Abstract

Time-dependent laser reflectometry measurements are presented as a means to rigorously characterize analyte diffusion dynamics of small molecules from mesoporous silicon (PSi) films for drug delivery and membrane physics applications. Calculations based on inclusion of a spatially and temporally dependent solute concentration profile in a one-dimensional Fickian diffusion flow model are performed to determine the diffusion coefficients for the selected prototypical polar species, sucrose (340 Da), exiting from PSi films. The diffusion properties of the molecules depend on both PSi pore size and film thickness. For films with average pore diameters between 1030nm and film thicknesses between 300900nm, the sucrose diffusion coefficient can be tuned between approximately 100 and 550μm2/s. Extensions of the real-time measurement and modeling approach for determining the diffusivity of small molecules that strongly interact with and corrode the internal surfaces of PSi films are also discussed.

© 2011 Optical Society of America

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  31. We note that the use of the Bruggeman approximation rather than the parallel approximation presented in Eq.  resulted in calculated diffusion coefficients that were within the expected error range of the technique (approximately ±50 μm2/s), as compared to the results presented. Additionally, there is no qualitatively different trend evident from use of the Bruggeman approximation, implying that the error of the technique is not sufficiently low to resolve the diffusion coefficients yielded by the different approximations.
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2010 (2)

J. D. Ryckman, M. Liscidini, J. E. Sipe, and S. M. Weiss, “Direct imprinting of porous substrates: a rapid and low-cost approach for patterning porous nanomaterials,” Nano Lett. 11, 1857–1862 (2010).
[CrossRef] [PubMed]

Y.-C. Li and P. J. Cleall, “Analytical solutions for contaminant diffusion in double-layered porous media,” J. Geotech. Geoenviron. Eng. 136, 1542–1554 (2010).
[CrossRef]

2009 (2)

G. Liu, L. Barbour, and B. C. Si, “Unified multilayer diffusion model and application to diffusion experiment in porous media by method of chambers,” Environ. Sci. Technol. 43, 2412–2416 (2009).
[CrossRef] [PubMed]

J.-H. Park, L. Gu, G. von Maltzahn, E. Ruoslahti, S. N. Bhatia, and M. J. Sailor, “Biodegradable luminescent porous silicon nanoparticles for in vivo applications,” Nat. Mater. 8, 331–336 (2009).
[CrossRef] [PubMed]

2008 (3)

E. J. Anglin, L. Cheng, W. R. Freeman, and M. J. Sailor, “Porous silicon in drug delivery devices and materials,” Adv. Drug Deliv. Rev. 60, 1266–1277 (2008).
[CrossRef] [PubMed]

G. Rong, A. Najmaie, J. E. Sipe, and S. M. Weiss, “Nanoscale porous silicon waveguide for label-free DNA sensing,” Biosens. Bioelectron. 23, 1572–1576 (2008).
[CrossRef] [PubMed]

A. Navid and L. Pilon, “Effect of polarization and morphology on the optical properties of absorbing nanoporous thin films,” Thin Solid Films 516, 4159–4167 (2008).
[CrossRef]

2006 (2)

C. Pacholski, C. Yu, G. M. Miskelly, D. Godin, and M. J. Sailor, “Reflective interferometric Fourier transform spectroscopy: a self-compensating label-free immunosensor using double-layers of porous SiO2,” J. Am. Chem. Soc. 128, 4250–4252(2006).
[CrossRef] [PubMed]

S. M. Weiss and P. M. Fauchet, “Porous silicon one-dimensional photonic crystals for optical signal modulation,” IEEE J. Quantum Electron. 12, 1514–1519 (2006).
[CrossRef]

2005 (3)

R. Valiullin, P. Kortunov, J. Kärger, and V. Timoshenko, “Concentration-dependent self-diffusion of adsorbates in mesoporous materials,” Magn. Reson. Imag. 23, 209–214 (2005).
[CrossRef]

J. Salonen, L. Laitinen, A. M. Kaukonen, J. Tuura, M. Björkqvist, T. Heikkilä, K. Vähä-Heikkilä, J. Hirvonen, and V. P. Lehto, “Mesoporous silicon microparticles for oral drug delivery: loading and release of five model drugs,” J. Controlled Release 108, 362–374 (2005).
[CrossRef]

S. Cruz, A. Honig-d’Orville, and J. Muller, “Fabrication and optimization of porous silicon substrates for diffusion membrane applications,” J. Electrochem. Soc. 152, C418–C424 (2005).
[CrossRef]

2004 (1)

A. Carbonaro, R. Walczak, P. M. Calderale, and M. Ferrari, “Nano-pore silicon membrane characterization by diffusion and electrical resistance,” J. Membr. Sci. 241, 249–255 (2004).
[CrossRef]

2003 (3)

V. Lemaire, J. Bélair, and P. Hildgen, “Structural modeling of drug release from biodegradable porous matrices based on a combined diffusion/erosion process,” Int. J. Pharm. 258, 95–107 (2003).
[CrossRef] [PubMed]

K. Malek and M.-O. Coppens, “Knudsen self- and Fickian diffusion in rough nanoporous media,” J. Chem. Phys. 119, 2801–2811 (2003).
[CrossRef]

V. Agarwal and J. A. del Rio, “Tailoring the photonic band gap of a porous silicon dielectric mirror,” Appl. Phys. Lett. 82, 1512–1514 (2003).
[CrossRef]

2002 (2)

H. Föll, M. Christophersen, J. Carstensen, and G. Hasse, “Formation and application of porous silicon,” Mat. Sci. Eng. R 39, 93–141 (2002).
[CrossRef]

K. Malek and M.-O. Coppens, “Pore roughness effects on self- and transport diffusion in nanoporous materials,” Colloids Surf. A 206, 335–348 (2002).
[CrossRef]

2001 (1)

P. Allcock and P. A. Snow, “Time-resolved sensing of organic vapors in low modulating porous silicon dielectric mirrors,” J. Appl. Phys. 90, 5052–5057 (2001).
[CrossRef]

2000 (1)

M. P. Stewart and J. M. Buriak, “Chemical and biological applications of porous silicon technology,” Adv. Mater. 12, 859–869 (2000).
[CrossRef]

1999 (1)

S. C. Bayliss, R. Heald, D. I. Fletcher, and L. D. Buckberry, “The culture of mammalian cells on nanostructured silicon,” Adv. Mater. 11, 318–321 (1999).
[CrossRef]

1998 (2)

E. Llobet, X. Vilanova, J. Brezmes, J. E. Sueiras, and X. Correig, “Transient response of thick-film tin oxide gas-sensors to multicomponent gas mixtures,” Sens. Actuators B 47, 104–112 (1998).
[CrossRef]

C. Liu and W. P. Ball, “Analytical modeling of diffusion-limited contamination and decontamination in a two-layer porous medium,” Adv. Water Res. 21, 297–313 (1998).
[CrossRef]

1997 (2)

A. G. Cullis, L. T. Canham, and P. D. J. Calcott, “The structural and luminescence properties of porous silicon,” J. Appl. Phys. 82, 909–965 (1997).
[CrossRef]

V. S. Y. Lin, K. Motesharei, K.-P. S. Dancil, M. J. Sailor, and M. R. Ghadiri, “A porous silicon-based optical interferometric biosensor,” Science 278, 840–843 (1997).
[CrossRef] [PubMed]

1996 (1)

A. Loni, L. T. Canham, M. G. Berger, R. Arens-Fischer, H. Munder, H. Luth, H. F. Arrand, and T. M. Benson, “Porous silicon multilayer optical waveguides,” Thin Solid Films 276, 143–146 (1996).
[CrossRef]

1993 (1)

K. S. Sorbie and C. J. Tomlinson, “Analytical method for evaluating the effective molecular diffusion coefficient within porous media,” Chem. Eng. Sci. 48, 1813–1818 (1993).
[CrossRef]

1992 (1)

R. L. Smith and S. D. Collins, “Porous silicon formation mechanisms,” J. Appl. Phys. 71, R1–R22 (1992).
[CrossRef]

1991 (1)

A. Richter, P. Steiner, F. Kozlowski, and W. Lang, “Current-induced light emission from a porous silicon device,” IEEE Electron. Dev. Lett. 12, 691–692 (1991).
[CrossRef]

1990 (1)

1988 (2)

M. G. Davidson and W. M. Deen, “Hindered diffusion of water-soluble macromolecules in membranes,” Macromolecules 21, 3474–3481 (1988).
[CrossRef]

W. M. b.-M. Yunus and A. b.-A. Rahman, “Refractive index of solutions at high concentrations,” Appl. Opt. 27, 3341–3343(1988).
[CrossRef] [PubMed]

1984 (1)

M. T. van Genuchten and J. C. Parker, “Boundary conditions for displacement experiments through short laboratory soil columns,” Soil Sci. Soc. Am. J. 48, 703–708 (1984).
[CrossRef]

1934 (1)

C. Barnes, “Diffusion through a membrane,” Physics 5, 4–8 (1934).
[CrossRef]

Agarwal, V.

V. Agarwal and J. A. del Rio, “Tailoring the photonic band gap of a porous silicon dielectric mirror,” Appl. Phys. Lett. 82, 1512–1514 (2003).
[CrossRef]

Allcock, P.

P. Allcock and P. A. Snow, “Time-resolved sensing of organic vapors in low modulating porous silicon dielectric mirrors,” J. Appl. Phys. 90, 5052–5057 (2001).
[CrossRef]

Anglin, E. J.

E. J. Anglin, L. Cheng, W. R. Freeman, and M. J. Sailor, “Porous silicon in drug delivery devices and materials,” Adv. Drug Deliv. Rev. 60, 1266–1277 (2008).
[CrossRef] [PubMed]

Arens-Fischer, R.

A. Loni, L. T. Canham, M. G. Berger, R. Arens-Fischer, H. Munder, H. Luth, H. F. Arrand, and T. M. Benson, “Porous silicon multilayer optical waveguides,” Thin Solid Films 276, 143–146 (1996).
[CrossRef]

Arrand, H. F.

A. Loni, L. T. Canham, M. G. Berger, R. Arens-Fischer, H. Munder, H. Luth, H. F. Arrand, and T. M. Benson, “Porous silicon multilayer optical waveguides,” Thin Solid Films 276, 143–146 (1996).
[CrossRef]

Ball, W. P.

C. Liu and W. P. Ball, “Analytical modeling of diffusion-limited contamination and decontamination in a two-layer porous medium,” Adv. Water Res. 21, 297–313 (1998).
[CrossRef]

Barbour, L.

G. Liu, L. Barbour, and B. C. Si, “Unified multilayer diffusion model and application to diffusion experiment in porous media by method of chambers,” Environ. Sci. Technol. 43, 2412–2416 (2009).
[CrossRef] [PubMed]

Barnes, C.

C. Barnes, “Diffusion through a membrane,” Physics 5, 4–8 (1934).
[CrossRef]

Bayliss, S. C.

S. C. Bayliss, R. Heald, D. I. Fletcher, and L. D. Buckberry, “The culture of mammalian cells on nanostructured silicon,” Adv. Mater. 11, 318–321 (1999).
[CrossRef]

Bélair, J.

V. Lemaire, J. Bélair, and P. Hildgen, “Structural modeling of drug release from biodegradable porous matrices based on a combined diffusion/erosion process,” Int. J. Pharm. 258, 95–107 (2003).
[CrossRef] [PubMed]

Benson, T. M.

A. Loni, L. T. Canham, M. G. Berger, R. Arens-Fischer, H. Munder, H. Luth, H. F. Arrand, and T. M. Benson, “Porous silicon multilayer optical waveguides,” Thin Solid Films 276, 143–146 (1996).
[CrossRef]

Berger, M. G.

A. Loni, L. T. Canham, M. G. Berger, R. Arens-Fischer, H. Munder, H. Luth, H. F. Arrand, and T. M. Benson, “Porous silicon multilayer optical waveguides,” Thin Solid Films 276, 143–146 (1996).
[CrossRef]

Bhatia, S. N.

J.-H. Park, L. Gu, G. von Maltzahn, E. Ruoslahti, S. N. Bhatia, and M. J. Sailor, “Biodegradable luminescent porous silicon nanoparticles for in vivo applications,” Nat. Mater. 8, 331–336 (2009).
[CrossRef] [PubMed]

Björkqvist, M.

J. Salonen, L. Laitinen, A. M. Kaukonen, J. Tuura, M. Björkqvist, T. Heikkilä, K. Vähä-Heikkilä, J. Hirvonen, and V. P. Lehto, “Mesoporous silicon microparticles for oral drug delivery: loading and release of five model drugs,” J. Controlled Release 108, 362–374 (2005).
[CrossRef]

Brezmes, J.

E. Llobet, X. Vilanova, J. Brezmes, J. E. Sueiras, and X. Correig, “Transient response of thick-film tin oxide gas-sensors to multicomponent gas mixtures,” Sens. Actuators B 47, 104–112 (1998).
[CrossRef]

Buckberry, L. D.

S. C. Bayliss, R. Heald, D. I. Fletcher, and L. D. Buckberry, “The culture of mammalian cells on nanostructured silicon,” Adv. Mater. 11, 318–321 (1999).
[CrossRef]

Buriak, J. M.

M. P. Stewart and J. M. Buriak, “Chemical and biological applications of porous silicon technology,” Adv. Mater. 12, 859–869 (2000).
[CrossRef]

Calcott, P. D. J.

A. G. Cullis, L. T. Canham, and P. D. J. Calcott, “The structural and luminescence properties of porous silicon,” J. Appl. Phys. 82, 909–965 (1997).
[CrossRef]

Calderale, P. M.

A. Carbonaro, R. Walczak, P. M. Calderale, and M. Ferrari, “Nano-pore silicon membrane characterization by diffusion and electrical resistance,” J. Membr. Sci. 241, 249–255 (2004).
[CrossRef]

Canham, L.

L. Canham, “Porous silicon as a therapeutic biomaterial,” in Proceedings of the 1st Annual International Conference on Microtechnologies in Medicine and Biology (IEEE, 2000), pp. 109–112.

L. Canham, Properties of Porous Silicon (Institution of Engineering and Technology, 1997).

Canham, L. T.

A. G. Cullis, L. T. Canham, and P. D. J. Calcott, “The structural and luminescence properties of porous silicon,” J. Appl. Phys. 82, 909–965 (1997).
[CrossRef]

A. Loni, L. T. Canham, M. G. Berger, R. Arens-Fischer, H. Munder, H. Luth, H. F. Arrand, and T. M. Benson, “Porous silicon multilayer optical waveguides,” Thin Solid Films 276, 143–146 (1996).
[CrossRef]

Carbonaro, A.

A. Carbonaro, R. Walczak, P. M. Calderale, and M. Ferrari, “Nano-pore silicon membrane characterization by diffusion and electrical resistance,” J. Membr. Sci. 241, 249–255 (2004).
[CrossRef]

Carstensen, J.

H. Föll, M. Christophersen, J. Carstensen, and G. Hasse, “Formation and application of porous silicon,” Mat. Sci. Eng. R 39, 93–141 (2002).
[CrossRef]

Cheng, L.

E. J. Anglin, L. Cheng, W. R. Freeman, and M. J. Sailor, “Porous silicon in drug delivery devices and materials,” Adv. Drug Deliv. Rev. 60, 1266–1277 (2008).
[CrossRef] [PubMed]

Christophersen, M.

H. Föll, M. Christophersen, J. Carstensen, and G. Hasse, “Formation and application of porous silicon,” Mat. Sci. Eng. R 39, 93–141 (2002).
[CrossRef]

Cleall, P. J.

Y.-C. Li and P. J. Cleall, “Analytical solutions for contaminant diffusion in double-layered porous media,” J. Geotech. Geoenviron. Eng. 136, 1542–1554 (2010).
[CrossRef]

Collins, S. D.

R. L. Smith and S. D. Collins, “Porous silicon formation mechanisms,” J. Appl. Phys. 71, R1–R22 (1992).
[CrossRef]

Coppens, M.-O.

K. Malek and M.-O. Coppens, “Knudsen self- and Fickian diffusion in rough nanoporous media,” J. Chem. Phys. 119, 2801–2811 (2003).
[CrossRef]

K. Malek and M.-O. Coppens, “Pore roughness effects on self- and transport diffusion in nanoporous materials,” Colloids Surf. A 206, 335–348 (2002).
[CrossRef]

Correig, X.

E. Llobet, X. Vilanova, J. Brezmes, J. E. Sueiras, and X. Correig, “Transient response of thick-film tin oxide gas-sensors to multicomponent gas mixtures,” Sens. Actuators B 47, 104–112 (1998).
[CrossRef]

Crank, J.

J. Crank, The Mathematics of Diffusion, 2nd ed. (Oxford Univ. Press, 1975).

Cruz, S.

S. Cruz, A. Honig-d’Orville, and J. Muller, “Fabrication and optimization of porous silicon substrates for diffusion membrane applications,” J. Electrochem. Soc. 152, C418–C424 (2005).
[CrossRef]

Cullis, A. G.

A. G. Cullis, L. T. Canham, and P. D. J. Calcott, “The structural and luminescence properties of porous silicon,” J. Appl. Phys. 82, 909–965 (1997).
[CrossRef]

Dancil, K.-P. S.

V. S. Y. Lin, K. Motesharei, K.-P. S. Dancil, M. J. Sailor, and M. R. Ghadiri, “A porous silicon-based optical interferometric biosensor,” Science 278, 840–843 (1997).
[CrossRef] [PubMed]

Davidson, M. G.

M. G. Davidson and W. M. Deen, “Hindered diffusion of water-soluble macromolecules in membranes,” Macromolecules 21, 3474–3481 (1988).
[CrossRef]

Deen, W. M.

M. G. Davidson and W. M. Deen, “Hindered diffusion of water-soluble macromolecules in membranes,” Macromolecules 21, 3474–3481 (1988).
[CrossRef]

del Rio, J. A.

V. Agarwal and J. A. del Rio, “Tailoring the photonic band gap of a porous silicon dielectric mirror,” Appl. Phys. Lett. 82, 1512–1514 (2003).
[CrossRef]

Fauchet, P. M.

S. M. Weiss and P. M. Fauchet, “Porous silicon one-dimensional photonic crystals for optical signal modulation,” IEEE J. Quantum Electron. 12, 1514–1519 (2006).
[CrossRef]

Ferrari, M.

A. Carbonaro, R. Walczak, P. M. Calderale, and M. Ferrari, “Nano-pore silicon membrane characterization by diffusion and electrical resistance,” J. Membr. Sci. 241, 249–255 (2004).
[CrossRef]

Fletcher, D. I.

S. C. Bayliss, R. Heald, D. I. Fletcher, and L. D. Buckberry, “The culture of mammalian cells on nanostructured silicon,” Adv. Mater. 11, 318–321 (1999).
[CrossRef]

Föll, H.

H. Föll, M. Christophersen, J. Carstensen, and G. Hasse, “Formation and application of porous silicon,” Mat. Sci. Eng. R 39, 93–141 (2002).
[CrossRef]

Freeman, W. R.

E. J. Anglin, L. Cheng, W. R. Freeman, and M. J. Sailor, “Porous silicon in drug delivery devices and materials,” Adv. Drug Deliv. Rev. 60, 1266–1277 (2008).
[CrossRef] [PubMed]

Ghadiri, M. R.

V. S. Y. Lin, K. Motesharei, K.-P. S. Dancil, M. J. Sailor, and M. R. Ghadiri, “A porous silicon-based optical interferometric biosensor,” Science 278, 840–843 (1997).
[CrossRef] [PubMed]

Godin, D.

C. Pacholski, C. Yu, G. M. Miskelly, D. Godin, and M. J. Sailor, “Reflective interferometric Fourier transform spectroscopy: a self-compensating label-free immunosensor using double-layers of porous SiO2,” J. Am. Chem. Soc. 128, 4250–4252(2006).
[CrossRef] [PubMed]

Gu, L.

J.-H. Park, L. Gu, G. von Maltzahn, E. Ruoslahti, S. N. Bhatia, and M. J. Sailor, “Biodegradable luminescent porous silicon nanoparticles for in vivo applications,” Nat. Mater. 8, 331–336 (2009).
[CrossRef] [PubMed]

Hasse, G.

H. Föll, M. Christophersen, J. Carstensen, and G. Hasse, “Formation and application of porous silicon,” Mat. Sci. Eng. R 39, 93–141 (2002).
[CrossRef]

Heald, R.

S. C. Bayliss, R. Heald, D. I. Fletcher, and L. D. Buckberry, “The culture of mammalian cells on nanostructured silicon,” Adv. Mater. 11, 318–321 (1999).
[CrossRef]

Heikkilä, T.

J. Salonen, L. Laitinen, A. M. Kaukonen, J. Tuura, M. Björkqvist, T. Heikkilä, K. Vähä-Heikkilä, J. Hirvonen, and V. P. Lehto, “Mesoporous silicon microparticles for oral drug delivery: loading and release of five model drugs,” J. Controlled Release 108, 362–374 (2005).
[CrossRef]

Hildgen, P.

V. Lemaire, J. Bélair, and P. Hildgen, “Structural modeling of drug release from biodegradable porous matrices based on a combined diffusion/erosion process,” Int. J. Pharm. 258, 95–107 (2003).
[CrossRef] [PubMed]

Hirvonen, J.

J. Salonen, L. Laitinen, A. M. Kaukonen, J. Tuura, M. Björkqvist, T. Heikkilä, K. Vähä-Heikkilä, J. Hirvonen, and V. P. Lehto, “Mesoporous silicon microparticles for oral drug delivery: loading and release of five model drugs,” J. Controlled Release 108, 362–374 (2005).
[CrossRef]

Honig-d’Orville, A.

S. Cruz, A. Honig-d’Orville, and J. Muller, “Fabrication and optimization of porous silicon substrates for diffusion membrane applications,” J. Electrochem. Soc. 152, C418–C424 (2005).
[CrossRef]

Ishida, H.

Kärger, J.

R. Valiullin, P. Kortunov, J. Kärger, and V. Timoshenko, “Concentration-dependent self-diffusion of adsorbates in mesoporous materials,” Magn. Reson. Imag. 23, 209–214 (2005).
[CrossRef]

Kaukonen, A. M.

J. Salonen, L. Laitinen, A. M. Kaukonen, J. Tuura, M. Björkqvist, T. Heikkilä, K. Vähä-Heikkilä, J. Hirvonen, and V. P. Lehto, “Mesoporous silicon microparticles for oral drug delivery: loading and release of five model drugs,” J. Controlled Release 108, 362–374 (2005).
[CrossRef]

Kortunov, P.

R. Valiullin, P. Kortunov, J. Kärger, and V. Timoshenko, “Concentration-dependent self-diffusion of adsorbates in mesoporous materials,” Magn. Reson. Imag. 23, 209–214 (2005).
[CrossRef]

Kozlowski, F.

A. Richter, P. Steiner, F. Kozlowski, and W. Lang, “Current-induced light emission from a porous silicon device,” IEEE Electron. Dev. Lett. 12, 691–692 (1991).
[CrossRef]

Laitinen, L.

J. Salonen, L. Laitinen, A. M. Kaukonen, J. Tuura, M. Björkqvist, T. Heikkilä, K. Vähä-Heikkilä, J. Hirvonen, and V. P. Lehto, “Mesoporous silicon microparticles for oral drug delivery: loading and release of five model drugs,” J. Controlled Release 108, 362–374 (2005).
[CrossRef]

Lang, W.

A. Richter, P. Steiner, F. Kozlowski, and W. Lang, “Current-induced light emission from a porous silicon device,” IEEE Electron. Dev. Lett. 12, 691–692 (1991).
[CrossRef]

Lehto, V. P.

J. Salonen, L. Laitinen, A. M. Kaukonen, J. Tuura, M. Björkqvist, T. Heikkilä, K. Vähä-Heikkilä, J. Hirvonen, and V. P. Lehto, “Mesoporous silicon microparticles for oral drug delivery: loading and release of five model drugs,” J. Controlled Release 108, 362–374 (2005).
[CrossRef]

Lemaire, V.

V. Lemaire, J. Bélair, and P. Hildgen, “Structural modeling of drug release from biodegradable porous matrices based on a combined diffusion/erosion process,” Int. J. Pharm. 258, 95–107 (2003).
[CrossRef] [PubMed]

Li, Y.-C.

Y.-C. Li and P. J. Cleall, “Analytical solutions for contaminant diffusion in double-layered porous media,” J. Geotech. Geoenviron. Eng. 136, 1542–1554 (2010).
[CrossRef]

Lin, V. S. Y.

V. S. Y. Lin, K. Motesharei, K.-P. S. Dancil, M. J. Sailor, and M. R. Ghadiri, “A porous silicon-based optical interferometric biosensor,” Science 278, 840–843 (1997).
[CrossRef] [PubMed]

Liscidini, M.

J. D. Ryckman, M. Liscidini, J. E. Sipe, and S. M. Weiss, “Direct imprinting of porous substrates: a rapid and low-cost approach for patterning porous nanomaterials,” Nano Lett. 11, 1857–1862 (2010).
[CrossRef] [PubMed]

Liu, C.

C. Liu and W. P. Ball, “Analytical modeling of diffusion-limited contamination and decontamination in a two-layer porous medium,” Adv. Water Res. 21, 297–313 (1998).
[CrossRef]

Liu, G.

G. Liu, L. Barbour, and B. C. Si, “Unified multilayer diffusion model and application to diffusion experiment in porous media by method of chambers,” Environ. Sci. Technol. 43, 2412–2416 (2009).
[CrossRef] [PubMed]

Llobet, E.

E. Llobet, X. Vilanova, J. Brezmes, J. E. Sueiras, and X. Correig, “Transient response of thick-film tin oxide gas-sensors to multicomponent gas mixtures,” Sens. Actuators B 47, 104–112 (1998).
[CrossRef]

Loni, A.

A. Loni, L. T. Canham, M. G. Berger, R. Arens-Fischer, H. Munder, H. Luth, H. F. Arrand, and T. M. Benson, “Porous silicon multilayer optical waveguides,” Thin Solid Films 276, 143–146 (1996).
[CrossRef]

Luth, H.

A. Loni, L. T. Canham, M. G. Berger, R. Arens-Fischer, H. Munder, H. Luth, H. F. Arrand, and T. M. Benson, “Porous silicon multilayer optical waveguides,” Thin Solid Films 276, 143–146 (1996).
[CrossRef]

Malek, K.

K. Malek and M.-O. Coppens, “Knudsen self- and Fickian diffusion in rough nanoporous media,” J. Chem. Phys. 119, 2801–2811 (2003).
[CrossRef]

K. Malek and M.-O. Coppens, “Pore roughness effects on self- and transport diffusion in nanoporous materials,” Colloids Surf. A 206, 335–348 (2002).
[CrossRef]

Miskelly, G. M.

C. Pacholski, C. Yu, G. M. Miskelly, D. Godin, and M. J. Sailor, “Reflective interferometric Fourier transform spectroscopy: a self-compensating label-free immunosensor using double-layers of porous SiO2,” J. Am. Chem. Soc. 128, 4250–4252(2006).
[CrossRef] [PubMed]

Motesharei, K.

V. S. Y. Lin, K. Motesharei, K.-P. S. Dancil, M. J. Sailor, and M. R. Ghadiri, “A porous silicon-based optical interferometric biosensor,” Science 278, 840–843 (1997).
[CrossRef] [PubMed]

Muller, J.

S. Cruz, A. Honig-d’Orville, and J. Muller, “Fabrication and optimization of porous silicon substrates for diffusion membrane applications,” J. Electrochem. Soc. 152, C418–C424 (2005).
[CrossRef]

Munder, H.

A. Loni, L. T. Canham, M. G. Berger, R. Arens-Fischer, H. Munder, H. Luth, H. F. Arrand, and T. M. Benson, “Porous silicon multilayer optical waveguides,” Thin Solid Films 276, 143–146 (1996).
[CrossRef]

Najmaie, A.

G. Rong, A. Najmaie, J. E. Sipe, and S. M. Weiss, “Nanoscale porous silicon waveguide for label-free DNA sensing,” Biosens. Bioelectron. 23, 1572–1576 (2008).
[CrossRef] [PubMed]

Navid, A.

A. Navid and L. Pilon, “Effect of polarization and morphology on the optical properties of absorbing nanoporous thin films,” Thin Solid Films 516, 4159–4167 (2008).
[CrossRef]

Ohta, K.

Pacholski, C.

C. Pacholski, C. Yu, G. M. Miskelly, D. Godin, and M. J. Sailor, “Reflective interferometric Fourier transform spectroscopy: a self-compensating label-free immunosensor using double-layers of porous SiO2,” J. Am. Chem. Soc. 128, 4250–4252(2006).
[CrossRef] [PubMed]

Park, J.-H.

J.-H. Park, L. Gu, G. von Maltzahn, E. Ruoslahti, S. N. Bhatia, and M. J. Sailor, “Biodegradable luminescent porous silicon nanoparticles for in vivo applications,” Nat. Mater. 8, 331–336 (2009).
[CrossRef] [PubMed]

Parker, J. C.

M. T. van Genuchten and J. C. Parker, “Boundary conditions for displacement experiments through short laboratory soil columns,” Soil Sci. Soc. Am. J. 48, 703–708 (1984).
[CrossRef]

Pilon, L.

A. Navid and L. Pilon, “Effect of polarization and morphology on the optical properties of absorbing nanoporous thin films,” Thin Solid Films 516, 4159–4167 (2008).
[CrossRef]

Rahman, A. b.-A.

Richter, A.

A. Richter, P. Steiner, F. Kozlowski, and W. Lang, “Current-induced light emission from a porous silicon device,” IEEE Electron. Dev. Lett. 12, 691–692 (1991).
[CrossRef]

Rong, G.

G. Rong, A. Najmaie, J. E. Sipe, and S. M. Weiss, “Nanoscale porous silicon waveguide for label-free DNA sensing,” Biosens. Bioelectron. 23, 1572–1576 (2008).
[CrossRef] [PubMed]

Ruoslahti, E.

J.-H. Park, L. Gu, G. von Maltzahn, E. Ruoslahti, S. N. Bhatia, and M. J. Sailor, “Biodegradable luminescent porous silicon nanoparticles for in vivo applications,” Nat. Mater. 8, 331–336 (2009).
[CrossRef] [PubMed]

Ryckman, J. D.

J. D. Ryckman, M. Liscidini, J. E. Sipe, and S. M. Weiss, “Direct imprinting of porous substrates: a rapid and low-cost approach for patterning porous nanomaterials,” Nano Lett. 11, 1857–1862 (2010).
[CrossRef] [PubMed]

Sailor, M. J.

J.-H. Park, L. Gu, G. von Maltzahn, E. Ruoslahti, S. N. Bhatia, and M. J. Sailor, “Biodegradable luminescent porous silicon nanoparticles for in vivo applications,” Nat. Mater. 8, 331–336 (2009).
[CrossRef] [PubMed]

E. J. Anglin, L. Cheng, W. R. Freeman, and M. J. Sailor, “Porous silicon in drug delivery devices and materials,” Adv. Drug Deliv. Rev. 60, 1266–1277 (2008).
[CrossRef] [PubMed]

C. Pacholski, C. Yu, G. M. Miskelly, D. Godin, and M. J. Sailor, “Reflective interferometric Fourier transform spectroscopy: a self-compensating label-free immunosensor using double-layers of porous SiO2,” J. Am. Chem. Soc. 128, 4250–4252(2006).
[CrossRef] [PubMed]

V. S. Y. Lin, K. Motesharei, K.-P. S. Dancil, M. J. Sailor, and M. R. Ghadiri, “A porous silicon-based optical interferometric biosensor,” Science 278, 840–843 (1997).
[CrossRef] [PubMed]

Salonen, J.

J. Salonen, L. Laitinen, A. M. Kaukonen, J. Tuura, M. Björkqvist, T. Heikkilä, K. Vähä-Heikkilä, J. Hirvonen, and V. P. Lehto, “Mesoporous silicon microparticles for oral drug delivery: loading and release of five model drugs,” J. Controlled Release 108, 362–374 (2005).
[CrossRef]

Si, B. C.

G. Liu, L. Barbour, and B. C. Si, “Unified multilayer diffusion model and application to diffusion experiment in porous media by method of chambers,” Environ. Sci. Technol. 43, 2412–2416 (2009).
[CrossRef] [PubMed]

Sipe, J. E.

J. D. Ryckman, M. Liscidini, J. E. Sipe, and S. M. Weiss, “Direct imprinting of porous substrates: a rapid and low-cost approach for patterning porous nanomaterials,” Nano Lett. 11, 1857–1862 (2010).
[CrossRef] [PubMed]

G. Rong, A. Najmaie, J. E. Sipe, and S. M. Weiss, “Nanoscale porous silicon waveguide for label-free DNA sensing,” Biosens. Bioelectron. 23, 1572–1576 (2008).
[CrossRef] [PubMed]

Smith, R. L.

R. L. Smith and S. D. Collins, “Porous silicon formation mechanisms,” J. Appl. Phys. 71, R1–R22 (1992).
[CrossRef]

Snow, P. A.

P. Allcock and P. A. Snow, “Time-resolved sensing of organic vapors in low modulating porous silicon dielectric mirrors,” J. Appl. Phys. 90, 5052–5057 (2001).
[CrossRef]

Sorbie, K. S.

K. S. Sorbie and C. J. Tomlinson, “Analytical method for evaluating the effective molecular diffusion coefficient within porous media,” Chem. Eng. Sci. 48, 1813–1818 (1993).
[CrossRef]

Steiner, P.

A. Richter, P. Steiner, F. Kozlowski, and W. Lang, “Current-induced light emission from a porous silicon device,” IEEE Electron. Dev. Lett. 12, 691–692 (1991).
[CrossRef]

Stewart, M. P.

M. P. Stewart and J. M. Buriak, “Chemical and biological applications of porous silicon technology,” Adv. Mater. 12, 859–869 (2000).
[CrossRef]

Sueiras, J. E.

E. Llobet, X. Vilanova, J. Brezmes, J. E. Sueiras, and X. Correig, “Transient response of thick-film tin oxide gas-sensors to multicomponent gas mixtures,” Sens. Actuators B 47, 104–112 (1998).
[CrossRef]

Timoshenko, V.

R. Valiullin, P. Kortunov, J. Kärger, and V. Timoshenko, “Concentration-dependent self-diffusion of adsorbates in mesoporous materials,” Magn. Reson. Imag. 23, 209–214 (2005).
[CrossRef]

Tomlinson, C. J.

K. S. Sorbie and C. J. Tomlinson, “Analytical method for evaluating the effective molecular diffusion coefficient within porous media,” Chem. Eng. Sci. 48, 1813–1818 (1993).
[CrossRef]

Tuura, J.

J. Salonen, L. Laitinen, A. M. Kaukonen, J. Tuura, M. Björkqvist, T. Heikkilä, K. Vähä-Heikkilä, J. Hirvonen, and V. P. Lehto, “Mesoporous silicon microparticles for oral drug delivery: loading and release of five model drugs,” J. Controlled Release 108, 362–374 (2005).
[CrossRef]

Vähä-Heikkilä, K.

J. Salonen, L. Laitinen, A. M. Kaukonen, J. Tuura, M. Björkqvist, T. Heikkilä, K. Vähä-Heikkilä, J. Hirvonen, and V. P. Lehto, “Mesoporous silicon microparticles for oral drug delivery: loading and release of five model drugs,” J. Controlled Release 108, 362–374 (2005).
[CrossRef]

Valiullin, R.

R. Valiullin, P. Kortunov, J. Kärger, and V. Timoshenko, “Concentration-dependent self-diffusion of adsorbates in mesoporous materials,” Magn. Reson. Imag. 23, 209–214 (2005).
[CrossRef]

van Genuchten, M. T.

M. T. van Genuchten and J. C. Parker, “Boundary conditions for displacement experiments through short laboratory soil columns,” Soil Sci. Soc. Am. J. 48, 703–708 (1984).
[CrossRef]

Vilanova, X.

E. Llobet, X. Vilanova, J. Brezmes, J. E. Sueiras, and X. Correig, “Transient response of thick-film tin oxide gas-sensors to multicomponent gas mixtures,” Sens. Actuators B 47, 104–112 (1998).
[CrossRef]

von Maltzahn, G.

J.-H. Park, L. Gu, G. von Maltzahn, E. Ruoslahti, S. N. Bhatia, and M. J. Sailor, “Biodegradable luminescent porous silicon nanoparticles for in vivo applications,” Nat. Mater. 8, 331–336 (2009).
[CrossRef] [PubMed]

Walczak, R.

A. Carbonaro, R. Walczak, P. M. Calderale, and M. Ferrari, “Nano-pore silicon membrane characterization by diffusion and electrical resistance,” J. Membr. Sci. 241, 249–255 (2004).
[CrossRef]

Weiss, S. M.

J. D. Ryckman, M. Liscidini, J. E. Sipe, and S. M. Weiss, “Direct imprinting of porous substrates: a rapid and low-cost approach for patterning porous nanomaterials,” Nano Lett. 11, 1857–1862 (2010).
[CrossRef] [PubMed]

G. Rong, A. Najmaie, J. E. Sipe, and S. M. Weiss, “Nanoscale porous silicon waveguide for label-free DNA sensing,” Biosens. Bioelectron. 23, 1572–1576 (2008).
[CrossRef] [PubMed]

S. M. Weiss and P. M. Fauchet, “Porous silicon one-dimensional photonic crystals for optical signal modulation,” IEEE J. Quantum Electron. 12, 1514–1519 (2006).
[CrossRef]

Yu, C.

C. Pacholski, C. Yu, G. M. Miskelly, D. Godin, and M. J. Sailor, “Reflective interferometric Fourier transform spectroscopy: a self-compensating label-free immunosensor using double-layers of porous SiO2,” J. Am. Chem. Soc. 128, 4250–4252(2006).
[CrossRef] [PubMed]

Yunus, W. M. b.-M.

Adv. Drug Deliv. Rev. (1)

E. J. Anglin, L. Cheng, W. R. Freeman, and M. J. Sailor, “Porous silicon in drug delivery devices and materials,” Adv. Drug Deliv. Rev. 60, 1266–1277 (2008).
[CrossRef] [PubMed]

Adv. Mater. (2)

S. C. Bayliss, R. Heald, D. I. Fletcher, and L. D. Buckberry, “The culture of mammalian cells on nanostructured silicon,” Adv. Mater. 11, 318–321 (1999).
[CrossRef]

M. P. Stewart and J. M. Buriak, “Chemical and biological applications of porous silicon technology,” Adv. Mater. 12, 859–869 (2000).
[CrossRef]

Adv. Water Res. (1)

C. Liu and W. P. Ball, “Analytical modeling of diffusion-limited contamination and decontamination in a two-layer porous medium,” Adv. Water Res. 21, 297–313 (1998).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

V. Agarwal and J. A. del Rio, “Tailoring the photonic band gap of a porous silicon dielectric mirror,” Appl. Phys. Lett. 82, 1512–1514 (2003).
[CrossRef]

Biosens. Bioelectron. (1)

G. Rong, A. Najmaie, J. E. Sipe, and S. M. Weiss, “Nanoscale porous silicon waveguide for label-free DNA sensing,” Biosens. Bioelectron. 23, 1572–1576 (2008).
[CrossRef] [PubMed]

Chem. Eng. Sci. (1)

K. S. Sorbie and C. J. Tomlinson, “Analytical method for evaluating the effective molecular diffusion coefficient within porous media,” Chem. Eng. Sci. 48, 1813–1818 (1993).
[CrossRef]

Colloids Surf. A (1)

K. Malek and M.-O. Coppens, “Pore roughness effects on self- and transport diffusion in nanoporous materials,” Colloids Surf. A 206, 335–348 (2002).
[CrossRef]

Environ. Sci. Technol. (1)

G. Liu, L. Barbour, and B. C. Si, “Unified multilayer diffusion model and application to diffusion experiment in porous media by method of chambers,” Environ. Sci. Technol. 43, 2412–2416 (2009).
[CrossRef] [PubMed]

IEEE Electron. Dev. Lett. (1)

A. Richter, P. Steiner, F. Kozlowski, and W. Lang, “Current-induced light emission from a porous silicon device,” IEEE Electron. Dev. Lett. 12, 691–692 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. M. Weiss and P. M. Fauchet, “Porous silicon one-dimensional photonic crystals for optical signal modulation,” IEEE J. Quantum Electron. 12, 1514–1519 (2006).
[CrossRef]

Int. J. Pharm. (1)

V. Lemaire, J. Bélair, and P. Hildgen, “Structural modeling of drug release from biodegradable porous matrices based on a combined diffusion/erosion process,” Int. J. Pharm. 258, 95–107 (2003).
[CrossRef] [PubMed]

J. Am. Chem. Soc. (1)

C. Pacholski, C. Yu, G. M. Miskelly, D. Godin, and M. J. Sailor, “Reflective interferometric Fourier transform spectroscopy: a self-compensating label-free immunosensor using double-layers of porous SiO2,” J. Am. Chem. Soc. 128, 4250–4252(2006).
[CrossRef] [PubMed]

J. Appl. Phys. (3)

P. Allcock and P. A. Snow, “Time-resolved sensing of organic vapors in low modulating porous silicon dielectric mirrors,” J. Appl. Phys. 90, 5052–5057 (2001).
[CrossRef]

R. L. Smith and S. D. Collins, “Porous silicon formation mechanisms,” J. Appl. Phys. 71, R1–R22 (1992).
[CrossRef]

A. G. Cullis, L. T. Canham, and P. D. J. Calcott, “The structural and luminescence properties of porous silicon,” J. Appl. Phys. 82, 909–965 (1997).
[CrossRef]

J. Chem. Phys. (1)

K. Malek and M.-O. Coppens, “Knudsen self- and Fickian diffusion in rough nanoporous media,” J. Chem. Phys. 119, 2801–2811 (2003).
[CrossRef]

J. Controlled Release (1)

J. Salonen, L. Laitinen, A. M. Kaukonen, J. Tuura, M. Björkqvist, T. Heikkilä, K. Vähä-Heikkilä, J. Hirvonen, and V. P. Lehto, “Mesoporous silicon microparticles for oral drug delivery: loading and release of five model drugs,” J. Controlled Release 108, 362–374 (2005).
[CrossRef]

J. Electrochem. Soc. (1)

S. Cruz, A. Honig-d’Orville, and J. Muller, “Fabrication and optimization of porous silicon substrates for diffusion membrane applications,” J. Electrochem. Soc. 152, C418–C424 (2005).
[CrossRef]

J. Geotech. Geoenviron. Eng. (1)

Y.-C. Li and P. J. Cleall, “Analytical solutions for contaminant diffusion in double-layered porous media,” J. Geotech. Geoenviron. Eng. 136, 1542–1554 (2010).
[CrossRef]

J. Membr. Sci. (1)

A. Carbonaro, R. Walczak, P. M. Calderale, and M. Ferrari, “Nano-pore silicon membrane characterization by diffusion and electrical resistance,” J. Membr. Sci. 241, 249–255 (2004).
[CrossRef]

Macromolecules (1)

M. G. Davidson and W. M. Deen, “Hindered diffusion of water-soluble macromolecules in membranes,” Macromolecules 21, 3474–3481 (1988).
[CrossRef]

Magn. Reson. Imag. (1)

R. Valiullin, P. Kortunov, J. Kärger, and V. Timoshenko, “Concentration-dependent self-diffusion of adsorbates in mesoporous materials,” Magn. Reson. Imag. 23, 209–214 (2005).
[CrossRef]

Mat. Sci. Eng. R (1)

H. Föll, M. Christophersen, J. Carstensen, and G. Hasse, “Formation and application of porous silicon,” Mat. Sci. Eng. R 39, 93–141 (2002).
[CrossRef]

Nano Lett. (1)

J. D. Ryckman, M. Liscidini, J. E. Sipe, and S. M. Weiss, “Direct imprinting of porous substrates: a rapid and low-cost approach for patterning porous nanomaterials,” Nano Lett. 11, 1857–1862 (2010).
[CrossRef] [PubMed]

Nat. Mater. (1)

J.-H. Park, L. Gu, G. von Maltzahn, E. Ruoslahti, S. N. Bhatia, and M. J. Sailor, “Biodegradable luminescent porous silicon nanoparticles for in vivo applications,” Nat. Mater. 8, 331–336 (2009).
[CrossRef] [PubMed]

Physics (1)

C. Barnes, “Diffusion through a membrane,” Physics 5, 4–8 (1934).
[CrossRef]

Science (1)

V. S. Y. Lin, K. Motesharei, K.-P. S. Dancil, M. J. Sailor, and M. R. Ghadiri, “A porous silicon-based optical interferometric biosensor,” Science 278, 840–843 (1997).
[CrossRef] [PubMed]

Sens. Actuators B (1)

E. Llobet, X. Vilanova, J. Brezmes, J. E. Sueiras, and X. Correig, “Transient response of thick-film tin oxide gas-sensors to multicomponent gas mixtures,” Sens. Actuators B 47, 104–112 (1998).
[CrossRef]

Soil Sci. Soc. Am. J. (1)

M. T. van Genuchten and J. C. Parker, “Boundary conditions for displacement experiments through short laboratory soil columns,” Soil Sci. Soc. Am. J. 48, 703–708 (1984).
[CrossRef]

Thin Solid Films (2)

A. Navid and L. Pilon, “Effect of polarization and morphology on the optical properties of absorbing nanoporous thin films,” Thin Solid Films 516, 4159–4167 (2008).
[CrossRef]

A. Loni, L. T. Canham, M. G. Berger, R. Arens-Fischer, H. Munder, H. Luth, H. F. Arrand, and T. M. Benson, “Porous silicon multilayer optical waveguides,” Thin Solid Films 276, 143–146 (1996).
[CrossRef]

Other (4)

L. Canham, “Porous silicon as a therapeutic biomaterial,” in Proceedings of the 1st Annual International Conference on Microtechnologies in Medicine and Biology (IEEE, 2000), pp. 109–112.

L. Canham, Properties of Porous Silicon (Institution of Engineering and Technology, 1997).

We note that the use of the Bruggeman approximation rather than the parallel approximation presented in Eq.  resulted in calculated diffusion coefficients that were within the expected error range of the technique (approximately ±50 μm2/s), as compared to the results presented. Additionally, there is no qualitatively different trend evident from use of the Bruggeman approximation, implying that the error of the technique is not sufficiently low to resolve the diffusion coefficients yielded by the different approximations.

J. Crank, The Mathematics of Diffusion, 2nd ed. (Oxford Univ. Press, 1975).

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Figures (9)

Fig. 1
Fig. 1

a) Schematic of the diffusion chamber apparatus used for measurement of optical reflectance during film diffusion. b) Magnified side view of the pre-experiment arrangement of a porous film under interrogation. Note that dimensions and spacing are exaggerated in the image and that the gate, film, and tensioned backing are all in intimate contact prior to experiment execution.

Fig. 2
Fig. 2

Optical and instrumentation setup for diffusion measurements utilizing the diffusion-chamber apparatus.

Fig. 3
Fig. 3

Typical SEM images of PSi films used to characterize film thickness and average pore Feret diameter. a) Cross-sectional image of a PSi film shows well-defined nanometer-scale pores terminating at a uniform depth. b) Top-view image of the same film shows clear, unobstructed pore openings. c) The pore-size distribution, as derived from a larger field of view top-view image ( 0.5 μm × 0.75 μm ) indicates an average pore diameter of approximately 26 nm .

Fig. 4
Fig. 4

Summary of a) average pore diameter, b) porosity, and c) etch rate of PSi films utilized in this work. Note that the error bars in a) are the standard deviation of the pore-diameter distribution.

Fig. 5
Fig. 5

The change in reflectance during sucrose outdiffusion for three films of different thickness (as labeled) but equal mean pore diameter ( 16.3 nm ). The modeled best-fit curve to the data (smooth) is shown for each curve along with the fitted effective diffusion coefficients. Note that the curves are shifted with respect to each other by 0.25% for image clarity.

Fig. 6
Fig. 6

Effective diffusion coefficients of 24 PSi samples with varying nominal thicknesses and average pore diameters. The dotted line indicates the diffusion coefficient of sucrose in water (at room temperature), 540 μm 2 / s .

Fig. 7
Fig. 7

An overlay of a modeled and experimental reflectance curves for the case of outdiffusion of sucrose from an 11.5 μm PSi film. Mean pore diameter of the films is 26.0 nm (70% porosity), and the diffusion coefficient is taken to be that of sucrose in ambient water, 540 μm 2 / s .

Fig. 8
Fig. 8

Representative examples of reflectance measurements of the outdiffusion of SDS from PSi films of varying mean pore diameters but equal nominal thickness of 300 nm .

Fig. 9
Fig. 9

Reflectance oscillations observed in the outdiffusion of SDS from a PSi sample of small average pore diameter, 11.2 nm . The initial 400 s of the data are well predicted by a linear change in refractive index of the film, consistent with a linear degrada tion rate.

Equations (10)

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L j = [ exp ( i β ) 0 0 exp ( i β ) ] ,
H j = 1 τ j [ 1 r j r j 1 ] ,
n eff ( x , t ) = n Si ( 1 p ) + n p ( x , t ) p ,
n p ( x , t ) = n v + ( n f n v ) C ( x , t ) ,
β = ( 2 π λ 0 ) [ n eff , v L + ( n eff , f n eff , v ) 0 L C ( x , t ) d x ] ,
C ( x , t ) t = D 2 x 2 C ( x , t ) .
C ( x , 0 ) = C 0 , 0 x < L ,
C ( x , t ) = 0 , x > 0 ,
t C ( x , t 0 ) = 0 , x > 0 .
C ( x , t ) = 1 2 C 0 [ erf ( L x 2 D t ) + erf ( L + x 2 D t ) ] ,

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