Abstract

The light propagation and scattering in monolithic transparent nanoporous materials such as Vycor glasses exhibit two optical turbidities, both of which are slightly deviated from the λ4 Rayleigh wavelength dependence in the visible region: one is a transient white turbidity τf, characterized by the convex-upward dependence on the inverse fourth power of wavelength, and the other is turbidity τsp inherent to the structural inhomogeneity, characterized by the convex-downward dependence. The former is attributed to a fractal-like percolation network of imbibed or drained pores as a consequence of transient imbibition or drainage of wetting fluid into or from the pore space. The latter is attributed to the structural inhomogeneities inherent to the original dry porous glass, which are produced by spinodal decomposition. In this paper, we develop a general scheme to estimate the transmittance spectra of Vycor through the turbidities τf and τsp in the visible region on the basis of the theory of dielectric constant fluctuations. We show the applicability and its limitation of the turbidity analysis of the photospectroscopically measured data as a method to study the correlation functions that characterize the pore space and the structural features of isotropic transparent nanoporous media, on the presupposition that there exists no light attenuation other than the scattering.

© 2017 Optical Society of America

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Optically observed imbibition and drainage of wetting fluid in nanoporous Vycor glass

Shigeo Ogawa and Jiro Nakamura
J. Opt. Soc. Am. A 32(12) 2397-2406 (2015)

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2015 (2)

S. Ogawa and J. Nakamura, “Optically observed imbibition and drainage of wetting fluid in nanoporous Vycor glass,” J. Opt. Soc. Am. A 32, 2397–2406 (2015).
[Crossref]

S. Ogawa and J. Nakamura, “Photospectroscopically observed pore-space correlations of a wetting fluid during the drying process in nanoporous Vycor glass,” J. Opt. Soc. Am. A32, 533–537 (2015).
[Crossref]

2013 (3)

2009 (2)

A. V. Kityk, K. Knorr, and P. Huber, “Liquid n-hexane condensed in silica nanochannels: a combined optical birefringence and vapor sorption isotherm study,” Phys. Rev. B 80, 035421 (2009).
[Crossref]

A. Ch. Mitropoulos, “Small-angle X-ray scattering studies of adsorption in Vycor glass,” J. Colloid Interface Sci. 336, 679–690 (2009).
[Crossref]

2008 (2)

Y. Y. Maruo, J. Nakamura, M. Uchiyama, M. Higuchi, and K. Izumi, “Development of formaldehyde sensing element using porous glass impregnated with Schiff’s reagent,” Sens. Actuators B 129, 544–550 (2008).
[Crossref]

A. A. Evstrapov and N. A. Esikova, “Study of porous glasses by the methods of optical spectroscopy,” J. Opt. Technol. 75, 266–270 (2008).
[Crossref]

2007 (2)

F. H. El-Batal, M. S. Selim, S. Y. Marzouk, and M. A. Azooz, “UV-vis absorption of the transition metal-doped SiO2-B2O-Na2O glasses,” Physica B 398, 126–134 (2007).
[Crossref]

Y. Y. Maruo, “Measurement of ambient ozone using newly developed porous glass sensor,” Sens. Actuators B 126, 485–491 (2007).
[Crossref]

2006 (1)

M.-H. Kim and C. J. Glinka, “Ultra small angle neutron scattering study of the nanometer to micrometer structure of porous Vycor,” Microporous Mesoporous Mater. 91, 305–311 (2006).
[Crossref]

2005 (1)

A. A. Evstrapov, N. Esikova, and T. V. Antropova, “Spectral characteristics and structure of porous glasses,” Opt. Appl. 35, 753–759 (2005).

2003 (1)

A. A. Evstrapov, T. V. Antropova, I. A. Drozdova, and S. G. Yastrebov, “Optical properties and structure of porous glasses,” Opt. Appl. 33, 45–54 (2003).

2000 (3)

T. Ohyama, Y. Y. Maruo, T. Tanaka, and T. Hayashi, “A ppb-level NO2 detection system using coloration reactions in porous glass and its humidity dependence,” Sens. Actuators B 64, 142–146 (2000).
[Crossref]

E. S. Kikkinides, M. E. Kainougiakis, K. L. Stefanopoulos, A. Ch. Mitropoulos, A. K. Stubos, and N. K. Kanellopoulos, “Combination of small angle scattering and three-dimensional stochastic reconstruction for the study of adsorption–desorption processes in Vycor porous glass,” J. Chem. Phys. 112, 9881–9887 (2000).
[Crossref]

G. N. Greaves, Y. Vaills, S. Sen, and R. Winter, “Density fluctuations, phase separation and micro-segregation in silicate glasses,” J. Optoelectron. Adv. Mater. 2, 299–316 (2000).

1999 (1)

T. Tanaka, A. Guilleux, T. Ohyama, Y. Y. Maruo, and T. Hayashi, “A ppb-level NO2 gas sensor using coloration reactions in porous glass,” Sens. Actuators B 56, 247–253 (1999).
[Crossref]

1998 (3)

T. Tanaka, T. Ohyama, Y. Y. Maruo, and T. Hayashi, “Coloration reactions between NO2 and organic compounds in porous glass for cumulative gas sensor,” Sens. Actuators B 47, 65–69 (1998).
[Crossref]

V. G. Vereshchagin, R. A. Dynich, and A. N. Ponyavina, “Effective optical parameters of porous dielectric structures,” Opt. Spectrosc. 84, 427–431 (1998).

L. Skuja, “Optically active oxygen-deficiency-related centers in amorphous silicon dioxide,” J. Non-Cryst. Solids 239, 16–48 (1998).
[Crossref]

1997 (4)

A. Burneau, J. Lepage, and G. Maurice, “Porous silica-water interactions, I. structural and dimensional changes induced by water adsorption,” J. Non-Cryst. Solids 217, 1–10 (1997).
[Crossref]

A. Ch. Mitropoulos, P. K. Makri, N. K. Kanellopoulos, U. Keiderling, and A. Wiedenmann, “The surface geometry of Vycor,” J. Colloid Interface Sci. 193, 137–139 (1997).
[Crossref]

M. Agamalian, J. M. Drake, S. K. Sinha, and J. D. Axe, “Neutron diffraction study of the pore surface layer of Vycor glass,” Phys. Rev. E 55, 3021–3027 (1997).
[Crossref]

F. Katsaros, P. Makri, A. Mitropoulos, N. Kanellopoulos, U. Keiderling, and A. Wiedenmann, “On the morphology and surface geometry of Vycor,” Physica B 234–236, 402–404 (1997).
[Crossref]

1996 (1)

R. J. Gehr and R. W. Boyd, “Optical properties of nanostructured optical materials,” Chem. Mater. 8, 1807–1819 (1996).
[Crossref]

1995 (3)

J. H. Page, J. Liu, B. Abeles, E. Herbolzheimer, H. W. Deckman, and D. A. Weitz, “Adsorption and desorption of a wetting fluid in Vycor studied by acoustic and optical techniques,” Phys. Rev. E 52, 2763–2777 (1995).
[Crossref]

A. Ch. Mitropoulos, J. M. Haynes, R. M. Richardson, and N. K. Kanellopoulos, “Characterization of porous glass by adsorption of dibromo-methane in conjunction with small-angle x-ray scattering,” Phys. Rev. B 52, 10035–10042 (1995).
[Crossref]

A. F. Novikov and V. I. Zemskii, “Glassy spectral gas sensors based on the immobilized indicators,” Proc. SPIE 2550, 119–129 (1995).
[Crossref]

1994 (2)

J.-C. Li and D. K. Ross, “Dynamical scaling for spinodal decomposition—a small-angle scattering study of porous Vycor glass with fractal properties,” J. Phys. Condens. Matter 6, 351–362 (1994).
[Crossref]

J.-C. Li, D. K. Ross, L. D. Howe, K. L. Stefanopoulos, J. P. A. Fairclough, R. Heenan, and K. Ibel, “Small-angle neutron-scattering studies of the fractal-like network formed during desorption and adsorption of water in porous materials,” Phys. Rev. B 49, 5911–5917 (1994).
[Crossref]

1993 (1)

J. H. Page, J. Liu, B. Abeles, E. Herbolzheimer, H. W. Deckman, and D. A. Weitz, “Pore–space correlations in capillary condensation in Vycor,” Phys. Rev. Lett. 71, 1216–1219 (1993).
[Crossref]

1991 (4)

J.-C. Li, D. K. Ross, and M. J. Benham, “Small-angle neutron scattering studies of water and ice in porous Vycor glass,” J. Appl. Crystallogr. 24, 794–802 (1991).
[Crossref]

P. Levitz, G. Ehret, S. K. Sinha, and J. M. Drake, “Porous Vycor glass: the microstructure as probed by electron microscopy, direct energy transfer, small-angle scattering, and molecular adsorption,” J. Chem. Phys. 95, 6151–6161 (1991).
[Crossref]

S. A. Kuchinskii, V. I. Sukhanov, M. V. Khazova, and A. V. Dotsenko, “Effective optical constants of porous glass,” Opt. Spectrosc. 70, 85–88 (1991).

H. Hosono, Y. Abe, H. Imagawa, H. Imai, and K. Arai, “Experimental evidence for the Si-Si bond model of the 7.6-eV band in SiO2 glass,” Phys. Rev. B 44, 12043–12045 (1991).
[Crossref]

1989 (4)

S. K. Sinha, “Scattering from fractal structures,” Physica D 38, 310–314 (1989).
[Crossref]

R. Tohmon, H. Mizuno, Y. Ohki, K. Sasagane, K. Nagasawa, and Y. Hama, “Correlation of the 5.0- and 7.6-eV absorption bands in SiO2 with oxygen vacancy,” Phys. Rev. B 39, 1337–1345 (1989).
[Crossref]

M. J. Benham, J. C. Cook, J.-C. Li, D. K. Ross, P. L. Hall, and B. Sarkissian, “Small-angle neutron scattering study of adsorbed water in porous Vycor glass: supercooling phase transition and interfacial structure,” Phys. Rev. B 39, 633–636 (1989).
[Crossref]

E. Cheng, M. W. Cole, and P. Pfeifer, “Defractalization of films adsorbed on fractal surfaces,” Phys. Rev. B 39, 12962–12965 (1989).
[Crossref]

1988 (2)

A. Hoehr, H.-B. Neumann, P. W. Schmidt, P. Pfeifer, and D. Avnir, “Fractal structure and cluster structure of controlled-pore glasses and Vycor porous glass as revealed by small-angle x-ray and neutron scattering,” Phys. Rev. B 38, 1462–1467 (1988).
[Crossref]

H. Imai, K. Arai, H. Imagawa, H. Hosono, and Y. Abe, “Two types of oxygen-deficient centers in synthetic silica glass,” Phys. Rev. B 38, 12772–12775 (1988).
[Crossref]

1987 (2)

D. W. Schaefer, B. C. Bunker, and J. P. Wilcoxon, “Are leached porous glass fractal?” Phys. Rev. Lett. 58, 284 (1987).
[Crossref]

P. Wiltzius, F. S. Bates, S. B. Dierker, and G. D. Wignall, “Structure of porous Vycor glass,” Phys. Rev. A 36, 2991–2994 (1987).
[Crossref]

1986 (3)

D. F. R. Mildner and P. L. Hall, “Small-angle scattering from porous solids with fractal geometry,” J. Phys. D 19, 1535–1545 (1986).
[Crossref]

D. L. Wood and E. M. Rabinovich, “Infrared studies of alkoxide gels,” J. Non-Cryst. Solids 82, 171–176 (1986).
[Crossref]

T. Freltoft, J. K. Kjems, and S. K. Sinha, “Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering,” Phys. Rev. B 33, 269–275 (1986).
[Crossref]

1985 (1)

D. L. Griscom, “Defect structure of glasses,” J. Non-Cryst. Solids 73, 51–77 (1985).
[Crossref]

1983 (1)

I. T. Godmanis, A. N. Trukhin, and K. Huebner, “Exciton-phonon interaction in crystalline and vitreous SiO2,” Phys. Status Solidi B 116, 279–287 (1983).
[Crossref]

1979 (1)

1976 (1)

S. T. Pantelides and W. A. Harrison, “Electronic structure, spectra, and properties of 4:2-coordinated materials, I. crystalline and amorphous SiO2 and GeO2,” Phys. Rev. B 13, 2667–2691 (1976).
[Crossref]

1971 (2)

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T. H. DiStefano and D. E. Eastman, “The band edge of amorphous SiO2 by photoinjection and photoconductivity measurements,” Solid State Commun. 9, 2259–2261 (1971).
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1961 (1)

J. W. Cahn, “On spinodal decomposition,” Acta Metall. 9, 795–801 (1961).
[Crossref]

1957 (1)

P. Debye, H. R. Anderson, and H. Brumberger, “Scattering by an inhomogeneous solid, II, the correlation function and its application,” J. Appl. Phys. 28, 679–683 (1957).
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1949 (1)

P. Debye and A. M. Bueche, “Scattering by an inhomogeneous solids,” J. Appl. Phys. 20, 518–525 (1949).
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1944 (1)

M. E. Nordberg, “Properties of some Vycor-brand glasses,” J. Am. Ceram. Soc. 27, 299–305 (1944).
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H. Hosono, Y. Abe, H. Imagawa, H. Imai, and K. Arai, “Experimental evidence for the Si-Si bond model of the 7.6-eV band in SiO2 glass,” Phys. Rev. B 44, 12043–12045 (1991).
[Crossref]

H. Imai, K. Arai, H. Imagawa, H. Hosono, and Y. Abe, “Two types of oxygen-deficient centers in synthetic silica glass,” Phys. Rev. B 38, 12772–12775 (1988).
[Crossref]

Abeles, B.

J. H. Page, J. Liu, B. Abeles, E. Herbolzheimer, H. W. Deckman, and D. A. Weitz, “Adsorption and desorption of a wetting fluid in Vycor studied by acoustic and optical techniques,” Phys. Rev. E 52, 2763–2777 (1995).
[Crossref]

J. H. Page, J. Liu, B. Abeles, E. Herbolzheimer, H. W. Deckman, and D. A. Weitz, “Pore–space correlations in capillary condensation in Vycor,” Phys. Rev. Lett. 71, 1216–1219 (1993).
[Crossref]

Agamalian, M.

M. Agamalian, J. M. Drake, S. K. Sinha, and J. D. Axe, “Neutron diffraction study of the pore surface layer of Vycor glass,” Phys. Rev. E 55, 3021–3027 (1997).
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Akai, T.

T. Akai and D. Chen, “Method for producing high silicate glass and high silicate glass,” U.S. patent7,975,508B2 (July 12, 2011).

Anderson, H. R.

P. Debye, H. R. Anderson, and H. Brumberger, “Scattering by an inhomogeneous solid, II, the correlation function and its application,” J. Appl. Phys. 28, 679–683 (1957).
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Antropova, T. V.

A. A. Evstrapov, N. Esikova, and T. V. Antropova, “Spectral characteristics and structure of porous glasses,” Opt. Appl. 35, 753–759 (2005).

A. A. Evstrapov, T. V. Antropova, I. A. Drozdova, and S. G. Yastrebov, “Optical properties and structure of porous glasses,” Opt. Appl. 33, 45–54 (2003).

Arai, K.

H. Hosono, Y. Abe, H. Imagawa, H. Imai, and K. Arai, “Experimental evidence for the Si-Si bond model of the 7.6-eV band in SiO2 glass,” Phys. Rev. B 44, 12043–12045 (1991).
[Crossref]

H. Imai, K. Arai, H. Imagawa, H. Hosono, and Y. Abe, “Two types of oxygen-deficient centers in synthetic silica glass,” Phys. Rev. B 38, 12772–12775 (1988).
[Crossref]

Avnir, D.

A. Hoehr, H.-B. Neumann, P. W. Schmidt, P. Pfeifer, and D. Avnir, “Fractal structure and cluster structure of controlled-pore glasses and Vycor porous glass as revealed by small-angle x-ray and neutron scattering,” Phys. Rev. B 38, 1462–1467 (1988).
[Crossref]

Axe, J. D.

M. Agamalian, J. M. Drake, S. K. Sinha, and J. D. Axe, “Neutron diffraction study of the pore surface layer of Vycor glass,” Phys. Rev. E 55, 3021–3027 (1997).
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Azooz, M. A.

F. H. El-Batal, M. S. Selim, S. Y. Marzouk, and M. A. Azooz, “UV-vis absorption of the transition metal-doped SiO2-B2O-Na2O glasses,” Physica B 398, 126–134 (2007).
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P. Wiltzius, F. S. Bates, S. B. Dierker, and G. D. Wignall, “Structure of porous Vycor glass,” Phys. Rev. A 36, 2991–2994 (1987).
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Benham, M. J.

J.-C. Li, D. K. Ross, and M. J. Benham, “Small-angle neutron scattering studies of water and ice in porous Vycor glass,” J. Appl. Crystallogr. 24, 794–802 (1991).
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M. J. Benham, J. C. Cook, J.-C. Li, D. K. Ross, P. L. Hall, and B. Sarkissian, “Small-angle neutron scattering study of adsorbed water in porous Vycor glass: supercooling phase transition and interfacial structure,” Phys. Rev. B 39, 633–636 (1989).
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Boyd, R. W.

R. J. Gehr and R. W. Boyd, “Optical properties of nanostructured optical materials,” Chem. Mater. 8, 1807–1819 (1996).
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Brumberger, H.

P. Debye, H. R. Anderson, and H. Brumberger, “Scattering by an inhomogeneous solid, II, the correlation function and its application,” J. Appl. Phys. 28, 679–683 (1957).
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Bueche, A. M.

P. Debye and A. M. Bueche, “Scattering by an inhomogeneous solids,” J. Appl. Phys. 20, 518–525 (1949).
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D. W. Schaefer, B. C. Bunker, and J. P. Wilcoxon, “Are leached porous glass fractal?” Phys. Rev. Lett. 58, 284 (1987).
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A. Burneau, J. Lepage, and G. Maurice, “Porous silica-water interactions, I. structural and dimensional changes induced by water adsorption,” J. Non-Cryst. Solids 217, 1–10 (1997).
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A. Burneau and J.-P. Gallas, The Surface Properties of Silicas, A. P. Legrand, ed. (Wiley, 1998), pp. 147–234.

Busch, M.

P. Huber, M. Busch, S. Calus, and A. V. Kityk, “Thermotropic nematic order under nanocapillary filling,” Phys. Rev. E 87, 042502 (2013).
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Cahn, J. W.

J. W. Cahn, “On spinodal decomposition,” Acta Metall. 9, 795–801 (1961).
[Crossref]

Calus, S.

P. Huber, M. Busch, S. Calus, and A. V. Kityk, “Thermotropic nematic order under nanocapillary filling,” Phys. Rev. E 87, 042502 (2013).
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Chen, D.

T. Akai and D. Chen, “Method for producing high silicate glass and high silicate glass,” U.S. patent7,975,508B2 (July 12, 2011).

Cheng, E.

E. Cheng, M. W. Cole, and P. Pfeifer, “Defractalization of films adsorbed on fractal surfaces,” Phys. Rev. B 39, 12962–12965 (1989).
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B. Chu, Laser Light Scattering, 2nd ed. (Academic, 1991), pp. 21–28.

Cole, M. W.

E. Cheng, M. W. Cole, and P. Pfeifer, “Defractalization of films adsorbed on fractal surfaces,” Phys. Rev. B 39, 12962–12965 (1989).
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Cook, J. C.

M. J. Benham, J. C. Cook, J.-C. Li, D. K. Ross, P. L. Hall, and B. Sarkissian, “Small-angle neutron scattering study of adsorbed water in porous Vycor glass: supercooling phase transition and interfacial structure,” Phys. Rev. B 39, 633–636 (1989).
[Crossref]

Debye, P.

P. Debye, H. R. Anderson, and H. Brumberger, “Scattering by an inhomogeneous solid, II, the correlation function and its application,” J. Appl. Phys. 28, 679–683 (1957).
[Crossref]

P. Debye and A. M. Bueche, “Scattering by an inhomogeneous solids,” J. Appl. Phys. 20, 518–525 (1949).
[Crossref]

Deckman, H. W.

J. H. Page, J. Liu, B. Abeles, E. Herbolzheimer, H. W. Deckman, and D. A. Weitz, “Adsorption and desorption of a wetting fluid in Vycor studied by acoustic and optical techniques,” Phys. Rev. E 52, 2763–2777 (1995).
[Crossref]

J. H. Page, J. Liu, B. Abeles, E. Herbolzheimer, H. W. Deckman, and D. A. Weitz, “Pore–space correlations in capillary condensation in Vycor,” Phys. Rev. Lett. 71, 1216–1219 (1993).
[Crossref]

Dierker, S. B.

P. Wiltzius, F. S. Bates, S. B. Dierker, and G. D. Wignall, “Structure of porous Vycor glass,” Phys. Rev. A 36, 2991–2994 (1987).
[Crossref]

DiStefano, T. H.

T. H. DiStefano and D. E. Eastman, “The band edge of amorphous SiO2 by photoinjection and photoconductivity measurements,” Solid State Commun. 9, 2259–2261 (1971).
[Crossref]

Dotsenko, A. V.

S. A. Kuchinskii, V. I. Sukhanov, M. V. Khazova, and A. V. Dotsenko, “Effective optical constants of porous glass,” Opt. Spectrosc. 70, 85–88 (1991).

Drake, J. M.

M. Agamalian, J. M. Drake, S. K. Sinha, and J. D. Axe, “Neutron diffraction study of the pore surface layer of Vycor glass,” Phys. Rev. E 55, 3021–3027 (1997).
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P. Levitz, G. Ehret, S. K. Sinha, and J. M. Drake, “Porous Vycor glass: the microstructure as probed by electron microscopy, direct energy transfer, small-angle scattering, and molecular adsorption,” J. Chem. Phys. 95, 6151–6161 (1991).
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Drozdova, I. A.

A. A. Evstrapov, T. V. Antropova, I. A. Drozdova, and S. G. Yastrebov, “Optical properties and structure of porous glasses,” Opt. Appl. 33, 45–54 (2003).

Dynich, R. A.

V. G. Vereshchagin, R. A. Dynich, and A. N. Ponyavina, “Effective optical parameters of porous dielectric structures,” Opt. Spectrosc. 84, 427–431 (1998).

Eastman, D. E.

T. H. DiStefano and D. E. Eastman, “The band edge of amorphous SiO2 by photoinjection and photoconductivity measurements,” Solid State Commun. 9, 2259–2261 (1971).
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Ehret, G.

P. Levitz, G. Ehret, S. K. Sinha, and J. M. Drake, “Porous Vycor glass: the microstructure as probed by electron microscopy, direct energy transfer, small-angle scattering, and molecular adsorption,” J. Chem. Phys. 95, 6151–6161 (1991).
[Crossref]

El-Batal, F. H.

F. H. El-Batal, M. S. Selim, S. Y. Marzouk, and M. A. Azooz, “UV-vis absorption of the transition metal-doped SiO2-B2O-Na2O glasses,” Physica B 398, 126–134 (2007).
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Elmer, T. H.

T. H. Elmer, “Porous and reconstructed glasses,” in Engineered Materials Handbook, Vol. 4 of Ceramics and Glasses (ASM International, 1992), pp. 427–431.

Esikova, N.

A. A. Evstrapov, N. Esikova, and T. V. Antropova, “Spectral characteristics and structure of porous glasses,” Opt. Appl. 35, 753–759 (2005).

Esikova, N. A.

Evstrapov, A. A.

A. A. Evstrapov and N. A. Esikova, “Study of porous glasses by the methods of optical spectroscopy,” J. Opt. Technol. 75, 266–270 (2008).
[Crossref]

A. A. Evstrapov, N. Esikova, and T. V. Antropova, “Spectral characteristics and structure of porous glasses,” Opt. Appl. 35, 753–759 (2005).

A. A. Evstrapov, T. V. Antropova, I. A. Drozdova, and S. G. Yastrebov, “Optical properties and structure of porous glasses,” Opt. Appl. 33, 45–54 (2003).

Fairclough, J. P. A.

J.-C. Li, D. K. Ross, L. D. Howe, K. L. Stefanopoulos, J. P. A. Fairclough, R. Heenan, and K. Ibel, “Small-angle neutron-scattering studies of the fractal-like network formed during desorption and adsorption of water in porous materials,” Phys. Rev. B 49, 5911–5917 (1994).
[Crossref]

Freltoft, T.

T. Freltoft, J. K. Kjems, and S. K. Sinha, “Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering,” Phys. Rev. B 33, 269–275 (1986).
[Crossref]

S. K. Sinha, T. Freltoft, and J. Kjems, “Observation of power-law correlations in silica-particle aggregates by small-angle neutron scattering,” in International Conference on Kinetics of Aggregation and Gelation, Athens, 1984, pp. 87–90.

Gallas, J.-P.

A. Burneau and J.-P. Gallas, The Surface Properties of Silicas, A. P. Legrand, ed. (Wiley, 1998), pp. 147–234.

Gehr, R. J.

R. J. Gehr and R. W. Boyd, “Optical properties of nanostructured optical materials,” Chem. Mater. 8, 1807–1819 (1996).
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Glinka, C. J.

M.-H. Kim and C. J. Glinka, “Ultra small angle neutron scattering study of the nanometer to micrometer structure of porous Vycor,” Microporous Mesoporous Mater. 91, 305–311 (2006).
[Crossref]

Godmanis, I. T.

I. T. Godmanis, A. N. Trukhin, and K. Huebner, “Exciton-phonon interaction in crystalline and vitreous SiO2,” Phys. Status Solidi B 116, 279–287 (1983).
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Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, A. Jeffrey, ed. (Academic, 1980).

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G. N. Greaves, Y. Vaills, S. Sen, and R. Winter, “Density fluctuations, phase separation and micro-segregation in silicate glasses,” J. Optoelectron. Adv. Mater. 2, 299–316 (2000).

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D. L. Griscom, “Defect structure of glasses,” J. Non-Cryst. Solids 73, 51–77 (1985).
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T. Tanaka, A. Guilleux, T. Ohyama, Y. Y. Maruo, and T. Hayashi, “A ppb-level NO2 gas sensor using coloration reactions in porous glass,” Sens. Actuators B 56, 247–253 (1999).
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Hall, P. L.

M. J. Benham, J. C. Cook, J.-C. Li, D. K. Ross, P. L. Hall, and B. Sarkissian, “Small-angle neutron scattering study of adsorbed water in porous Vycor glass: supercooling phase transition and interfacial structure,” Phys. Rev. B 39, 633–636 (1989).
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D. F. R. Mildner and P. L. Hall, “Small-angle scattering from porous solids with fractal geometry,” J. Phys. D 19, 1535–1545 (1986).
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Hama, Y.

R. Tohmon, H. Mizuno, Y. Ohki, K. Sasagane, K. Nagasawa, and Y. Hama, “Correlation of the 5.0- and 7.6-eV absorption bands in SiO2 with oxygen vacancy,” Phys. Rev. B 39, 1337–1345 (1989).
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S. T. Pantelides and W. A. Harrison, “Electronic structure, spectra, and properties of 4:2-coordinated materials, I. crystalline and amorphous SiO2 and GeO2,” Phys. Rev. B 13, 2667–2691 (1976).
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Hayashi, T.

T. Ohyama, Y. Y. Maruo, T. Tanaka, and T. Hayashi, “A ppb-level NO2 detection system using coloration reactions in porous glass and its humidity dependence,” Sens. Actuators B 64, 142–146 (2000).
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T. Tanaka, A. Guilleux, T. Ohyama, Y. Y. Maruo, and T. Hayashi, “A ppb-level NO2 gas sensor using coloration reactions in porous glass,” Sens. Actuators B 56, 247–253 (1999).
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T. Tanaka, T. Ohyama, Y. Y. Maruo, and T. Hayashi, “Coloration reactions between NO2 and organic compounds in porous glass for cumulative gas sensor,” Sens. Actuators B 47, 65–69 (1998).
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Haynes, J. M.

A. Ch. Mitropoulos, J. M. Haynes, R. M. Richardson, and N. K. Kanellopoulos, “Characterization of porous glass by adsorption of dibromo-methane in conjunction with small-angle x-ray scattering,” Phys. Rev. B 52, 10035–10042 (1995).
[Crossref]

Heenan, R.

J.-C. Li, D. K. Ross, L. D. Howe, K. L. Stefanopoulos, J. P. A. Fairclough, R. Heenan, and K. Ibel, “Small-angle neutron-scattering studies of the fractal-like network formed during desorption and adsorption of water in porous materials,” Phys. Rev. B 49, 5911–5917 (1994).
[Crossref]

Herbolzheimer, E.

J. H. Page, J. Liu, B. Abeles, E. Herbolzheimer, H. W. Deckman, and D. A. Weitz, “Adsorption and desorption of a wetting fluid in Vycor studied by acoustic and optical techniques,” Phys. Rev. E 52, 2763–2777 (1995).
[Crossref]

J. H. Page, J. Liu, B. Abeles, E. Herbolzheimer, H. W. Deckman, and D. A. Weitz, “Pore–space correlations in capillary condensation in Vycor,” Phys. Rev. Lett. 71, 1216–1219 (1993).
[Crossref]

Higuchi, M.

Y. Y. Maruo, J. Nakamura, M. Uchiyama, M. Higuchi, and K. Izumi, “Development of formaldehyde sensing element using porous glass impregnated with Schiff’s reagent,” Sens. Actuators B 129, 544–550 (2008).
[Crossref]

Hoehr, A.

A. Hoehr, H.-B. Neumann, P. W. Schmidt, P. Pfeifer, and D. Avnir, “Fractal structure and cluster structure of controlled-pore glasses and Vycor porous glass as revealed by small-angle x-ray and neutron scattering,” Phys. Rev. B 38, 1462–1467 (1988).
[Crossref]

Hosono, H.

H. Hosono, Y. Abe, H. Imagawa, H. Imai, and K. Arai, “Experimental evidence for the Si-Si bond model of the 7.6-eV band in SiO2 glass,” Phys. Rev. B 44, 12043–12045 (1991).
[Crossref]

H. Imai, K. Arai, H. Imagawa, H. Hosono, and Y. Abe, “Two types of oxygen-deficient centers in synthetic silica glass,” Phys. Rev. B 38, 12772–12775 (1988).
[Crossref]

Howe, L. D.

J.-C. Li, D. K. Ross, L. D. Howe, K. L. Stefanopoulos, J. P. A. Fairclough, R. Heenan, and K. Ibel, “Small-angle neutron-scattering studies of the fractal-like network formed during desorption and adsorption of water in porous materials,” Phys. Rev. B 49, 5911–5917 (1994).
[Crossref]

Huber, P.

P. Huber, M. Busch, S. Calus, and A. V. Kityk, “Thermotropic nematic order under nanocapillary filling,” Phys. Rev. E 87, 042502 (2013).
[Crossref]

A. V. Kityk, K. Knorr, and P. Huber, “Liquid n-hexane condensed in silica nanochannels: a combined optical birefringence and vapor sorption isotherm study,” Phys. Rev. B 80, 035421 (2009).
[Crossref]

Huebner, K.

I. T. Godmanis, A. N. Trukhin, and K. Huebner, “Exciton-phonon interaction in crystalline and vitreous SiO2,” Phys. Status Solidi B 116, 279–287 (1983).
[Crossref]

Ibel, K.

J.-C. Li, D. K. Ross, L. D. Howe, K. L. Stefanopoulos, J. P. A. Fairclough, R. Heenan, and K. Ibel, “Small-angle neutron-scattering studies of the fractal-like network formed during desorption and adsorption of water in porous materials,” Phys. Rev. B 49, 5911–5917 (1994).
[Crossref]

Imagawa, H.

H. Hosono, Y. Abe, H. Imagawa, H. Imai, and K. Arai, “Experimental evidence for the Si-Si bond model of the 7.6-eV band in SiO2 glass,” Phys. Rev. B 44, 12043–12045 (1991).
[Crossref]

H. Imai, K. Arai, H. Imagawa, H. Hosono, and Y. Abe, “Two types of oxygen-deficient centers in synthetic silica glass,” Phys. Rev. B 38, 12772–12775 (1988).
[Crossref]

Imai, H.

H. Hosono, Y. Abe, H. Imagawa, H. Imai, and K. Arai, “Experimental evidence for the Si-Si bond model of the 7.6-eV band in SiO2 glass,” Phys. Rev. B 44, 12043–12045 (1991).
[Crossref]

H. Imai, K. Arai, H. Imagawa, H. Hosono, and Y. Abe, “Two types of oxygen-deficient centers in synthetic silica glass,” Phys. Rev. B 38, 12772–12775 (1988).
[Crossref]

Izumi, K.

Y. Y. Maruo, J. Nakamura, M. Uchiyama, M. Higuchi, and K. Izumi, “Development of formaldehyde sensing element using porous glass impregnated with Schiff’s reagent,” Sens. Actuators B 129, 544–550 (2008).
[Crossref]

K. Izumi, M. Utiyama, and Y. Y. Maruo, “Evaluation of air quality with simple and easy chemical sensors: development of porous glass-based elements,” in International Conference on Control, Automation and Systems (IEEE, 2008), pp. 2756–2760.

Kainougiakis, M. E.

E. S. Kikkinides, M. E. Kainougiakis, K. L. Stefanopoulos, A. Ch. Mitropoulos, A. K. Stubos, and N. K. Kanellopoulos, “Combination of small angle scattering and three-dimensional stochastic reconstruction for the study of adsorption–desorption processes in Vycor porous glass,” J. Chem. Phys. 112, 9881–9887 (2000).
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Kanellopoulos, N.

F. Katsaros, P. Makri, A. Mitropoulos, N. Kanellopoulos, U. Keiderling, and A. Wiedenmann, “On the morphology and surface geometry of Vycor,” Physica B 234–236, 402–404 (1997).
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Kanellopoulos, N. K.

E. S. Kikkinides, M. E. Kainougiakis, K. L. Stefanopoulos, A. Ch. Mitropoulos, A. K. Stubos, and N. K. Kanellopoulos, “Combination of small angle scattering and three-dimensional stochastic reconstruction for the study of adsorption–desorption processes in Vycor porous glass,” J. Chem. Phys. 112, 9881–9887 (2000).
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A. Ch. Mitropoulos, P. K. Makri, N. K. Kanellopoulos, U. Keiderling, and A. Wiedenmann, “The surface geometry of Vycor,” J. Colloid Interface Sci. 193, 137–139 (1997).
[Crossref]

A. Ch. Mitropoulos, J. M. Haynes, R. M. Richardson, and N. K. Kanellopoulos, “Characterization of porous glass by adsorption of dibromo-methane in conjunction with small-angle x-ray scattering,” Phys. Rev. B 52, 10035–10042 (1995).
[Crossref]

Katsaros, F.

F. Katsaros, P. Makri, A. Mitropoulos, N. Kanellopoulos, U. Keiderling, and A. Wiedenmann, “On the morphology and surface geometry of Vycor,” Physica B 234–236, 402–404 (1997).
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Keiderling, U.

A. Ch. Mitropoulos, P. K. Makri, N. K. Kanellopoulos, U. Keiderling, and A. Wiedenmann, “The surface geometry of Vycor,” J. Colloid Interface Sci. 193, 137–139 (1997).
[Crossref]

F. Katsaros, P. Makri, A. Mitropoulos, N. Kanellopoulos, U. Keiderling, and A. Wiedenmann, “On the morphology and surface geometry of Vycor,” Physica B 234–236, 402–404 (1997).
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M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), p. 470.

Khazova, M. V.

S. A. Kuchinskii, V. I. Sukhanov, M. V. Khazova, and A. V. Dotsenko, “Effective optical constants of porous glass,” Opt. Spectrosc. 70, 85–88 (1991).

Kikkinides, E. S.

E. S. Kikkinides, M. E. Kainougiakis, K. L. Stefanopoulos, A. Ch. Mitropoulos, A. K. Stubos, and N. K. Kanellopoulos, “Combination of small angle scattering and three-dimensional stochastic reconstruction for the study of adsorption–desorption processes in Vycor porous glass,” J. Chem. Phys. 112, 9881–9887 (2000).
[Crossref]

Kim, M.-H.

M.-H. Kim and C. J. Glinka, “Ultra small angle neutron scattering study of the nanometer to micrometer structure of porous Vycor,” Microporous Mesoporous Mater. 91, 305–311 (2006).
[Crossref]

Kityk, A. V.

P. Huber, M. Busch, S. Calus, and A. V. Kityk, “Thermotropic nematic order under nanocapillary filling,” Phys. Rev. E 87, 042502 (2013).
[Crossref]

A. V. Kityk, K. Knorr, and P. Huber, “Liquid n-hexane condensed in silica nanochannels: a combined optical birefringence and vapor sorption isotherm study,” Phys. Rev. B 80, 035421 (2009).
[Crossref]

Kjems, J.

S. K. Sinha, T. Freltoft, and J. Kjems, “Observation of power-law correlations in silica-particle aggregates by small-angle neutron scattering,” in International Conference on Kinetics of Aggregation and Gelation, Athens, 1984, pp. 87–90.

Kjems, J. K.

T. Freltoft, J. K. Kjems, and S. K. Sinha, “Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering,” Phys. Rev. B 33, 269–275 (1986).
[Crossref]

Knorr, K.

A. V. Kityk, K. Knorr, and P. Huber, “Liquid n-hexane condensed in silica nanochannels: a combined optical birefringence and vapor sorption isotherm study,” Phys. Rev. B 80, 035421 (2009).
[Crossref]

Kuchinskii, S. A.

S. A. Kuchinskii, V. I. Sukhanov, M. V. Khazova, and A. V. Dotsenko, “Effective optical constants of porous glass,” Opt. Spectrosc. 70, 85–88 (1991).

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Lepage, J.

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Figures (6)

Fig. 1.
Fig. 1. (a) Dependence of anisotropic correlation volume ω f ( q ) on scattering wave vector q , illustrating the fractal correlations of half and integral numbers of fractal dimension D . The curve with D = 5 / 2 well approximates the slope of experimental data depicted by open circles, which represent scattered light intensity as a function of q with a monochromatic incident light from a laser source (514.2 nm) [37,38]. This comparison is justified by considering the fact that ω f ( q ) is proportional to the scattered intensity. For this comparison with data in [37,38], we have to intentionally set the correlation length ( ξ ) equal to 10 μm, which is about 10 3 times longer than our estimated value [40]. (b) Dependence of integrated dissymmetry factors ( Ω D ) for various values of D , where Ω D is proportional to the fractal turbidity τ f on 1 / λ 0 4 , indicating both τ f and Ω D are convex-upward functions of 1 / λ 0 4 .
Fig. 2.
Fig. 2. (a) Dependence of integrated spinodal correlation volume Ω s p on 1 / λ 0 4 , indicating Ω s p is a convex-downward function of 1 / λ 0 4 , with the parameter β = 0.251    nm 1 [2729] and the correlation length ζ = 19.95    nm , which was determined to fit best our data described in the text. (b) Dependence of corresponding turbidity τ s p on 1 / λ 0 4 , indicating τ s p is also a convex-downward function of 1 / λ 0 4 .
Fig. 3.
Fig. 3. (a) Dependence of spinodal correlation function γ s p on the radius r in nanometers. (b) Dependence of the corresponding correlation volume ω s p ( q ) on the scattering vector q . Both curves are estimated from the respective models, namely, Eq. (11) for (a) and Eq. (12) for (b), with parameters β = 0.251    nm 1 and ζ = 19.95    nm , both of which were determined to fit best the measured τ s p in the range of 350 to 850 nm.
Fig. 4.
Fig. 4. (a) Changes in the turbidity (estimated from the logarithm of the observed transmittance) as a function of the inverse fourth power of wavelength in air ( 1 / λ 0 4 ). The slight deviations of the turbidities for 1 and 3 from the λ 4 Rayleigh wavelength dependence are well fitted by the spinodal-decomposed turbidity curves, while that for 2 is well fitted by the one-parameter theoretical curve based on the fractal scattering with fractal dimension of 2.5. (b) Corresponding change in UV-Vis-light transmittance spectra of a porous Vycor glass after drying for 0, 45, and 165 min immediately after removal from ultrapure water immersion for 2 h at room temperature. In both (a) and (b), the previous results (Fig. 1 in [8]) are re-examined on the basis of the theory of dielectric constant fluctuations, and a comparison between the measured and fitted data is shown by solid and dotted lines, respectively.
Fig. 5.
Fig. 5. (a) Effects of changes in parameters ( β , ζ ) on the correlation volume ω s p ( q ) as a function of q . Each ω s p ( q ) curve corresponds to the parameter pair denoted by A: ( β [ nm 1 ], ζ [nm]) = (0.251, 19.95), B: (0.251, 10.2), C: (0.251, 29.0), D: (0.200, 19.95), and E: (0.300, 19.95), respectively. (b) Corresponding changes in the estimated UV-Vis-light transmittance spectra of a porous Vycor glass.
Fig. 6.
Fig. 6. (a) Measured and fitted UV-Vis light transmittance spectra of a porous Vycor in the dry state at 165 min, on the basis of the theory of dielectric constant fluctuations. The fitted curve is drawn with the parameters derived from the best fitting of τ s p in the range of 350 to 850 nm. (b) Difference between measured and fitted UV-Vis light absorbances, estimated from Abs ( λ 0 ) = log 10 { T fit ( λ 0 ) / T meas ( λ 0 ) } , as a function of the photon energy. The observed optical absorption spectrum is best fitted by three Gaussian curves, denoted as A, B, and C.

Tables (3)

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Table 1. Structural Information on Vycor 7930 Obtained by SAS Measurements a

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Table 2. Optical Absorption Bands Separated by Three-Gaussian Fitting

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Table 3. Correlation Volumes and Dissymmetrical Factors of Fractal Correlation Functions

Equations (61)

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I ( q ) η 2 A V π 2 λ 0 4 V ω ( q ) ,
ω ( q ) = 0 4 π r 2 γ ( r ) sin ( q r ) q r d r ,
τ = η 2 A V π 2 λ 0 4 4 π d Ω 1 + cos 2 θ 2 ω ( q ) ,
γ f ( r ) = C f r d D exp ( r ξ ) ,
ω f ( q ) = C f 4 π ξ D Γ ( D 1 ) [ 1 + q 2 ξ 2 ] D / 2 [ 1 + q 2 ξ 2 ( q ξ ) 2 ] 1 / 2 sin [ ( D 1 ) tan 1 ( q ξ ) ] ,
f 5 / 2 ( q ) = 2 3 [ 1 + q 2 ξ 2 1 ] 1 / 2 [ 1 + q 2 ξ 2 + 2 ] q ξ [ 1 + q 2 ξ 2 ] 3 / 2 .
τ f = η f 2 A V k 4 ( 2 π ) 2 ε A V 2 ω 0 ( 5 / 2 ) · Ω 5 / 2 ,
Ω 5 / 2 = 2 π 0 π d θ sin θ 1 + cos 2 θ 2 f 5 / 2 ( 2 k sin ( θ 2 ) ) = 16 2 π 3 b 3 { h ( 1 + 1 + b , b ) h ( 2 , b ) } ,
h ( z , b ) = 2 z { z 3 7 z 2 5 z 3 ( 2 + b ) ( z 3 + 1 ) 1 + b + b 2 / 2 z 1 } .
g ( r ) 1 = A · sin ( β m r ) β m r exp ( r / ζ ) ,
γ s p ( r ) = C s p sin ( β r ) β r exp ( r / ζ ) ,
ω s p ( q ) = 8 π ζ 3 q 4 ξ 4 2 ( 1 β 2 ζ 2 ) q 2 ζ 2 + ( 1 + β 2 ζ 2 ) 2 ,
τ s p = η s p 2 A V k 4 ( 4 π ) 2 ε A V 2 · Ω s p ,
Ω s p = 2 π 0 π d θ sin θ 1 + cos 2 θ 2 ω s p ( 2 k sin ( θ 2 ) ) = ( 8 π b ) 2 ζ 3 [ 1 ( 1 2 + 1 a b ) ln { 1 + b 2 + 2 b ( 1 a ) ( 1 + a ) 2 } + b 8 a { 1 + ( 1 + 2 ( 1 b ) b ) 2 16 a b 2 } { tan 1 ( 1 a + b 2 a ) tan 1 ( 1 a 2 a ) } ] ,
η ( 0 ) η ( r ) = ( ε A V ρ ) S 2 ρ ( 0 ) ρ ( r ) .
T = I out I in = ( 1 r ) 2 exp ( τ ext · d opt ) ,
r = ( n air n pG n air + n pG ) 2 ,
T = exp ( τ f · d opt ) · ( 1 r ) 2 exp ( τ s p · d opt ) .
T r ( λ 0 ) T ( λ 0 , t ) / T ( λ 0 , t e ) .
τ f ( λ 0 ) = 1 d opt ln { 1 T r ( λ 0 ) } = η f 2 A V π 2 λ 0 4 ω 0 Ω 5 / 2 ,
τ s p ( λ 0 ) = 1 d opt ln { 1 T ( λ 0 , t e ) } + 1 d opt ln ( 1 r ) ,
η s p 2 A V = ( ε p ε s ) 2 φ ( 1 φ ) ,
0 = f · ε water ε p ( f ) ε water + 2 ε p ( f ) + ( 1 f ) · ε air ε p ( f ) ε air + 2 ε p ( f ) ,
ε pG ( f ) = ε SiO 2 [ ε p ( f ) + 2 ε SiO 2 + 2 φ { ε p ( f ) ε SiO 2 } ε p ( f ) + 2 ε SiO 2 φ { ε p ( f ) ε SiO 2 } ] ,
η f 2 A V = ( ε air ε water ) 2 f ( 1 f ) · φ ( 1 φ ) .
V ( Δ ρ ) 2 A V / ρ 0 2 = k B T A β T ,
( ε A V ρ ) S 2 ( Δ ρ ) 2 A V = ε A V 4 p · k B T A β T ,
n eff = n [ 1 i 2 π N k 3 S ( 0 ) ] ,
n eff = n [ 1 i 2 π N k 3 ( i k 3 α + 2 3 k 6 α 2 ) ] ,
exp { i k 0 ( c t n eff z ) } = exp ( i ω t ) exp { i k ( 1 + 2 π N α ) } exp ( 4 π 3 N k 4 α 2 · z ) ,
Abs ( E ) = μ = A , B , C I μ exp { ( E E μ ) 2 2 σ μ 2 } ,
γ ( r 1 , r 2 ) η ( r 1 ) η ( r 2 ) / η 2 A V
E in ( R , t ) = n ^ i E 0 exp [ i ( k i · R ω i t ) ] ,
E s ( R , t ) = k s × ( k s × n ^ i ) E 0 ε A V exp [ i ( k s R ω i t ) ] 4 π R × V d 3 r η ( r ) exp [ i ( k i k s ) · r ] ,
I s ( k i k s ) = 1 2 ε A V μ | E 0 | 2 | k s × ( k s × n ^ i ) | 2 ( 4 π R ) 2 ε A V 2 × v d 3 r 1 d 3 r 2 exp [ i ( k i k s ) · ( r 1 r 2 ) ] η ( r 1 ) η ( r 2 ) ,
I s ( q ) = I in k s 4 sin 2    χ ( 4 π R ) 2 · η 2 A V ε A V 2 V 0 4 π r 2 γ ( r ) sin ( q r ) q r d r ,
d P d Ω = 1 2 Re [ R 2 R ^ · E s × H s * ] = I s ( q ) R 2 I in V ,
τ = η 2 A V k s 4 ( 4 π ) 2 ε A V 2 4 π d Ω 1 + cos 2 θ 2 ω ( q ) = η 2 A V π 2 λ 0 4 4 π d Ω 1 + cos 2 θ 2 ω ( q ) ,
ω ( q ) = 0 4 π r 2 γ ( r ) sin ( q r ) q r d r .
f D ( q ) = ω f ( q ) ω 0 ( D ) = Γ ( D 1 ) Γ ( D ) [ 1 + q 2 ξ 2 ( q ξ ) 2 ] 1 / 2 sin [ ( D 1 ) tan 1 ( q ξ ) ] [ 1 + q 2 ξ 2 ] D / 2 ,
ω 0 ( D ) = 4 π ξ D · Γ ( D ) .
2 [ 1 + q 2 ξ 2 1 ] 1 / 2 q ξ
tan 1 ( q ξ ) q ξ
2 q ξ [ 1 + q 2 ξ 2 1 1 + q 2 ξ 2 ] 1 / 2
1 1 + q 2 ξ 2
2 3 [ 1 + q 2 ξ 2 1 ] 1 / 2 [ 1 + q 2 ξ 2 + 2 ] q ξ [ 1 + q 2 ξ 2 ] 3 / 2
1 [ 1 + q 2 ξ 2 ] 2
τ f ( D ) = η f 2 A V k 4 ( 4 π ) 2 ε A V 2 ω 0 ( D ) · Ω D ,
Ω D = 2 π 0 π d θ sin θ 1 + cos 2 θ 2 f D ( 2 k sin ( θ 2 ) ) .
Ω 1 / 2 = 16 2 π b 3 { f ( 1 + 1 + b , b ) f ( 2 , b ) } ,
f ( z , b ) = 2 z { z 5 11 5 z 4 9 + 8 z 3 7 4 z 2 5 b ( z 3 7 3 z 2 5 + 2 z 3 ) + b 2 2 ( z 3 1 ) } .
Ω 1 / 2 = 16 2 π 3465 b 3 [ 2 { 256 + 33 b ( 16 + 35 b ) } 1 + 1 + b { 128 ( 1 + 1 + b ) + 8 b ( 31 + 23 1 + b ) + b 2 ( 1822 795 1 + b ) } ] .
Ω 3 / 2 = 16 2 π b 3 { g ( 1 + 1 + b , b ) g ( 2 , b ) } ,
g ( z , b ) = 2 z { z 4 9 4 z 3 7 + 4 z 2 5 b ( z 2 5 2 z 3 ) + b 2 2 } .
Ω 3 / 2 = 16 2 π 315 b 3 [ 2 { 256 + 21 b ( 16 + 15 b ) } + 1 + 1 + b { 128 ( 1 + 1 + b ) + 8 b ( 19 + 11 1 + b ) + 259 b 2 } ] .
Ω 5 / 2 = 16 2 π 3 b 3 { h ( 1 + 1 + b , b ) h ( 2 , b ) } ,
h ( z , b ) = 2 z { z 3 7 z 2 5 z 3 ( 2 + b ) ( z 3 + 1 ) 1 + b + b 2 / 2 z 1 } .
Ω 5 / 2 = 16 2 π 315 b 3 [ 2 ( 768 + 560 b + 105 b 2 ) 1 + 1 + b 1 + b { 384 ( 1 + 1 + b ) + 8 b ( 53 + 29 1 + b ) + 145 b 2 } ] .
Ω 1 = 4 π 15 [ 22 b tan 1 ( b ) + 6 + 7 b b 2 ( 6 + 10 b + 15 b 2 ) b 3 ln ( 1 + b ) ]
Ω 2 = 4 π b 2 { ( 2 + b ) + 2 + 2 b + b 2 b ln ( 1 + b ) } .
Ω 3 = 4 π b 2 { ( 2 + b ) 2 1 + b 2 ( 2 + b ) b ln ( 1 + b ) } ,

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