Abstract

Sudden rearrangement of the tightly packed bubbles in a foam can be induced by application of shear and by diffusion of gas from smaller to larger bubbles. Using diffusing-wave spectroscopy, we measure dynamics at times that are short in comparison with the rate of these events, where the relative bubble configurations are fixed. We find a distinctly nonexponential contribution to the autocorrelation function that spans several decades in delay time prior to the full decay caused by rearrangements. The short-time dynamics are independent of changes in the rearrangement rate brought about by alteration of the coarsening rate or by application of shear and are therefore attributed to thermal fluctuations of the bubbles. The magnitude of these microscopic fluctuations can be understood in terms of the macroscopic shear modulus.

© 1997 Optical Society of America

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References

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  1. J. M. Aubert, A. M. Kraynik, and P. B. Rand, “Aqueous foams,” Sci. Am. 254, 74–82 (1986).
    [CrossRef]
  2. A. M. Kraynik, “Foam flows,” Annu. Rev. Fluid Mech. 20, 325–357 (1988).
  3. T. Okuzono and K. Kawasaki, “Intermittent flow behavior of random foams: a computer experiment on foam rheology,” Phys. Rev. E 51, 1246–1253 (1995).
    [CrossRef]
  4. D. J. Durian, “Foam mechanics at the bubble scale,” Phys. Rev. Lett. 75, 4780–4783 (1995).
    [CrossRef] [PubMed]
  5. A. D. Gopal and D. J. Durian, “Nonlinear bubble dynamics in a slowly driven foam,” Phys. Rev. Lett. 75, 2610–2613 (1995).
    [CrossRef] [PubMed]
  6. G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987); D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988); D. A. Weitz and D. J. Pine, in Dynamic Light Scattering, W. Brown, ed. (Oxford U. Press, Oxford, 1992), pp. 652–720.
    [CrossRef] [PubMed]
  7. Gillette Foamy Regular shave cream (Gillette Company, Box 61, Boston, Mass. 02199).
  8. D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
    [CrossRef] [PubMed]
  9. M. U. Vera and D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
    [CrossRef]
  10. D. J. Durian, D. A. Weitz, and D. J. Pine, “Scaling behavior in shaving cream,” Phys. Rev. A 44, R7902–R7905 (1991).
    [CrossRef] [PubMed]
  11. B. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Krieger, Malabar, Fla., 1990).
  12. J. C. Earnshaw and M. Wilson, “A diffusing wave spectroscopy study of constrictive flow of foam,” J. Phys. (Paris) II 6, 713–722 (1996); J. C. Earnshaw and M. Wilson, “Strain-induced dynamics of a flowing foam: an experimental study,” J. Phys. Condens. Matter. 7, L49–L53 (1995); J. C. Earnshaw and A. H. Jaafar, “Diffusing-wave spectroscopy of a flowing foam,” Phys. Rev. E PLEEE8 49, 5408–5411 (1994).
    [CrossRef]
  13. H. Gang, A. H. Krall, and D. A. Weitz, “Thermal fluctuations of the shapes of droplets in dense and compressed emulsions,” Phys. Rev. E 52, 6289–6302 (1995).
    [CrossRef]
  14. A. H. Krall, Z. Huang, and D. A. Weitz, in “Dynamics of fractal colloidal gels,” presented at the 1995 Fall Meeting of the Materials Research Society, Boston, Mass., 1995.
  15. J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
    [CrossRef] [PubMed]
  16. P. N. Pusey and W. van Megen, “Dynamic light scattering by non-ergodic media,” Physica A 157, 705–741 (1989).
    [CrossRef]
  17. T. G. Mason and D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250–1253 (1995).
    [CrossRef] [PubMed]
  18. D. A. Weitz, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pa. 19104 (personal communication, 1996).
  19. We used commercial rheometers, with a sample cell that consists of parallel disks notched to preclude wall slip, and obtain consistent values of the shear modulus by two different measurement methods. The first method applies a controlled oscillatory shear stress and measures the resulting shear strain as functions of oscillation frequency and strain amplitude. Extrapolating to zero frequency and amplitude, we extract the elastic shear modulus of the foam. The second method applies a continuous shear strain at a controlled strain rate and measures the resulting shear stress. Extrapolating to zero strain rate, we extract the yield stress of the foam. We combine this with an independent measurement of the yield strain (Ref. 5) and estimate the shear modulus as G=σyieldyield. These mechanical measurements are done on the same foam, but at an earlier age, when the bubbles are 20 μm in diameter. Since the modulus scales inversely with bubble diameter (Refs. 20 and 21), we divide the rheometer results by 3 to obtain the shear modulus for the 60-μm bubble-diameter foam.
  20. H. M. Princen, “Rheology of foams and highly concentrated emulsions,” J. Colloid Interface Sci. 91, 160–175 (1983).
    [CrossRef]
  21. T. G. Mason, J. Bibette, and D. A. Weitz, “Elasticity of compressed emulsions,” Phys. Rev. Lett. 75, 2051–2054 (1995).
    [CrossRef] [PubMed]

1996 (1)

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

1995 (6)

H. Gang, A. H. Krall, and D. A. Weitz, “Thermal fluctuations of the shapes of droplets in dense and compressed emulsions,” Phys. Rev. E 52, 6289–6302 (1995).
[CrossRef]

T. Okuzono and K. Kawasaki, “Intermittent flow behavior of random foams: a computer experiment on foam rheology,” Phys. Rev. E 51, 1246–1253 (1995).
[CrossRef]

D. J. Durian, “Foam mechanics at the bubble scale,” Phys. Rev. Lett. 75, 4780–4783 (1995).
[CrossRef] [PubMed]

A. D. Gopal and D. J. Durian, “Nonlinear bubble dynamics in a slowly driven foam,” Phys. Rev. Lett. 75, 2610–2613 (1995).
[CrossRef] [PubMed]

T. G. Mason and D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250–1253 (1995).
[CrossRef] [PubMed]

T. G. Mason, J. Bibette, and D. A. Weitz, “Elasticity of compressed emulsions,” Phys. Rev. Lett. 75, 2051–2054 (1995).
[CrossRef] [PubMed]

1992 (1)

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[CrossRef] [PubMed]

1991 (2)

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

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

1989 (1)

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

1986 (1)

J. M. Aubert, A. M. Kraynik, and P. B. Rand, “Aqueous foams,” Sci. Am. 254, 74–82 (1986).
[CrossRef]

1983 (1)

H. M. Princen, “Rheology of foams and highly concentrated emulsions,” J. Colloid Interface Sci. 91, 160–175 (1983).
[CrossRef]

Aubert, J. M.

J. M. Aubert, A. M. Kraynik, and P. B. Rand, “Aqueous foams,” Sci. Am. 254, 74–82 (1986).
[CrossRef]

Bibette, J.

T. G. Mason, J. Bibette, and D. A. Weitz, “Elasticity of compressed emulsions,” Phys. Rev. Lett. 75, 2051–2054 (1995).
[CrossRef] [PubMed]

Chaikin, P. M.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[CrossRef] [PubMed]

Durian, D. J.

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

A. D. Gopal and D. J. Durian, “Nonlinear bubble dynamics in a slowly driven foam,” Phys. Rev. Lett. 75, 2610–2613 (1995).
[CrossRef] [PubMed]

D. J. Durian, “Foam mechanics at the bubble scale,” Phys. Rev. Lett. 75, 4780–4783 (1995).
[CrossRef] [PubMed]

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

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

Gang, H.

H. Gang, A. H. Krall, and D. A. Weitz, “Thermal fluctuations of the shapes of droplets in dense and compressed emulsions,” Phys. Rev. E 52, 6289–6302 (1995).
[CrossRef]

Gopal, A. D.

A. D. Gopal and D. J. Durian, “Nonlinear bubble dynamics in a slowly driven foam,” Phys. Rev. Lett. 75, 2610–2613 (1995).
[CrossRef] [PubMed]

Kawasaki, K.

T. Okuzono and K. Kawasaki, “Intermittent flow behavior of random foams: a computer experiment on foam rheology,” Phys. Rev. E 51, 1246–1253 (1995).
[CrossRef]

Krall, A. H.

H. Gang, A. H. Krall, and D. A. Weitz, “Thermal fluctuations of the shapes of droplets in dense and compressed emulsions,” Phys. Rev. E 52, 6289–6302 (1995).
[CrossRef]

Kraynik, A. M.

J. M. Aubert, A. M. Kraynik, and P. B. Rand, “Aqueous foams,” Sci. Am. 254, 74–82 (1986).
[CrossRef]

Mason, T. G.

T. G. Mason and D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250–1253 (1995).
[CrossRef] [PubMed]

T. G. Mason, J. Bibette, and D. A. Weitz, “Elasticity of compressed emulsions,” Phys. Rev. Lett. 75, 2051–2054 (1995).
[CrossRef] [PubMed]

Milner, S. T.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[CrossRef] [PubMed]

Okuzono, T.

T. Okuzono and K. Kawasaki, “Intermittent flow behavior of random foams: a computer experiment on foam rheology,” Phys. Rev. E 51, 1246–1253 (1995).
[CrossRef]

Pine, D. J.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[CrossRef] [PubMed]

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

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

Princen, H. M.

H. M. Princen, “Rheology of foams and highly concentrated emulsions,” J. Colloid Interface Sci. 91, 160–175 (1983).
[CrossRef]

Pusey, P. N.

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

Rand, P. B.

J. M. Aubert, A. M. Kraynik, and P. B. Rand, “Aqueous foams,” Sci. Am. 254, 74–82 (1986).
[CrossRef]

van Megen, W.

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

Vera, M. U.

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

Weitz, D. A.

T. G. Mason, J. Bibette, and D. A. Weitz, “Elasticity of compressed emulsions,” Phys. Rev. Lett. 75, 2051–2054 (1995).
[CrossRef] [PubMed]

T. G. Mason and D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250–1253 (1995).
[CrossRef] [PubMed]

H. Gang, A. H. Krall, and D. A. Weitz, “Thermal fluctuations of the shapes of droplets in dense and compressed emulsions,” Phys. Rev. E 52, 6289–6302 (1995).
[CrossRef]

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

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

Wu, X.-L.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[CrossRef] [PubMed]

Xue, J.-Z.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[CrossRef] [PubMed]

J. Colloid Interface Sci. (1)

H. M. Princen, “Rheology of foams and highly concentrated emulsions,” J. Colloid Interface Sci. 91, 160–175 (1983).
[CrossRef]

Phys. Rev. A (2)

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

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (3)

H. Gang, A. H. Krall, and D. A. Weitz, “Thermal fluctuations of the shapes of droplets in dense and compressed emulsions,” Phys. Rev. E 52, 6289–6302 (1995).
[CrossRef]

T. Okuzono and K. Kawasaki, “Intermittent flow behavior of random foams: a computer experiment on foam rheology,” Phys. Rev. E 51, 1246–1253 (1995).
[CrossRef]

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

Phys. Rev. Lett. (4)

D. J. Durian, “Foam mechanics at the bubble scale,” Phys. Rev. Lett. 75, 4780–4783 (1995).
[CrossRef] [PubMed]

A. D. Gopal and D. J. Durian, “Nonlinear bubble dynamics in a slowly driven foam,” Phys. Rev. Lett. 75, 2610–2613 (1995).
[CrossRef] [PubMed]

T. G. Mason and D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250–1253 (1995).
[CrossRef] [PubMed]

T. G. Mason, J. Bibette, and D. A. Weitz, “Elasticity of compressed emulsions,” Phys. Rev. Lett. 75, 2051–2054 (1995).
[CrossRef] [PubMed]

Physica A (1)

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

Sci. Am. (1)

J. M. Aubert, A. M. Kraynik, and P. B. Rand, “Aqueous foams,” Sci. Am. 254, 74–82 (1986).
[CrossRef]

Science (1)

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

Other (8)

A. M. Kraynik, “Foam flows,” Annu. Rev. Fluid Mech. 20, 325–357 (1988).

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987); D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988); D. A. Weitz and D. J. Pine, in Dynamic Light Scattering, W. Brown, ed. (Oxford U. Press, Oxford, 1992), pp. 652–720.
[CrossRef] [PubMed]

Gillette Foamy Regular shave cream (Gillette Company, Box 61, Boston, Mass. 02199).

D. A. Weitz, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pa. 19104 (personal communication, 1996).

We used commercial rheometers, with a sample cell that consists of parallel disks notched to preclude wall slip, and obtain consistent values of the shear modulus by two different measurement methods. The first method applies a controlled oscillatory shear stress and measures the resulting shear strain as functions of oscillation frequency and strain amplitude. Extrapolating to zero frequency and amplitude, we extract the elastic shear modulus of the foam. The second method applies a continuous shear strain at a controlled strain rate and measures the resulting shear stress. Extrapolating to zero strain rate, we extract the yield stress of the foam. We combine this with an independent measurement of the yield strain (Ref. 5) and estimate the shear modulus as G=σyieldyield. These mechanical measurements are done on the same foam, but at an earlier age, when the bubbles are 20 μm in diameter. Since the modulus scales inversely with bubble diameter (Refs. 20 and 21), we divide the rheometer results by 3 to obtain the shear modulus for the 60-μm bubble-diameter foam.

A. H. Krall, Z. Huang, and D. A. Weitz, in “Dynamics of fractal colloidal gels,” presented at the 1995 Fall Meeting of the Materials Research Society, Boston, Mass., 1995.

B. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Krieger, Malabar, Fla., 1990).

J. C. Earnshaw and M. Wilson, “A diffusing wave spectroscopy study of constrictive flow of foam,” J. Phys. (Paris) II 6, 713–722 (1996); J. C. Earnshaw and M. Wilson, “Strain-induced dynamics of a flowing foam: an experimental study,” J. Phys. Condens. Matter. 7, L49–L53 (1995); J. C. Earnshaw and A. H. Jaafar, “Diffusing-wave spectroscopy of a flowing foam,” Phys. Rev. E PLEEE8 49, 5408–5411 (1994).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Square of the normalized field autocorrelation function for a quiescent foam with an average bubble size of 60 µm and for a semidilute dispersion of polystyrene spheres in water. The full decays due to bubble rearrangements and polyball diffusion, respectively, are evident at late τ. The semilogarithmic plot of the same data in the inset shows that this part of the decay is approximately exponential. The dashed curve is fitted to the late-τ foam dynamics. Note the nonexponential decay of the foam data, extending over several decades of delay time.

Fig. 2
Fig. 2

Top plot: time evolution due to coarsening at different ambient temperatures, as labeled, of the average bubble diameter. Note that the bubble diameter grows as a power law of age with exponent 1/2 for all three foams. Bottom plot: average amount that a bubble radius grows in the time τ0 between rearrangements; the symbols indicate temperature, as in the top plot. Note that this growth is approximately 50 nm, independent of foam age and temperature.

Fig. 3
Fig. 3

Top plot: square of the normalized field autocorrelation function for an L= 8 mm sample during, before, and after the application of a shear strain. Bottom plot: similar data for quiescent 7-mm samples at different ambient temperatures, as labeled. Notice that the partial decay of the correlation data at short τ is independent of the bubble rearrangement rate, for all the cases. The dashed curve is a fit than assumes no short-τ bubble dynamics.

Fig. 4
Fig. 4

Inversion parameter Y/k2, defined by Eq. (4), as a function of delay time τ for the data of Fig. 3 and for similar shear data measured in a 10-mm-thick sample. Note that the thermal fluctuations are independent of the rearrangement dynamics.

Fig. 5
Fig. 5

Inversion data of Fig. 4, replotted on linear axes to emphasize the rearrangement dynamics. The rearrangement data extrapolates to δ2 = 2.0 ± 0.5 nm2, at zero τ, and sets δ=14±3 Å as the amplitude of the thermal fluctuations.

Fig. 6
Fig. 6

Square of the normalized field autocorrelation function for quiescent 20°C foams with an average bubble diameter of 60 µm. From left to right, the solid curves denote samples of thickness 10, 7, 4, and 2 mm. The dashed curves are fits to the data by use of Eqs. (3) and (5). The dotted curve is a fit that assumes that thermal fluctuations do not saturate.

Fig. 7
Fig. 7

Amplitude of thermal fluctuations, δ, as a function of bubble diameter, at different ambient temperatures and for various sample thicknesses. The single solid line and the two dashed lines correspond to the average value of δ and the uncertainty in δ, respectively, for the 60-µm bubble foams discussed in the text.

Equations (5)

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

I(τ)I(0)/I2=1+β|g1(τ)|2,
g1(τ)=(L/l*)+2ze1+ze×sinh(x)+zex cosh(x)(1+ze2x)sinh[(L/l*)x]+2zex cosh[(L/l*)x],
g1(τ)=(L/l*)Xsinh[(L/l*)X](L/l*)6τ/τ0sinh[(L/l*)6τ/τ0],
g1(τ)=(L/l*)Ysinh[(L/l*)Y],
X(τ)=k2δ2[1-exp(-fτ)]α,

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