Abstract

Diffusion-controlled triplet–triplet annihilation is analyzed in terms of non-Fickian diffusion. The description is based on the second Fick’s law with a time-dependent diffusion coefficient. We introduce the time-dependent diffusion coefficient because of the interaction of two associated relaxation phenomena: first-order decay and diffusion-controlled annihilation. The equation for the time-dependent rate parameter k2A(t) obtained from the model proposed is compared with the standard Smoluchowski expression. The new equation is applied for the evaluation of the kinetic data of the diffusion-controlled triplet–triplet annihilation of anthracene. The limits of the applicability of the proposed model are discussed.

© 2005 Optical Society of America

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  6. C. Chen, B. Han, J. Li, T. Shang, J. Zou, and W. Jiang, "A new model on the diffusion of small molecule penetrants in dense polymer membranes," J. Membr. Sci. 187, 109-118 (2001).
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  9. R. Ash and S. E. Espenhahn, "Transport through a slab membrane governed by a concentration-dependent diffusion coefficient. Part I. The four time-lags: some general considerations," J. Membr. Sci. 154, 105-119 (1999).
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  27. R. M. Noyes, "Effects of diffusion rates on chemical kinetics," Prog. React. Kinet. 1, 129-160 (1961).
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  29. G. Wilemski and M. Fixman, "General theory of diffusion-controlled reactions," J. Chem. Phys. 58, 4009-4019 (1973).
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  30. P. R. Butler and M. J. Pilling, "Long range mechanism for the temperature dependence of the ratio of delayed monomer and delayed excimer fluorescence following triplet-triplet annihilation in liquids," J. Chem. Soc., Faraday Trans. 2 73, 886-894 (1977).
    [CrossRef]
  31. J. P. Pilling and S. A. Rice, "Theoretical model for diffusion controlled reactions of solvated electrons, incorporating a tunnelling mechanism," J. Chem. Soc., Faraday Trans. 2 71, 1563-1571 (1975).
    [CrossRef]
  32. J. P. Pilling and S. A. Rice, "Long range energy transfer by dipole-dipole and exchange interactions in rigid media and in liquids," J. Chem. Soc., Faraday Trans. 2 72, 792-801 (1976).
    [CrossRef]
  33. S. A. Adelman, "Focker-Planck equations for simple non-Markovian systems," J. Chem. Phys. 64, 124-130 (1976).
    [CrossRef]
  34. W. Dong, F. Baros, and J. C. André, "Non-Markovian effect on diffusion-controlled reactions," Ber. Bunsenges. Phys. Chem. 94, 269-274 (1990).
    [CrossRef]
  35. W. Dong and J. C. André, "Diffusion-controlled reactions.II. An approach based on generalized diffusion equation," J. Chem. Phys. 101, 299-306 (1994).
    [CrossRef]
  36. W. Dong, F. Baros, and J. C. André, "Diffusion-controlled reactions. I. Molecular dynamics simulation of a noncontinuum model," J. Chem. Phys. 91, 4643-4650 (1989).
    [CrossRef]
  37. J. Jasny, B. Nickel, and P. Borowicz, "Wavelength- and temperature-dependent measurement of the refractive indices," J. Opt. Soc. Am. B 21, 729-738 (2004).
    [CrossRef]
  38. E. G. Meyer and B. Nickel, "Diffusion coefficients of aromatic hydrocarbons in their lowest triplet state: anthracene in hexane, octane, hexadecane, perfluorohexane, and methylcyclohexane; pyrene and 9,10-diphenylanthracene in hexane," Z. Naturforsch. Teil A 35, 503-520 (1980).
  39. A. A. Ruth, B. Nickel, and H. Lesche, "Temperature dependence of viscosity and density of glass-forming alkenes," Z. Phys. Chem. (Munich) 175, 91-108 (1992).
    [CrossRef]
  40. P. R. Butler and M. J. Pilling, "Long range quenching of triplet phenantrene by copper ions in the liquid phase," Chem. Phys. 39, 33-36 (1979).
    [CrossRef]
  41. P. R. Butler and M. J. Pilling, "The breakdown of Förster kinetics in low viscosity liquids. An approximate analytical form for the time-dependent rate constant," Chem. Phys. 41, 239-243 (1979).
    [CrossRef]
  42. V. C. Sinclair, J. Monteath Robertson, and A. McL. Mathieson, "The crystal and molecular structure of anthracene. II. Structure investigation by the triple Fourier series method," Acta Crystallogr. 3, 251-256 (1950).
    [CrossRef]
  43. A. Bondi, "Van der Waals volumes and radii," J. Phys. Chem. 68, 441-451 (1964).
    [CrossRef]
  44. W. Liptay, J. Becker, D. Wehning, W. Lang, and Z. O. Burkhard, "The determination of molecular quantities from measurements on macroscopic systems II. The determination of electric dipole moments," Z. Naturforsch. Teil A 37, 1396-1408 (1982).
  45. K. Rotkiewicz and Z. R. Grabowski, "Excited states of aminoanthracenes: an experimental approach to electron density distribution," Trans. Faraday Soc. 65, 3263-3278 (1969).
    [CrossRef]
  46. J. T. Edward, "Molecular volumes and the Stokes-Einstein equation," J. Chem. Educ. 47, 261-270 (1970).
    [CrossRef]
  47. M. Litniewski and J. Górecki, "On the applicability of the step function nonradiative lifetime model for diffusion controlled reactions," J. Chem. Phys. 119, 8464-8472 (2003).
    [CrossRef]
  48. M. Litniewski and J. Górecki, "Molecular dynamics tests of the Smoluchowski-Collins-Kimball model for fluorescence quenching of spherical molecules," Phys. Chem. Chem. Phys. 6, 72-83 (2004).
    [CrossRef]

2004 (2)

J. Jasny, B. Nickel, and P. Borowicz, "Wavelength- and temperature-dependent measurement of the refractive indices," J. Opt. Soc. Am. B 21, 729-738 (2004).
[CrossRef]

M. Litniewski and J. Górecki, "Molecular dynamics tests of the Smoluchowski-Collins-Kimball model for fluorescence quenching of spherical molecules," Phys. Chem. Chem. Phys. 6, 72-83 (2004).
[CrossRef]

2003 (2)

M. Litniewski and J. Górecki, "On the applicability of the step function nonradiative lifetime model for diffusion controlled reactions," J. Chem. Phys. 119, 8464-8472 (2003).
[CrossRef]

M. A. Zahran, E. M. Abulwafa, and S. A. Elwakil, "The fractional Fokker-Planck equation on comb-like model," Physica A 323, 237-248 (2003).
[CrossRef]

2002 (2)

S. Zhou, Z. Weng, Z. Huang, and Z. Pan, "Sorption kinetics of 1-fluoro-1,1-dichloroethane in vinylidene chloride-acrylonitrile-styrene terpolymer," Eur. Polym. J. 38, 211-217 (2002).
[CrossRef]

R. Kimmich, "Strange kinetics, porous media and NMR," Chem. Phys. 284, 253-285 (2002).
[CrossRef]

2001 (3)

C. Chen, B. Han, J. Li, T. Shang, J. Zou, and W. Jiang, "A new model on the diffusion of small molecule penetrants in dense polymer membranes," J. Membr. Sci. 187, 109-118 (2001).
[CrossRef]

M. Sanopoulou and D. F. Stamatialis, "Study of the transition from Fickian to Case II sorption kinetics in the system poly(ethyl methacrylate) - liquid methyl alcohol," Polymer 42, 1429-1439 (2001).
[CrossRef]

A. V. Barzykin, K. Seki, and M. Tachiya, "Kinetics of diffusion-assisted reactions in microheterogeneous systems," Adv. Colloid Interface Sci. 89-90, 47-140 (2001).
[CrossRef] [PubMed]

2000 (2)

M. S. Vicente and J. C. Y. Gottifredi, "Effect of volume changes due to absorption in polymer membranes," J. Membr. Sci. 169, 249-254 (2000).
[CrossRef]

T. Ishii, "Theoretical investigation of anomalous diffusion in a random lattice," Solid State Commun. 116, 327-331 (2000).
[CrossRef]

1999 (3)

P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, and A. Vulpiani, "On strong anomalous diffusion," Physica D 134, 75-93 (1999).
[CrossRef]

R. Ash and S. E. Espenhahn, "Transport through a slab membrane governed by a concentration-dependent diffusion coefficient. Part I. The four time-lags: some general considerations," J. Membr. Sci. 154, 105-119 (1999).
[CrossRef]

L. Massaro and X. X. Zhu, "Physical model of diffusion for polymer solutions, gels and solids," Prog. Polym. Sci. 24, 731-775 (1999).
[CrossRef]

1998 (1)

S. J. Huang, C. J. Durning, and B. D. Freeman, "Modeling weakly non-linear two-stage sorption kinetics in glassy polymer films," J. Membr. Sci. 143, 1-11 (1998).
[CrossRef]

1997 (1)

B. Nickel, H. E. Wilhelm, and C. P. Jänsch, "Effect of the Förster energy transfers S1 + S1 --> S0 + Sn and S1 + T1 --> S0 + Tm on the time dependence of the delayed fluorescence from aromatic compounds: anti-Smoluchowski and Smoluchowski temporal behaviour," Opt. Spectrosc. 83, 541-556 (1997).

1996 (1)

S. Zou, J. Xia, and A. D. Koussis, "Analytical solutions to non-Fickian subsurface in uniform groundwater flow," J. Hydrol. 179, 237-258 (1996).
[CrossRef]

1995 (1)

P. F. Nealey, R. E. Cohen, and A. S. Argon, "Limited-supply non-Fickian diffusion in glassy polymers," Polymer 36, 3687-3695 (1995).
[CrossRef]

1994 (2)

B. Nickel, H. E. Wilhelm, and A. A. Ruth, "Anti-Smoluchowski time dependence of the delayed fluorescence from anthracene in viscous solutions due to triplet-triplet annihilation. Effect of Förster energy transfer S1 + T1 --> S0 + Tn on the initial spatial distribution of molecules in T1," Chem. Phys. 188, 267-287 (1994).
[CrossRef]

W. Dong and J. C. André, "Diffusion-controlled reactions.II. An approach based on generalized diffusion equation," J. Chem. Phys. 101, 299-306 (1994).
[CrossRef]

1992 (1)

A. A. Ruth, B. Nickel, and H. Lesche, "Temperature dependence of viscosity and density of glass-forming alkenes," Z. Phys. Chem. (Munich) 175, 91-108 (1992).
[CrossRef]

1990 (1)

W. Dong, F. Baros, and J. C. André, "Non-Markovian effect on diffusion-controlled reactions," Ber. Bunsenges. Phys. Chem. 94, 269-274 (1990).
[CrossRef]

1989 (1)

W. Dong, F. Baros, and J. C. André, "Diffusion-controlled reactions. I. Molecular dynamics simulation of a noncontinuum model," J. Chem. Phys. 91, 4643-4650 (1989).
[CrossRef]

1988 (1)

J. Saltiel and B. W. Atwater, "Spin-statistical factors in diffusion-controlled reactions," Adv. Photochem. 14, 1-90 (1988).

1983 (1)

B. Dick and B. Nickel, "Accessibility of the lowest quintet state of organic molecules through triplet-triplet annihilation; an INDO CI study," Chem. Phys. 78, 1-16 (1983).
[CrossRef]

1982 (1)

W. Liptay, J. Becker, D. Wehning, W. Lang, and Z. O. Burkhard, "The determination of molecular quantities from measurements on macroscopic systems II. The determination of electric dipole moments," Z. Naturforsch. Teil A 37, 1396-1408 (1982).

1980 (1)

E. G. Meyer and B. Nickel, "Diffusion coefficients of aromatic hydrocarbons in their lowest triplet state: anthracene in hexane, octane, hexadecane, perfluorohexane, and methylcyclohexane; pyrene and 9,10-diphenylanthracene in hexane," Z. Naturforsch. Teil A 35, 503-520 (1980).

1979 (2)

P. R. Butler and M. J. Pilling, "Long range quenching of triplet phenantrene by copper ions in the liquid phase," Chem. Phys. 39, 33-36 (1979).
[CrossRef]

P. R. Butler and M. J. Pilling, "The breakdown of Förster kinetics in low viscosity liquids. An approximate analytical form for the time-dependent rate constant," Chem. Phys. 41, 239-243 (1979).
[CrossRef]

1977 (1)

P. R. Butler and M. J. Pilling, "Long range mechanism for the temperature dependence of the ratio of delayed monomer and delayed excimer fluorescence following triplet-triplet annihilation in liquids," J. Chem. Soc., Faraday Trans. 2 73, 886-894 (1977).
[CrossRef]

1976 (2)

J. P. Pilling and S. A. Rice, "Long range energy transfer by dipole-dipole and exchange interactions in rigid media and in liquids," J. Chem. Soc., Faraday Trans. 2 72, 792-801 (1976).
[CrossRef]

S. A. Adelman, "Focker-Planck equations for simple non-Markovian systems," J. Chem. Phys. 64, 124-130 (1976).
[CrossRef]

1975 (1)

J. P. Pilling and S. A. Rice, "Theoretical model for diffusion controlled reactions of solvated electrons, incorporating a tunnelling mechanism," J. Chem. Soc., Faraday Trans. 2 71, 1563-1571 (1975).
[CrossRef]

1973 (1)

G. Wilemski and M. Fixman, "General theory of diffusion-controlled reactions," J. Chem. Phys. 58, 4009-4019 (1973).
[CrossRef]

1970 (1)

J. T. Edward, "Molecular volumes and the Stokes-Einstein equation," J. Chem. Educ. 47, 261-270 (1970).
[CrossRef]

1969 (2)

K. Rotkiewicz and Z. R. Grabowski, "Excited states of aminoanthracenes: an experimental approach to electron density distribution," Trans. Faraday Soc. 65, 3263-3278 (1969).
[CrossRef]

C. A. Parker, "Triplet state processes in fluid solutions," Ber. Bunsenges. Phys. Chem. 73, 764-772 (1969).

1966 (1)

T. Alfrey, Jr., E. F. Gurnee, and W. G. Lloyd, "Diffusion in glassy polymers," J. Polym. Sci., Part C: Polym. Lett. 12, 249-261 (1966).
[CrossRef]

1964 (1)

A. Bondi, "Van der Waals volumes and radii," J. Phys. Chem. 68, 441-451 (1964).
[CrossRef]

1961 (1)

R. M. Noyes, "Effects of diffusion rates on chemical kinetics," Prog. React. Kinet. 1, 129-160 (1961).

1953 (1)

D. L. Dexter, "A theory of sensitized luminescence in solids," J. Phys. Chem. 21, 836-850 (1953).
[CrossRef]

1950 (1)

V. C. Sinclair, J. Monteath Robertson, and A. McL. Mathieson, "The crystal and molecular structure of anthracene. II. Structure investigation by the triple Fourier series method," Acta Crystallogr. 3, 251-256 (1950).
[CrossRef]

1949 (1)

F. C. Collins and G. E. Kimball, "Diffusion-controlled reaction rates," J. Colloid Sci. 4, 425-437 (1949).
[CrossRef]

1917 (1)

M. von Smoluchowski, "Versuch einer mathematischen theorie der koagulationskinetik kolloider Lösungen," Z. Phys. Chem. (Frankfurt am Main) 92, 129-168 (1917).

Abulwafa, E. M.

M. A. Zahran, E. M. Abulwafa, and S. A. Elwakil, "The fractional Fokker-Planck equation on comb-like model," Physica A 323, 237-248 (2003).
[CrossRef]

Adelman, S. A.

S. A. Adelman, "Focker-Planck equations for simple non-Markovian systems," J. Chem. Phys. 64, 124-130 (1976).
[CrossRef]

Alfrey, T.

T. Alfrey, Jr., E. F. Gurnee, and W. G. Lloyd, "Diffusion in glassy polymers," J. Polym. Sci., Part C: Polym. Lett. 12, 249-261 (1966).
[CrossRef]

André, J. C.

W. Dong and J. C. André, "Diffusion-controlled reactions.II. An approach based on generalized diffusion equation," J. Chem. Phys. 101, 299-306 (1994).
[CrossRef]

W. Dong, F. Baros, and J. C. André, "Non-Markovian effect on diffusion-controlled reactions," Ber. Bunsenges. Phys. Chem. 94, 269-274 (1990).
[CrossRef]

W. Dong, F. Baros, and J. C. André, "Diffusion-controlled reactions. I. Molecular dynamics simulation of a noncontinuum model," J. Chem. Phys. 91, 4643-4650 (1989).
[CrossRef]

Argon, A. S.

P. F. Nealey, R. E. Cohen, and A. S. Argon, "Limited-supply non-Fickian diffusion in glassy polymers," Polymer 36, 3687-3695 (1995).
[CrossRef]

Ash , R.

R. Ash and S. E. Espenhahn, "Transport through a slab membrane governed by a concentration-dependent diffusion coefficient. Part I. The four time-lags: some general considerations," J. Membr. Sci. 154, 105-119 (1999).
[CrossRef]

Atwater, B. W.

J. Saltiel and B. W. Atwater, "Spin-statistical factors in diffusion-controlled reactions," Adv. Photochem. 14, 1-90 (1988).

Baros, F.

W. Dong, F. Baros, and J. C. André, "Non-Markovian effect on diffusion-controlled reactions," Ber. Bunsenges. Phys. Chem. 94, 269-274 (1990).
[CrossRef]

W. Dong, F. Baros, and J. C. André, "Diffusion-controlled reactions. I. Molecular dynamics simulation of a noncontinuum model," J. Chem. Phys. 91, 4643-4650 (1989).
[CrossRef]

Barzykin, A. V.

A. V. Barzykin, K. Seki, and M. Tachiya, "Kinetics of diffusion-assisted reactions in microheterogeneous systems," Adv. Colloid Interface Sci. 89-90, 47-140 (2001).
[CrossRef] [PubMed]

Becker, J.

W. Liptay, J. Becker, D. Wehning, W. Lang, and Z. O. Burkhard, "The determination of molecular quantities from measurements on macroscopic systems II. The determination of electric dipole moments," Z. Naturforsch. Teil A 37, 1396-1408 (1982).

Bondi, A.

A. Bondi, "Van der Waals volumes and radii," J. Phys. Chem. 68, 441-451 (1964).
[CrossRef]

Borowicz, P.

Burkhard, Z. O.

W. Liptay, J. Becker, D. Wehning, W. Lang, and Z. O. Burkhard, "The determination of molecular quantities from measurements on macroscopic systems II. The determination of electric dipole moments," Z. Naturforsch. Teil A 37, 1396-1408 (1982).

Butler , P. R.

P. R. Butler and M. J. Pilling, "Long range quenching of triplet phenantrene by copper ions in the liquid phase," Chem. Phys. 39, 33-36 (1979).
[CrossRef]

P. R. Butler and M. J. Pilling, "The breakdown of Förster kinetics in low viscosity liquids. An approximate analytical form for the time-dependent rate constant," Chem. Phys. 41, 239-243 (1979).
[CrossRef]

P. R. Butler and M. J. Pilling, "Long range mechanism for the temperature dependence of the ratio of delayed monomer and delayed excimer fluorescence following triplet-triplet annihilation in liquids," J. Chem. Soc., Faraday Trans. 2 73, 886-894 (1977).
[CrossRef]

Castiglione, P.

P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, and A. Vulpiani, "On strong anomalous diffusion," Physica D 134, 75-93 (1999).
[CrossRef]

Chen, C.

C. Chen, B. Han, J. Li, T. Shang, J. Zou, and W. Jiang, "A new model on the diffusion of small molecule penetrants in dense polymer membranes," J. Membr. Sci. 187, 109-118 (2001).
[CrossRef]

Cohen, R. E.

P. F. Nealey, R. E. Cohen, and A. S. Argon, "Limited-supply non-Fickian diffusion in glassy polymers," Polymer 36, 3687-3695 (1995).
[CrossRef]

Collins , F. C.

F. C. Collins and G. E. Kimball, "Diffusion-controlled reaction rates," J. Colloid Sci. 4, 425-437 (1949).
[CrossRef]

Dexter, D. L.

D. L. Dexter, "A theory of sensitized luminescence in solids," J. Phys. Chem. 21, 836-850 (1953).
[CrossRef]

Dick , B.

B. Dick and B. Nickel, "Accessibility of the lowest quintet state of organic molecules through triplet-triplet annihilation; an INDO CI study," Chem. Phys. 78, 1-16 (1983).
[CrossRef]

Dong , W.

W. Dong and J. C. André, "Diffusion-controlled reactions.II. An approach based on generalized diffusion equation," J. Chem. Phys. 101, 299-306 (1994).
[CrossRef]

Dong, W.

W. Dong, F. Baros, and J. C. André, "Non-Markovian effect on diffusion-controlled reactions," Ber. Bunsenges. Phys. Chem. 94, 269-274 (1990).
[CrossRef]

W. Dong, F. Baros, and J. C. André, "Diffusion-controlled reactions. I. Molecular dynamics simulation of a noncontinuum model," J. Chem. Phys. 91, 4643-4650 (1989).
[CrossRef]

Durning, C. J.

S. J. Huang, C. J. Durning, and B. D. Freeman, "Modeling weakly non-linear two-stage sorption kinetics in glassy polymer films," J. Membr. Sci. 143, 1-11 (1998).
[CrossRef]

Edward, J. T.

J. T. Edward, "Molecular volumes and the Stokes-Einstein equation," J. Chem. Educ. 47, 261-270 (1970).
[CrossRef]

Elwakil, S. A.

M. A. Zahran, E. M. Abulwafa, and S. A. Elwakil, "The fractional Fokker-Planck equation on comb-like model," Physica A 323, 237-248 (2003).
[CrossRef]

Espenhahn, S. E.

R. Ash and S. E. Espenhahn, "Transport through a slab membrane governed by a concentration-dependent diffusion coefficient. Part I. The four time-lags: some general considerations," J. Membr. Sci. 154, 105-119 (1999).
[CrossRef]

Fixman, M.

G. Wilemski and M. Fixman, "General theory of diffusion-controlled reactions," J. Chem. Phys. 58, 4009-4019 (1973).
[CrossRef]

Freeman, B. D.

S. J. Huang, C. J. Durning, and B. D. Freeman, "Modeling weakly non-linear two-stage sorption kinetics in glassy polymer films," J. Membr. Sci. 143, 1-11 (1998).
[CrossRef]

Górecki, J.

M. Litniewski and J. Górecki, "Molecular dynamics tests of the Smoluchowski-Collins-Kimball model for fluorescence quenching of spherical molecules," Phys. Chem. Chem. Phys. 6, 72-83 (2004).
[CrossRef]

M. Litniewski and J. Górecki, "On the applicability of the step function nonradiative lifetime model for diffusion controlled reactions," J. Chem. Phys. 119, 8464-8472 (2003).
[CrossRef]

Gottifredi, J. C. Y.

M. S. Vicente and J. C. Y. Gottifredi, "Effect of volume changes due to absorption in polymer membranes," J. Membr. Sci. 169, 249-254 (2000).
[CrossRef]

Grabowski, Z. R.

K. Rotkiewicz and Z. R. Grabowski, "Excited states of aminoanthracenes: an experimental approach to electron density distribution," Trans. Faraday Soc. 65, 3263-3278 (1969).
[CrossRef]

Gurnee, E. F.

T. Alfrey, Jr., E. F. Gurnee, and W. G. Lloyd, "Diffusion in glassy polymers," J. Polym. Sci., Part C: Polym. Lett. 12, 249-261 (1966).
[CrossRef]

Han, B.

C. Chen, B. Han, J. Li, T. Shang, J. Zou, and W. Jiang, "A new model on the diffusion of small molecule penetrants in dense polymer membranes," J. Membr. Sci. 187, 109-118 (2001).
[CrossRef]

Huang, S. J.

S. J. Huang, C. J. Durning, and B. D. Freeman, "Modeling weakly non-linear two-stage sorption kinetics in glassy polymer films," J. Membr. Sci. 143, 1-11 (1998).
[CrossRef]

Huang, Z.

S. Zhou, Z. Weng, Z. Huang, and Z. Pan, "Sorption kinetics of 1-fluoro-1,1-dichloroethane in vinylidene chloride-acrylonitrile-styrene terpolymer," Eur. Polym. J. 38, 211-217 (2002).
[CrossRef]

Ishii, T.

T. Ishii, "Theoretical investigation of anomalous diffusion in a random lattice," Solid State Commun. 116, 327-331 (2000).
[CrossRef]

Jänsch, C. P.

B. Nickel, H. E. Wilhelm, and C. P. Jänsch, "Effect of the Förster energy transfers S1 + S1 --> S0 + Sn and S1 + T1 --> S0 + Tm on the time dependence of the delayed fluorescence from aromatic compounds: anti-Smoluchowski and Smoluchowski temporal behaviour," Opt. Spectrosc. 83, 541-556 (1997).

Jasny, J.

Jiang, W.

C. Chen, B. Han, J. Li, T. Shang, J. Zou, and W. Jiang, "A new model on the diffusion of small molecule penetrants in dense polymer membranes," J. Membr. Sci. 187, 109-118 (2001).
[CrossRef]

Kimball, G. E.

F. C. Collins and G. E. Kimball, "Diffusion-controlled reaction rates," J. Colloid Sci. 4, 425-437 (1949).
[CrossRef]

Kimmich, R.

R. Kimmich, "Strange kinetics, porous media and NMR," Chem. Phys. 284, 253-285 (2002).
[CrossRef]

Koussis, A. D.

S. Zou, J. Xia, and A. D. Koussis, "Analytical solutions to non-Fickian subsurface in uniform groundwater flow," J. Hydrol. 179, 237-258 (1996).
[CrossRef]

Lang, W.

W. Liptay, J. Becker, D. Wehning, W. Lang, and Z. O. Burkhard, "The determination of molecular quantities from measurements on macroscopic systems II. The determination of electric dipole moments," Z. Naturforsch. Teil A 37, 1396-1408 (1982).

Lesche, H.

A. A. Ruth, B. Nickel, and H. Lesche, "Temperature dependence of viscosity and density of glass-forming alkenes," Z. Phys. Chem. (Munich) 175, 91-108 (1992).
[CrossRef]

Li, J.

C. Chen, B. Han, J. Li, T. Shang, J. Zou, and W. Jiang, "A new model on the diffusion of small molecule penetrants in dense polymer membranes," J. Membr. Sci. 187, 109-118 (2001).
[CrossRef]

Liptay, W.

W. Liptay, J. Becker, D. Wehning, W. Lang, and Z. O. Burkhard, "The determination of molecular quantities from measurements on macroscopic systems II. The determination of electric dipole moments," Z. Naturforsch. Teil A 37, 1396-1408 (1982).

Litniewski , M.

M. Litniewski and J. Górecki, "Molecular dynamics tests of the Smoluchowski-Collins-Kimball model for fluorescence quenching of spherical molecules," Phys. Chem. Chem. Phys. 6, 72-83 (2004).
[CrossRef]

M. Litniewski and J. Górecki, "On the applicability of the step function nonradiative lifetime model for diffusion controlled reactions," J. Chem. Phys. 119, 8464-8472 (2003).
[CrossRef]

Lloyd, W. G.

T. Alfrey, Jr., E. F. Gurnee, and W. G. Lloyd, "Diffusion in glassy polymers," J. Polym. Sci., Part C: Polym. Lett. 12, 249-261 (1966).
[CrossRef]

Massaro , L.

L. Massaro and X. X. Zhu, "Physical model of diffusion for polymer solutions, gels and solids," Prog. Polym. Sci. 24, 731-775 (1999).
[CrossRef]

Mathieson, A. McL.

V. C. Sinclair, J. Monteath Robertson, and A. McL. Mathieson, "The crystal and molecular structure of anthracene. II. Structure investigation by the triple Fourier series method," Acta Crystallogr. 3, 251-256 (1950).
[CrossRef]

Mazzino, A.

P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, and A. Vulpiani, "On strong anomalous diffusion," Physica D 134, 75-93 (1999).
[CrossRef]

Meyer , E. G.

E. G. Meyer and B. Nickel, "Diffusion coefficients of aromatic hydrocarbons in their lowest triplet state: anthracene in hexane, octane, hexadecane, perfluorohexane, and methylcyclohexane; pyrene and 9,10-diphenylanthracene in hexane," Z. Naturforsch. Teil A 35, 503-520 (1980).

Muratore-Ginanneschi, P.

P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, and A. Vulpiani, "On strong anomalous diffusion," Physica D 134, 75-93 (1999).
[CrossRef]

Nealey, P. F.

P. F. Nealey, R. E. Cohen, and A. S. Argon, "Limited-supply non-Fickian diffusion in glassy polymers," Polymer 36, 3687-3695 (1995).
[CrossRef]

Nickel, B.

J. Jasny, B. Nickel, and P. Borowicz, "Wavelength- and temperature-dependent measurement of the refractive indices," J. Opt. Soc. Am. B 21, 729-738 (2004).
[CrossRef]

B. Nickel, H. E. Wilhelm, and C. P. Jänsch, "Effect of the Förster energy transfers S1 + S1 --> S0 + Sn and S1 + T1 --> S0 + Tm on the time dependence of the delayed fluorescence from aromatic compounds: anti-Smoluchowski and Smoluchowski temporal behaviour," Opt. Spectrosc. 83, 541-556 (1997).

B. Nickel, H. E. Wilhelm, and A. A. Ruth, "Anti-Smoluchowski time dependence of the delayed fluorescence from anthracene in viscous solutions due to triplet-triplet annihilation. Effect of Förster energy transfer S1 + T1 --> S0 + Tn on the initial spatial distribution of molecules in T1," Chem. Phys. 188, 267-287 (1994).
[CrossRef]

A. A. Ruth, B. Nickel, and H. Lesche, "Temperature dependence of viscosity and density of glass-forming alkenes," Z. Phys. Chem. (Munich) 175, 91-108 (1992).
[CrossRef]

B. Dick and B. Nickel, "Accessibility of the lowest quintet state of organic molecules through triplet-triplet annihilation; an INDO CI study," Chem. Phys. 78, 1-16 (1983).
[CrossRef]

E. G. Meyer and B. Nickel, "Diffusion coefficients of aromatic hydrocarbons in their lowest triplet state: anthracene in hexane, octane, hexadecane, perfluorohexane, and methylcyclohexane; pyrene and 9,10-diphenylanthracene in hexane," Z. Naturforsch. Teil A 35, 503-520 (1980).

Noyes, R. M.

R. M. Noyes, "Effects of diffusion rates on chemical kinetics," Prog. React. Kinet. 1, 129-160 (1961).

Pan, Z.

S. Zhou, Z. Weng, Z. Huang, and Z. Pan, "Sorption kinetics of 1-fluoro-1,1-dichloroethane in vinylidene chloride-acrylonitrile-styrene terpolymer," Eur. Polym. J. 38, 211-217 (2002).
[CrossRef]

Parker, C. A.

C. A. Parker, "Triplet state processes in fluid solutions," Ber. Bunsenges. Phys. Chem. 73, 764-772 (1969).

Pilling , J. P.

J. P. Pilling and S. A. Rice, "Long range energy transfer by dipole-dipole and exchange interactions in rigid media and in liquids," J. Chem. Soc., Faraday Trans. 2 72, 792-801 (1976).
[CrossRef]

J. P. Pilling and S. A. Rice, "Theoretical model for diffusion controlled reactions of solvated electrons, incorporating a tunnelling mechanism," J. Chem. Soc., Faraday Trans. 2 71, 1563-1571 (1975).
[CrossRef]

Pilling, M. J.

P. R. Butler and M. J. Pilling, "The breakdown of Förster kinetics in low viscosity liquids. An approximate analytical form for the time-dependent rate constant," Chem. Phys. 41, 239-243 (1979).
[CrossRef]

P. R. Butler and M. J. Pilling, "Long range quenching of triplet phenantrene by copper ions in the liquid phase," Chem. Phys. 39, 33-36 (1979).
[CrossRef]

P. R. Butler and M. J. Pilling, "Long range mechanism for the temperature dependence of the ratio of delayed monomer and delayed excimer fluorescence following triplet-triplet annihilation in liquids," J. Chem. Soc., Faraday Trans. 2 73, 886-894 (1977).
[CrossRef]

Rice, S. A.

J. P. Pilling and S. A. Rice, "Long range energy transfer by dipole-dipole and exchange interactions in rigid media and in liquids," J. Chem. Soc., Faraday Trans. 2 72, 792-801 (1976).
[CrossRef]

J. P. Pilling and S. A. Rice, "Theoretical model for diffusion controlled reactions of solvated electrons, incorporating a tunnelling mechanism," J. Chem. Soc., Faraday Trans. 2 71, 1563-1571 (1975).
[CrossRef]

Robertson, J. Monteath

V. C. Sinclair, J. Monteath Robertson, and A. McL. Mathieson, "The crystal and molecular structure of anthracene. II. Structure investigation by the triple Fourier series method," Acta Crystallogr. 3, 251-256 (1950).
[CrossRef]

Rotkiewicz , K.

K. Rotkiewicz and Z. R. Grabowski, "Excited states of aminoanthracenes: an experimental approach to electron density distribution," Trans. Faraday Soc. 65, 3263-3278 (1969).
[CrossRef]

Ruth, A. A.

B. Nickel, H. E. Wilhelm, and A. A. Ruth, "Anti-Smoluchowski time dependence of the delayed fluorescence from anthracene in viscous solutions due to triplet-triplet annihilation. Effect of Förster energy transfer S1 + T1 --> S0 + Tn on the initial spatial distribution of molecules in T1," Chem. Phys. 188, 267-287 (1994).
[CrossRef]

A. A. Ruth, B. Nickel, and H. Lesche, "Temperature dependence of viscosity and density of glass-forming alkenes," Z. Phys. Chem. (Munich) 175, 91-108 (1992).
[CrossRef]

Saltiel , J.

J. Saltiel and B. W. Atwater, "Spin-statistical factors in diffusion-controlled reactions," Adv. Photochem. 14, 1-90 (1988).

Sanopoulou , M.

M. Sanopoulou and D. F. Stamatialis, "Study of the transition from Fickian to Case II sorption kinetics in the system poly(ethyl methacrylate) - liquid methyl alcohol," Polymer 42, 1429-1439 (2001).
[CrossRef]

Seki, K.

A. V. Barzykin, K. Seki, and M. Tachiya, "Kinetics of diffusion-assisted reactions in microheterogeneous systems," Adv. Colloid Interface Sci. 89-90, 47-140 (2001).
[CrossRef] [PubMed]

Shang, T.

C. Chen, B. Han, J. Li, T. Shang, J. Zou, and W. Jiang, "A new model on the diffusion of small molecule penetrants in dense polymer membranes," J. Membr. Sci. 187, 109-118 (2001).
[CrossRef]

Sinclair, V. C.

V. C. Sinclair, J. Monteath Robertson, and A. McL. Mathieson, "The crystal and molecular structure of anthracene. II. Structure investigation by the triple Fourier series method," Acta Crystallogr. 3, 251-256 (1950).
[CrossRef]

Stamatialis, D. F.

M. Sanopoulou and D. F. Stamatialis, "Study of the transition from Fickian to Case II sorption kinetics in the system poly(ethyl methacrylate) - liquid methyl alcohol," Polymer 42, 1429-1439 (2001).
[CrossRef]

Tachiya, M.

A. V. Barzykin, K. Seki, and M. Tachiya, "Kinetics of diffusion-assisted reactions in microheterogeneous systems," Adv. Colloid Interface Sci. 89-90, 47-140 (2001).
[CrossRef] [PubMed]

Vicente , M. S.

M. S. Vicente and J. C. Y. Gottifredi, "Effect of volume changes due to absorption in polymer membranes," J. Membr. Sci. 169, 249-254 (2000).
[CrossRef]

von Smoluchowski, M.

M. von Smoluchowski, "Versuch einer mathematischen theorie der koagulationskinetik kolloider Lösungen," Z. Phys. Chem. (Frankfurt am Main) 92, 129-168 (1917).

Vulpiani, A.

P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, and A. Vulpiani, "On strong anomalous diffusion," Physica D 134, 75-93 (1999).
[CrossRef]

Wehning, D.

W. Liptay, J. Becker, D. Wehning, W. Lang, and Z. O. Burkhard, "The determination of molecular quantities from measurements on macroscopic systems II. The determination of electric dipole moments," Z. Naturforsch. Teil A 37, 1396-1408 (1982).

Weng, Z.

S. Zhou, Z. Weng, Z. Huang, and Z. Pan, "Sorption kinetics of 1-fluoro-1,1-dichloroethane in vinylidene chloride-acrylonitrile-styrene terpolymer," Eur. Polym. J. 38, 211-217 (2002).
[CrossRef]

Wilemski , G.

G. Wilemski and M. Fixman, "General theory of diffusion-controlled reactions," J. Chem. Phys. 58, 4009-4019 (1973).
[CrossRef]

Wilhelm, H. E.

B. Nickel, H. E. Wilhelm, and C. P. Jänsch, "Effect of the Förster energy transfers S1 + S1 --> S0 + Sn and S1 + T1 --> S0 + Tm on the time dependence of the delayed fluorescence from aromatic compounds: anti-Smoluchowski and Smoluchowski temporal behaviour," Opt. Spectrosc. 83, 541-556 (1997).

B. Nickel, H. E. Wilhelm, and A. A. Ruth, "Anti-Smoluchowski time dependence of the delayed fluorescence from anthracene in viscous solutions due to triplet-triplet annihilation. Effect of Förster energy transfer S1 + T1 --> S0 + Tn on the initial spatial distribution of molecules in T1," Chem. Phys. 188, 267-287 (1994).
[CrossRef]

Xia, J.

S. Zou, J. Xia, and A. D. Koussis, "Analytical solutions to non-Fickian subsurface in uniform groundwater flow," J. Hydrol. 179, 237-258 (1996).
[CrossRef]

Zahran, M. A.

M. A. Zahran, E. M. Abulwafa, and S. A. Elwakil, "The fractional Fokker-Planck equation on comb-like model," Physica A 323, 237-248 (2003).
[CrossRef]

Zhou, S.

S. Zhou, Z. Weng, Z. Huang, and Z. Pan, "Sorption kinetics of 1-fluoro-1,1-dichloroethane in vinylidene chloride-acrylonitrile-styrene terpolymer," Eur. Polym. J. 38, 211-217 (2002).
[CrossRef]

Zhu, X. X.

L. Massaro and X. X. Zhu, "Physical model of diffusion for polymer solutions, gels and solids," Prog. Polym. Sci. 24, 731-775 (1999).
[CrossRef]

Zou, J.

C. Chen, B. Han, J. Li, T. Shang, J. Zou, and W. Jiang, "A new model on the diffusion of small molecule penetrants in dense polymer membranes," J. Membr. Sci. 187, 109-118 (2001).
[CrossRef]

Zou, S.

S. Zou, J. Xia, and A. D. Koussis, "Analytical solutions to non-Fickian subsurface in uniform groundwater flow," J. Hydrol. 179, 237-258 (1996).
[CrossRef]

Acta Crystallogr. (1)

V. C. Sinclair, J. Monteath Robertson, and A. McL. Mathieson, "The crystal and molecular structure of anthracene. II. Structure investigation by the triple Fourier series method," Acta Crystallogr. 3, 251-256 (1950).
[CrossRef]

Adv. Colloid Interface Sci. (1)

A. V. Barzykin, K. Seki, and M. Tachiya, "Kinetics of diffusion-assisted reactions in microheterogeneous systems," Adv. Colloid Interface Sci. 89-90, 47-140 (2001).
[CrossRef] [PubMed]

Adv. Photochem. (1)

J. Saltiel and B. W. Atwater, "Spin-statistical factors in diffusion-controlled reactions," Adv. Photochem. 14, 1-90 (1988).

Ber. Bunsenges. Phys. Chem. (2)

W. Dong, F. Baros, and J. C. André, "Non-Markovian effect on diffusion-controlled reactions," Ber. Bunsenges. Phys. Chem. 94, 269-274 (1990).
[CrossRef]

C. A. Parker, "Triplet state processes in fluid solutions," Ber. Bunsenges. Phys. Chem. 73, 764-772 (1969).

Chem. Phys. (5)

R. Kimmich, "Strange kinetics, porous media and NMR," Chem. Phys. 284, 253-285 (2002).
[CrossRef]

B. Dick and B. Nickel, "Accessibility of the lowest quintet state of organic molecules through triplet-triplet annihilation; an INDO CI study," Chem. Phys. 78, 1-16 (1983).
[CrossRef]

B. Nickel, H. E. Wilhelm, and A. A. Ruth, "Anti-Smoluchowski time dependence of the delayed fluorescence from anthracene in viscous solutions due to triplet-triplet annihilation. Effect of Förster energy transfer S1 + T1 --> S0 + Tn on the initial spatial distribution of molecules in T1," Chem. Phys. 188, 267-287 (1994).
[CrossRef]

P. R. Butler and M. J. Pilling, "Long range quenching of triplet phenantrene by copper ions in the liquid phase," Chem. Phys. 39, 33-36 (1979).
[CrossRef]

P. R. Butler and M. J. Pilling, "The breakdown of Förster kinetics in low viscosity liquids. An approximate analytical form for the time-dependent rate constant," Chem. Phys. 41, 239-243 (1979).
[CrossRef]

Eur. Polym. J. (1)

S. Zhou, Z. Weng, Z. Huang, and Z. Pan, "Sorption kinetics of 1-fluoro-1,1-dichloroethane in vinylidene chloride-acrylonitrile-styrene terpolymer," Eur. Polym. J. 38, 211-217 (2002).
[CrossRef]

J. Chem. Educ. (1)

J. T. Edward, "Molecular volumes and the Stokes-Einstein equation," J. Chem. Educ. 47, 261-270 (1970).
[CrossRef]

J. Chem. Phys. (5)

M. Litniewski and J. Górecki, "On the applicability of the step function nonradiative lifetime model for diffusion controlled reactions," J. Chem. Phys. 119, 8464-8472 (2003).
[CrossRef]

S. A. Adelman, "Focker-Planck equations for simple non-Markovian systems," J. Chem. Phys. 64, 124-130 (1976).
[CrossRef]

W. Dong and J. C. André, "Diffusion-controlled reactions.II. An approach based on generalized diffusion equation," J. Chem. Phys. 101, 299-306 (1994).
[CrossRef]

W. Dong, F. Baros, and J. C. André, "Diffusion-controlled reactions. I. Molecular dynamics simulation of a noncontinuum model," J. Chem. Phys. 91, 4643-4650 (1989).
[CrossRef]

G. Wilemski and M. Fixman, "General theory of diffusion-controlled reactions," J. Chem. Phys. 58, 4009-4019 (1973).
[CrossRef]

J. Chem. Soc., Faraday Trans. 2 (3)

P. R. Butler and M. J. Pilling, "Long range mechanism for the temperature dependence of the ratio of delayed monomer and delayed excimer fluorescence following triplet-triplet annihilation in liquids," J. Chem. Soc., Faraday Trans. 2 73, 886-894 (1977).
[CrossRef]

J. P. Pilling and S. A. Rice, "Theoretical model for diffusion controlled reactions of solvated electrons, incorporating a tunnelling mechanism," J. Chem. Soc., Faraday Trans. 2 71, 1563-1571 (1975).
[CrossRef]

J. P. Pilling and S. A. Rice, "Long range energy transfer by dipole-dipole and exchange interactions in rigid media and in liquids," J. Chem. Soc., Faraday Trans. 2 72, 792-801 (1976).
[CrossRef]

J. Colloid Sci. (1)

F. C. Collins and G. E. Kimball, "Diffusion-controlled reaction rates," J. Colloid Sci. 4, 425-437 (1949).
[CrossRef]

J. Hydrol. (1)

S. Zou, J. Xia, and A. D. Koussis, "Analytical solutions to non-Fickian subsurface in uniform groundwater flow," J. Hydrol. 179, 237-258 (1996).
[CrossRef]

J. Membr. Sci. (4)

C. Chen, B. Han, J. Li, T. Shang, J. Zou, and W. Jiang, "A new model on the diffusion of small molecule penetrants in dense polymer membranes," J. Membr. Sci. 187, 109-118 (2001).
[CrossRef]

S. J. Huang, C. J. Durning, and B. D. Freeman, "Modeling weakly non-linear two-stage sorption kinetics in glassy polymer films," J. Membr. Sci. 143, 1-11 (1998).
[CrossRef]

R. Ash and S. E. Espenhahn, "Transport through a slab membrane governed by a concentration-dependent diffusion coefficient. Part I. The four time-lags: some general considerations," J. Membr. Sci. 154, 105-119 (1999).
[CrossRef]

M. S. Vicente and J. C. Y. Gottifredi, "Effect of volume changes due to absorption in polymer membranes," J. Membr. Sci. 169, 249-254 (2000).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. (2)

D. L. Dexter, "A theory of sensitized luminescence in solids," J. Phys. Chem. 21, 836-850 (1953).
[CrossRef]

A. Bondi, "Van der Waals volumes and radii," J. Phys. Chem. 68, 441-451 (1964).
[CrossRef]

J. Polym. Sci., Part C: Polym. Lett. (1)

T. Alfrey, Jr., E. F. Gurnee, and W. G. Lloyd, "Diffusion in glassy polymers," J. Polym. Sci., Part C: Polym. Lett. 12, 249-261 (1966).
[CrossRef]

Opt. Spectrosc. (1)

B. Nickel, H. E. Wilhelm, and C. P. Jänsch, "Effect of the Förster energy transfers S1 + S1 --> S0 + Sn and S1 + T1 --> S0 + Tm on the time dependence of the delayed fluorescence from aromatic compounds: anti-Smoluchowski and Smoluchowski temporal behaviour," Opt. Spectrosc. 83, 541-556 (1997).

Phys. Chem. Chem. Phys. (1)

M. Litniewski and J. Górecki, "Molecular dynamics tests of the Smoluchowski-Collins-Kimball model for fluorescence quenching of spherical molecules," Phys. Chem. Chem. Phys. 6, 72-83 (2004).
[CrossRef]

Physica A (1)

M. A. Zahran, E. M. Abulwafa, and S. A. Elwakil, "The fractional Fokker-Planck equation on comb-like model," Physica A 323, 237-248 (2003).
[CrossRef]

Physica D (1)

P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, and A. Vulpiani, "On strong anomalous diffusion," Physica D 134, 75-93 (1999).
[CrossRef]

Polymer (2)

P. F. Nealey, R. E. Cohen, and A. S. Argon, "Limited-supply non-Fickian diffusion in glassy polymers," Polymer 36, 3687-3695 (1995).
[CrossRef]

M. Sanopoulou and D. F. Stamatialis, "Study of the transition from Fickian to Case II sorption kinetics in the system poly(ethyl methacrylate) - liquid methyl alcohol," Polymer 42, 1429-1439 (2001).
[CrossRef]

Prog. Polym. Sci. (1)

L. Massaro and X. X. Zhu, "Physical model of diffusion for polymer solutions, gels and solids," Prog. Polym. Sci. 24, 731-775 (1999).
[CrossRef]

Prog. React. Kinet. (1)

R. M. Noyes, "Effects of diffusion rates on chemical kinetics," Prog. React. Kinet. 1, 129-160 (1961).

Solid State Commun. (1)

T. Ishii, "Theoretical investigation of anomalous diffusion in a random lattice," Solid State Commun. 116, 327-331 (2000).
[CrossRef]

Trans. Faraday Soc. (1)

K. Rotkiewicz and Z. R. Grabowski, "Excited states of aminoanthracenes: an experimental approach to electron density distribution," Trans. Faraday Soc. 65, 3263-3278 (1969).
[CrossRef]

Z. Naturforsch. Teil A (2)

W. Liptay, J. Becker, D. Wehning, W. Lang, and Z. O. Burkhard, "The determination of molecular quantities from measurements on macroscopic systems II. The determination of electric dipole moments," Z. Naturforsch. Teil A 37, 1396-1408 (1982).

E. G. Meyer and B. Nickel, "Diffusion coefficients of aromatic hydrocarbons in their lowest triplet state: anthracene in hexane, octane, hexadecane, perfluorohexane, and methylcyclohexane; pyrene and 9,10-diphenylanthracene in hexane," Z. Naturforsch. Teil A 35, 503-520 (1980).

Z. Phys. Chem. (Frankfurt am Main) (1)

M. von Smoluchowski, "Versuch einer mathematischen theorie der koagulationskinetik kolloider Lösungen," Z. Phys. Chem. (Frankfurt am Main) 92, 129-168 (1917).

Z. Phys. Chem. (Munich) (1)

A. A. Ruth, B. Nickel, and H. Lesche, "Temperature dependence of viscosity and density of glass-forming alkenes," Z. Phys. Chem. (Munich) 175, 91-108 (1992).
[CrossRef]

Other (4)

H. Wilhelm, "Anti-Smoluchowski-Zeitverlauf der verzögerten fluoreszenz aromatischer verbindungen," Ph.D. thesis (Cuvillier Verlag, Göttingen, Germany, 1995), p. 6.

C. Jänsch, "Anwendung von Smoluchowskis Theorie auf die Kinetik der diffusionskontrollierten Triplett-Triplett-Annihilation aromatischer Verbindungen," Ph.D. thesis (Cuvillier Verlag, Göttingen, Germany, 1997), p. 2.

A. H. Alwattar, M. D. Lumb, and J. B. Birks, "Diffusion-controlled rate processes," in Organic Molecular Photophysics , J. B. Birks, ed. (Wiley-Interscience, London, 1973), Vol. 1, p. 417.

J. Crank, The Mathematics of Diffusion , 2nd ed. (Clarendon, Oxford, UK, 1979), pp. 254-257.

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Figures (6)

Fig. 1
Fig. 1

Comparison of the local density as a function of distance: dashed curve, Smoluchowski-type distribution after t=2.5×τT; solid curve, TTA after t=2.5×τT. The initial concentration of the particles ρ(r, t=0) equals 0 for rRA and ρ0 for r>RA. The initial conditions, the spatially random distribution of the particles, are the same for both processes.

Fig. 2
Fig. 2

Experimental setup: L, Ar, Ar+ laser; BP, Brewster prisms; B1–B4, blends; L1–L6, lenses; M1–M13, mirrors; Ch1–Ch3, choppers; BS, beam splitter; PTM1 and PTM2, photomultipliers; MCS and PC, multichannel scaler and computer.

Fig. 3
Fig. 3

Schematic representation of the important instants of a decay: 1 is the end of the anti-Smoluchowski time range; 2 is the final instant of the short-time effect. The time is represented in logarithmic scale to expand the initial part of the decay, illustrating important time ranges. In the insert, time is plotted in linear scale.

Fig. 4
Fig. 4

Results of our fitting procedure of the delayed fluorescence measured at 143 K obtained for different starting points of the evaluation, t0, with the generalized Smoluchowski model. (a) The first-order rate constant kT obtained from the fit of the decay of delayed fluorescence kTdf (points), together with the first-order rate constant obtained from a monoexponential fit of phosphorescence decay kTph (the solid line represents the value of kTph; the dashed lines represent its limits resulting from the accuracy of the measurement and the accuracy of the fitting procedure). (b) The annihilation radius RA dependent on the staring point of the evaluation, t0.

Fig. 5
Fig. 5

Temperature dependence of the first-order decay constant kT measured directly from phosphorescence kTph (monoexponential function) and from delayed fluorescence kTdf (generalized Smoluchowski’s model) in the temperature range 132–147 K.

Fig. 6
Fig. 6

Temperature dependence of the annihilation radius RA obtained from the generalized Smoluchowski model with application of the extrapolated diffusion coefficients (for details, see Section 3). The temperature range is 132–147 K.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

 3M*+3M*1(MM)**1M*+1M1M+1M+hν,
IDF(t)=(C0)2k2A(t)exp(-2kTt),
ρS(r, t)=ρ01-RAr+RAr erfr-RA2DTt.
k2AS(t)=4πRADT1+RA(πDTt)1/2.
 3M*+3M*3(MM)**3M**+1M3M*+1M
 3M*+3M*5D*,
ρ(r, t)t=-kTρ(r, t)+DT exp(-2kTt)2ρ(r, t),
ρ(r, t)=ρ0 exp(-kTt)f(r, t),
f(r, t)t=DT exp(-2kTt)2f(r, t).
f(r, t)=1-RAr+RAr erf2kT(r-RA)2{DT[1-exp(-2kTt)]}1/2,
k2A(t)=4πDTRA1+2kTRA{πDT[1-exp(-2kTt)]}1/2.
f(r, t)t1-RAr+RAr erf2kT(r-RA)2DTr1,
k2A(t)r4πDTRA1+2kTRAπDT.
IDFA(t)=P0+P1 exp(-2t/P3),
IDFM(t)=P0+P11+2πP2P3[1-exp(-2t/P3)]1/2×exp(-2t/P3),

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