can be lowered by 7 orders of magnitude without any significant change to the fitted pKa,np. It is not until negative values of pKTPA are reached that TPA+ association becomes significant (see Supporting Information, Section III.A). Therefore, on the basis of the pKTPA reported in the literature, the structure-director does not affect the dissociation of surface silanol sites. This result, although based on simple thermodynamic analysis, is not unexpected when one considers that the phase behavior (i.e., pH and CAC) for solutions with inorganic cations (Na+, Cs+), tetramethylammonium (TMA+), and tetraethylammonium (TEA+) are identical to that of TPA+.5 To summarize, we find that a condensation model that is based on silica solubility and acid-base chemistry can predict the CAC and phase behavior. Analyses in region I indicate that oligomeric species are present at high pH. The ion effects on model predictions, from the inclusion of both activity coefficients and TPA+ adsorption, are negligible. Last, we find that pKa,np can vary by ∼5% depending on the degree of internal deprotonation and that pKa,np is an effective constant that accounts for both the formation and breaking of chemical bonds and the surface charge. This last point is discussed in the next section. 2. Surface Complexation Model. Complexation models have been employed in the study of zeolite and mica surfaces to predict surface charge and gain insight into organic cation adsorption.15,32,43 Here, we are interested in using such models to obtain an estimate of the surface potential of the silica nanoparticle. Within the range of surface potentials for silicalite-1 (less than -80 mV), interparticle forces between nanoparticles, as calculated by the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, cannot predict their stability in solution. It has been shown that the inclusion of a Stern layer from the adsorbed template may result in the steric stabilization of the particles;13,45 however, nanoparticles form in the absence of organic cations.5 For instance, solutions of 40 SiO2/9 NaOH/9500 H2O/160 EtOH result in nanoparticles (45) Schoeman, B. J. Microporous Mesoporous Mater. 1998, 22, 9.
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Figure 7. (a) Complexation model results for 9 TPAOH/9500 H2O solutions with various numbers of silicon atoms per particle (n ) 286, 356, and 426). (b) Plots of the surface potentials, ψo, that were obtained from the model as a function of pH.49
that have approximately the same size as those in the 9 TPAOH/9500 H2O solution. Therefore, there must be additional factors that lead to the enhanced stability of the nanoparticles. Electrophoretic mobility measurements show that, at pH 11-12, silicalite-1 crystals have a ζ potential of approximately -60 to -80 mV.15 These measurements cannot be extended to study the nanoparticles because of their small size (∼3 nm), and thus we use a complexation model to estimate the magnitude of surface charge and potential. The surface charge density, σo, of the nanoparticle is given by
σo ) e[tSiO-]SA
(27)
in which [tSiO-]SA is the density of surface sites (sites/ area), which is related to the volumetric concentration through the following expression:
[tSiO-]SA )
δ[tSiO-]NA SA,npnp
(28)
in which δ ) 0.76 and SA,np) 42 nm2 for a 9 TPAOH/9500 H2O solution. The number density of the nanoparticles, np (particles/liter of solution), is a function of the total number of silicon atoms per particle, n, using np ) NA[SiO2]np/n. The nanoparticle has an oblate ellipsoid (or disklike) shape that places it somewhere between that of a flat plate and a sphere. For these calculations, we approximate the surface as a flat plate, for which the surface charge density (charge/area) and the surface potential, ψo, are related through the Grahame equation: 46
σo ) (8okBTNAI)1/2 sinh(zeψo/2kBT)
(29)
in which I is the ionic strength in the bulk liquid (mol/m3), o is the permittivity of free space, and is the dielectric constant ( ) 70.31 for a 9500 H2O/160 EtOH solution6). The flat plate was chosen because eq 29 is derived from an exact solution to the Poisson-Boltzmann equation (PBE).31,46 Approximate analytical expressions for spheres47 and cylinders48 have been derived by Ohshima et al.; however, it was found that the differences in model predictions between the various geometries are negligible, (46) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed. (revised and expanded); Marcel Dekker: New York, 1997. (47) Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 90, 17. (48) Ohshima, H. J. Colloid Interface Sci. 1998, 200, 291.
and thus the flat plate equation is used (which is typical for complexation model analysis). Equations 16 and 27-29 are coupled with the condensation model to solve for the surface potential, allowing all of the silica to deprotonate (i.e., using eq 10). Figure 7a shows the model results for the 9 TPAOH/9500 H2O curve in which pKoa,np ) 8.4 provides the best fit for a nanoparticle with n ) 356. Note that this is the true equilibrium constant given in eq 17; therefore, the incorporation of a double layer results in pKoa,np e 9.0, which is within the range expected for large oligomers. There is an n dependence in the complexation model (eq 28), and thus the results are shown for (20% of the experimental value (n ) 356) to illustrate the variation that may arise from errors in SAXS measurements. The surface potentials that are obtained from the complexation model are shown in Figure 7b. Analogous to zeolites, we observe that the nanoparticle surface potential is pH dependent; however, unlike zeolites, we find that the ψo for the nanoparticles is much higher (-120 to -170 mV), which could explain their stability in solution. Why do the nanoparticles have higher surface potentials than that of zeolites? One must first recall that the potential measured in electrophoretic mobility studies is the ζ potential, which is slightly less than the surface potential because of the Stern layer of adsorbed TPA+. The thickness of the Stern layer is the approximate length of the hydrated TPA+, and thus the positive charge of the template is located ∼1-2 nm from the surface of the particle, depending on the degree of hydration. As previously shown, the TPA+ within the Stern layer does not affect the surface chemistry, nor does it alter the surface potential. However, the TPA+ that adsorbs in the zeolite pores has a large effect on the surface potential. Nikolakis et al. report a pK of -5.81 for TPA+ adsorption in the pore.15 If we consider a solution at a pH of 11, the complexation model accounting for both tSiO- and SporeTPA+ sites provides ψo ≈ -70 mV. Removing the pore association from the model results in ψo ≈ -150 mV, which is the same order of magnitude as that obtained for the nanoparticles in Figure 7. Therefore, the high ψo of the nanoparticle may be attributed to the absence of an ordered pore structure, which allows for the strong adsorption of TPA+ and the subsequent reduction of the surface potential. Charge Distribution and Potential of Porous Silica Nanoparticles Here, we determine the gain in free energy per particle and per mole of SiO2 associated with the formation of an electric double layer. The particles are treated as porous
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spheres, and thus the potential distributions in the liquid phase and in the internal porous phase of the nanoparticle are determined. From this analysis, we can explain the lower solubility of the nanoparticles and offer a plausible explanation for why the self-assembly of particles of size
