Anal. Chem. 2008, 80, 3229-3243
Theory and Simulation of Diffusion-Reaction into Nano- and Mesoporous Structures. Experimental Application to Sequestration of Mercury(II) Christian Amatore,*,† Alexander Oleinick,†,‡ Oleksiy V. Klymenko,‡ Cyril Delacoˆte,§,| Alain Walcarius,§ and Irina Svir*,†,‡
De´ partement de Chimie, Ecole Normale Supe´ rieure, UMR CNRS-ENS-UPMC 8640 “PASTEUR”, 24 rue Lhomond, 75231 Paris Cedex 05, France, Mathematical and Computer Modelling Laboratory, Kharkov National University of Radioelectronics, 14 Lenin Avenue, 61166 Kharkov, Ukraine, and Laboratoire de Chimie Physique et Microbiologie pour l’Environnement, Nancy University, UMR 7564 CNRS, 405 rue de Vandoeuvre, F-54600 Villers-le´ s-Nancy, France
The complex problem of diffusion-reaction inside of bundles of nanopores assembled into microspherical particles is investigated theoretically based on the numerical solutions of the physicochemical equations that describe the kinetics and the thermodynamics of the phenomena taking place. These theoretical results enable the delineation of the main factors that control the system reactivity and examination of their thermodynamic and kinetic effects to afford quantitative predictions for the optimization of the particles’ dimensional characteristics for a targeted application. The validity and usefulness of the theoretical approach disclosed here are established by the presentation of the complete analysis of the performance of thiol-functionalized microspheres aimed for sequestration of Hg(II) ions from solutions to be remediated. This allows the comparison of the microparticles’ performance at two different pH (2 and 4) and the rationalization of the observed changes in terms of the main microscopic parameters that define the system. Increasing attention has been devoted over the past decade to the sol-gel synthesis of micrometric structures formed of bundles of nano- and mesopores decorated by chemically functional sites.1-31 In particular, decorating a nanopore wall with * To whom correspondence should be addressed. E-mail: Christian.Amatore@ ens.fr.
[email protected]. † Ecole Normale Supe´rieure. ‡ Kharkov National University of Radioelectronics. § Nancy University. | Present address: University of Illinois, 600 S. Mattews Ave. RAL, Box 91-5, M/C 712, Urbana, IL. (1) (a) Kresge, C. T.; Leonowicz, M. E.; Vartuli, W. J.; Beck, J. S. Nature 1992, 359, 710-712. (b) Beck, J. S.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L.; Vartuli, J. C.; Roth, W. J.; Leonowicz, M. E.; Sheppard, E. W. J. Am. Chem. Soc. 1992, 114, 10834-10843. (c) Vartuli, J. C.; Olson, D. H.; Sheppard, E. W.; Schmitt, K. D.; Kresge, C. T.; Roth, W. J.; Leonowicz, M. E.; Schlenker, J. L. Chem. Mater. 1994, 6, 2317-2326. (d) Vartuli, J. C.; Kresge, C. T.; Leonowicz, M. E.; Chu, A. S.; McCullen, S. B.; Johnson, I. D.; Sheppard, E. W. Chem. Mater. 1994, 6, 2070-2077. (e) Beck, J. S.; Vartuli, J. C.; Kennedy, G. J.; Kresge, C. T.; Roth, W. J.; Schramm, S. E. Chem. Mater. 1994, 6, 1816-1821. (f) Yang, H.; Ozin, G. A.; Kresge, C. T. Adv. Mater. 1998, 10, 883-887. (g) Yang, H.; Coombs, N.; Ozin, G. A. J. Mater. Chem. 1998, 8, 1205-1211. (h) Hunks, W. J.; Ozin, G. A. Chem. Mater. 2004, 16, 5465-5472. (2) Wu, J.; Gross, A. F.; Tolbert, S. H. J. Phys. Chem. B 1999, 103, 2374-2384. 10.1021/ac702420p CCC: $40.75 Published on Web 03/27/2008
© 2008 American Chemical Society
appropriate chemical ligands controls selectively the flux of a target ion inside the particles32 while the chelating properties of the ligands allow their selective trapping by the nanopores walls.17a-f Owing to the large developed surface area of a nanopore (viz., ∼2πRporeL, where Rpore is its radius and L its length; see Figure 1) relative to its overall volume (viz., ∼π(Rpore + ω)2L, where ω is (3) (a) Wakayama, H.; Fukushima, Y. Chem. Mater. 2000, 12, 756-761. (b) Itoh, T.; Yano, K.; Kajino, T.; Itoh, S.; Shibata, Y.; Mino, H.; Miyamoto, R.; Inada, Y.; Iwai, S.; Fukushima, Y. J. Phys. Chem. B 2004, 108, 13683-13687. (c) Wakayama, H.; Fukushima, Y. Ind. Eng. Chem. Res. 2006, 45, 33283331. (d) Oda, I.; Hirata, K.; Watanabe, S.; Shibata, Y.; Kajino, T.; Fukushima, Y.; Iwai, S.; Itoh, S. J. Phys. Chem. B 2006, 110, 1114-1120. (4) Han, S.; Sohn, K.; Hyeon, T. Chem. Mater. 2000, 12, 3337-3341. (5) Scott, B. J.; Wirnsberger, G.; Stucky, G. D. Chem. Mater. 2001, 13, 31403150. (6) (a) Lee, J.-S.; Joo, S. H.; Ryoo, R. J. Am. Chem. Soc. 2002, 124, 1156-1157. (b) Choi, M.; Kleitz, F.; Liu, D.; Lee, H. Y.; Ahn, W.-S.; Ryoo, R. J. Am. Chem. Soc. 2005, 127, 1924-1932. (7) Sun, J.-H.; Shan, Z.; Maschmeyer, T.; Coppens, M.-O. Langmuir 2003, 19, 8395-8402. (8) Braun, P. V. In Nanocomposite Science and Technology; Ajayan, P. M., Schadler, L. S., Braun, P. V., Eds.; Wiley-VCH Verlag GmbH & Co. KGaA: Wenheim, 2003; pp. 155-214. (9) Burleigh, M. C.; Marcowitz, M. A.; Jayasundera, S.; Spector, M. S.; Thomas, C. W.; Gaber, B. P. J. Phys. Chem. B 2003, 107, 12628-12634. (10) Kim, S. H.; Liu, B. Y. H.; Zachariah, M. R. Langmuir 2004, 20, 2523-2526. (11) Villaescusa, L. A.; Mihi, A.; Rodrigues, I.; Garcia-Bennett, A. E.; Miguez, H. J. Phys. Chem. B 2005, 109, 19643-19649. (12) Sanchez, C.; Julian, B.; Belleville, P.; Popall, M. J. Mater. Chem. 2005, 15, 3559-3592. (13) Hartmann, M. Chem. Mater. 2005, 17, 4577-4593. (14) Hoffman, F.; Cornelius, M.; Morell, J.; Froeba, M. Angew. Chem., Int. Ed. 2006, 45, 3216-3251. (15) Cheng, C.-F.; Cheng, H.-H.; Cheng, P.-W.; Lee, Y.-J. Macromolecules 2006, 39, 7583-7590. (16) Wang, Y.; Caruso, F. Chem. Mater. 2006, 18, 4089-4100. (17) (a) Walcarius, A.; Delacoˆte, C. Chem. Mater. 2003, 15, 4181-4192. (b) Walcarius, A.; Delacoˆte, C. Anal. Chim. Acta 2005, 547, 3-13. (c) Etienne, M.; Lebeau, B.; Walcarius, A. New J. Chem. 2002, 26, 384-386. (d) Walcarius, A.; Etienne, M.; Bessie`re, J. Chem. Mater. 2002, 14, 27572766. (e) Walcarius, A.; Etienne, M.; Lebeau, B. Chem. Mater. 2003, 15, 2161-2173. (f) Walcarius, A.; Sibottier, E.; Etienne, M.; Ghanbaja, J. Nat. Mater. 2007, 6, 602-608. (g) Etienne, M.; Quach, A.; Grosso, D.; Nicole, L.; Sanchez, C.; Walcarius, A. Chem. Mater. 2007, 19, 844-856. (18) (a) Ko´nya, Z.; Puntes, V. F.; Kiricsi, I.; Zhu, J.; Ager, J. W., III; Ko, M. K.; Frei, H.; Somorjai, G. A. Chem. Mater. 2003, 15, 1242-1248. (b) Nakamura, R.; Frei, H. J. Am. Chem. Soc. 2006, 128, 10668-10669. (c) Han, H.; Frei, H. Microporous Mesoporous Mater. 2007, 103, 265-272. (d) Wasylenko, W.; Frei, H. J. Phys. Chem. C 2007, 111, 9884-9890.
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the width of its sol-gel wall, generally much smaller than Rpore), the final loaded ion concentration (i.e., its quantity per volume of particle) may reach extremely high values though the target ion may be present at very low concentrations in the external solution.17b Hence, such particles present high interest for selective filtration through sequestration of undesirable ions leading to new devices for the control of quality of fluids for everyday and industrial applications. Another important application is for environmental purposes, allowing remediation of metal-contaminated waters, in particular those related to the secure disposal of dilute radioactive wastes in nuclear plants exhausts of coater cooling circuits. In both series of applications, the main interest is that the target species is trapped at extremely high concentrations inside a sol-gel structure, which may then be vitrified and stored securely.31a Up to now, most of the synthetic works, and hence the particles produced in this area, have relied on empirical views mostly based on chemical activity (viz., focused on the ligands) and engineering (viz., fixing pore size and length) rationales aimed at increasing the thermodynamic loading properties of the designed particles. However, the diffusion-reaction patterns created inside such nanoor mesoporous materials condition their filling kinetics through controlling their cross-communications with the bulk solution,32 henceforth crucially control the effective efficiency of the systems. Indeed, for any of the applications recalled above, the crucial factor is not the value of the storage ability at infinite time (i.e., the thermodynamic equilibrium value) but the effective storage after a given residence time in the macroscopic filter or reactor, which contains the particles and through which the fluid to be remediated or cleaned circulates. However, when any structure is made smaller and smaller, the usual ratios between size and surface, on one hand, and between surface and volume, on the other hand, vary so that our “macroscopic” chemical knowledge and views may not apply. Surface effects tend to dominate over volumetric ones, and sizerelated effects dominate surface ones, evidencing readily that (19) (a) Kidder, M. K.; Britt, P. F.; Zhang, Z.; Dai, S.; Hagaman, E. W.; Chaffee, A. L.; Buchanan, A. C., III. J. Am. Chem. Soc. 2005, 127, 6353-6360. (b) Kidder, M. K.; Britt, P. F.; Chaffee, A. L.; Buchanan, A. C., III. Chem. Commun. 2007, 1, 52-54. (20) Ji, X.; Herle, P. S.; Rho, Y.; Nazar, L. F. Chem. Mater. 2007, 19, 374-383. (21) Bhattacharya, S.; Gubbins, K. E. Langmuir 2006, 22, 7726-7731. (22) Sayari, A.; Yang, Y. Chem. Mater. 2005, 17, 6108-6113. (23) Alsyouri, H. M.; Lin, J. Y. S. J. Phys. Chem. B 2005, 109, 13623-13629. (24) Lei, C.; Shin, Y. J.; Liu, E. J.; Ackerman, J. Am. Chem. Soc. 2002, 124, 11242-11243. (25) Coasne, B.; Galarneau, A.; Renzo, F. D.; Pellenq, R. J. M. Langmuir 2006, 22, 11097-11105. (26) Zimmerman, A.; Chorover, J.; Goyne, K. W.; Brantley, S. L. Environ. Sci. Technol. 2004, 38, 4542-4548. (27) Vinu, A.; Miyahara, M.; Ariga, K. J. Phys. Chem. B 2005, 109, 6436-6441. (28) (a) Hernandez, R.; Tseng, H.-R.; Wong, J. W.; Stoddart, J. F.; Zink, J. I. J. Am. Chem. Soc. 2004, 126, 3370-3371. (b) Nguyen, T. D.; Liu, Y.; Saha, S.; Leung, K. C.-F.; Stoddart, J. F.; Zink, J. I. J. Am. Chem. Soc. 2007, 129, 626-634. (29) Gao, X.; Nie, S. J. Phys. Chem. B 2003, 107, 11575-11578. (30) Shui, W.; Fan, J.; Yang, P.; Liu, C.; Zhai, J.; Lei, J.; Yan, Y.; Zhao, D.; Chen, X. Anal. Chem. 2006, 78, 4811-4819. (31) (a) Cuenot, F.; Meyer, M.; Bucaille, A.; Guilard, R. J. Mol. Liq. 2005, 118, 89-99. (b) Brande`s, S.; David, G.; Suspe`ne, C.; Corriu, R. J. P.; Guilard, R. Chem. Eur. J. 2007, 13, 3480-3490. (c) Barbe, J.-M.; Canard, G.; Brande`s, S.; Guilard, R. Chem. Eur. J. 2007, 13, 2118-2129. (c) Barbe, J.-M.; Canard, G.; Brande`s, S.; Guilard, R. Angew. Chem, 2005, 117, 3163-3166. (32) Wang, G.; Zhang, B.; Wayment, J. R.; Harris, J. M.; White, H. S. J. Am. Chem. Soc. 2006, 128, 7679-7686.
3230 Analytical Chemistry, Vol. 80, No. 9, May 1, 2008
Figure 1. Schematic representation of a spherical mesoporous particle consisting of a dense bundle of nanopores. Rpart is the particle radius, Rpore that of the nanopores, L the nanopore length, ω the halfthickness of the solid wall separating two adjacent nanopores, and δ the thickness of the steady-state diffusion layer surrounding each particle into the solution. (a) Schematic view showing one nanopore placement inside the particle and its opening at the particle surface for a hexagonal packing (as in the experimental system investigated in this work), together with its diffusional “projection” into the solution. (b) Cross section of one nanopore and “its” solution diffusion layer along a plane containing the nanopore axis (shown by the central dashed line); note that the origin of the abscissa axis, x, is fixed onto the particle surface with the convention that positive values describe the position inside the nanopore while negative ones that in solution; ordinates are measured orthogonally from the nanopore axis.
dynamic interfacing between nano- and macroworlds obeys its own physics so that use of pure macroscopic considerations may lead to wrong decisions. The general laws that govern the diffusion-reaction patterns created inside such nano- or mesoporous materials during their filling kinetics and that control their cross-communications with the bulk solution have been presented and discussed in a recent paper.33 This general physico-mathematical treatment has delineated the complex kinetic situations that may arise and outlined the general kinetic laws obeyed under each kinetic range. Its main results are presented in Figure 2 as a function of the two main dimensionless parameters that govern primarily the complex kinetics at hand.33 However, despite the great conceptual utility of such a general approach, any precise application to a particular experimental situation requires the consideration of many other (33) Amatore, C. Chem. Eur. J. In press
Figure 3. X-ray diffractogram obtained for the thiol-functionalized mesoporous silica sample.
Figure 2. (a) Kinetic zone diagram illustrating the different behaviors experienced by 1D nanoporous system as a function of its main dimensionless parameters characterizing its dynamics. Ξ0 ) (2πRporeL) × Γsite/[(πRpore2L)Cb0] is the ratio of the quantity of species storable by the nanopore wall sites (Γsite is the surface concentration of active sites) and that stored by the solution inside the nanopore at initial solution concentration Cb0; η ) Dpore/Dbulk is the ratio of diffusivities inside the pore and in the bulk solution; and λ0 ) kadsL2Cb0/Dbulk is the dimensionless adsorption rate constant. In (b) are represented the schematic time dependences of the extraction parameter, f (τ), on dimensionless time, τ, in each zone delineated in (a). (adapted from ref 33; see below for definitions of parameters in the Glossary).
parameters that influence the system time constant, though they play minor role on the general kinetic laws. It is the purpose of this work to examine the role of these factors and present a complete numerical approach for the physicochemical problem at hand. This will be developed following two examples of complete rationalization of actual experimental systems since these will serve to validate the model and offer an experimental introduction to the problem at hand. The investigated system was designed for extraction of mercury ions from contaminated water,34, 35 and its geometrical parameters suggest that the model elaborated for one-dimensional pores33 may apply. Yet, whenever the conclusions achieved for the 1D model are partial or do not apply to two-dimensional nanopores, the 2D model will be considered in addition. RESULTS AND DISCUSSION Experimental Results. The thiol-functionalized mesoporous silica microspheres aimed for sequestration of Hg(II) ions were prepared as described earlier and as recalled in the Computational and Experimental Section (see below).17a,c These conditions led
to the formation of mesoporous organosilica spheres of narrow size distribution centered at 0.39 µm (as determined using a light scattering analyzer (model LA920, Horiba) based on the Mie scattering theory). The material was mesostructured with mesopore channels organized in hexagonal geometry, as determined by X-ray diffraction (XRD; see Figure 3). BET analysis of N2 adsorption-desorption experiments yielded a specific surface area of 1538 m2 g-1, a total pore volume of 0.84 cm3 g-1, and a mean pore diameter (BJH method) of 27.8 Å. The material contained 0.75 mmol of SH moieties/g of material, as determined by elemental analysis. Taking into account the d100 value calculated from XRD data (33.5 Å) and considering the pore geometry (hexagonal) and its diameter (27.8 Å), the wall thickness 2ω was estimated as equal to 11 Å. A solution (0.118 mM) of Hg(NO3)2 (Prolabo) was prepared in a 5-mL beaker, and 5 mg of mesoporous particles (viz., 9.3 × 109 particles/mL) diluted into a minimal amount of water and already fully wetted were introduced, resulting into a 0.10 mM concentration of Hg(II), at the beginning of the sequestration experiment. Kinetic experiments performed to characterize the rate of mercury(II) binding to the thiol-functionalized mesoporous silica sample were carried out at two pH values (2 and 4, as adjusted by appropriate amounts of HNO3) and monitored continuously through the measurement of the solution-phase Hg(II) directly in adsorbent suspensions.17d,e This was achieved by hydrodynamic voltammetry at a rotating disk electrode (RDE; 4 mm in diameter; applied potential -0.5 V vs SCE; 2000 rd min-1) immersed in the mercury(II) solution. Constant stirring was ensured by the RDE rotation to ensure the homogeneity of the bulk solution (note that this was responsible for the noisy structure of experimental currents as reflected in the plots shown in Figure 4). The amount of Hg(II) bonded to the adsorbent was deduced from its decreasing concentration in the bulk solution, as monitored amperometrically in situ by recording the corresponding current decrease at the RDE). This enabled us to monitor the sequestration kinetics in the form of the variation of f (t) ) Q/Q∞ ratios with time, where Q ) ([Hg(II)]t)0 - [Hg(II)]t)Vsoln and Q∞ ) ([Hg(II)]t)0 - [Hg(II)]t)∞)Vsoln represent the quantity of Hg(II) that has been extracted from the bulk solution by the (34) Mercier, L.; Pinnavaia, T. J. Adv. Mater. 1997, 9, 500-503. (35) Brown, J.; Richer, R.; Mercier, L. Microporous Mesoporous Mater. 2000, 37, 41-48.
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Figure 4. (a, b) Kinetic and fitting results for sequestration of Hg(II) cations (0.1 mM in 5 mL of water) at (a) pH ) 4, (b) pH ) 2, and (c) kinetic zone diagram for 1D model33 showing the positions of bestfit parameters for pH ) 2 (b) and pH ) 4 (9). The best-fit parameters are as follows: for (a) λ0 ) 3.6 × 10-3, η ) 0.02, κ ) 10-7, φ ) 0.91, υ ) 1.8 × 10-4 (which corresponds to the set of dimensioned values Dbulk ) 5 × 10-6 cm2 s-1, Dpore ) 9.5 × 10-8 cm2 s-1, kads ) 109 cm3 mol-1 s-1, kdes ) 10-5 s-1, δ ) 1 µm); for (b) λ0 ) 7.8 × 10-5, η ) 0.0012, κ ) 0.5, φ ) 0.94, υ ) 1.8 × 10-4 (i.e., Dbulk ) 5 × 10-6 cm2 s-1, Dpore ) 6.1 × 10-9 cm2 s-1, kads ) 2.2 × 107 cm3 mol-1 s-1, kdes ) 1.1 s-1, δ ) 0.87 µm). In (a, b) Npart ) 4.66 × 1010. The horizontal segments near each symbol (b or 9) show the corresponding titration positions, i.e., υΞ0 ) 1, owing to the fitted values of Ξ0, υ and η. 3232 Analytical Chemistry, Vol. 80, No. 9, May 1, 2008
solid particles, respectively, after a time duration t and when the system ceased to evolve anymore (i.e., as determined after t ) 24 h in our experiments). Considering that at each instant the RDE current, I(t), was proportional to [Hg(II)]t, it ensued that f (t) ) [I(0) - I(t)]/[I(0) - I(24h)] so that these variations could then readily be deduced from those of the RDE current. Panels a and b in Figure 4 describe the results of these kinetic procedures as obtained when the mesoporous organosilica spheres described above operated at two different pH. As apparent from these plots, one observes a general behavior akin to that predicted within the general theoretical framework recalled above (see Figure 2).33 Furthermore, it is observed that a change of pH affects drastically the sequestration kinetics, though qualitative inspection of the curves, even based on the general theory,33 does not evidence any sound behavioral difference. Even if it is obvious that a pH change must affect the very nature of the mercury-SH assemblies inside the mesoporous organosilica spheres,17b,36 and therefore the binding ability toward Hg(II) cations and diffusive properties within the nanopores, the relationship between such effect and the observed kinetics remains elusive. Theory and Simulations of the Experimental Systems. In the following, we wish to evaluate quantitatively the function f (t) ) Q(t)/Q∞ ) (C0 - C(t))/(C0 - C∞), where C(t) is the concentration of target species in solution at a given time t, C0 and C∞ being those at initial time and at infinite times. In this task, we will rely on the general formulation presented in our previous work,33 though our former analysis needs to be specifically adapted in order to represent quantitatively the systems investigated experimentally here. This will enable us to extract the main parameters that govern these systems. Basically, many parameters command the overall kinetics. Some are external and relate to the hydrodynamic conditions prevailing in the solution in which the thiol-functionalized mesoporous silica spheres are stirred. Others are geometrical, and define the nanospace inside the spheres, in which diffusion/ sequestration of mercury cations occur. Finally, the last series of parameters control the kinetics and thermodynamics of the diffusion/sequestration processes. Evidently, based on such a large series of parameters, few of them being known independently, one may easily fit any smooth curve such as any of the experimental ones presented in Figure 4. One of the main outcomes of the mathematical formalism developed previously was to show that the complete theoretical treatment of the system is amenable to the “simple” treatment of the diffusion/sequestration processes occurring within an “average” nanopore surrounded by the fraction of the stagnant layer through which it is fed and which is wrapped around each microsphere due to the combined effects of spherical diffusion and hydrodynamic regime (see Figure 1).33 1. Geometrical Parameters Describing the “Average” Pore. The geometrical parameters describing the average pore in Figure 1 are of two kinds. For the experimental system investigated here, parameters Rpore ≈ 14 Å and ω ) 5.5 Å, which define the inner pore radius and the half-thickness of the wall separating two adjacent nanopores, are known independently from spectroscopic and microscopic controls and apply to each individual nanopore.
The average nanopore length, L, may be estimated from the requirement of dense hexagonal packing of linear nanopores into a spherical volume. This problem was addressed in our previous work, and it ensues that L ) Rpart/3,33 where Rpart ≈ 0.4 µm is the radius of the monodisperse thiol-functionalized mesoporous silica spheres (see above). Since Rpore is known, L may also be estimated indirectly from the total pore volume, 0.84 cm3 g-1, as deduced from BET analysis (see above), thus producing a comparable value as inferred by the geometrical method. The thickness, δ, of the stagnant layer cannot be known independently. Since Rpart ≈ 0.4 µm, δ is at most a few micrometers as would be spontaneously imposed by steady-state spherical diffusion around the microspheres in a still solution. Owing to the vigorous stirring and convective effects due to the continuous collisions of a large number of microspheres in the experimental solutions, it is presumable that δ is even smaller being in the 1-µm range, though a precise evaluation may be made only by fitting the experimental curves. Nevertheless, the range for δ is sufficiently narrow for the very value of this parameter not to affect significantly the outcome of the simulations and, hence, of the quality of the other parameters, which will be determined by fitting. 2. Nature of Hg(II) Species Stored within the Particles. The amount of Hg(II) ions sequestrated inside the microsphere nanopores consists a priori of solution-contained ones and of those bound to the thiol moieties, which line the nanopore wall. At infinite time (t ) 24 h), the solution concentration C∞ is sufficiently small versus the initial one, C0, for the amount stored in solution form inside the nanopores to be neglected. However, this may be not true at any time, especially when C(t) ≈ C0. For evaluating this fraction, we proceed as follows. First we express the overall quantity stored in all nanoparticles as in eq 1:
Q(t) ) Npart
4Rpart2 (Rpore + ω)
2
[πRpore2
∫
L
0
Cpore dx +
2ΓsiteπRpore
∫
L
0
θ dx] (1)
where Npart is the total number of particles, 4R2part/(Rpore + ω)2 is the number of average pores per microsphere,33 Cpore is the concentration of Hg(II) ions at a distance x from the entrance of an average pore at time t, θ is the fraction of thiol groups bound by Hg(II) ions inside the nanopore at a distance x from its entrance at time t, and Γsite is the surface coverage of the thiol groups lining each nanopore wall at t ) 0. Rearranging eq 1, and noting Ξ0 ) 2Γsite/(RporeC0), it follows that
Q(t) ) 2ΓsiteπRporeL × 4Rpart2
[
1 Npart 2 Ξ (Rpore + ω) 0
∫
1
0
Cpore dy + C0
∫
1
0
]
θ dy (2)
In this expression, the factor 2ΓsiteπRporeL × Npart 4R2part/(Rpore + ω)2 represents the total quantity of mercury that may be sequestrated by the assembly of microspheres walls when all their thiol sites are occupied. The first term in the brackets represents the normalized quantity of Hg(II) ions confined into
the solution filling all nanopores, while the second term is the overall fraction of thiol groups bound to Hg moieties. Remarking that at any time Cpore is necessarily smaller than or equal to the concentration C(t) in solution at the same time and that C(t) is necessarily smaller than C0, it ensues that the ratio, qwall(t)/qsoln(t), between bound species and solution species in the average nanopore at time t is such as
qwall(t) soln
q
(t)
∫
1
) Ξ0 ×
0
∫
1
0
θ dy
Cpore dy C0
g Ξ0
C0 C(t)
Ξ0 ×
×
∫
∫
1
0
1
0
θ dy g
θ dy ) Ξ0θav(t) (3)
where θav(t) is the average fraction of mercury-bound thiol groups at time t considering the whole assembly. The values of the active site loading of the particles (7.5 × 10-4 mol of SH groups per gram of material) and the specific surface area of 1538 m2 g-1 afford a surface concentration of active sites of Γsite ) 4.88 × 10-11mol cm-2. On the other hand, one knows independently that Rpart ) 0.4 µm, Rpore ) 1.4 nm, and ω ) 0.55 nm(see above). Taking into account the initial bulk concentration of the mercury salt in the solution, i.e., C0 ) 0.1 mM, the value of parameter Ξ0 can be evaluated to be Ξ0 ) 2Γsite/(RporeCb0 ) ≈ 7 × 103 under our conditions. The double inequality in eq 3 then confirms that the ratio qwall(t)/qsoln(t) exceeds unity considerably as soon as θav(t) is larger than a few 10-4, viz., as soon as the system has displayed visible activity. In other words, one has Q(t)/Q∞ ) θav(t) over the whole time range of interest in Figure 4a and b, this being valid with a precision of better than 1% for any t except the very beginning of the experiment, when Q(t)/ Q∞ values are below ∼10-2. Yet this range is below our experimental precision due to the convection-induced noise (see Figure 1). This has two main consequences. First, this shows that the solution contained by each nanopore plays almost no role in the storage capacity but acts only to feed the sites lining up the nanopore walls. Second, based on our former work,33 this establishes that the systems at hand behave physically as a collection of 1D nanopores, a fact that will simplify our following analysis of its kinetics. 3. Theoretical Formulation of the 1D Problem. As described in ref 33 for all cases amenable to a 1D formulation such as those considered here, the diffusion-adsorption problem may be cast in dimensionless form by the set of eqs 4-6:
∂a ∂ 2a ) η 2 - (λ0Ξ0)[(1 - θ)a - θκ] ∂τ ∂y
(4)
∂θ ) λ0[(1 - θ)a - θκ] ∂τ
(5)
dc ) - ϑ(c - ay ) 0) dτ
(6)
which is associated with the following initial conditions (τ ) 0):
c)1 0 < y e1:
a ) 0,
(7a) θ)0
Analytical Chemistry, Vol. 80, No. 9, May 1, 2008
(7b) 3233
a ) Cb0,
y ) 0:
θ)0
(7c)
and boundary conditions (τ > 0):
y ) 0:
(∂a∂y )
y)0
y ) 1:
φ ) - (c - ay)0) η
(∂a∂y )
y)1
)0
(8a) (8b)
In the above set of equations, we resorted directly to dimensionless formulations rather than experimental ones since this is more useful to restrict the simulations to be performed with really significant parameters rather than with a collection of experimental ones. Indeed a dimensionless formulation allows the integration of parameters that act on the system into the few master ones, which ultimately control the kinetics of the systems under scrutiny. All the dimensionless quantities and parameters introduced in eqs 4-8 have been defined elsewhere33 and are also reproduced here in the Glossary. Equations 4-5 describe the evolution of the dimensionless concentration of Hg(II) species, a, inside the 1D solution filling the nanopore, and the local fraction, θ, of thiol sites being bound. The concentration inside the nanopore is coupled to that in the homogeneous bulk solution, c, through eq 6, which accounts for the steady-state diffusion regime inside of the stagnant layer surrounding each particle (see Figure 1). Through such formulation, concentrations are in fact acting as “transparent” variables whose only role is to obtain the time dependence of the function f (t) ) Q(t)/Q∞, which is the only accessible experimental kinetic information. Indeed, taking into consideration eq 2 and remarking that, for the experimental cases to be evaluated, Ξ0 . 1, one has
Q∞ ≈ 2ΓsiteπRporeL × Npart
4Rpart2 (Rpore + ω)
2
∫
1
0
θ(y, τ f ∞) dy (9)
so that the function f is ultimately given in dimensionless formulation by
(∫ θ(y) dy) (∫ θ(y) dy) 1
Q(τ) f (τ) ) ) Q∞
0
τ
1
0
(10)
τf∞
which involves only the time and space dependencies of the fraction θ of thiol groups bound by mercury atoms. Alternatively, f may be expressed according to its initial definition, which is rewritten in dimensionless formulation as
f (τ) )
Q(τ) 1 - c(τ) ) Q∞ 1 - c∞
(11)
which depends only on the dimensionless bulk concentration, c(τ) ) C(τ)/C0. The interest of this dual expression for f is to provide an internal check of coherence of the approximations that have led to eq 3 at each time step of the simulations. This was verified 3234
Analytical Chemistry, Vol. 80, No. 9, May 1, 2008
in the following numerical treatments, hence validates further the above sequence of approximations. The formulation in eqs 4-8 establishes that the kinetics exhibited by the systems at hand depend on few master dimensionless parameters, which incorporate numerous individual kinetic and thermodynamic effectors: η ) Dpore/Dbulk, the ratio between diffusivities of Hg(II) cations inside the 1D nanopore and in the bulk solution, λ0 ) kadsL2C0/Dbulk, the dimensionless rate constant of adsorption (kads being the real rate constant), Ξ0 ) 2Γsite/(RporeC0), which compares the solution versus thiol sites storage ability of nanopores, κ ) Kdes/C0, the dimensionless dissociation constant of mercury-bound thiols (Kdes being the real equilibrium constant), and φ ) (L/δ)(1 + δ/Rpart)(1 + ω/Rpore)2, which is a geometric parameter linking the diffusion lengths in solution (δ) to the system geometrical features.33 Among these, only Ξ0 ) 7 × 103 is known independently (see above), the four other ones, viz., λ0, η, κ, and φ, requiring their determination through the fitting of the experimental shapes of f (t) curves (Figure 4a and b). However, the range of possible variations of φ is minimal since δ ≈ 1 µm (see above) and the system geometrical features are known independently. Similarly, though κ is not known, the experimental efficiency of the system is such to ensure that this parameter is smaller than unity, a fact that greatly minimizes its influence. Therefore, the experimental shapes of f (t) curves are mostly determined by the values of λ0 and η, which are the only master parameters for efficient sequestrating systems whenever Ξ0 is known independently.33 4. Simulations and Fitting of the Experimental Kinetics. Because the above formulations are expressed in dimensionless units, the simulations outcome is expressed under the form of the dependence of simulated values of f (τ) versus the dimensionless time. However, since Dbulk ) 5 × 10-6 cm2 s-1 from the amperometric measurement at the RDE, and L ≈ 0.13 µm as deduced from Rpart ≈ 0.4 µm (see above), a simple scaling affords the predicted f (t) function in the real time scale: τ ) Dbulkt/L2 ) 2.9 × 104t when t is expressed in seconds. In the simulation process, a broad agreement was initially sought between simulated and experimental f (τ) curves through adjusting the key parameters λ0 and η and using δ ≈ 1 µm and κ ) 0 as fixed entries. Note that η governs primarily the f (τ) curves at short times, while λ0 becomes a major player afterward.33 This duality is for example directly evident in the experimental curves through the drastic change of curvatures between the initial and final time ranges.33 This peculiar characteristic ensured that both λ0 and η may be determined with high certainty by the fitting procedure. Then the initial fits were finely adjusted around this set of values by allowing the whole set of λ0, η, δ, and κ values to be varied automatically in order to minimize the least-squared sum of residuals. Note that δ and κ do not play simultaneously on the f (τ) curve shapes since κ becomes influent only when f (τ) approaches its maximum, viz., when the concentration in the bulk approaches its final limit. Again, this characteristic ensured the accuracy of the overall fitting procedure, each parameter having a stronger influence in different time ranges. The first experimental data set depicted in Figure 4a corresponds to pH ) 4. The fitting curve shown in the same figure by the solid smooth line is in excellent agreement with the experimental data. It corresponds to the following best-fit dimensionless
parameters: λ0 ) 3.6 × 10-3, η ) 0.02, κ ) 10-7, φ ) 0.91. This set of dimensionless parameters, considering the other ones known independently, allowed the extraction of the following dimensioned parameters: Dpore ) 9.5 × 10-8 cm2 s-1, kads ) 109 cm3 mol-1s-1, kdes ) 10-5 s-1 (being the unbinding rate constant of the mercury-thiol assemblies), and δ ) 1 µm. The ensuing values log (λ0) ) -2.45 and log (Ξ0/η) ) 5.54 indicate that, in this case, the experimental system characteristics place it in the zone II of the schematic kinetic zone diagram depicted in Figure 4c to summarize the system reactivity status as a function of these two master parameters.33 The second data set (Figure 4b) was recorded under the same conditions as the previous one, but with pH ) 2. Again, the fitting procedure led to an excellent agreement, corresponding now to the following values of the dimensionless parameters: λ0 ) 7.8 × 10-5, η ) 0.0012, κ ) 0.5, φ ) 0.94. As above, from this set of dimensionless parameters and the other ones known independently, we obtained the values of the following real parameters: Dpore ) 6.1 × 10-9 cm2 s-1, kads ) 2.2 × 107 cm3 mol-1s-1, kdes ) 1.1 s-1, and δ ) 0.87 µm. The values log (λ0) ) -4.1, log (Ξ0/η) ) 6.77 show that this experimental behavior corresponds still to zone II in the kinetic zone diagram presented in Figure 4c. Note that the continuous and slow rise of f (t) for t > 20 s reflects the poor thermodynamics of the system at pH ) 2 (viz., κ ) 0.5) associated with the reduced diffusion coefficient. The comparison between the outcomes of the two above kinetic treatments clearly shows that upon decreasing pH the kinetics and thermodynamics of the mercury ions binding to the thiol groups lining the nanopore walls considerably deteriorated. This is as expected chemically because the nature of the two binding complexes should be different. Indeed, at pH ) 4, a neutral complex (matrix - S - Hg - OH) is anticipated, whereas at pH ) 2 an ionic complex (matrix - S - Hg +••NO3-) is expected.17b, 36 It is of interest to observe that this change of binding complex structure is also found to affect Dpore, the apparent diffusion coefficient inside the pore. This drops dramatically when passing from pH ) 4 to pH ) 2. This is in perfect agreement with the larger size of the “Hg +••NO3-” moiety versus that of the “HgOH” one, and also with the drastic change in electrostatic interactions since the dipole “Hg+••NO3-” is anticipated to “poise” somewhat the mobility of Hg(II) cations, which have to flow above the already occupied sites to reach deeper ranges within the nanopore. So our results are perfectly consistent with chemical expectations, though the present theory and fitting procedure allow a full quantification of the microscopic phenomena in the presence and of their changes based on a rather simple experimental procedure. To conclude this section, we wish to discuss another issue that is evidenced by the location of the two systems in the zone diagram in Figure 4c. This establishes that both of them correspond to the same zone II in the kinetic zone diagram. This zone is characteristic of systems exhibiting sufficiently large binding rate (viz., the value of λ0Ξ0/η is greater than 1) for being of experimental interest but not for leading to almost immediate binding of any Hg(II) ion that has reached an unoccupied thiol (36) Chen, C.-C.; Mckimmy, E. J.; Pinnavaia, T. J.; Hayes, K. F. Environ. Sci. Technol. 2004, 38, 4758-4762.
site inside the nanopore. At the same time, diffusion is not so slow compared to the kinetic rate as in zone I. Therefore, the concentration profile inside the nanotube is linear (though its maximum reduces with time due to the exhaustion of the bulk solution) throughtout the major part of the diffusion layer, followed by a smooth transition to zero at its end. When the diffusion layer has reached the bottom of the tube, the concentration profile becomes virtually constant and smoothly tends to the limiting value c∞ over the whole length of the tube. The active site coverage behaves as a propagating wave with a sufficiently wide transition between occupied and unoccupied sites as opposed to the behavior in zone I where the coverage has a very sharp transition (see section Analytical Formulation of 1D System Kinetics at Large λ0 Values). This characteristic behavior produces overall the peculiar aspects of the curves shown in Figure 4a and b, where fast initial rises of the f (t) functions are followed by smoother variations that resemble a combination of classical homogeneous kinetics though the system is essentially heterogeneous. Theoretical Support for Optimizing Sequestration Kinetics in Real Systems. The above theoretical and simulation section has established the experimental validity of the conceptual approach developed previously33 and elaborated on much quantitative ground here. Furthermore, the extraction of rate constants and diffusion coefficients that characterize microscopically the reactivity of nanoporous sequestrating materials based on simple chronoamperometric experiments performed at a RDE has demonstrated the experimental usefulness of simulation procedures based on this theoretical framework. In the following, we wish to expose another significant outcome of the present theoretical and analytical/numerical formulation of the complex interplay between diffusional feeding and surface kinetics whose outcomes condition the efficiency of many systems of interest. This goes beyond the two specific experimental cases that have been investigated above. Indeed, the ensuing theoretical views may be of great usefulness when serving as educated bases in the optimization of experimental systems designed for covering a wide range of applications.1-32 We will analyze the complex interplay between diffusional feeding and surface kinetics in the case when the surface kinetics involve sequestration of a target species to exemplify quantitatively the following analyzes. Nonetheless, this will serve as a paradigm for many other situations since many interactions between dissolved substrates and surface-borne sites are amenable to the same kinetic formulation. Finally, though we have specialized our above analyses for 1D nanopores to treat the two above experimental systems, several experimental situations may involve pores with larger volume/surface area ratio though remaining in the nanoscale range. Our general analysis, reported previously, has evidenced that systems with larger volume/surface area ratio behave as 2D nanopores and hence require a more sophisticated analysis.33 We wish to take advantage of the fact that the full mathematical description of the equations at hand (viz., the counterparts of eqs 4-8 describing the 1D case) is published elsewhere also for the 2D system,33 for simplifying our presentation and restricting it to a comparison between 1D and 2D behaviors. Similarly, we wish to focus our comparative presentation onto several aspects of experimental importance. Indeed, if above we Analytical Chemistry, Vol. 80, No. 9, May 1, 2008
3235
have centered our work onto the fitting of time dependences of the sequestration functions (Figure 4), from an operative, viz., chemical engineering, point of view, the main quantities that matter are the following: (i) the final solution concentration achieved when an assembly of particles operates in a solution to be cleaned or remediated and (ii) the time duration of the sequestration process since this is an important parameter that ought to be adjusted so that it is compatible with a reasonable residence time of the microparticles into the solution. Our above analyses or those published elsewhere33 have evidenced that the overall kinetics of sequestration may be quite complex, involving a succession of different time regimes with different time constants. Therefore, characterizing the above quantities by their values at half sequestration time may not be truly useful for performance optimization. Conversely, characterizing a system by the values of these quantities at 90% of completion seems to offer a better interest in terms of evaluating the effective efficiency and for educated optimization. Finally, the objective of this section is dealing with systems initially designed for extremely high efficiencies in terms of their thermodynamics. For this reason and to simplify the following analysis, we will restrict to the evaluation of systems with κ , 1, i.e., which lead to irreversible sequestration under the conditions where they are applied. 1. General Constraint of Efficiency: Titration Behavior. Though the sequestration is considered largely irreversible, the overall efficiency depends on the number of sites and internal volume (see eq 1) represented by Npart particles with respective to the amount of target species at concentration C0 in a solution of initial volume Vb. Indeed, the situation is formally akin to a titration. It was established in ref 33 that when κ f 0, c∞, the final dimensionless concentration (viz., C/C0 when τ f ∞) of the target species in the bulk solution is given by
c∞ )
1 - υ Ξ0 + |1 - υΞ0| 2(1 + υ)
(13)
This may be refined by evaluating the condition on υ and Ξ0 in 3236
Analytical Chemistry, Vol. 80, No. 9, May 1, 2008
order to reach a final concentration value lower than a threshold , required to achieve a given purification operative one, crequired ∞ level. Thus, from eq 12
1 - crequired 1 - crequired ∞ ∞ υ g required ≈ Ξ c∞ + Ξ0 0
(14)
since Ξ0 . 1 (see Figure 5). Note that the inequality (13) may also be derived upon considering the requirement for a successful titration, viz., that the number of adsorptive sites in the mesoporous particles introduced into the solution must be not less than the number of molecules or ions of target species in the initial solution. Indeed
υΞ0 ) Npart
(12)
where υ ) 4πR3partNpart (Rpore/(Rpore + ω))2/(3Vb) compares the cumulated nanopore internal volume to that of the solution, while Ξ0 was shown above to weight the overall amount of species sequestrated by active sites against that which may be confined into the solution filling the nanopores. Note that the case when υ is large has no useful experimental meaning, since υ > 1 implies that the total internal volume displayed by the pores exceeds that of the solution to be cleaned. This corresponds then to a simple mechanical scooping of the initial solution into the microspheres, so that we consider hereafter only cases where υ , 1. Figure 5 shows the dependence of c∞ predicted by eq 12 on υ and Ξ0 for the ranges of experimental interest of these parameters. The highest efficiency of the system is achieved when c∞ becomes sufficiently close to zero, i.e., when most of the target species is extracted from the solution by the mesoporous particles. Figure 5 and eq 12 show that this demands the following inequality to be respected:
υ gΞ0-1
Figure 5. Limiting bulk concentration c∞ dependence on υ and Ξ0 at infinite time.
Npart
4πRpart3
(
3V
b
(
Rpore Rpore + ω
)
4πRpart2
π(Rpore + ω)2
)
2
2Γsite ) RporeC0
(2πRporeL)Γsite
C0Vb
) Npart
Qsites (15) Q0
where Qsites is the number of active sites borne by one particle, while Q0 is the total quantity present initially in the solution. Note that eq 2 establishes that Qsites represents closely the whole storage ability of one microsphere when Ξ0 . 1, viz., under the situations of experimental interest (Figure 5). This titration condition may be illustrated through comparing the evolution of the overall fraction Θ90% of unoccupied sites at τ90% (viz., the dimensionless time at which f (τ) reaches 90% of its maximum) to the value of c∞ when υΞ0 varies. This is shown for the 1D and 2D systems at selected values of λ0 in the series of Figure 6. Though the variations differ as a function of λ0 (which controls the kinetics efficiency of the system), a common feature is the presence of a titration end point occurring at υΞ0 ) 1 in phase with a sudden rise of Θ90% . In other words, if one needs to respect as much as possible the inequality in eq 13, reaching situations where υΞ0 > 1 corresponds to a waste of the sequestrating material, though in Figure 6a and b, it is seen that this is a way to decrease the sequestration time (see below). Since υΞ0 is
Figure 6. Distributions of c∞, Θ90%, and τ90% as functions of υΞ0 for (a) 1D model with λ0 ) 0.1, η ) 0.01, φ ) 2.82, κ ) 10-6, ) 104, σ ) 1.17, (b) 2D model with the same parameter values except λ0 ) 0.01 and η ) 1, and (c) 2D model with λ0 ) 100, other parameters being the same as for (b). Insets show detailed variations of each quantity around the sharp transitions at υΞ0 ) 1.
proportional to Npart as evidenced by eq 15, υΞ0 ≈ 1 may be achieved by adjusting adequately the number of particles for a given application, possibly through exchanging an excess of material for a gain in kinetics. 2. Kinetics. The previous conclusions are based essentially on thermodynamics. However, kinetics dictates also the applicative efficiency of sequestrating systems. Indeed, not only the thermodynamic ability should be correct (viz., υΞ0 ≈ 1) but also the
duration of the sequestration process should be compatible with the residence time of the solution to be remediated into the reactor where the particles are active. For this reason, we wish to examine how the dimensionless time τ90% at which f (τ) reaches 0.90 depends on the parameters controlling the system efficiency. Figure 6 compares the variations of τ90% with υΞ0 to those of Θ90% and c∞, for the 1D (Figure 6a) or the 2D case (Figure 6b and c). Interestingly, in each case, τ90% experiences a sharp Analytical Chemistry, Vol. 80, No. 9, May 1, 2008
3237
Figure 7. Dependences of τ90% (a, b) and τ50% (c, d) on λ0 and Ξ0 for the two models: (a) τ90% , 1D model; (b) τ90% , 2D model; (c) τ50%, 1D model; (d) τ50%, 2D model. Data for the 1D model correspond to the following fixed dimensionless parameters: η ) 0.01, φ ) 2.82, υ ) 9.35 × 10-4, κ ) 10-6, ) 104, σ ) 1.17. Data for the 2D model were obtained with the following set of dimensionless parameters: η ) 1, φ ) 2.82, υ ) 9.35 × 10-4, κ ) 10-6, ) 104, σ ) 1.17.
maximum at the titration point. The rising branch observed at υΞ0 values less than unity features essentially the increasing difficulty to pump out the target species from the bulk solution when it tends to saturate the nanopore storage ability (compare the variations of Θ90% and see below). Conversely, the decrease of τ90% when υΞ0 > 1 reflects only the fact that when υΞ0 increases above unity an increasing number of sites remain unoccupied at τ90% (compare the variations of Θ90%). Thus, although this may be surprising at first glance, the sharp slope transition of τ90% at υΞ0 ) 1 is perfectly expected, being a consequence of the titration occurring at this point. Close inspection of the variations (see, e.g., insets in Figure 6a) around υΞ0 ) 1 indicates that these transitions are continuous over a limited range of υΞ0 values as occurs in any titration case. Moreover, the width of this transition enlarges when the thermodynamics of the binding reaction becomes less favorable (data not shown), a property common to any titration example in chemistry (note that κ ) 10-6 in Figure 6). 3238 Analytical Chemistry, Vol. 80, No. 9, May 1, 2008
More interesting are the variations of τ90% with Ξ0 and λ0 since, as established above (Figure 4c), these are the parameters that mostly control the kinetic behavior of the sequestrating system when the value of η is imposed. Note that one always has η ) 1 for 2D systems since the internal radii of pores are sufficient for classical diffusion to occur as in the bulk solution even if the nanopores have nanometric dimensions.33 Panels a and b in Figure 7 show that the variations of τ90% with Ξ0 and λ0 are very similar irrespective of the 1D or 2D nature of the case at hand. The most significant difference concerns the relative time scales, the 1D systems being considerably slower, reflecting the much slower rate of diffusion inside the nanopores. This evidences that even if 1D systems present a manifest advantage over 2D ones in terms of surface-to-volume ratio, hence in the amount of active surface groups per gram of the sequestrating material, this is obtained at the price of much longer durations for the reaction. Interestingly, the same relative variation is observed for the variations of τ50% with Ξ0 and λ0 (Figure 7c and
d), which confirms our previous conclusion that, though they fundamentally differ from a mechanistic point of view, 1D and 2D situations produce very similar f (τ) shapes.33 The second difference concerns the shape of τ90% spiked variations when Ξ0 varies at a constant λ0 (note that Ξ0 ≈ 103 in Figure 6 corresponds to υΞ0 ≈ 1 owing to the values of υ ≈ 10-3 used in constructing these plots) whose origin has been explained above when discussing the variations of τ90% in Figure 6. As is already apparent from the comparison of Figure 6a (1D case) to Figure 6b and c (2D case), the spike shape is extremely unsymmetrical for the 2D situation, τ90% tending toward a plateaulike regime more and more pronounced when both Ξ0 and λ0 are large. This difference reflects the comparatively higher rate of diffusion inside the 2D nanopores, which is equal to that in the bulk solution (viz., η ) 1) compared to the very small ones in 1D nanopores ( η , 1). On the other hand, when Ξ0 and λ0 are large, the value of τ90% is controlled almost exclusively by the rate of transport of target species. For the 2D cases, this is not coupled with the kinetics of sequestration because the two phenomena occur in different spaces,33 so τ90% tends toward a limit essentially imposed by diffusion (the situation is akin to Nernstian ones in electrochemistry). In the 1D case, large Ξ0 and λ0 values imply that the target species is blocked during its diffusional transport inside the nanopore whenever the wall sites are not occupied.33 Therefore, though the system is under rapid diffusional regime, this coupling imposes that the sharp limit, which is then created between fully occupied and fully empty sites, may propagate inside the nanopore rather than the diffusion of the target species. This phenomenon was qualitatively rationalized in our previous work.33 Yet, owing to the importance of this kinetic regime in terms of experimental applications, we wish to quantify it further below. 3. Analytical Formulation of 1D System Kinetics at Large λ0 Values. To simplify its presentation, the following section will be developed for situations where κ , 1. This allows neglecting the last term in eq 5, which can then be solved analytically, so that, upon taking into account the initial condition θ(τ ) 0) ) 0, one obtains readily
θ ) 1 - exp[ - λ0
∫ a dτ]
and
{
10,
y , 0 ey eδfront δfront L gy > δfront
(18)
being associated with
θ)
{
1, 0 ey eδfront 0, L gy > δfront
(19)
It ensues that the solution of the problem amounts to obtaining the rate of propagation, δfront(τ), of the sharp front created at y ) δfront between occupied and unoccupied sites, rather than the diffusion of the target species itself. Since λ0 is large, the variation of θ in the vicinity of y ) δfront is necessarily extremely sharp and localized, so that the expression for adsorptive coverage in eq 16 may conveniently be approximated by a Heaviside step function,37 viz., θ ) H(δfront(τ) - y). On the other hand, we consider now the matter conservation equation, which expresses that the overall quantity of the target species, which has entered inside the nanopore since the beginning of the experiment, equals the sum of those bound and unbound inside the nanopore. This is formulated in dimensionless form through eq 20:
-η
∫
∂a | dτ ) ∂y y)0
τ
0
∫
1
0
a dy + Ξ0
∫
1
0
θ dy
(20)
which, upon taking into account eqs 17-19, is rewritten as
η
∫
τ
0
1 1 dτ ) + Ξ0 δfront(τ) 2 δfront(τ)
(
)
(21)
Differentiating both terms of eq 21 vs the time affords
η δfront
)
(21 + Ξ ) 0
dδfront dt
(22)
(16) so that taking into account the initial condition δfront ) 0 at τ ) 0 finally yields
Since λ0 is considered to be extremely large in this regime (Figures 7a and 6a), eq 16 shows that whenever a (i.e., the dimensionless target species concentration within the nanopore) is different from zero, θ ) 1 since then [λ0 ∫ a dτ] f ∞. Conversely, when θ ) 0, one has a , (1/λ0) f 0. In other words, the target species progression inside the 1D nanopore is poised by its irreversible consumption by the first unoccupied sites it meets at y ) δfront. Since before this point all sites are occupied (see above), the target species diffuses freely over the range 0 e y e δfront, being under steady state between its fast consumption at y ) δfront and its slower supply at the entrance of the pore. Thus
a)
{
1 , 0 ey eδfront ∂a δfront ) ∂y L gy > δfront 0,
(17)
δfront )
x
4η τ 1 + 2Ξ0
(23)
Equation 23 establishes that although the system kinetics results from a complex interplay between diffusion-adsorption processes, the overall behavior obeys a classical diffusive behavior though ruled by an apparent diffusion coefficient R ) η/(1 + 2Ξ0), viz., D ) Dpore/(1 + 2Ξ0) in dimensioned form. The fact that this situation is created by the high site density is illustrated by the fact that when Ξ0 f 0 (which is equivalent to low adsorptive capacity and is therefore of no great interest for applications envisioned here) the diffusion layer thickness becomes δfront f 2xητ corresponding to linear diffusion, viz., D ) Dpore. Con(37) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C: the Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, UK, 1992.
Analytical Chemistry, Vol. 80, No. 9, May 1, 2008
3239
versely, for the conditions envisioned here, i.e., when Ξ0 is large, the rate of diffusional progression within the pore is severely limited by adsorption kinetics, and δfront ≈ x2ητ/Ξ0. Introducing the result in eq 23 into eq 16 provides the time dependence of the wall coverage, θ, at any point 0 e y e δfront ) 2 xRτ within the nanopore
[
∫
θ ) 1 - exp - λ0
τ
y2/R
(
1-
y
) ] [ (
2xRζ
dζ )
1 - exp - λ0τ 1 -
y 2xRτ
)] 2
(24)
while θ is zero for 2 xRτ e y e L. Integrating eq 24 as a function of τ finally affords the time evolution of the overall coverage, Θ(τ) for τ e(1 + 2Ξ0)/4η
Θ(τ) )
∫
1
0
θ dy ) 2 xRτ -
x
πR erf(xλ0τ) λ0
(25)
where erf denotes the error function (note that Θ(τ) ) 1 for τ > (1 + 2Ξ0)/4η). Since erf(xλ0τ) e1 and λ0 f ∞ when the present regime holds, one obtains finally
Θ(τ) )
{
2 1,
x
ητ , τ e(1 + 2Ξ0)/4η 1 + 2Ξ0 τ > (1 + 2Ξ0)/4η
(26)
Note that the expression in eq 26 predicts a rupture of slope at τ ) (1 + 2Ξ0)/4η. However, this is an artifact due to the fact that our model does not consider any area for the bottom of the pore (viz., no site may be populated in the cross section of the nanopore at y ) L). Furthermore, since the above prediction is valid for a single average nanopore representative of the distribution of nanopores forming the microspherical particles,33 consideration of the real distribution as disclosed in our previous work33 shows that for a real experimental system this sharp transition is smoothed and cannot be observable. Finally, to conclude this section, we wish to take advantage that this situation is the most complex one to solve numerically due to the large range of λ0 values where this limit applies, and to the sharp concentration transition, which occurs in phase with the sharp limit between occupied and unoccupied sites around y ) 2 xRτ, for testing the performance of the general simulation program developed in this work. This is performed in Figure 8, which superimposes the numerical results obtained at several values of τ to the analytical predictions achieved above for a(y)/ a(0) (eq 17), θ(y) (eq 24), and Θ(τ) (eq 26). The perfect agreement (better than 0.3%) between analytical predictions and numerical solutions demonstrates the extreme quality of the program even in these most difficult conditions for its application. This in turn ensures the validity of the numerical analyses presented here (or used in the simulations shown in Figures 4a and 4b). 4. Effect of Other Parameters: η, φ, and E. In the above analyses, we have focused our investigations on the main parameters that govern the system, viz., λ0 and Ξ0 (or Ξ0/η and Ξ0υ whenever this applied).33 In this section, we examine the 3240 Analytical Chemistry, Vol. 80, No. 9, May 1, 2008
Figure 8. Comparison between analytical predictions (symbols) and simulated values (lines) of (a) concentration, (b) coverage, and (c) average coverage. Dimensionless parameters used for the computations are as follows: λ0 ) 10, Ξ0 ) 200, η ) 0.01, φ ) 2.82, κ ) 10-8, ) 104, υ ) 9.35 × 10-4, σ ) 1.17. In (a, b) the status of the variable is shown at different dimensionless times: τ ) 21.9 (1), τ ) 593.2 (2), τ ) 2477.2 (3), τ ) 5566.3 (4), τ ) 9839.8 (5).
system dependence on the other parameters that govern its reactivity, viz., η, φ, and , both to evaluate their role and to establish that though significant under certain extreme circumstances they affect only in a minor way the reactivity of the system. Keeping in mind the kinetics, we will restrict this analysis to the variations of τ90%.
Note that any variation of ) (L/Rpore)2 may be interpreted as a direct consequence of changing the nanopore radius, Rpore, and length, L. Changes in φ ) (1 + δ/Rpart)(1 + ω/Rpore)2(L/δ) may be considered as modeling variations of the rate of target species supply from the solution bulk (through the involvement of δ, the thickness of the stagnant layer wrapping around each particle) scaled versus characteristic dimensions of the nanopore. Finally, η ) Dpore/Dbulk represents the effect of nonconventional transport within nanometric pores due to their molecular scales and to the interactions created by the sites lining up the nanopore walls (note that this applies only to 1D nanopores, η being unity for 2D ones). However, in evaluating meaningfully the role of these parameters, one needs to ensure that the ensuing variations of τ90% are effectively comparable. For this reason, the actual density of active sites on the pore walls was kept the same for all values of . This was achieved by fixing the value of the dimensionless parameter γsite ) Γsite/(LCb0 ); this required the values of Ξ0 to be modified when is varied: Ξ0 ) 2γsite x. Hence, υ ) Ξ0-1 was adjusted accordingly to ensure that the system performed always at its maximum performance ( υΞ0 ) 1, compare Figure 5), i.e., always keeping the number of available sites equal to the number of target ions in the solution. Changes in η did not require any coupled change in Ξ0 or υ since these parameters are fully independent (though in a real system they are intrinsically related through the synthesis of the material). Figure 9a presents contour plots of τ90% in a (η, φ) diagram for a constant λ0 ) 0.1 value. This was performed for 1D nanopores since this is only in this case where η may be less than unity. It is evident from the figure that the dependences of τ90% on both parameters are smoother than those observed when λ0, Ξ0/η and Ξ0υ vary (see Figures 5-7). The dependence on φ is negligible when this parameter is high enough. Conversely, when φ is small, a significant effect is observed but only for the highest range of η values, i.e., when the diffusion coefficient inside the nanopore becomes comparable to that in the bulk solution, a fact that is approached only when the system tends toward a 2D situation (see below for the effect of φ on 2D systems). Under these conditions, one observes a decreasing trend of τ90% upon increasing φ, viz., upon decreasing the diffusion layer thickness around the mesoporous particles. In other words, this confirms that at large η values enhancing the flux of target species at the nanopore entrance favors the overall sequestration process. As expected, the largest variations of τ90% are observed with variations in η. Figure 9a shows that this dependence is very strong regardless of the value of φ. This confirms that the difference in diffusion rates inside and outside of the nanopores composing mesoporous particles is a major factor determining the kinetic reactivity of the 1D systems. Yet, the effect of this parameter is integrated with that of Ξ0, which is a major actor through the combination of both parameters, viz., through the parameter Ξ0/η.33 The variations of τ90% with and φ are evaluated in Figure 9b for 2D nanopores since this is when φ displays its most significant effects (see Figure 9a). This plot confirms the smoother changes in reactivity associated with these parameters compared to those related to λ0 and Ξ0υ. As expected, τ90% is lower at low values since this corresponds to larger pores, a fact that facilitates transport, and for high values of φ, which corresponds to faster
supply of target species to the nanopore entrance through the decrease of the thickness of the thin diffusion layer surrounding the particles. In addition, Figure 9b shows an interesting property of the system. Indeed, at large φ values (sufficiently thin diffusion layers), τ90% becomes virtually independent of φ showing that under such operative conditions the hydrodynamic regime in the solution to be remediated has no real influence. Conversely, when the value of φ is not very high, the rate of diffusion inside the nanopore is much faster than the rate of supply of target species toward the nanopore entrance. In these circumstances, the value of τ90% is mostly imposed by the rate of target species supply from the solution and that controlling its adsorption at the pore walls. More quantitatively, the boundary conditions, which define mathematically 2D systems,33 determine that under these conditions one has τ90% ∝ λ0Ξ0φ/(2). Bearing in mind that the relationship Ξ0 ) 2γsite x is imposed, it ensues that at a constant λ0 the value τ90% is constant whenever
log φ )
1 log + const 2
(27)
This is precisely what is observed in the lower part of Figure 9b, viz., at low φ values, since when plotted in a (log , log φ) diagram the contour lines of τ90% tend toward straight lines with a slope 1/2. COMPUTATIONAL AND EXPERIMENTAL SECTION Experimental Details. The thiol-functionalized mesoporous silica microspheres were prepared as described earlier.17a,c Briefly, 2.4 g of cetyltrimethylammonium bromide (CTAB) was dissolved in 50 mL of water, 50 mL of ethanol, and 13 mL of 28% ammoniac solution. In a second solution, 0.19 g of mercaptopropyltrimethoxysilane was mixed with 3.0 g of tetraethoxysilane. The resulting mixture was added to the surfactant solution under magnetic stirring. A white precipitate was observed after a few minutes of reaction. Stirring was maintained for 2 h, and the product was filtered, washed with ethanol, and dried under vacuum (
