pubs.acs.org/Langmuir © 2009 American Chemical Society
Intraparticle Transport and Release of Dextran in Silica Spheres with Cylindrical Mesopores Jovice B. S. Ng,† Padideh Kamali-Zare,‡ Malin S€orensen,§ Hjalmar Brismar,‡ Niklas Hedin,† and Lennart Bergstr€om*,† † Materials Chemistry Research Group, Department of Physical, Inorganic and Structural Chemistry, Arrhenius Laboratory, Stockholm University, SE-106 91 Stockholm, Sweden, ‡Division of Cell Physics, Department of Applied Physics, Royal Institute of Technology, AlbaNova, SE-106 91 Stockholm, Sweden, and § YKI, Institute of Surface Chemistry, Box 5607, SE-114 86 Stockholm, Sweden
Received June 11, 2009. Revised Manuscript Received August 30, 2009 The transport of oligomeric molecules in silica spheres with cylindrical mesopores has been quantified and related to the structural features of the spherical particles and the interactions at the solid-liquid interface. An emulsion-solvent evaporation method was used to produce silica spheres having cylindrical mesopores with an average pore diameter of 6.5 nm. The transport of dextran molecules (fluorescently tagged) with molecular weights of 3000 and 10 000 g/mol was quantified using confocal laser scanning microscopy (CLSM). The intraparticle concentration profiles in the dextrancontaining spheres were flat at all times, suggesting that the release is not isotropic and not limited by diffusion. The release of dextran into the solution is characterized by an initial burst, followed by long-term sustained release. The release follows a logarithmic time dependency, which was rationalized by coupling concentration-dependent effective diffusion constants with adsorption/desorption.
1. Introduction Immobilization, uptake, and controlled release of molecules from carrier materials are the basis of a large number of applications (e.g., controlled drug delivery,1 biocatalysis,2,3 and bioseparation4,5). Porous materials of various forms are already commercially established as flexible platforms to separate and dispense guest molecules. Surfactant-templated inorganic materials,6,7 such as mesoporous silica, are a relatively new class of porous materials characterized by well-controlled and tunable pore size, porosity, and pore structure. Recent studies have demonstrated how these materials can be used for the adsorption/separation of macromolecules3-5 and as carrier materials for drug delivery8-11 and gene delivery.12 However, the studies rarely go beyond investigations of bulk uptake and release, and studies *Corresponding author. E-mail:
[email protected]. Tel: þ46 8 674 7023. Fax: þ46 8 15 21 87.
(1) Balas, F.; Manzano, M.; Colilla, M.; Vallet-Regı´ , M. Acta Biomater. 2008, 4, 514–522. (2) Dı´ az, J. F.; Balkus, K. J. J. Mol. Catal. B 1996, 2, 115–126. (3) Reis, P.; Witula, T.; Holmberg, K. Microporous Mesoporous Mater. 2008, 110, 355–362. (4) Katiyar, A.; Yadav, S.; Smirniotis, P. G.; Pinto, N. G. J. Chromatogr., A 2006, 1122, 13–20. (5) Ma, Y.; Qi, L.; Ma, J.; Wu, Y.; Liu, O.; Cheng, H. Colloids Surf., A 2003, 229, 1–8. (6) Yanagisawa, T.; Shimizu, T.; Kuroda, K.; Kato, C. Bull. Chem. Soc. Jpn. 1990, 63, 988–992. (7) Beck, J. S.; Vartuli, J. C.; Roth, W. J.; Leonowicz, M. E.; Kresge, C. T.; Schmitt, K. D.; Chu, C. T. W.; Olson, D. H.; Sheppard, E. W.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. J. Am. Chem. Soc. 1992, 114, 10834–10843. (8) Vallet-Regi, M.; Ramila, A.; del Real, R. P.; Perez-Pariente, J. Chem. Mater. 2001, 13, 308–311. (9) Zhu, Y. F.; Shi, J. L.; Li, Y. S.; Chen, H. R.; Shen, W. H.; Dong, X. P. J. Mater. Res. 2005, 20, 54–61. (10) Goyne, K. W.; Chorover, J.; Kubicki, J. D.; Zimmerman, A. R.; Brantley, S. L. J. Colloid Interface Sci. 2005, 283, 160–170. (11) Qu, F.; Zhu, G.; Huang, S.; Li, S.; Sun, J.; Zhang, D.; Qiu, S. Microporous Mesoporous Mater. 2006, 92, 1–9. (12) Lai, C. Y.; Trewyn, B. G.; Jeftinija, D. M.; Jeftinija, K.; Xu, S.; Jeftinija, S.; Lin, V. S. Y. J. Am. Chem. Soc. 2003, 125, 4451–4459.
466 DOI: 10.1021/la902092e
where the molecular distribution and hindered transport within the material are analyzed to offer a better understanding of the release mechanisms are sparse. Cheng and Landry13 have shown how confocal laser scanning microscopy (CLSM)14 can be used to section mesoporous spheres optically to obtain the transient intraparticle concentration profile of adsorbing/desorbing fluorescent molecules. We have recently demonstrated that CLSM can be used to investigate and follow the release of anionic and cationic fluorescent dyes.15 The time and spatially dependent variations in the concentration of dyes within these mesoporous silica spheres (generated by an aerosol method) were linked to the specific interactions with the negatively charged walls.15 The aim of the present study is to explore the molecular transport of macromolecules in mesoporous silica spheres. In this study, we have used fluorescently tagged-dextran molecules (originating from the bacterial strain Leuconostoc mesenteroides) of two different molecular weights. The hydrodynamic radii of the macromolecules are similar to the diameter of the cylindrical pores (6.5 nm) of the mesoporous particles that were produced by a novel emulsion-solvent evaporation method. CLSM was used to follow the transport of the tagged molecules within the spheres over an extended period of time. Various diffusion and adsorption models were employed to analyze the time-dependent concentration profiles that were obtained from the intensity data after correcting for both photobleaching and light attenuation. Describing the bulk release with a simple exponential relation showed that the transport of the dextrans within the cylindrical mesopores is orders of magnitude slower than in the bulk. The pronounced flatness of the concentration profiles within the mesoporous (13) Cheng, K.; Landry, C. C. J. Am. Chem. Soc. 2007, 129, 9674–9685. (14) Sheppard, C. J. R.; Shotton, D. M. Confocal Laser Scanning Microscopy; Bios Scientific Publishers: Singapore, 1997; p 106. (15) Ng, J. B. S.; Kamali-Zare, P.; Brismar, H.; Bergstrom, L. Langmuir 2008, 24, 11096–11102.
Published on Web 09/21/2009
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Figure 1. Images of the calcined spherical ESE particles. The SEM image reveals the polydispersity of the spheres (left). The CLSM optical slice of the spheres (right) shows that Dex3k has penetrated the spheres to their cores.
spheres suggests that the release is not limited by diffusion and the time dependence of the release is related to the specifics in the adsorption and desorption of dextrans in the mesopores.
2. Experimental Section 2.1. Materials. A series of Texas red-labeled dextran, a hydrophilic polysaccharide synthesized from the Leuconostoc mesenteroides bacteria, were obtained from Molecular Probes Europe BV (Leiden, The Netherlands). Conjugates of fluorescent dextrans are frequently used as a suitable molecule for modeling large biological molecules.16 Texas red-tagged dextrans of two different weights, 3000 (Dex3k) and 10 000 g/mol (Dex10k), were used in this study. Tetraethoxysilane TEOS (Sigma Purum >98%) was used as the silica source, Pluronic triblock copolymer F127 ([(EO)106(PO)70(EO)106], BASF) was used to template aggregates, poly(propylene glycol) PPG (3000 g/mol, Alfa Aesar) was used as a swelling agent, and Arlacel P135 (Avecia) was used as a polymeric dispersant. Dulbecco’s phosphate-buffered saline (PBS), (135 mmol/dm3, pH 7.2) was obtained from Sigma, and ethanols in two purity classes (99% and 99.7%) were used. In all experiments and syntheses, Millipore-grade water was used. 2.2. Synthesis of Mesoporous Spheres. Mesostructured silica spheres with large pores were produced with a recently developed method: the emulsion and solvent evaporation (ESE) method.17 This method consists of five steps: (i) the preparation of a precursor solution; (ii) the emulsification of the precursor solution; (iii) the evaporation of ethanol and water by reducing the pressure; (iv) the separation of particles; and finally (v) removing the surfactant by a thermal treatment in air (calcination). The synthesis was done according to the procedures described in an earlier publication.17 The particles were washed in ethanol, and the organic template was removed by heat treatment in air at 550 °C for 4 h to produce large-pore mesoporous silica spheres. The details of the characterization of the particles are available in the Supporting Information.
Table 1. Size and Bulk Diffusion Coefficients of Dextran average molecular weight (g/mol)a
dye molecules/ dextrana
dg (nm)
dh (nm)
Dbulk (10-7 cm2/s)b
3000 0.5-1 5.220 2.920 2321 10 000 1-2 8.920 5.320 1421 a b Data from the supplier, Invitrogen, Molecular Probe. In 0.3 wt % agarose gel, 150 mmol/dm3 NaCl.
spheres were deposited in glass-bottomed microwell dishes (MatTek Corporation) filled with 4 mL of PBS. Stock solutions of the dextrans were prepared at concentrations of 0.10 mmol/dm3
(Dex3k) and 0.01 mmol/dm3 (Dex10k). The silica particles (ESE) were loaded with Dex3k or Dex10k by incubation in a dextran PBS-based solution of 0.25 μmol/dm3 and 250 nmol/dm3, respectively. The particles were incubated for 24 h (Dex3k) or 48 h (Dex10). The release of the dextrans from the spheres was initiated by exchanging the dextran-containing solution with pure PBS. Following two rinses in PBS, the dextran-containing mesoporous spheres were left in clean PBS, and the system was allowed to equilibrate for 30 ( 10 min before CLSM imaging was started, defined as the observation time, τ = 0. The initial equilibration time ensured that any potential excess dextran on the exterior surfaces of the spheres was removed. Because of the slow release, the initial equilibration did not interfere with the release of dextran from the sphere’s interior. The dextran-containing spheres were optically sectioned into 20 equidistant slices separated by approximately 1 μm. By considering the geometry of the particles, the slice with the largest diameter was taken as the equatorial slice. The glass-bottomed dish was covered to minimize evaporation. The spheres were imaged every hour for 18 h. The position of the equatorial slice was verified each time the spheres were imaged. Confocal laser scanning microscopy imaging was performed on an Axiovert 200 inverted microscope with a Zeiss LSM 510 scanner. A 543 nm HeNe laser (100% power) was used together with a 60/1.4 NA oil-immersion objective lens and a long-pass 560 nm filter to image the dextran-containing spheres. All profiles were stored as eight-bit line scans with a resolution of 512 pixels 512 pixels representing an area of 146.2 μm 146.2 μm. Control experiments were conducted to evaluate the photobleaching of the dyes attached to the dextrans and the light attenuation during the experiment. (See Supporting Information for details.)
(16) Kwon, K. D.; Green, H.; Bjoorn, P.; Kubicki, J. D. Environ. Sci. Technol. 2006, 40, 7739–7744. (17) S€orensen, M. H.; Zhu, J.; Corkery, R. W.; Hayward, R. C.; Alberius, P. C. A. Microporous Mesoporous Mater. 2009, 120, 359–367.
Figure 1a shows that the calcined silica particles are spherical with an average particle diameter of 30 μm. The nitrogen
2.3. Confocal Laser Scanning Microscopy Imaging of Molecular Transport within Mesoporous Spheres. The ESE
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3. Results and Discussions
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Figure 2. CLSM optical sections at the equator of the sphere imaged at different times. Image series of Dex3k (left) and Dex10k (right) showing decreasing fluorescence intensity with release time. Note that these intensity data have not been adjusted for photobleaching.
Figure 3. Intraparticle intensity profile (center to edge of slice) as a function of time, extracted from the sphere equatorial slices of Dex3k-loaded ESE spheres. Note that the intensity has been corrected for both photobleaching and light attenuation.
isotherm, X-ray patterns, and TEM images of the spheres are available in the Supporting Information (SI Figure 1). According to the IUPAC nomenclature, the nitrogen isotherm can be classified as a type IV isotherm with a type H2 adsorption-desorption hysteresis loop, which is typically for mesoporous materials with cylindrical or ink-bottle-shaped pores.18 The specific surface area, SBET, and the pore volume of the mesoporous spheres from the nitrogen sorption are 460 m2/g and 0.37 cm3/ g, respectively. The average pore size calculated using the cylindrical BdB-FHH pore model19 is 6.5 nm. The TEM and the XRD analyses indicate that the spheres exhibit domains of 2D hexagonal pore structure with a p6mm symmetry (i.e., channels of cylindrical pores packed together in a hexagonal arrangement). A series of dextran from the bacterial strain Leuconostoc mesenteroides were used to study the molecular transport in the mesoporous spheres. The diameter of gyration, dg, and the diameter of hydration (or Stokes-Einstein diameters), dh, and the bulk diffusion coefficient, D, of the dextrans with different molecular weights are summarized in Table 1. (18) Kruk, M.; Jaroniec, M. Chem. Mater. 2001, 13, 3169–3183. (19) Lukens, W. W.; Schmidt-Winkel, P.; Zhao, D.; Feng, J.; Stucky, G. D. Langmuir 1999, 15, 5403–5409. (20) Kloster, C.; Bica, C.; Rochas, C.; Samios, D.; Geissler, E. Macromolecules 2000, 33, 6372–6377. (21) Nicholson, C.; Tao, L. Biophys. J. 1993, 65, 2277–2290.
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The time evolution of the fluorescence within the Dex3k- and Dex10k-loaded ESE particles (Figure 2) shows that the dextrans have penetrated to the core of the spheres. No evidence of a hollow core or any other unavailable or blocked regions has been observed. (The dark spots in Figure 2 are large pores that stem from the evaporation-driven synthesis process.22) The effect of photobleaching was quantified by repeatedly scan the same section of the Dex3k- and Dex10k-containing ESE spheres (SI section 1.2). The scans were performed in sequence without any time delay between the image stacks. This rapid scanning ensures a short measurement time with negligible Dextran release, and the drop in the signal intensity can be attributed solely to photobleaching. After correcting for photobleaching, the (corrected) fluorescence intensity in the spheres is a quantitative measure of the dextran concentration. The total amount of tagged polymer within the observed slice (at the equator of the sphere) can be obtained by integration of the fluorescence intensity over the entire measurement volume. The time-dependent release of the fluorescent dextrans can thus be estimated by integrating the intensities within a specific slice at different times. This information is similar to the data that can be obtained from traditional bulk release studies, where the change in polymer concentration of the liquid media is followed as a function of time. We have minimized the sum of the composite squared deviation of the data to yield a curve that can be fitted to a diffusion model based on Ficks second law15 (details in SI section 1.6), and this simplified analysis results in apparent hindered diffusion coefficients of Dex3k and Dex10k that are at least 6 orders of magnitude lower than their bulk diffusion coefficients in free solution (Table 1). Whereas it is clear that the transport of dextran in the cylindrical mesopores is hindered, relating the fluorescence information only to an exponential model for the overall release of the molecules does not give much information on the mechanisms of molecular transport. The information obtained by CLSM allows us both to quantify the time-dependent concentration variations inside the porous carrier and to follow the bulk release. Quantification requires that the intensity data be adjusted for the effects of light attenuation, which depends on the optical path length. We have corrected for light attenuation by using the simplified short-cut method proposed by Susanto et al.23 This method is based on the Lambert-Beer equation for the intensity loss along the excitation path (details in SI section 1.3). After appropriate corrections, we (22) Andersson, N.; Kronberg, B.; Corkery, R.; Alberius, P. Langmuir 2007, 23, 1459–1464. (23) Susanto, A.; Herrmann, T.; Hubbuch, J. J. Chromatogr., A 2006, 1136, 29–38.
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Figure 4. Release profiles of Dex3k-containing spheres as a function of observation time, plotted in (a) linear form and (b) logarithmic form.
Figure 5. Release profiles of Dex10k-containing spheres as a function of observation time, plotted in (a) linear form and (b) logarithmic form.
obtain concentration profiles inside the ESE spheres that essentially do not vary with the distance to the external surface. The flat concentration profile in the Dex3k-loaded (Figure 3) and Dex10kloaded (SI Figure 6) particles that is preserved at all times strongly indicates that the release of dextran from the ESE spheres cannot be well described by a Fickian-type model (characterized by Gaussian-shaped intraparticle concentration profiles). Instead, the flat intraparticle concentration profiles at all observation times suggest that the release is not fully isotropic and not limited by diffusion. Intraparticle concentration profiles that do not show any significant spatial dependence is a signature of systems where the transport resistance at the surface is much higher than in the interior of the particle. Indeed, recent findings that have shown that the pore channels close to the external surface of mesoporous crystals17,22,24 may be much more tortous and may even bend back into the sphere at the surface, which could be related to enhanced surface resistance. (24) Che, S.; Lund, K.; Tatsumi, T.; Iijima, S.; Joo, S. H.; Ryoo, R.; Terasaki, O. Angew. Chem., Int. Ed. 2003, 42, 2182–2185.
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However, further analyses of the integrated fluorescence intensities revealed that the release of dextran molecules exhibits an initial burst, which was followed by a truly sustained release (Figures 4a and 5a). The release behavior is neither exponential nor biexponential in nature. The lack of a typical steplike-form intraparticle profile with a sharp change in concentration at the surface, together with the nonexponential time-dependent release of our system, suggests that the release is controlled by factors other than enhanced surface resistance. If the release is determined by the search probability of an exit at the surface for a molecule to leave the particle, then the release profile generally exhibits an exponential form.25 We have made an attempt to rationalize the release kinetics by accounting for the adsorption/desorption of dextran onto the silica pore walls. An important consequence when the molecules can adsorb to the pore walls is that the efficient diffusion coefficients become concentration-dependent, as described in detail by Reyes et al.26 for gas diffusion in porous materials. In short, the overall diffusion rate is predicted to be higher at high concentrations than at low concentrations because of the limited number of surface sites that become progressively occupied with an increasing concentration. Hence, because the concentration of dextran decreases with time, a release mechanism that is controlled by adsorption/desorption is expected to result in a sustained release that is no longer monoexponential in time. Indeed, Figures 4b and 5b show that the release kinetics of both dextrans conform well to a logarithmic time dependency. The similarity of the release curves for Dex3k and Dex10k indicates a similar release mechanism for the two macromolecules of different molecular weights. Logarithmic rate laws are often observed for chemisorptions and the formation of, for example, oxide films on metals. For gas molecules in porous media, logarithmic time dependencies have also been found.27 Landsberg28 has rationalized this time (25) Grigoriev, I. V.; Makhnovskii, Y. A.; Berezhkovskii, A. M.; Zitserman, V. Y. J. Chem. Phys. 2002, 116, 9574–9577. (26) Reyes, S. C.; Sinfelt, J. H.; DeMartin, G. J. J. Phys. Chem. B 2000, 104, 5750–5761. (27) Freund, T. J. Chem. Phys. 1957, 26, 713–713. (28) Landsberg, P. T. J. Chem. Phys. 1955, 23, 1079–1087.
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dependency for chemisorption; however, because of the limited observation time in this study we choose not to present a detailed model for this dependency but simply describe a general form of the rate law (eq 1) q ¼ q0 -
1 t ln B t00
ð1Þ
where q* is the time-dependent concentration, t is the observation time, t00 is the starting observation time, and B is a constant. It is interesting that the reversible adsorption of proteins, for which dextrans are frequently used as model compounds, onto surfaces has been reported to obey the Temkin isotherm,29 which indicates that there is a distribution of protein-surface binding energies. Although many materials have surface sites that display a range of interaction energies,30 the mesoporous silicas used in this study are also known to have a nonhomogeneous spatial distribution of surface silanol groups.31,32 This uneven distribution leads to a system that is analogous to the heterogeneous protein adsorption system previously mentioned. Adsorption isotherms similar to the Temkin model are expected to have an approximate logarithmic time dependency for the release over a few decades of observation time. Indeed, eq 1 can be deduced from and is analogous to the Temkin model.33 The adsorption (energy) of macromolecules commonly displays a variation with surface coverage.34 Previous work has also shown that flexible macromolecules such as dextran may transform from a flatter confirmation to a conformation having a significant fraction of segments extending into the solution with increasing coverage.16 The conformation and the total adsorption energy also depend on the molecular weight, which often results in a slower desorption rate for high-molecular-weight polymers compared to that for lower-molecular-weight polymers.35 Please note that both the burst and sustained release observed in Figures 4 and 5 are natural consequences of the logarithmic time dependency. Although the logarithmic dependency can be rationalized by coupling concentration-dependent effective diffusion constants with certain models for desorption, it is clear that most adsorption/desorption behaviors (with their respective isotherms) will lead to a quick burst followed by a more sustained release of adsorbates from the porous carrier. For many porous materials, effects of molecular crowding have been observed (e.g., single-file diffusion).36,37 For this kind of (29) Johnson, R. D.; Arnold, F. H. Biochim. Biophys. Acta 1995, 1247, 293–297. (30) Zhmud, B. V.; Sonnefeld, J.; Bergstr€om, L. Colloids Surf., A 1999, 158, 327– 341. (31) Ottaviani, M. F.; Galarneau, A.; Desplantier-Giscard, D.; Di Renzo, F.; Fajula, F. Microporous Mesoporous Mater. 2001, 44-45, 1–8. (32) Galarneau, A.; Desplantier-Giscard, D.; Di Renzo, F.; Fajula, F. Catal. Today 2001, 68, 191–200. (33) Temkin, M.; Pyzhev, V. J. Phys. Chem. (U.S.S.R.) 1939, 13, 851–867. (34) Ball, V.; Schaaf, P.; Voegel, J.-C. Mechanism of Interfacial Exchange Phenomena for Proteins Adsorbed at Solid-Liquid Interfaces. In Biopolymers at Interfaces, 2nd ed.; Malmsten, M., Ed.; Marcel Dekker: New York, 2003; Vol. 110, p 295. (35) Myers, D. Surfaces, Interfaces and Colloids: Principles and Applications, 2nd ed.; John Wiley & Sons: New York, 1999. (36) Kukla, V.; Kornatowski, J.; Demuth, D.; Girnus, I.; Pfeifer, H.; Rees, L. V. C.; Schunk, S.; Unger, K. K.; K€arger, J. Science 1996, 272, 702–704. (37) Nelissen, K.; Misko, V. R.; Peeters, F. M. Lett. J. Explor. Front. Phys. 2007, 80, 1–5.
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diffusion, a slower diffusion rate is expected at higher solute concentrations. This jamming should result in an accelerated time dependence of the release because the effective diffusion constant should increase with decreasing solute concentration. Because both the Dex3k and Dex10k systems display sustained rather than accelerated release behavior, this suggests that the effects of blockage or single-file diffusion are insignificant. In addition, if single-file diffusion were important, then one would expect a very strong dependency of the release rate on the molecular size. The half-life for Dex10k is about 2.5 times longer than that for Dex3k, whereas the free diffusion of Dex3k is about 1.6 times faster than that for Dex10k. The difference is partially related to the slower free diffusion of Dex10k (Table 1), and the small remaining difference is probably related to the difference in adsorption characteristics for the two dextrans.
4. Summary and Conclusions We have used confocal laser scanning microscopy to evaluate the details of the release of dextran from mesoporous spheres as well as the time-dependent spatial distribution. The mesoporous silica spheres were produced with an emulsion-solvent evaporation (ESE) method and exhibit cylindrical pores having a 2D hexagonal arrangement with a narrow pore size distribution and an average pore size of 6.5 nm. Various models were employed to analyze the time-dependent concentration profiles that were obtained from the intensity data after correcting for both photobleaching and light attenuation. Describing the bulk release with a simple exponential relation showed that the transport of dextrans within the cylindrical mesopores is several orders of magnitude slower than in the bulk. The pronounced flatness of the concentration profiles within the mesoporous spheres suggests that the release is not limited by diffusion. A logarithmic time dependency was observed for the release of fluorescently tagged dextran molecules. The initial burst and the sustained release can be rationalized by the combined action of adsorption/desorption and the diffusion of dextran within the porous framework. Acknowledgment. This work was supported by the Swedish Science Council (VR), the Centre for Controlled Delivery and Release (CODIRECT), which is supported by VINNOVA, the Knowledge Foundation, and the Foundation for Strategic Research (SSF) and Industry. Special thanks go to Dr. Peter Oleynikov for extending his expertise in Mathcad for the light attenuation correction work. Supporting Information Available: Detailed information on the characterization of the mesoporous spheres, details on the photobleaching compensation for the release study of the dextran-containing spheres, optical sections of dextranloaded spheres, details of the light attenuation correction for the release studies of both Dex3k- and 10k-containing spheres, intraparticle concentration profile of Dex10k-containing spheres, and details on the dextran bulk release analysis using a diffusion model based on Ficks second law. This material is available free of charge via the Internet at http://pubs.acs.org.
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