Skip to main content
SpringerLink
Log in

Luminescence Decays with Underlying Distributions of Rate Constants: General Properties and Selected Cases

  • Chapter
  • First Online:
Fluorescence of Supermolecules, Polymers, and Nanosystems

Part of the book series: Springer Series on Fluorescence ((SS FLUOR,volume 4))

Abstract

The mathematical properties of the general luminescence decay law are described. Special attentionis paid to cases represented by continuous distributions of decay rate constants. Six important decay functionsare described in detail: stretched exponential (Kohlrausch), compressed hyperbola (Becquerel), Mittag–Leffler,Heaviside, Weibull, and truncated Gaussian.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. James DR, Ware WR (1985) A fallacy in the interpretation of fluorescence decay parameters. Chem Phys Lett 120:455–459

    Article  CAS  Google Scholar 

  2. Liu YS, Ware WR (1993) Photophysics of polycyclic aromatic hydrocarbons adsorbed on silica gel surfaces 1. Fluorescence lifetime distribution analysis: An ill-conditioned problem. J Phys Chem 97:5980–5986

    Article  CAS  Google Scholar 

  3. Mollay B, Kauffmann HF (1994) In: Richert R, Blumen A (eds) Disorder Effects on Relaxational Processes. Springer, Berlin, pp 509–541

    Chapter  Google Scholar 

  4. Berberan-Santos MN, Choppinet P, Fedorov A, Jullien L, Valeur B (1999) Multichromophoric cyclodextrins, 6. Investigation of excitation energy hopping by Monte Carlo simulations and time-resolved anisotropy. J Am Chem Soc 121:2526–2533

    Article  CAS  Google Scholar 

  5. Berberan-Santos MN, Choppinet P, Fedorov A, Jullien L, Valeur B (2000) Multichromophoric cyclodextrins, 8. Dynamics of homo- and heterotransfer of excitation energy in inclusion complexes with fluorescent dyes. J Am Chem Soc 122:11876–11886

    Article  CAS  Google Scholar 

  6. Bodunov EN, Berberan-Santos MN, Nunes Pereira EJ, Martinho JMG (2000) Eigenvalue spectrum of the survival probability of excitation in nonradiative energy transport. Chem Phys 259:49–61

    Article  CAS  Google Scholar 

  7. Wagner BD, Ware WR (1990) Recovery of fluorescence lifetime distributions: Application to Förster transfer in rigid and viscous media. J Phys Chem 94:3489–3494

    Article  Google Scholar 

  8. Lee M, Tang J, Hochstrasser RM (2001) Fluorescence lifetime distribution of single molecules undergoing Förster energy transfer. Chem Phys Lett 344:501–508

    Article  CAS  Google Scholar 

  9. Livesey AK, Brochon JC (1987) Analyzing the distribution of decay constants in pulse-fluorimetry using the maximum entropy method. Biophys J 52:693–706

    Article  CAS  Google Scholar 

  10. Siemiarczuk A, Wagner BD, Ware WR (1990) Comparison of the maximum entropy and exponential series methods for the recovery of distributions of lifetimes from fluorescence lifetime data. J Phys Chem 94:1661–1666

    Article  CAS  Google Scholar 

  11. Berberan-Santos MN, Valeur B (2007) Luminescence decays with underlying distributions: General properties and analysis with mathematical functions. J Lumin 126:263–272

    Article  CAS  Google Scholar 

  12. Verbeek G, Vaes A, Van der Auweraer M, De Schryver FC, Geelen C, Terrell D, De Meutter S (1993) Gaussian distributions of the decay times of the singlet excited state of aromatic amines dispersed in polymer films. Macromolecules 26:472–478

    Article  CAS  Google Scholar 

  13. Berberan-Santos MN, Bodunov EN, Valeur B (2005) Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential). Chem Phys 315:171–182

    Article  CAS  Google Scholar 

  14. Berberan-Santos MN, Bodunov EN, Valeur B (2005) Mathematical functions for the analysis of luminescence decays with underlying distributions 2. Becquerel (compressed hyperbola) and related decay functions. Chem Phys 317:57–62

    Article  CAS  Google Scholar 

  15. Valeur B (2002) Molecular fluorescence. Principles and applications. Wiley-VCH, Weinheim

    Google Scholar 

  16. Jameson DM, Gratton E, Hall RD (1984) The measurement and analysis of heterogeneous emissions by multifrequency phase and modulation fluorometry. Appl Spectrosc Rev 20:55–106

    Article  CAS  Google Scholar 

  17. Böttcher CJF, Bordewijk (1978) Theory of electric polarization vol II. Elsevier, Amsterdam

    Google Scholar 

  18. Widder DV (1946) The Laplace transform. Princeton University Press, Princeton

    Google Scholar 

  19. Feller W (1971) An introduction to Probability Theory and its applications vol 2, 2nd edn. Wiley, New York

    Google Scholar 

  20. Koppel DE (1972) Analysis of macromolecular polydispersity in intensity correlation spectroscopy: The method of cumulants. J Chem Phys 57:4814–4820

    Article  CAS  Google Scholar 

  21. Frisken BJ (2001) Revisiting the method of cumulants for the analysis of dynamic light-scattering data. Appl Opt 40:4087–4091

    Article  CAS  Google Scholar 

  22. Fritz R, Kungl A, Rettig W, Springer J (1996) Photochemical fluorescence probes: rate distribution in solid polymers. Chem Phys Lett 260:409–417

    Article  CAS  Google Scholar 

  23. Berberan-Santos MN (2005) Analytical inversion of the Laplace transform without contour integration. Application to luminescence decay laws and other relaxation functions. J Math Chem 38:165–173

    Article  CAS  Google Scholar 

  24. Berberan-Santos MN (2007) Computation of one-sided probability density functions from their cumulants. J Math Chem 41:71–77

    Article  CAS  Google Scholar 

  25. Gardiner CW (1985) Handbook of Stochastic Methods, 2nd edn. Springer-Verlag, Berlin

    Google Scholar 

  26. James DR, Liu Y-S, de Mayo P, Ware WR (1985) Distributions of fluorescence lifetimes: Consequences for the photophysics of molecules adsorbed on surfaces. Chem Phys Lett 120:460–465

    Article  CAS  Google Scholar 

  27. James DR, Ware WR (1986) Recovery of underlying distributions of lifetimes from fluorescence decay data. Chem Phys Lett 126:7–11

    Article  CAS  Google Scholar 

  28. Alcala JR, Gratton E, Prendergast FG (1987) Resolvability of fluorescence lifetime distributions using phase fluorometry. Biophys J 51:587–596

    Article  CAS  Google Scholar 

  29. Montroll EW, Bendler JT (1984) On Levy (or stable) distributions and the Williams-Watts model of dielectric relaxation. J Stat Phys 34:129–162

    Article  Google Scholar 

  30. Kohlrausch R (1854) Theorie des elektrischen Rückstandes in der Leidener Flasche II. Ann Phys Chem (Poggendorff) 91:179–214

    Google Scholar 

  31. Kohlrausch R (1854) Theorie des elektrischen Rückstandes in der Leidener Flasche. Ann Phys Chem (Poggendorff) 91:56–82

    Google Scholar 

  32. Werner A (1907) Quantitative Messungen der An- und Abklingung getrennter Phosphorescenzbanden. Ann Phys 24:164–190

    Article  Google Scholar 

  33. Rice SA (1985) In: Bamford CH, Tipper CFH, Compton RG (eds) Chemical Kinetics, vol 25. Elsevier, Amsterdam

    Google Scholar 

  34. Förster T (1949) Experimentelle und theoretische Untersuchung des zwischenmolekularen Übergangs von Elektronenanregungsenergie. Z Naturforsch 4a:321–327

    Google Scholar 

  35. Sveshnikov BY, Shirokov VI (1962) Change in the mean life-time and yield of luminescence in the process of quenching as a function of the law of molecular interaction. Opt Spectrosc 12:576–581

    CAS  Google Scholar 

  36. Inokuti M, Hirayama F (1965) Influence of energy transfer by exchange mechanism on donor luminescence. J Chem Phys 43:1978–1989

    Article  CAS  Google Scholar 

  37. Huber DL, Hamilton DS, Barnett B (1977) Time-dependent effects in fluorescent line narrowing. Phys Rev B 16:4642–4650

    Article  CAS  Google Scholar 

  38. Huber DL (1981) Dynamics of incoherent transfer. Top Appl Phys 49:83–111

    Article  CAS  Google Scholar 

  39. Fredrickson GH, Andersen HC, Frank CW (1983) Electronic excited-state transport on isolated polymer-chains. Macromolecules 16:1456–1464

    Article  CAS  Google Scholar 

  40. Fredrickson GH, Andersen HC, Frank CW (1985) Macromolecular pair correlation-functions from fluorescence depolarization experiments. J Polym Sci: Polym Phys Ed 23:591–599

    Article  CAS  Google Scholar 

  41. Peterson KA, Fayer MD (1986) Electronic excitation transport on isolated, flexible polymer-chains in the amorphous solid-state randomly tagged or end tagged with chromophores. J Chem Phys 85:4702–4711

    Article  CAS  Google Scholar 

  42. Roy AK, Blumen A (1989) On the direct energy transfer from donors to acceptors in chainlike polymer systems. J Chem Phys 91:4353–4359

    Article  CAS  Google Scholar 

  43. Bodunov EN, Berberan-Santos MN, Martinho JMG (2001) Electronic energy transfer in linear polymers randomly labelled with chromophores. Chem Phys 274:243–253

    Article  CAS  Google Scholar 

  44. Bodunov EN, Berberan-Santos MN, Martinho JMG (2002) Electronic energy transfer in polymers labelled at both ends with fluorescent groups. J Luminescence 96:269–278

    Article  CAS  Google Scholar 

  45. Bodunov EN, Berberan-Santos MN, Martinho JMG (2002) Electronic energy transfer in linear polymer chains. High Energ Chem 36:245–249

    Article  CAS  Google Scholar 

  46. Linnros J, Lalic N, Galeckas A, Grivickas V (1999) Analysis of the stretched exponential photoluminescence decay from nanometer-sized silicon crystals in SiO2. J Appl Phys 86:6128–6134

    Article  CAS  Google Scholar 

  47. Chen R (2003) Apparent stretched-exponential luminescence decay in crystalline solids. J Luminescence 102–103:510–518

    Article  Google Scholar 

  48. Soloviev VN, Eichhöfer A, Fenske D, Banin U (2001) Size-dependent optical spectroscopy of a homologous series of CdSe cluster molecules. J Am Chem Soc 123:2354–2364

    Article  CAS  Google Scholar 

  49. Tang J, Marcus RA (2005) Single particle versus ensemble average: From power-law intermittency of a single quantum dot to quasistretched exponential fluorescence decay of an ensemble. J Chem Phys 123:204511

    Article  Google Scholar 

  50. Hof M, Schleicher J, Schneider FW (1989) Time resolved fluorescence in doped aerogels and organosilicate glasses. Ber Bunsenges Phys Chem 93:1377–1381

    Article  CAS  Google Scholar 

  51. Wong AL, Harris JM, Marshall DB (1990) Measurements of energy dispersion at liquid-solid interfaces: Fluorescence quenching of pyrene bound to fumed silica. Can J Phys 68:1027–1034

    Article  CAS  Google Scholar 

  52. Métivier R, Leray I, Lefèvre J-P, Roy-Auberger M, Zaner-Szydlowski N, Valeur B (2003) Characterization of alumina surfaces by fluorescence spectroscopy, Part 2. Photophysics of a bound pyrene derivative as a probe of the spatial distribution of reactive hydroxyl groups. Phys Chem Chem Phys 5:758–766

    Article  Google Scholar 

  53. Lee J, Lee J, Lee M, Lee K-J-B, Ko D-S (2004) Scanning confocal fluorescence microscopy of single DNA-EtBr complexes dispersed in polymer. Chem Phys Lett 394:49–53

    Article  CAS  Google Scholar 

  54. Siegel J, Elson DS, Webb SED, Benny Lee KC, Vlandas A, Gambaruto GL, Lévêque-Fort S, John Lever M, Tadrous PJ, Stamp GWH, Wallace AL, Sandison A, Watson TF, Alvarez F, French PMW (2003) Studying biological tissue with fluorescence lifetime imaging: microscopy, endoscopy, and complex decay profiles. Appl Optics 42:2995–3004

    Article  Google Scholar 

  55. Huber DL (1996) Dynamical model for stretched exponential relaxation in solids. Phys Rev E 53:6544–6546

    Article  CAS  Google Scholar 

  56. Jurlewicz A, Weron K (1999) A general probabilistic approach to the universal relaxation response of complex systems. Cellul Mol Biol Lett 4:55–86

    Google Scholar 

  57. Huber DL (2000) Two-state model for sub-exponential fluorescence. J Luminescence 86:95–99

    Article  CAS  Google Scholar 

  58. García-Adeva AJ, Huber DL (2001) Two-state model for sub-exponential fluorescence revisited. J Luminescence 92:65–72

    Article  Google Scholar 

  59. Williams G, Watts DC (1970) Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function. Trans Faraday Soc 66:80–85

    Article  CAS  Google Scholar 

  60. Collins FC, Kimball GE (1949) Diffusion-controlled reaction rates. J Colloid Sci 4:425–437

    Article  CAS  Google Scholar 

  61. Pilling MJ, Rice SA (1976) 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

    Article  CAS  Google Scholar 

  62. Macdonald JR (1987) Linear relaxation: Distributions, thermal activation, structure, and ambiguity. J Appl Phys 62:R51–R62

    Article  CAS  Google Scholar 

  63. Pollard H (1946) The representation of e −xλ as a Laplace integral. Bull Am Math Soc 52:908–910

    Google Scholar 

  64. Berberan-Santos MN (2005) Relation between the inverse Laplace transforms of I(t β) and I(t): Application to the Mittag–Leffler and asymptotic inverse power law relaxation functions. J Math Chem 38:265–270

    Article  CAS  Google Scholar 

  65. Humbert CP (1945) Nouvelles correspondances symboliques. Bull Soc Math France 69:121–129

    Google Scholar 

  66. Lindsey CP, Patterson GD (1980) Detailed comparison of the Williams–Watts and Cole–Davidson functions. J Chem Phys 73:3348–3357

    Article  CAS  Google Scholar 

  67. Becquerel E (1867) La lumière; ses causes et ses effets, vol 1. Firmin Didot, Paris

    Google Scholar 

  68. Curie D (1963) Luminescence in Crystals. Methuen, London

    Google Scholar 

  69. Wlodarczyk J, Kierdaszuk B (2003) Interpretation of fluorescence decays using a power-like model. Biophys J 85:589–598

    Article  CAS  Google Scholar 

  70. Mittag-Leffler GM (1903) Une généralisation de l'intégrale de Laplace–Abel. CR Acad Sci Paris Sér II 136:537–539

    Google Scholar 

  71. Berberan-Santos MN (2005) Properties of the Mittag–Leffler relaxation function. J Math Chem 38:629–635

    Article  CAS  Google Scholar 

  72. Wintner A (1959) On Heaviside's and Mittag–Leffler's generalizations of the exponential function, the symmetric stable distributions of Cauchy–Lévy, and a property of the Γ-functions. J Math Pur Appl IX 38:165–182

    Google Scholar 

  73. Pollard H (1948) The completely monotonic character of the Mittag-Leffler function E α (− x). Bull Am Math Soc 54:1115–1116

    Article  Google Scholar 

  74. Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18:293–297

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centro de Química-Física Molecular, Instituto Superior Técnico, 1049-001, Lisboa, Portugal

    Mário N. Berberan-Santos

  2. Physical Department, Petersburg State Transport University, 190031, St. Petersburg, Russia

    Evgeny N. Bodunov

  3. CNRS UMR 8531, Laboratoire de Chimie Générale, CNAM, 292 rue Saint-Martin, 75141, Paris cedex 03, France

    Bernard Valeur

  4. Laboratoire PPSM, ENS-Cachan, 61 avenue du Président Wilson, 94235, Cachan cedex, France

    Bernard Valeur

Authors
  1. Mário N. Berberan-Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Evgeny N. Bodunov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bernard Valeur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mário N. Berberan-Santos .

Editor information

M. N. Berberan-Santos

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Berberan-Santos, M.N., Bodunov, E.N., Valeur, B. (2007). Luminescence Decays with Underlying Distributions of Rate Constants: General Properties and Selected Cases. In: Berberan-Santos, M.N. (eds) Fluorescence of Supermolecules, Polymers, and Nanosystems. Springer Series on Fluorescence, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/4243_2007_001

Download citation

Publish with us

Policies and ethics

Search

Navigation