Abstract

The multiple scattering of light by aqueous foams is systematically studied as a function of wavelength, bubble size, and liquid fraction. Results are analyzed in terms of the transport mean free path of the photons and an extrapolation length ratio for the diffuse photon concentration field. The wavelength dependence is minimal and may be attributed entirely to the wavelength dependence of the refractive index of water rather than thin-film interference effects. The transport mean free path is found to be proportional to the bubble diameter and the reciprocal of the square root of liquid fraction. The extrapolation length ratio varies almost linearly with liquid fraction between the values for water–glass–air and air–glass–air interfaces.

© 2001 Optical Society of America

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References

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  1. D. Weaire, S. Hutzler, The Physics of Foams (Oxford U. Press, New York, 1999).
  2. D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
    [CrossRef] [PubMed]
  3. C. Monnereau, M. Vignes-Adler, “Dynamics of 3D real foam coarsening,” Phys. Rev. Lett. 80, 5228–5231 (1998).
    [CrossRef]
  4. M. R. Fetterman, E. Tan, L. Ying, R. A. Stack, D. L. Marks, S. Feller, E. Cull, J. M. Sullivan, D. C. Munson, S. T. Thoroddsen, D. J. Brady, “Tomographic imaging of foam,” Opt. Express 7, 186–197 (2000), http://www.opticsexpress.org/oearchive/source/23156.htm .
  5. D. J. Durian, D. A. Weitz, D. J. Pine, “Scaling behavior in shaving cream,” Phys. Rev. A 44, R7902–R7905 (1991).
    [CrossRef] [PubMed]
  6. J. C. Earnshaw, A. H. Jaafar, “Diffusing-wave spectroscopy of a flowing foam,” Phys. Rev. E 49, 5408–5411 (1994).
    [CrossRef]
  7. R. Hohler, S. Cohen-Addad, H. Hoballah, “Periodic nonlinear bubble motion in aqueous foam under oscillating shear strain,” Phys. Rev. Lett. 79, 1154–1157 (1997).
    [CrossRef]
  8. A. D. Gopal, D. J. Durian, “Shear-induced ‘melting’ of an aqueous foam,” J. Coll. I. Sci. 213, 169–178 (1999).
    [CrossRef]
  9. P. D. Kaplan, A. D. Dinsmore, A. G. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
    [CrossRef]
  10. D. A. Weitz, D. J. Pine, in Dynamic Light Scattering: The Method and Some Applications, W. Brown, ed. (Claredon, Oxford, 1993), pp. 652–720.
  11. G. Maret, “Diffusing-wave spectroscopy,” Curr. Opin. Colloid Interface Sci. 2, 251–257 (1997).
    [CrossRef]
  12. A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
    [CrossRef]
  13. J. G. Fujimoto, M. S. Patterson, eds., Advances in Optical Imaging and Photon Migration Vol. 21 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 1998).
  14. M. U. Vera, D. J. Durian, “Gas and liquid transport in foam: the coarsening equation,” submitted to Phys. Rev. Lett.
  15. A. Saint-Jalmes, M. U. Vera, D. J. Durian, “Uniform foam production by turbulent mixing: new results on free drainage vs liquid content,” Eur. Phys. J. B 12, 67–73 (1999).
    [CrossRef]
  16. P. B. Rand, “Stabilized aqueous foam systems and concentrate and method for making them,” U.S. Patent no. 4,442,018 (10April1984).
  17. A. Saint-Jalmes, D. J. Durian, “Vanishing elasticity for wet foams: equivalence with emulsions and role of polydispersity,” J. Rheol. 43, 1411–1422 (1999).
    [CrossRef]
  18. M. U. Vera, D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
    [CrossRef]
  19. P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
    [CrossRef]
  20. S. A. Koehler, S. Hilgenfeldt, H. A. Stone, “A generalized view of foam drainage: experiment and theory,” Langmuir 16, 6327–6341 (2000).
    [CrossRef]

2000

1999

A. Saint-Jalmes, M. U. Vera, D. J. Durian, “Uniform foam production by turbulent mixing: new results on free drainage vs liquid content,” Eur. Phys. J. B 12, 67–73 (1999).
[CrossRef]

A. Saint-Jalmes, D. J. Durian, “Vanishing elasticity for wet foams: equivalence with emulsions and role of polydispersity,” J. Rheol. 43, 1411–1422 (1999).
[CrossRef]

A. D. Gopal, D. J. Durian, “Shear-induced ‘melting’ of an aqueous foam,” J. Coll. I. Sci. 213, 169–178 (1999).
[CrossRef]

1998

C. Monnereau, M. Vignes-Adler, “Dynamics of 3D real foam coarsening,” Phys. Rev. Lett. 80, 5228–5231 (1998).
[CrossRef]

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

1997

G. Maret, “Diffusing-wave spectroscopy,” Curr. Opin. Colloid Interface Sci. 2, 251–257 (1997).
[CrossRef]

R. Hohler, S. Cohen-Addad, H. Hoballah, “Periodic nonlinear bubble motion in aqueous foam under oscillating shear strain,” Phys. Rev. Lett. 79, 1154–1157 (1997).
[CrossRef]

1996

M. U. Vera, D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

1995

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

1994

J. C. Earnshaw, A. H. Jaafar, “Diffusing-wave spectroscopy of a flowing foam,” Phys. Rev. E 49, 5408–5411 (1994).
[CrossRef]

P. D. Kaplan, A. D. Dinsmore, A. G. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

1991

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

D. J. Durian, D. A. Weitz, D. J. Pine, “Scaling behavior in shaving cream,” Phys. Rev. A 44, R7902–R7905 (1991).
[CrossRef] [PubMed]

Brady, D. J.

Chance, B.

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

Cohen-Addad, S.

R. Hohler, S. Cohen-Addad, H. Hoballah, “Periodic nonlinear bubble motion in aqueous foam under oscillating shear strain,” Phys. Rev. Lett. 79, 1154–1157 (1997).
[CrossRef]

Cull, E.

Dinsmore, A. D.

P. D. Kaplan, A. D. Dinsmore, A. G. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

Durian, D. J.

A. D. Gopal, D. J. Durian, “Shear-induced ‘melting’ of an aqueous foam,” J. Coll. I. Sci. 213, 169–178 (1999).
[CrossRef]

A. Saint-Jalmes, M. U. Vera, D. J. Durian, “Uniform foam production by turbulent mixing: new results on free drainage vs liquid content,” Eur. Phys. J. B 12, 67–73 (1999).
[CrossRef]

A. Saint-Jalmes, D. J. Durian, “Vanishing elasticity for wet foams: equivalence with emulsions and role of polydispersity,” J. Rheol. 43, 1411–1422 (1999).
[CrossRef]

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

M. U. Vera, D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

D. J. Durian, D. A. Weitz, D. J. Pine, “Scaling behavior in shaving cream,” Phys. Rev. A 44, R7902–R7905 (1991).
[CrossRef] [PubMed]

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

M. U. Vera, D. J. Durian, “Gas and liquid transport in foam: the coarsening equation,” submitted to Phys. Rev. Lett.

Earnshaw, J. C.

J. C. Earnshaw, A. H. Jaafar, “Diffusing-wave spectroscopy of a flowing foam,” Phys. Rev. E 49, 5408–5411 (1994).
[CrossRef]

Feller, S.

Fetterman, M. R.

Gopal, A. D.

A. D. Gopal, D. J. Durian, “Shear-induced ‘melting’ of an aqueous foam,” J. Coll. I. Sci. 213, 169–178 (1999).
[CrossRef]

Hilgenfeldt, S.

S. A. Koehler, S. Hilgenfeldt, H. A. Stone, “A generalized view of foam drainage: experiment and theory,” Langmuir 16, 6327–6341 (2000).
[CrossRef]

Hoballah, H.

R. Hohler, S. Cohen-Addad, H. Hoballah, “Periodic nonlinear bubble motion in aqueous foam under oscillating shear strain,” Phys. Rev. Lett. 79, 1154–1157 (1997).
[CrossRef]

Hohler, R.

R. Hohler, S. Cohen-Addad, H. Hoballah, “Periodic nonlinear bubble motion in aqueous foam under oscillating shear strain,” Phys. Rev. Lett. 79, 1154–1157 (1997).
[CrossRef]

Hutzler, S.

D. Weaire, S. Hutzler, The Physics of Foams (Oxford U. Press, New York, 1999).

Jaafar, A. H.

J. C. Earnshaw, A. H. Jaafar, “Diffusing-wave spectroscopy of a flowing foam,” Phys. Rev. E 49, 5408–5411 (1994).
[CrossRef]

Kaplan, P. D.

P. D. Kaplan, A. D. Dinsmore, A. G. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

Koehler, S. A.

S. A. Koehler, S. Hilgenfeldt, H. A. Stone, “A generalized view of foam drainage: experiment and theory,” Langmuir 16, 6327–6341 (2000).
[CrossRef]

Lemieux, P.-A.

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Maret, G.

G. Maret, “Diffusing-wave spectroscopy,” Curr. Opin. Colloid Interface Sci. 2, 251–257 (1997).
[CrossRef]

Marks, D. L.

Monnereau, C.

C. Monnereau, M. Vignes-Adler, “Dynamics of 3D real foam coarsening,” Phys. Rev. Lett. 80, 5228–5231 (1998).
[CrossRef]

Munson, D. C.

Pine, D. J.

P. D. Kaplan, A. D. Dinsmore, A. G. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

D. J. Durian, D. A. Weitz, D. J. Pine, “Scaling behavior in shaving cream,” Phys. Rev. A 44, R7902–R7905 (1991).
[CrossRef] [PubMed]

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

D. A. Weitz, D. J. Pine, in Dynamic Light Scattering: The Method and Some Applications, W. Brown, ed. (Claredon, Oxford, 1993), pp. 652–720.

Rand, P. B.

P. B. Rand, “Stabilized aqueous foam systems and concentrate and method for making them,” U.S. Patent no. 4,442,018 (10April1984).

Saint-Jalmes, A.

A. Saint-Jalmes, D. J. Durian, “Vanishing elasticity for wet foams: equivalence with emulsions and role of polydispersity,” J. Rheol. 43, 1411–1422 (1999).
[CrossRef]

A. Saint-Jalmes, M. U. Vera, D. J. Durian, “Uniform foam production by turbulent mixing: new results on free drainage vs liquid content,” Eur. Phys. J. B 12, 67–73 (1999).
[CrossRef]

Stack, R. A.

Stone, H. A.

S. A. Koehler, S. Hilgenfeldt, H. A. Stone, “A generalized view of foam drainage: experiment and theory,” Langmuir 16, 6327–6341 (2000).
[CrossRef]

Sullivan, J. M.

Tan, E.

Thoroddsen, S. T.

Vera, M. U.

A. Saint-Jalmes, M. U. Vera, D. J. Durian, “Uniform foam production by turbulent mixing: new results on free drainage vs liquid content,” Eur. Phys. J. B 12, 67–73 (1999).
[CrossRef]

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

M. U. Vera, D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

M. U. Vera, D. J. Durian, “Gas and liquid transport in foam: the coarsening equation,” submitted to Phys. Rev. Lett.

Vignes-Adler, M.

C. Monnereau, M. Vignes-Adler, “Dynamics of 3D real foam coarsening,” Phys. Rev. Lett. 80, 5228–5231 (1998).
[CrossRef]

Weaire, D.

D. Weaire, S. Hutzler, The Physics of Foams (Oxford U. Press, New York, 1999).

Weitz, D. A.

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

D. J. Durian, D. A. Weitz, D. J. Pine, “Scaling behavior in shaving cream,” Phys. Rev. A 44, R7902–R7905 (1991).
[CrossRef] [PubMed]

D. A. Weitz, D. J. Pine, in Dynamic Light Scattering: The Method and Some Applications, W. Brown, ed. (Claredon, Oxford, 1993), pp. 652–720.

Ying, L.

Yodh, A. G.

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

P. D. Kaplan, A. D. Dinsmore, A. G. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

Curr. Opin. Colloid Interface Sci.

G. Maret, “Diffusing-wave spectroscopy,” Curr. Opin. Colloid Interface Sci. 2, 251–257 (1997).
[CrossRef]

Eur. Phys. J. B

A. Saint-Jalmes, M. U. Vera, D. J. Durian, “Uniform foam production by turbulent mixing: new results on free drainage vs liquid content,” Eur. Phys. J. B 12, 67–73 (1999).
[CrossRef]

J. Coll. I. Sci.

A. D. Gopal, D. J. Durian, “Shear-induced ‘melting’ of an aqueous foam,” J. Coll. I. Sci. 213, 169–178 (1999).
[CrossRef]

J. Rheol.

A. Saint-Jalmes, D. J. Durian, “Vanishing elasticity for wet foams: equivalence with emulsions and role of polydispersity,” J. Rheol. 43, 1411–1422 (1999).
[CrossRef]

Langmuir

S. A. Koehler, S. Hilgenfeldt, H. A. Stone, “A generalized view of foam drainage: experiment and theory,” Langmuir 16, 6327–6341 (2000).
[CrossRef]

Opt. Express

Phys. Rev. A

D. J. Durian, D. A. Weitz, D. J. Pine, “Scaling behavior in shaving cream,” Phys. Rev. A 44, R7902–R7905 (1991).
[CrossRef] [PubMed]

Phys. Rev. E

J. C. Earnshaw, A. H. Jaafar, “Diffusing-wave spectroscopy of a flowing foam,” Phys. Rev. E 49, 5408–5411 (1994).
[CrossRef]

P. D. Kaplan, A. D. Dinsmore, A. G. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

M. U. Vera, D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Phys. Rev. Lett.

R. Hohler, S. Cohen-Addad, H. Hoballah, “Periodic nonlinear bubble motion in aqueous foam under oscillating shear strain,” Phys. Rev. Lett. 79, 1154–1157 (1997).
[CrossRef]

C. Monnereau, M. Vignes-Adler, “Dynamics of 3D real foam coarsening,” Phys. Rev. Lett. 80, 5228–5231 (1998).
[CrossRef]

Phys. Today

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

Science

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

Other

D. Weaire, S. Hutzler, The Physics of Foams (Oxford U. Press, New York, 1999).

D. A. Weitz, D. J. Pine, in Dynamic Light Scattering: The Method and Some Applications, W. Brown, ed. (Claredon, Oxford, 1993), pp. 652–720.

J. G. Fujimoto, M. S. Patterson, eds., Advances in Optical Imaging and Photon Migration Vol. 21 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 1998).

M. U. Vera, D. J. Durian, “Gas and liquid transport in foam: the coarsening equation,” submitted to Phys. Rev. Lett.

P. B. Rand, “Stabilized aqueous foam systems and concentrate and method for making them,” U.S. Patent no. 4,442,018 (10April1984).

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Figures (3)

Fig. 1
Fig. 1

Extrapolation length ratio versus liquid fraction. Results are obtained from the angular dependence of the diffusely transmitted light. Filled circles are for the AOS foams, and the open square is the value found in Ref. 18 for Gillette Foamy Regular.

Fig. 2
Fig. 2

Ratio of transport mean free path to bubble diameter, versus wavelength, for several different liquid fractions as labeled. The foam samples are made from aqueous solution of AOS and have an average bubble diameter of ∼100 µm. The error bars shown for ε = 0.98 indicate systematic errors and are representative of all samples.

Fig. 3
Fig. 3

Ratio of transport mean free path to bubble diameter, versus liquid fraction, for both homemade foams of AOS and Gillette Foamy Regular as labeled. The open plus is for the value observed in Ref. 2. The solid line is the universal scaling function l*/D = 1/ε, which is predicted to hold for any aqeuous foam in which scattering is dominated by the Plateau borders. The short-dashed curve holds for the Mie regime in which the gas bubbles are spherical and distant from nearest neighbors. The long-dashed curve is an empirical form, l*/D = 1.5 + 0.4/ε, which describes the full range of data quite well.

Equations (3)

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

Pμeμe=32neni2ze+μi1-Rμi,
TdBdsinh3μa/3μa+ze cosh3μasinhL-13μa/3μa+ze coshL-13μa,
l*/D=1/ε,

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