J. Phys. Chem. C 2008, 112, 14769–14775
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Modeling Silica Nanoparticle Dissolution in TPAOH-TEOS-H2O Solutions John L. Provis,*,† John D. Gehman,‡ Claire E. White,† and Dionisios G. Vlachos*,§ Department of Chemical and Biomolecular Engineering, and School of Chemistry, Bio21 Institute, UniVersity of Melbourne, Victoria 3010, Australia, and Center for Catalytic Science and Technology, Department of Chemical Engineering, UniVersity of Delaware, Newark, Delaware 19716 ReceiVed: April 22, 2008; ReVised Manuscript ReceiVed: July 1, 2008
A model previously developed for the description of silica nanoparticle formation in the TPAOH-TEOS-H2O system, as a preliminary step in the clear-solution synthesis of silicalite-1, is here applied to nanoparticle dissolution. The model is found to give a reasonable description of the dissolution process of as-synthesized nanoparticles. Nanoparticles heated to 70 °C have previously been observed to develop increased structural ordering, and this is incorporated into the model in a way that correlates directly with new 29Si NMR data presented here. Dissolution of both as-synthesized and heated nanoparticles in alkaline solutions at different temperatures was modeled using activation energies obtained from the literature. Structural evolution occurring upon heating and dissolution of nanoparticles and/or spatial heterogeneity in nanoparticle ordering is apparent. Introduction The synthesis of pure-silica zeolites via clear-solution techniques is an area of not only theoretical1,2 but also practical3 interest. In particular, the growth of silicalite-1 products from tetrapropylammonium hydroxide (TPAOH)-tetraethyl orthosilicate (TEOS)-H2O mixtures has been shown to occur via assembly of less-ordered nanoparticles, whose structural rearrangement appears as a prerequisite to generating the final crystalline zeolite structure.1,4–10 A model has also been presented describing the evolution of structure within the nanoparticles as they gradually evolve toward the point at which they are able to crystallize.1,8 Similar nanoparticles have also been observed in other silica-containing systems, including those containing tetra-alkylammonium (TAA) cations other than TPA+11,12 and in aluminosilicate zeolite syntheses,13–15 as well as in analogous GeO2-based systems.16 The nanoparticles formed in TEOS-TAAOH-H2O systems are observed to have initially a core-shell structure,11 with a silica core surrounded by a shell of organocations. While the general mechanism of the crystallization of silicalite-1 from nanoparticles appears now to be relatively well established, the preliminary step whereby these nanoparticles are formed has been studied in less detail. Lattice-based simulations have shown that the emergence of nanoparticles in this system is to be expected from a fundamental standpoint,17 and a chemical equilibrium model has been shown to describe the phase behavior of TPAOH-TEOS-H2O and related systems with a good degree of accuracy.5,13 Although these approaches are of immense value in understanding the chemistry of the systems of interest, neither is able to provide detailed kinetic information relating the rate of nanoparticle formation to other system parameters. A modified population balance model was developed, based on the * Corresponding authors. J.L.P.: phone, +61 3 8344 8755; e-mail, jprovis@ unimelb.edu.au. D.G.V.: phone, +1 302 831 2830; e-mail, vlachos@ udel.edu. † Department of Chemical and Biomolecular Engineering, University of Melbourne. ‡ School of Chemistry, University of Melbourne. § University of Delaware.
Becker-Do¨ring formulation18 of growth and fragmentation by monomer addition and detachment. This model provides for the first time the ability to analyze in detail the mechanisms of the very early stages of clear solution zeolite synthesis.19 By incorporating equilibrium relationships and a particle stabilization mechanism, the model was able to qualitatively match experimental observations and provided a conceptual basis for further analysis. However, due to the growth of nanoparticles during mixing of TPAOH and TEOS in aqueous solution, direct comparison of the growth model with experimental data was somewhat restricted (e.g., the time scales for growth were unknown). The primary aim of this Article is therefore to extend the applicability of the nanoparticle growth model to the study of nanoparticle dissolution. As the Becker-Do¨ring population balance model is based on a set of reversible reactions describing monomer attachment and detachment, it must necessarily be able to describe dissolution as well as growth processes. Also, none of the modifications that were made to the basic Becker-Do¨ring formulation to enable its application to silicate nanoparticle synthesis19 will violate this reversibility. Rimer et al.20 have recently presented extensive data on the dissolution rates of zeolite precursor nanoparticles in alkaline solutions, in particular using small-angle X-ray scattering (SAXS) to measure the change in particle dimensions as a function of time, and these data will be utilized in this Article. A comparison between model parameters and experimental connectivity distribution data obtained by new 29Si nuclear magnetic resonance (NMR) investigation of heated nanoparticle suspensions will also be used to provide a stronger conceptual basis for some of the model parameters. As an added outcome of this work, the more experimentally accessible rate of the dissolution process, when compared to nanoparticle growth, allows a mapping of the model time scale onto real time. NMR Experiments 29Si NMR spectra for suspensions of TPA-silica nanoparticles were obtained at 59.647 MHz on a Varian (Palo Alto, CA) Inova 300 NMR spectrometer (7.05 T). Samples of composition 25TEOS:9TPAOH:152H2O:290D2O were used; this composi-
10.1021/jp803506f CCC: $40.75 2008 American Chemical Society Published on Web 08/27/2008
14770 J. Phys. Chem. C, Vol. 112, No. 38, 2008 tion is significantly more concentrated than the samples used in dissolution experiments; however, an increased silica concentration was necessary to increase the signal-to-noise ratio of the spectra, and this solution composition has previously been shown to give nanoparticles comparable to those obtained in more dilute systems.21 Solutions were prepared a minimum of 24 h in advance of obtaining spectra and were held at room temperature (20-25 °C) during this period. Heat treatment of samples took place in a water bath at 70 °C for designated periods of time, and samples were returned to room temperature prior to study by NMR at 25 °C. Samples were contained in purpose-built 7 mm PTFE capsules under static conditions in a Varian (Palo Alto, CA) broadband MAS probe. Spectra were referenced to tetramethylsilane (TMS) at 0 ppm. The minimal silicon background meant that the spectra could be quantified without the need to subtract broad components due to silica tubes or glass spectrometer components; the stator material did not interfere with the spectral region of interest here. Each spectrum is presented as the sum of a minimum of 1000 transients, using a single π/4 (2.5 µs) pulse, an optimized preacquisition delay that provided for zero linear phase correction, and a recycle delay of 90 s. Selected spectra were also obtained with longer recycle delays to ensure that no components with longer relaxation times were attenuated by saturation. 200 Hz of Gaussian line broadening was applied to all spectra. For analysis, spectra were processed in magnitude mode, as small changes in zero-order phase correction provided equally acceptable phased spectra but unacceptably variable relative peak intensities. Resolution did not appear to be compromised by this processing. Model Summary of the Nanoparticle Growth Model. The full formulation of the modified Becker-Do¨ring model for silica nanoparticle growth was presented and justified in detail previously,19 and so will be presented here in summary form only. All nomenclature here is identical to that used in the previous paper. The full mathematical formulation of the model is presented as Supporting Information. The model is based on the fundamental assumption that nanoparticle growth and fragmentation occurs via addition or subtraction of monomers according to eq 1 and its deprotonated equivalents. ai
(≡SiOH)i + Si(OH)4 {\} (≡SiOH)i+1 + 3⁄2H2O
(1)
2⁄ 3
(
bi ) kpi zs 1 +
1
(4π⁄3) ⁄3Γ 1
i ⁄3
)
Provis et al.
(4)
Here, zs is the concentration of monomers that would be in equilibrium with an infinite flat plate as calculated from the equilibrium-based model of Rimer et al.,5 and Γ is a parameter dependent on the surface properties of the clusters as discussed in detail in ref 19. These expressions differ from those in ref 19 only by the explicit inclusion of the adjustable parameter kp, which was set to 10-3 throughout the earlier paper for computational convenience. It was estimated in ref 19 that this kp value referred to dissolution experiments conducted at around 35 °C. Further analysis of this parameter in the application of the model to dissolution has in fact shown that a value of 10-3 corresponds very well to a temperature of 30 °C, and so this should be used to refine the estimate presented in the previous paper (note that kp affects the time scale). Here, kp will be used as a means of introducing temperature effects directly into the model via the Arrhenius expression, kp ) A exp(-Ea/RT). Values of zs ) 1.05 × 10-3 and Γ ) 1.0 are used throughout this Article. The cluster cutoff size imax is specified for each set of experimental conditions as outlined below. The second major modification to the traditional BeckerDo¨ring model is that, following each time step in the kinetic population balance model, the monomer and dimer populations are equilibrated. This accounts for the observed tendency of low-coordinated silicate species to exchange silicate units much more rapidly than more highly condensed oligomeric species.23 Full details of the equilibrium expressions are given in ref 19 and as Supporting Information. Model Modification for Dissolution. Figure 1 represents the different nanoparticle-forming system compositions studied, as well as the final compositions of the systems in which they were dissolved. All initial compositions were below the critical aggregation concentration (CAC) as defined by Fedeyko et al.,21 and all final compositions except S3 and S5 were above the CAC.20 Systems are named following the conventions of Rimer et al.20 It must also be noted that the as-synthesized system S4 contained a 1:1 ratio TPABr:TPAOH; however, the presence of bromide was not found to be significant in the application of the kinetic model and so has been neglected in calculations. To apply the nanoparticle growth model to dissolution, the only major change required is that, instead of initializing the system with an appropriate concentration of TEOS, the model
bi+1
The rate of each condensation reaction is then modified by a factor f reflecting the known preference for condensation reactions to involve a singly charged and an uncharged silicate site.22 Writing a population balance equation for the number ni of clusters of size i, including the factor f (denoted fmo for the interaction between a monomer and an oligomer in this example), gives eq 2. The cluster size is expressed in terms of the number of silicon atoms comprising the cluster.
dni ) fmoai-1n1ni-1 - fmoain1ni + bi+1ni+1 - bini, i g 3 (2) dt The equivalent expressions for the monomer and dimer (i ) 1 and 2) are given in ref 19 and in the Supporting Information. The terms ai and bi+1 are the growth and dissolution rate constants defined in eq 1 and are given by: 2
(
ai ) kpi ⁄3 1 -
)
1 , i < imax imax - i
ai ) 0, i g imax
(3)
Figure 1. Initial (2) and final (0) compositions of systems S2-S6 for dissolution experiments at 30 °C. The dashed line represents the approximate position of the CAC, at [TPAOH]/[SiO2] ) 1.
Silica Nanoparticle Dissolution in TPAOH-TEOS-H2O
Figure 2. Dissolution of nanoparticles from systems S2 (40SiO2: 9TPAOH:9500H2O), S3 (60SiO2:9TPAOH:9500H2O), and S4 (40SiO2: 4.5TPAOH:4.5TPABr:9500H2O) in 0.3 M TPAOH: comparison of experiment (points) and model (lines).
was instead initialized with a “realistic” cluster size distribution. This was in turn generated simply by “growing” the nanoparticles from a TEOS precursor at the appropriate alkalinity, using a value of imax chosen to give the desired mean cluster size, and for a sufficiently long simulated growth time to allow the cluster size distribution to approximately stabilize. This kinetically stable cluster size distribution was then used as the initial point for the dissolution simulations. To describe the dissolution process, the alkalinity of the model system was increased to match the alkalinity used in the specific dissolution experiment of interest. The value of imax chosen here is 2, and its effect at short and intermediate dissolution times is insignificant (see Supporting Information). This means that the model used here is essentially describing fragmentation plus monomerdimer equilibrium, rather than having growth and fragmentation occurring simultaneously at all particle sizes. However, given that there is very little change in model output over a range of imax values as shown in the Supporting Information, the inferences drawn from the fragmentation-equilibration model results are equally applicable to the fragmentation-growthequilibration case. To model dissolution at different temperatures, an Arrhenius expression for kp was applied. An activation energy of 90 kJ/ mol is used throughout this Article for both nanoparticle dissolution and growth, following the work and detailed discussion of Rimer et al.20 This is also roughly consistent with the silicalite-1 nucleation activation energy of 94 kJ/mol reported by Twomey et al.,24 and while it is most likely that these two parameters represent different aspects of the role of nanoparticles in silicalite-1 synthesis, it is still interesting to note the similarity in Ea values. The Arrhenius frequency factor was calculated so as to give a value kp ) 1.0 × 10-3 at 30 °C. Results and Discussion Modeling of As-Synthesized Nanoparticles. Figures 2 and 3 present the results of the application of the dissolution model to systems S2-S6. The model “time” was related to the experimental time using a scale factor τ ) 1.8 × 105, where real time (in minutes) is given by the model time divided by τ. This value of τ will be used unchanged throughout this Article and may also be used in conjunction with the data presented in ref 19 to provide a more specific time scaling for the work
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Figure 3. Dissolution of nanoparticles from systems S5 and S6 (80SiO2:9TPAOH:9500H2O, in 0.3 M TPAOH and in 0.5 M TPAOH, respectively): comparison of experiment (points) and model (line).
presented in that paper. This value of τ is consistent with the earlier estimate, but enables more quantitative comparison between the model and experimental data when such data become available for the growth process. Note also that Figures 2 and 3 are plotted in terms of the number of silicate units comprising a nanoparticle rather than in radius units, to better emphasize the changes in cluster size upon dissolution. The conversion from the measured (SAXS) cluster radii to the number of silicate units assumes a nanoparticle density of 1.70 g/cm3 and a silica mass fraction in the nanoparticles of 58%, consistent with previous work on this system.5,19 The model is seen from these plots to describe the kinetics of dissolution of nanoparticles over a significant range of synthesis compositions and at different alkalinities with reasonable accuracy. The only parameter that was fitted here is τ, the conversion from model time to real time as noted above; kp was fixed at 10-3, and the appropriate initial cluster size was selected for each system. Only one model line is depicted in Figure 3 because the model predictions for the rate of change of the mean particle size in systems S5 and S6 did not differ significantly. The difference between these two systems is purely in the concentration of the TPAOH solution into which the nanoparticles are dissolving, and the experimental data also show that there is little difference in kinetics. A higher TPAOH concentration obviously increases pH. Figure 3 shows that the dissolution process occurs sufficiently far from equilibrium that this does not have a significant effect. Modeling of As-Synthesized Nanoparticle Dissolution at Different Temperatures. Rimer et al.20 also presented dissolution data for S4 nanoparticles in 0.3 M TPAOH at different temperatures. Comparison of these data to model predictions requires only the use of the Arrhenius temperature expression as outlined above, with the initial cluster size of each run selected to best match each specific set of experimental data. The effect of temperature on dissolution in the system S4 is then shown in Figure 4. This plot describes the dissolution rate in terms of Reff, the effective (geometric mean) radius of the nanoparticles observed in SAXS experiments.20 From Figure 4, the model underestimates the dissolution rate in the early part of the process in the lower-temperature data. This may be due to the fragmentation of the nanoparticles by the removal of silicate clusters in the experimental system and
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Figure 4. Temperature effects in dissolution of S4 nanoparticles: model predictions with Ea ) 90 kJ/mol (lines) and experimental data (points). Each model run was initialized using a different mean nanoparticle size, to match the different starting particle sizes in the experimental data sets.
is also observed (although to a lesser extent) in Figures 2 and 3. The good agreement between model and experiment should be expected, as the Ea value was calculated from these same experimental data by Rimer et al.20 on the basis of the assumption of a simple linear decrease in radius as a function of dissolution time. However, the fact that such good agreement with experiment is obtained using this highly nonlinear model and the same activation energy adds weight to the applicability of the model as presented. Modeling Dissolution of Heat-Treated Nanoparticles at 30 °C. Detailed analysis of the effects of heating on nanoparticle structure has previously been conducted.20,25,26 It was found that heating alters the nanoparticle size and structure as they evolve toward crystallinity, and Ostwald ripening takes place both during heating25 and during room-temperature aging.19 During heating, TPA+ cations become incorporated into the developing zeolitic pore network.25 This is not explicitly described in the population balance model presented here, but must be taken into account in comparing the model fit to experimental data. The heating periods modeled in this work are not sufficient to induce complete crystallization of the nanoparticles,20 but distinct developments in structure were observed calorimetrically,20 as well as via NMR in the present study. To model this change in structure, a phenomenological factor, c, is incorporated into the dissolution rate expression, which is now given by eq 5. 2⁄ 3
(
bi ) kp · c · i · zs 1 +
1
(4π⁄3) ⁄3Γ 1
i ⁄3
)
(5)
This factor c attempts in a linear way to describe the effects of structural variations within the nanoparticles. A possible physical meaning of c is discussed below. Increased structural order in the nanoparticles has been observed to lead to increased thermodynamic stability and therefore decreased dissolution rates.20 To represent this in the model, c takes values less than 1. Figure 5 compares the model predictions and experimental data for these experiments. The initial particle size distribution was computed using the growth model, without taking into account the effect of heating on this distribution. Jorge et al.26 have calculated cluster size distributions for nanoparticle suspensions heated to different temperatures via a lattice Monte Carlo model, and they found that the major characteristics of the distributions do not change dramati-
Figure 5. Effects of duration of heating at 70 °C in the system S2. Dissolution at 30 °C in 0.3 M TPAOH: model predictions (lines) and experimental data (points).
TABLE 1: Values of Parameter c Corresponding to Each Heating Time in Figure 5 time at 70 °C (h)
c
0 1 3 7
1.0 0.50 0.40 0.32
cally during Ostwald ripening in the temperature range of interest here (approximately 0.24 < T* < 0.28 in the dimensionless temperature scale used in their model26). The values of c corresponding to each data set in Figure 5 are given in Table 1. The unheated nanoparticles have a c value of 1.0 (data are the same as those presented in Figure 2). The rate of dissolution is effectively halved in the first hour of heat treatment, with much slower changes over the following 6 h. This may be compared to the calorimetric studies of Rimer et al.,20 who showed that one hour at 70 °C was sufficient to transform the nanoparticle structure from what correlates best with the energetics of amorphous silica to one that, while still disordered, correlates linearly with crystalline silicalite-1 and the samples that had been heated for longer periods. The first hour of heating is therefore clearly a time in which significant changes occur in nanoparticle structure, with subsequent changes being more gradual. 29Si NMR Study of the Effect of Heating on Nanoparticle Structure. To more clearly elucidate the changes occurring upon heating the nanoparticle suspensions, 29Si NMR spectra were obtained, with careful precautions taken to ensure that all of the silica sites were accurately measured as outlined in the experimental methods. The spectra are presented in Figure 6, and the results of quantification are given in Table 2. Previous investigations of similar solutions contain some debate regarding the interpretation of the Q4 peak, with Fedeyko et al.21 and Cheng and Shantz27 each assigning this peak purely to the probe background. Aerts et al.28 observed a small nonbackground Q4 peak, contributing less than 20% of the total intensity observed in a sample of comparable TEOS/TPAOH ratio to those studied here. However, the 7 s recycle delay used in that work may not have been sufficient for full relaxation of (in particular) Q4 Si sites. For our samples and spectrometer, increasing the relaxation delay incrementally from 4 to 60 s for the unheated nanoparticle solution showed a significant and monotonic increase in the measured Q4/Q3 ratio. A relaxation delay of 90 s was finally selected to ensure that all sites were measured accurately, and
Silica Nanoparticle Dissolution in TPAOH-TEOS-H2O
Figure 6. 29Si NMR spectra of nanoparticle suspensions, as-synthesized and after heating to 70 °C for 1, 3, 5, and 7 h (as marked). Dashed vertical lines show the division of regions for quantification purposes.
the care taken to ensure the absence of a silicon background in the Q4 region of the spectra obtained here brings confidence that the quantification presented in Table 2 is representative of the samples as prepared. Quantification was done simply by integrating the intensity in each region, with uncertainties assigned by moving the region boundaries by (0.5 ppm and calculating the maximum and minimum possible extents of depolymerization on the basis of these assignments of region boundaries. Regions were assigned according to the classification of Engelhardt and Michel.29 Table 2 also presents “core fraction” data, calculated as the fraction of sites that are not located within 0.3 nm of the surface of a sphere of the same size as the initial (experimental) mean particle size for each heating duration. This is most probably an overestimate given that the particles may be ellipsoidal (rather than spherical), they have internal porosity/defects, and the distance of 0.3 nm is arbitrary and ignores defects near the nanoparticle/solution interface that develop during dissolution. Therefore, this core fraction represents the maximum possible Q4 fraction of a particle of this size and should be viewed only as an upper estimate. It is interesting to note that the particles are in general relatively close to their maximum achievable cross-link density even prior to heating. While NMR data elucidate for the first time that the fraction of Q4 sites increases with heating time during growth, one cannot conclude from these data alone that the nanoparticles become more zeolitelike due to the associated increase in their size. Kragten et al.4 computed Qn distributions for MFI-structured building blocks of various geometries and defect concentrations that are similar to those obtained here by NMR for the 3, 5, and 7 h-heated nanoparticles, confirming that these nanoparticles are evolving toward a more crystalline state and giving validation for the interpretation of the c parameter. Figure 7 shows a plot of the mean connectivity (∑n n · fraction(Qn)) of the silicate sites in the nanoparticles (filled symbols) against the inverses of the c values given in Table 1 (hollow symbols). This shows that there appears to be a relatively strong correlation between these values, but it is also clear that there are changes taking place that are not represented in the connectivity distribution. In particular, the difference between the unheated and 1 h samples is significantly more than
J. Phys. Chem. C, Vol. 112, No. 38, 2008 14773 can be explained by the difference in connectivities. Rimer et al.20 showed significant changes in the energetics of silica nanoparticle dissolution after one hour of heating, and the results presented here show that these changes are significant in determining the dissolution rate. The growth of the nanoparticles during heating is taken into account in the initialization of the model with the correct initial particle size distribution, so this will not be a major factor here. However, the densification of the silica nanoparticle cores and incorporation of TPA+ cations reported by Rimer et al.25 upon heating is likely to be responsible for this change. The fact that the c parameter changes significantly in the first hour of heating, more so than can be explained simply by connectivity changes, adds further weight to this argument. It must also be noted that the model formulation as originally presented assumes that nanoparticles are comprised of Q3 silicate sites, rather than a mixture of Q2, Q3, and Q4 sites. The only significant impact of this assumption is in the expression for the deprotonation equilibrium of oligomeric and larger species, which is written in terms of a Q3 site losing its proton. Clearly, a Q4 site will not be able to be deprotonated, and the first pKa of a Q2 site will differ from that of a Q3 site. However, in the absence of accurate in situ data describing the variation of Qn fractions during dissolution (which will be very difficult to obtain due to the very long times taken for obtention of 29Si NMR spectra), the simplifying assumption that the nanoparticles are on average Q3 has been retained. Modeling Dissolution of Heated Nanoparticles at Various Temperatures. A further set of experiments were conducted by Rimer et al.20 whereby the S2 system was heated to 70 °C for 7 h, and then the nanoparticles were dissolved in 0.3 M TPAOH at different temperatures (30-50 °C). Rimer et al.20 calculated an Ea value of 88 ((5) kJ/mol from these data, which is not significantly different from the 90 kJ/mol used throughout this paper. An initial attempt was made to model these data using a single c parameter for all data sets; however, the model fit to the data proved unsatisfactory. Adopting the value c ) 0.32, as used in the previous section, resulted in a significant overestimation of the dissolution rates at higher temperatures, while fitting a single c value to the entire data set gave a value of 0.22 but a poor fit to the 30 °C experiment. Therefore, it was deemed necessary to fit a different c value for dissolution at different temperatures. Figure 8 shows the results of this fitting, and the c values used are shown in Table 3. It is clear that the dissolution rate does not increase with temperature as fast as expected, based on an apparent activation energy of 90 kJ/mol. Variations in the dissolution rate and thus in value of the parameter c by a factor of 2-3 in all experiments (Tables 1 and 3) may be associated with a change in the pre-exponential factor, or the activation energy, or both. The model here is comprised of active Q3 silicate sites only, and no interconversion between different coordination types is modeled. However, because we account for variation in the total number of sites with particle size, we do not expect this to be a dominant factor. It may be suggested that the decrease in c with nanoparticle heating (Table 1) and when increasing the dissolution temperature (e.g., 50 vs 30 °C in Table 3) might be related to structure development. A small increase of the apparent activation energy from 90 to ∼92 kJ/mol can lead to a reduction in c by more than a factor of 2. Given the large effect of this rather small change in Ea, it appears possible that during heating (the data in Table 1), the structural evolution of nanoparticles results in an increase of the apparent activation energy due to a net
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TABLE 2: Q-Site Distributions in Nanoparticles Heated at 70 °C for Different Timesa time at 70 °C (h) site
0
1
3
5
7
Q3 Q4 core fraction
0.193 (0.018) 0.374 (0.022) 0.432 (0.022) 0.532
0.184 (0.016) 0.381 (0.021) 0.435 (0.021) 0.574
0.113 (0.013) 0.368 (0.022) 0.519 (0.024) 0.617
0.128 (0.011) 0.362 (0.019) 0.510 (0.020)
0.111 (0.010) 0.376 (0.017) 0.513 (0.017) 0.637
Q2
a Uncertainties in all values are given in parentheses. The “core fraction” represents the theoretical maximum Q4 fraction for a spherical particle of a size corresponding to the experimental particle size.
heterogeneous ordering of nanoparticles that have a more ordered core than their surface. These phenomena can be quantitatively described via a rather small increase in Ea. Conclusions
Figure 7. Comparison of the mean connectivity of the silicate sites (filled symbols) with 1/c (hollow symbols), showing a strong correlation between the nanoparticle structure and the c parameter.
Figure 8. Dissolution of 70 °C/7 h heat-treated S2 nanoparticles in 0.3 M TPAOH at different temperatures: model predictions (lines) and experimental data (points).
TABLE 3: Values of c Used in Figure 8 dissolution temperature (°C)
c
30 35 40 45 50
0.32 0.28 0.24 0.20 0.16
increase in average site coordination number. In addition, during dissolution at higher temperatures (values of c in Table 3 and data in Figure 8), it seems that the chemical driving force decreases, possibly due to reorganization of silicate species near the surface as nanoparticles dissolve or because of spatial
Application of a reversible silica nanoparticle growth model in the TPAOH-TEOS-H2O system to the description of nanoparticle dissolution kinetics provides key insights into the mechanisms underlying dissolution. The model is generally able to describe the kinetics of nanoparticle dissolution using experimentally determined activation energies and connectivity distributions as model inputs. Correlation of the model “time” with real time has been feasible, enabling dynamic modeling of both growth and dissolution experiments. Heating the silicate nanoparticles to 70 °C leads to the gradual development of a more ordered yet nonperfectly crystalline structure for the heating times considered. The first hour of heating induces particularly significant structural changes. Dissolution of heated nanoparticles in solutions at different temperatures shows a significant increase in the stability of 70 °C-heated nanoparticles in solutions at 50 °C as compared to those at 30 °C. It appears that, during dissolution, structural reorganization may still be happening (especially at higher dissolution temperatures), or the nanoparticles are spatially nonuniform and expose a more well-ordered core with a higher coordination number and a slightly larger activation energy for dissolution. In closing, we should mention that the population balance model presented here has the potential to capture nonlinear behavior seen at long times in some of the experiments of ref 20 and also possible bimodality; however, comparison of such predictions to experiments becomes difficult because deconvolution of scattering data to extract a bimodal distribution is nontrivial. Acknowledgment. This work was funded in part through the Particulate Fluids Processing Centre, a Special Research Centre of the Australian Research Council, as well as by the U.S. Department of Energy (DE-FG-02-05ER25702). We would also like to thank Dr. J. Rimer and Professors F. Separovic and R. Lobo for useful discussions. Supporting Information Available: A comparison of model output for different imax values, and a summary of the full model formulation. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Davis, T. M.; Drews, T. O.; Ramanan, H.; He, C.; Schnablegger, H.; Katsoulakis, M. A.; Kokkoli, E.; Penn, R. L.; Tsapatsis, M. Nat. Mater. 2006, 5, 400. (2) Schoeman, B. J.; Regev, O. Zeolites 1996, 17, 447. (3) Lai, Z.-P.; Bonilla, G.; Diaz, I.; Nery, J. G.; Sujaoti, K.; Amat, M. A.; Kokkoli, E.; Terasaki, O.; Thompson, R. W.; Tsapatsis, M.; Vlachos, D. G. Science 2003, 300, 456.
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