Abstract

Microscopic movements of latex particles in the water-filled voids of a glass bead packing are investigated with a variant of dynamic light-scattering, diffusive wave illumination. We are able to measure simultaneously diffusive and convective motion of the scatterers within the pore spaces of the completely opaque medium. The influence of increased seepage velocity on the diffusion coefficient and on the speed distribution of the particles is studied. We find an exponential distribution of the convective speed. From the average speed we determine the tortuosity of the packing. Using this investigation system as an illustration, we address the methodological aspects of diffusive wave illumination and discuss its possibilities and limitations: (1) the possibility of tuning the amount of local oscillator mixed to the dynamic signal, (2) the inclusion of residual dynamics of the porous medium into the fitting model, and (3) a study of the influence of increased particle concentration on the signal correlation function.

© 1998 Optical Society of America

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  1. P. C. Carman, “Fluid flow through granular beds,” Trans. Inst. Chem. Eng. 15, 150–156 (1937).
  2. E. Charlaix, J. P. Hulin, T. J. Plona, “Experimental study of tracer dispersion in sintered glass porous materials of variable compaction,” Phys. Fluids 30, 1690–1698 (1987).
    [CrossRef]
  3. Y. D. Yan, M. Borkovec, H. Sticher, “Deposition and release of colloidal particles in porous media,” Prog. Colloid Polym. Sci. 98, 132–135 (1995).
    [CrossRef]
  4. W. Johnston, A. Dybbs, “Measurement of fluid velocity inside porous media with a laser anemometer,” Phys. Fluids 18, 913–914 (1975).
    [CrossRef]
  5. M. T. Bishop, K. H. Langley, F. E. Karasz, “Diffusion of a flexible polymer in a random porous material,” Phys. Rev. Lett. 57, 1741–1744 (1986).
    [CrossRef] [PubMed]
  6. M. T. Bishop, K. H. Langley, F. E. Karasz, “Dynamic light-scattering studies of polymer diffusion in porous materials: linear polystyrene in porous glass,” Macromolecules 22, 1220–1231 (1989).
    [CrossRef]
  7. R. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097–2107 (1981).
    [CrossRef] [PubMed]
  8. D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Classical Wave Localisation, P. Sheng, ed. (World Scientific, London, 1989).
  9. J. Rička, “Brownian dynamics in strongly scattering porous media: dynamic light scattering with single-mode matching,” Makromol. Chem. Macromol. Symp. 79, 45–55 (1994).
    [CrossRef]
  10. J. Rička, I. Flammer, W. Leutz, “Single-mode DLS: colloids in opaque porous media,” Prog. Colloid Polym. Sci. 104, 49–58 (1997).
  11. I. Flammer, J. Rička, “Dynamic light scattering with single-mode receivers: partial heterodyning regime,” Appl. Opt. 36, 7508–7517 (1997).
    [CrossRef]
  12. J. Rička, “Dynamic light scattering with single-mode and multimode receivers,” Appl. Opt. 32, 2860–2875 (1993).
    [CrossRef] [PubMed]
  13. Recent developments on DWS and related methods are found in the feature issue on diffusing photons in turbid media, J. Opt. Soc. Am. A 14, 136–342 (1997).
  14. G. Maret, P. Wolf, “Multiple light scattering from disordered media: the effect of Brownian motion on scatterers,” Z. Phys. B 65, 409–413 (1987).
    [CrossRef]
  15. D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101–2127 (1990).
    [CrossRef]
  16. P. N. Pusey, “Dynamic light scattering by non-ergodic media,” Physica A 157, 705–741 (1989).
    [CrossRef]
  17. K. Schätzel, “Dead time correction of photon correlation functions,” Appl. Phys. B 41, 95–102 (1986).
    [CrossRef]
  18. F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
    [CrossRef]
  19. H. Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  20. R. T. Foister, T. G. van de Ven, “Diffusion of Brownian particles in shear flows,” J. Fluid Mech. 96, 105–132 (1980).
    [CrossRef]
  21. T. G. M. V. de Ven, Colloidal Hydrodynamics (Academic, London, 1989).
  22. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
  23. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford Science Publications, Oxford, UK, 1959), Chap. XIV.
  24. W. Leutz, J. Rička, “On light propagation through glass bead packings,” Opt. Commun. 126, 260–268 (1996).
    [CrossRef]
  25. D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
    [CrossRef]
  26. L. R. Snyder, J. J. Kirkland, Introduction to Modern Liquid Chromatography (Wiley, New York, 1979).
  27. J. Bear, Dynamics of Fluids in Porous Media (Dover, New York, 1988).
  28. P. D. Kaplan, M. H. Kao, A. G. Yodh, D. J. Pine, “Geometric constraints for the design of diffusing-wave spectroscopy experiments,” Appl. Opt. 32, 3828–3836 (1993).
    [CrossRef] [PubMed]
  29. Y. E. Kutsovsky, L. E. Scriven, H. T. Davis, B. E. Hammer, “NMR imaging of velocity profiles and velocity distribution in bead packs,” Phys. Fluids 8, 863–871 (1996).
    [CrossRef]
  30. L. Lebon, L. Oger, J. Leblond, J. P. Hulin, N. S. Martys, L. M. Schwartz, “Pulsed gradient NMR measurements and numerical simulation of flow velocity distribution in sphere packing,” Phys. Fluids 8, 293–301 (1996).
    [CrossRef]

1997

1996

W. Leutz, J. Rička, “On light propagation through glass bead packings,” Opt. Commun. 126, 260–268 (1996).
[CrossRef]

Y. E. Kutsovsky, L. E. Scriven, H. T. Davis, B. E. Hammer, “NMR imaging of velocity profiles and velocity distribution in bead packs,” Phys. Fluids 8, 863–871 (1996).
[CrossRef]

L. Lebon, L. Oger, J. Leblond, J. P. Hulin, N. S. Martys, L. M. Schwartz, “Pulsed gradient NMR measurements and numerical simulation of flow velocity distribution in sphere packing,” Phys. Fluids 8, 293–301 (1996).
[CrossRef]

1995

Y. D. Yan, M. Borkovec, H. Sticher, “Deposition and release of colloidal particles in porous media,” Prog. Colloid Polym. Sci. 98, 132–135 (1995).
[CrossRef]

1994

J. Rička, “Brownian dynamics in strongly scattering porous media: dynamic light scattering with single-mode matching,” Makromol. Chem. Macromol. Symp. 79, 45–55 (1994).
[CrossRef]

1993

1990

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101–2127 (1990).
[CrossRef]

1989

P. N. Pusey, “Dynamic light scattering by non-ergodic media,” Physica A 157, 705–741 (1989).
[CrossRef]

M. T. Bishop, K. H. Langley, F. E. Karasz, “Dynamic light-scattering studies of polymer diffusion in porous materials: linear polystyrene in porous glass,” Macromolecules 22, 1220–1231 (1989).
[CrossRef]

1987

E. Charlaix, J. P. Hulin, T. J. Plona, “Experimental study of tracer dispersion in sintered glass porous materials of variable compaction,” Phys. Fluids 30, 1690–1698 (1987).
[CrossRef]

G. Maret, P. Wolf, “Multiple light scattering from disordered media: the effect of Brownian motion on scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

1986

K. Schätzel, “Dead time correction of photon correlation functions,” Appl. Phys. B 41, 95–102 (1986).
[CrossRef]

M. T. Bishop, K. H. Langley, F. E. Karasz, “Diffusion of a flexible polymer in a random porous material,” Phys. Rev. Lett. 57, 1741–1744 (1986).
[CrossRef] [PubMed]

1981

1980

R. T. Foister, T. G. van de Ven, “Diffusion of Brownian particles in shear flows,” J. Fluid Mech. 96, 105–132 (1980).
[CrossRef]

1975

W. Johnston, A. Dybbs, “Measurement of fluid velocity inside porous media with a laser anemometer,” Phys. Fluids 18, 913–914 (1975).
[CrossRef]

1972

D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

1971

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

1937

P. C. Carman, “Fluid flow through granular beds,” Trans. Inst. Chem. Eng. 15, 150–156 (1937).

Arecchi, F. T.

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

Bear, J.

J. Bear, Dynamics of Fluids in Porous Media (Dover, New York, 1988).

Bishop, M. T.

M. T. Bishop, K. H. Langley, F. E. Karasz, “Dynamic light-scattering studies of polymer diffusion in porous materials: linear polystyrene in porous glass,” Macromolecules 22, 1220–1231 (1989).
[CrossRef]

M. T. Bishop, K. H. Langley, F. E. Karasz, “Diffusion of a flexible polymer in a random porous material,” Phys. Rev. Lett. 57, 1741–1744 (1986).
[CrossRef] [PubMed]

Bonner, R.

Borkovec, M.

Y. D. Yan, M. Borkovec, H. Sticher, “Deposition and release of colloidal particles in porous media,” Prog. Colloid Polym. Sci. 98, 132–135 (1995).
[CrossRef]

Carman, P. C.

P. C. Carman, “Fluid flow through granular beds,” Trans. Inst. Chem. Eng. 15, 150–156 (1937).

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford Science Publications, Oxford, UK, 1959), Chap. XIV.

Chaikin, P. M.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Classical Wave Localisation, P. Sheng, ed. (World Scientific, London, 1989).

Charlaix, E.

E. Charlaix, J. P. Hulin, T. J. Plona, “Experimental study of tracer dispersion in sintered glass porous materials of variable compaction,” Phys. Fluids 30, 1690–1698 (1987).
[CrossRef]

Corti, M.

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

Davis, H. T.

Y. E. Kutsovsky, L. E. Scriven, H. T. Davis, B. E. Hammer, “NMR imaging of velocity profiles and velocity distribution in bead packs,” Phys. Fluids 8, 863–871 (1996).
[CrossRef]

de Ven, T. G. M. V.

T. G. M. V. de Ven, Colloidal Hydrodynamics (Academic, London, 1989).

Degiorgio, V.

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

Donati, S.

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

Dybbs, A.

W. Johnston, A. Dybbs, “Measurement of fluid velocity inside porous media with a laser anemometer,” Phys. Fluids 18, 913–914 (1975).
[CrossRef]

Flammer, I.

J. Rička, I. Flammer, W. Leutz, “Single-mode DLS: colloids in opaque porous media,” Prog. Colloid Polym. Sci. 104, 49–58 (1997).

I. Flammer, J. Rička, “Dynamic light scattering with single-mode receivers: partial heterodyning regime,” Appl. Opt. 36, 7508–7517 (1997).
[CrossRef]

Foister, R. T.

R. T. Foister, T. G. van de Ven, “Diffusion of Brownian particles in shear flows,” J. Fluid Mech. 96, 105–132 (1980).
[CrossRef]

Hammer, B. E.

Y. E. Kutsovsky, L. E. Scriven, H. T. Davis, B. E. Hammer, “NMR imaging of velocity profiles and velocity distribution in bead packs,” Phys. Fluids 8, 863–871 (1996).
[CrossRef]

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101–2127 (1990).
[CrossRef]

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Classical Wave Localisation, P. Sheng, ed. (World Scientific, London, 1989).

Hulin, J. P.

L. Lebon, L. Oger, J. Leblond, J. P. Hulin, N. S. Martys, L. M. Schwartz, “Pulsed gradient NMR measurements and numerical simulation of flow velocity distribution in sphere packing,” Phys. Fluids 8, 293–301 (1996).
[CrossRef]

E. Charlaix, J. P. Hulin, T. J. Plona, “Experimental study of tracer dispersion in sintered glass porous materials of variable compaction,” Phys. Fluids 30, 1690–1698 (1987).
[CrossRef]

Hulst, H.

H. Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford Science Publications, Oxford, UK, 1959), Chap. XIV.

Johnston, W.

W. Johnston, A. Dybbs, “Measurement of fluid velocity inside porous media with a laser anemometer,” Phys. Fluids 18, 913–914 (1975).
[CrossRef]

Kao, M. H.

Kaplan, P. D.

Karasz, F. E.

M. T. Bishop, K. H. Langley, F. E. Karasz, “Dynamic light-scattering studies of polymer diffusion in porous materials: linear polystyrene in porous glass,” Macromolecules 22, 1220–1231 (1989).
[CrossRef]

M. T. Bishop, K. H. Langley, F. E. Karasz, “Diffusion of a flexible polymer in a random porous material,” Phys. Rev. Lett. 57, 1741–1744 (1986).
[CrossRef] [PubMed]

Kirkland, J. J.

L. R. Snyder, J. J. Kirkland, Introduction to Modern Liquid Chromatography (Wiley, New York, 1979).

Koppel, D. E.

D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

Kutsovsky, Y. E.

Y. E. Kutsovsky, L. E. Scriven, H. T. Davis, B. E. Hammer, “NMR imaging of velocity profiles and velocity distribution in bead packs,” Phys. Fluids 8, 863–871 (1996).
[CrossRef]

Langley, K. H.

M. T. Bishop, K. H. Langley, F. E. Karasz, “Dynamic light-scattering studies of polymer diffusion in porous materials: linear polystyrene in porous glass,” Macromolecules 22, 1220–1231 (1989).
[CrossRef]

M. T. Bishop, K. H. Langley, F. E. Karasz, “Diffusion of a flexible polymer in a random porous material,” Phys. Rev. Lett. 57, 1741–1744 (1986).
[CrossRef] [PubMed]

Leblond, J.

L. Lebon, L. Oger, J. Leblond, J. P. Hulin, N. S. Martys, L. M. Schwartz, “Pulsed gradient NMR measurements and numerical simulation of flow velocity distribution in sphere packing,” Phys. Fluids 8, 293–301 (1996).
[CrossRef]

Lebon, L.

L. Lebon, L. Oger, J. Leblond, J. P. Hulin, N. S. Martys, L. M. Schwartz, “Pulsed gradient NMR measurements and numerical simulation of flow velocity distribution in sphere packing,” Phys. Fluids 8, 293–301 (1996).
[CrossRef]

Leutz, W.

J. Rička, I. Flammer, W. Leutz, “Single-mode DLS: colloids in opaque porous media,” Prog. Colloid Polym. Sci. 104, 49–58 (1997).

W. Leutz, J. Rička, “On light propagation through glass bead packings,” Opt. Commun. 126, 260–268 (1996).
[CrossRef]

Maret, G.

G. Maret, P. Wolf, “Multiple light scattering from disordered media: the effect of Brownian motion on scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Classical Wave Localisation, P. Sheng, ed. (World Scientific, London, 1989).

Martys, N. S.

L. Lebon, L. Oger, J. Leblond, J. P. Hulin, N. S. Martys, L. M. Schwartz, “Pulsed gradient NMR measurements and numerical simulation of flow velocity distribution in sphere packing,” Phys. Fluids 8, 293–301 (1996).
[CrossRef]

Nossal, R.

Oger, L.

L. Lebon, L. Oger, J. Leblond, J. P. Hulin, N. S. Martys, L. M. Schwartz, “Pulsed gradient NMR measurements and numerical simulation of flow velocity distribution in sphere packing,” Phys. Fluids 8, 293–301 (1996).
[CrossRef]

Pine, D. J.

P. D. Kaplan, M. H. Kao, A. G. Yodh, D. J. Pine, “Geometric constraints for the design of diffusing-wave spectroscopy experiments,” Appl. Opt. 32, 3828–3836 (1993).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101–2127 (1990).
[CrossRef]

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Classical Wave Localisation, P. Sheng, ed. (World Scientific, London, 1989).

Plona, T. J.

E. Charlaix, J. P. Hulin, T. J. Plona, “Experimental study of tracer dispersion in sintered glass porous materials of variable compaction,” Phys. Fluids 30, 1690–1698 (1987).
[CrossRef]

Pusey, P. N.

P. N. Pusey, “Dynamic light scattering by non-ergodic media,” Physica A 157, 705–741 (1989).
[CrossRef]

Ricka, J.

I. Flammer, J. Rička, “Dynamic light scattering with single-mode receivers: partial heterodyning regime,” Appl. Opt. 36, 7508–7517 (1997).
[CrossRef]

J. Rička, I. Flammer, W. Leutz, “Single-mode DLS: colloids in opaque porous media,” Prog. Colloid Polym. Sci. 104, 49–58 (1997).

W. Leutz, J. Rička, “On light propagation through glass bead packings,” Opt. Commun. 126, 260–268 (1996).
[CrossRef]

J. Rička, “Brownian dynamics in strongly scattering porous media: dynamic light scattering with single-mode matching,” Makromol. Chem. Macromol. Symp. 79, 45–55 (1994).
[CrossRef]

J. Rička, “Dynamic light scattering with single-mode and multimode receivers,” Appl. Opt. 32, 2860–2875 (1993).
[CrossRef] [PubMed]

Schätzel, K.

K. Schätzel, “Dead time correction of photon correlation functions,” Appl. Phys. B 41, 95–102 (1986).
[CrossRef]

Schwartz, L. M.

L. Lebon, L. Oger, J. Leblond, J. P. Hulin, N. S. Martys, L. M. Schwartz, “Pulsed gradient NMR measurements and numerical simulation of flow velocity distribution in sphere packing,” Phys. Fluids 8, 293–301 (1996).
[CrossRef]

Scriven, L. E.

Y. E. Kutsovsky, L. E. Scriven, H. T. Davis, B. E. Hammer, “NMR imaging of velocity profiles and velocity distribution in bead packs,” Phys. Fluids 8, 863–871 (1996).
[CrossRef]

Snyder, L. R.

L. R. Snyder, J. J. Kirkland, Introduction to Modern Liquid Chromatography (Wiley, New York, 1979).

Sticher, H.

Y. D. Yan, M. Borkovec, H. Sticher, “Deposition and release of colloidal particles in porous media,” Prog. Colloid Polym. Sci. 98, 132–135 (1995).
[CrossRef]

van de Ven, T. G.

R. T. Foister, T. G. van de Ven, “Diffusion of Brownian particles in shear flows,” J. Fluid Mech. 96, 105–132 (1980).
[CrossRef]

van Kampen, N. G.

N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).

Weitz, D. A.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101–2127 (1990).
[CrossRef]

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Classical Wave Localisation, P. Sheng, ed. (World Scientific, London, 1989).

Wolf, P.

G. Maret, P. Wolf, “Multiple light scattering from disordered media: the effect of Brownian motion on scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

Wolf, P. E.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Classical Wave Localisation, P. Sheng, ed. (World Scientific, London, 1989).

Yan, Y. D.

Y. D. Yan, M. Borkovec, H. Sticher, “Deposition and release of colloidal particles in porous media,” Prog. Colloid Polym. Sci. 98, 132–135 (1995).
[CrossRef]

Yodh, A. G.

Zhu, J. X.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101–2127 (1990).
[CrossRef]

Appl. Opt.

Appl. Phys. B

K. Schätzel, “Dead time correction of photon correlation functions,” Appl. Phys. B 41, 95–102 (1986).
[CrossRef]

J. Chem. Phys.

D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

J. Fluid Mech.

R. T. Foister, T. G. van de Ven, “Diffusion of Brownian particles in shear flows,” J. Fluid Mech. 96, 105–132 (1980).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. (France)

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101–2127 (1990).
[CrossRef]

Macromolecules

M. T. Bishop, K. H. Langley, F. E. Karasz, “Dynamic light-scattering studies of polymer diffusion in porous materials: linear polystyrene in porous glass,” Macromolecules 22, 1220–1231 (1989).
[CrossRef]

Makromol. Chem. Macromol. Symp.

J. Rička, “Brownian dynamics in strongly scattering porous media: dynamic light scattering with single-mode matching,” Makromol. Chem. Macromol. Symp. 79, 45–55 (1994).
[CrossRef]

Opt. Commun.

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

W. Leutz, J. Rička, “On light propagation through glass bead packings,” Opt. Commun. 126, 260–268 (1996).
[CrossRef]

Phys. Fluids

Y. E. Kutsovsky, L. E. Scriven, H. T. Davis, B. E. Hammer, “NMR imaging of velocity profiles and velocity distribution in bead packs,” Phys. Fluids 8, 863–871 (1996).
[CrossRef]

L. Lebon, L. Oger, J. Leblond, J. P. Hulin, N. S. Martys, L. M. Schwartz, “Pulsed gradient NMR measurements and numerical simulation of flow velocity distribution in sphere packing,” Phys. Fluids 8, 293–301 (1996).
[CrossRef]

E. Charlaix, J. P. Hulin, T. J. Plona, “Experimental study of tracer dispersion in sintered glass porous materials of variable compaction,” Phys. Fluids 30, 1690–1698 (1987).
[CrossRef]

W. Johnston, A. Dybbs, “Measurement of fluid velocity inside porous media with a laser anemometer,” Phys. Fluids 18, 913–914 (1975).
[CrossRef]

Phys. Rev. Lett.

M. T. Bishop, K. H. Langley, F. E. Karasz, “Diffusion of a flexible polymer in a random porous material,” Phys. Rev. Lett. 57, 1741–1744 (1986).
[CrossRef] [PubMed]

Physica A

P. N. Pusey, “Dynamic light scattering by non-ergodic media,” Physica A 157, 705–741 (1989).
[CrossRef]

Prog. Colloid Polym. Sci.

J. Rička, I. Flammer, W. Leutz, “Single-mode DLS: colloids in opaque porous media,” Prog. Colloid Polym. Sci. 104, 49–58 (1997).

Y. D. Yan, M. Borkovec, H. Sticher, “Deposition and release of colloidal particles in porous media,” Prog. Colloid Polym. Sci. 98, 132–135 (1995).
[CrossRef]

Trans. Inst. Chem. Eng.

P. C. Carman, “Fluid flow through granular beds,” Trans. Inst. Chem. Eng. 15, 150–156 (1937).

Z. Phys. B

G. Maret, P. Wolf, “Multiple light scattering from disordered media: the effect of Brownian motion on scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

Other

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Classical Wave Localisation, P. Sheng, ed. (World Scientific, London, 1989).

H. Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

L. R. Snyder, J. J. Kirkland, Introduction to Modern Liquid Chromatography (Wiley, New York, 1979).

J. Bear, Dynamics of Fluids in Porous Media (Dover, New York, 1988).

T. G. M. V. de Ven, Colloidal Hydrodynamics (Academic, London, 1989).

N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford Science Publications, Oxford, UK, 1959), Chap. XIV.

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Fig. 1
Fig. 1

Weighting function w(q) for three scatterers of different sizes illuminated with a wavelength of 386 nm (Ar+ 514-nm line in water, wave number k=1.63×107 m-1). The dotted curve corresponds to an infinitely small dipole scatterer with a form factor of P(q)=1. The dashed curve holds for the scatterers used in the experiments presented in this paper. The solid curve represents the distribution for large particles with a radius of gyration of 100 nm. The abscissa goes from 0 to 2k, which corresponds to a scattering angle interval from 0° to 180°. The size of the scatterers influences the experimental probing length scale; the inverse probing length scale is q=1.4k=1/47 nm-1 for the dipole scatterer, q=1.26k=1/49 nm-1 for our experiments, and q=0.83k=1/74 nm-1 for the large particles.

Fig. 2
Fig. 2

Dependence of the intercept of the correlogram on the mean count rate R for different positions of the receiver, i.e., different amounts of local oscillator. The solid curves are the fits with Eq. (35). On the left-hand side the solvent-filled porous medium is investigated, and on the right-hand side latex particles (4 mg/l) were added. At positions with single-mode matching, the intercept of the correlogram is seen to amount to 2, which is possible only with the use of optical single-mode receivers (cf. Refs. 11 and 12).

Fig. 3
Fig. 3

Time scale separation between the matrix dynamics g1f(τ) and the dynamics of the colloidal scatterers, g1s(τ). The data points show the signal correlation function measured without any colloids in the porous medium. The solid curve shows the expected signal correlation function for the moving scatterers inside the glass bead packing, assuming bulk diffusion and no flow. The matrix dynamics is well approximated up to 10 ms by a line (not shown) in the lin–log representation, which corresponds to single exponential decay. The inset illustrates the same information on a log-lin scale.

Fig. 4
Fig. 4

Two examples of measured signal correlation functions for pure thermal motion of the latex particles in the porous medium. Two differing amounts of local oscillator contributions are shown (h=0.23 for the filled circles, h=0.90 for the unfilled circles). The solid curves are the fits to the data. The inset depicts the same data on a log-lin scale.

Fig. 5
Fig. 5

Fitted diffusion coefficient for different values of the heterodyne parameter h. Measurements are presented for two illumination wavelengths. The diffusion coefficient does not depend on the heterodyne parameter h; its value is equal to the bulk diffusion coefficient (solid line) of D=5.11 µm2/s. The concentration of latex particles was 3 mg/l only.

Fig. 6
Fig. 6

Influence of the latex concentration on the measured diffusion coefficient. We see the comparison of the effective diffusion coefficient of the particles in the porous medium measured with DWI (filled circles) compared with simple bulk measurements (filled squares) at equal particle concentration. Repeated-scattering effects are much more pronounced in DWI measurements than in conventional DLS. The measured DWI data were corrected for repeated scattering by using Eq. (34); the result is shown as open circles. At 100 mg/l, deviations become apparent, indicating the limited range of validity of the approximation in Subsection 2.D.

Fig. 7
Fig. 7

Comparison of different speed distributions fitted to the measured data. The experimental seepage velocity was 800 µm/s. We compare four models: (1) short-time cumulant expansion [Eq. (23)] (dotted curves), (2) step distribution (dotted–dashed curves), (3) half-Gaussian distribution (dashed curves), (4) exponential distribution (solid curves). On the right-hand side, the initial behavior of the same data is shown.

Fig. 8
Fig. 8

Correlogram for latex particles flowing with a seepage velocity of 800 µm/s through the glass bead column. The solid curve is the fit function assuming an exponential speed distribution of the latex particles; the fit parameters are Deff=4.6 µm2/s, v=1020 µm/s, and h=0.715. For the matrix dynamics we find that m=0.11 and τc=62 ms.

Fig. 9
Fig. 9

Dependence of the diffusion coefficient on increasing flow rate. The seepage velocity vz is ranging from 0 µm/s to 800 µm/s. The bulk value of the diffusion coefficient is 5.11 µm2/s. The latex concentration is 3 mg/l, corresponding to a particle number density of c=7.6×1015 m-3. The solid circles are measured with λ=514 nm and the open circles with λ=488 nm.

Fig. 10
Fig. 10

Dependence of the average speed of the particles on increasing flow rate, measured at two illumination wavelengths. The slope of the linear dependence is fitted to be 1.20±0.05.

Equations (41)

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J(t)=E(t)E*(t),
E(t)=Em+Ep(t).
E(t)=Ec+Ef(t)+Ep(t).
g2(τ)=1+2h(1-h)g1(τ)+(1-h)2[g1(τ)]2,
g1(τ)=(1-m)g1p(τ)+mg1f(τ).
h=JcJ,m=Jf(t)Jf(t)+Jp(t),
Jc=EcEc*,Jf=EfEf*,andJp=EpEp*.
R=νts=αJ[1-αJθ(2-h2)],
ν0ντν2=1+2h(1-h)g1(τ){1-2Rθ[1+(1-h)2]}+(1-h)2[g1(τ)]2[1-2Rθ(2+h2)].
g1s(τ)(ei·eo)2P(q)F(q, τ)ΩiΩo.
F(q, τ)=exp[iq·ρ(τ)]=d3ρ exp(iq·ρ)p(ρ, τ).
F(q, τ)=d3ρ sin(qρ)qρp(ρ, τ),
g1s(τ)0πdθ02πdϕ02πdψ(sin θ)(ei·eo)2×P(q)F(q, τ).
g1s(τ)0πdθ(sin θ)(1+cos2 θ)P(q(θ))F(q(θ), τ).
g1s(τ)02kdq qk21+1-q22k22×exp(-q2Rg2/3)F(q, τ).
w(q)=qk21+1-q22k22exp(-q2Rg2/3).
w(q)qk2exp(-q2Rg2/3).
F(q, τ)=Fb(q, τ)Fv(q, τ).
Fb(q, τ)=exp(-q2Deffτ).
Fv(q, τ)0dv p(v) sin(qvτ)qvτ,
g1s(τ)02kdq qk21+1-q22k22×exp[-q2(Deffτ+Rg2/3)] arctan(qvτ)qvτ.
Fv(q, τ0)=1-16q2v2τ2+O(τ3).
Fv(q, τ)exp[-(q2v2τ2)/6].
g1s(τ0)=u2+u+12u3[1-exp(-2u)]-1u2,
u=2k2(Rg2/3+Deffτ+v2τ2/6).
g1s(τ)1u[1-exp(-2u)],
u=2k2(Rg2/3+Deffτ+v2τ2/6).
g1p(τ)=11-P(0)n=1P(n)[g1s(τ)]n.
p(n|s)=[μ(s)]nn!exp[-μ(s)],μ(s)=s/lc.
lc=1σϕc.
P(n)=0p(s) [μ(s)]nn!exp[-μ(s)]ds.
g1p(τ)=exp{[g1s(τ)-1]s/lc}s-exp(-s/lc)s1-exp(-s/lc)s,
g1p(τ)exp{[g1s(τ)-1]s/lc}-exp(-s/lc)1-exp(-s/lc).
p1(s)1sj=1 exp-j2π2l*s3L2sinjπl*L×sinjπ(L-l*)Lexp-sla.
p2(s)1s3/2 exp-(L-2l*)24l*s/3exp-sla.
JptJpt+JcE=n=1P(n)=1-exp(-s/lc)s1-exp(-s/lc).
Γ=Γsn1-exp(-n).
g2(0)=2-(1-Rf/R)2,
ν0ντν2=1+2h(1-h)g1(τ){1-2Rθ[1+(1-h)2]}+(1-h)2[g1(τ)]2[1-2Rθ(2+h2)],
g1(τ)=(1-m)g1s+mg1f,
g1s(τ)02kdq qk21+1-q22k22×exp[-q2(Deffτ+Rg2/3)] arctan(qvτ)qvτ.

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