Abstract

We present a new technique for measuring velocity gradients for laminar shear flow, using dynamic light scattering in the strongly multiple-scattering regime. We derive temporal autocorrelation functions for multiply scattered light, taking into account particle displacements arising from deterministic shear flow and random Brownian motion. The laminar shear flow and Brownian motion are characterized by the relaxation rates τS1=Γ¯k0l*/30 and τB−1 = Dk02, respectively, where Γ¯ is the mean shear rate of the scatterers, k0 = 2πn/λ is the wave number in the scattering medium, l* is the transport mean free path of the photons, and D is the diffusion coefficient of the scatterers. We obtain excellent agreement between theory and experiment over a wide range of shear rates, 0.5sec1<Γ¯<200sec1. In addition, the autocorrelation function for forward scattering is independent of the scattering properties of the medium and depends only on the mean shear rate and sample thickness when τS is much less than τB. Thus the mean shear rate can be simply determined by a single measurement.

© 1990 Optical Society of America

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References

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  1. G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555 (1980).
    [CrossRef]
  2. P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon-correlation spectroscopy,” Phys. Rev. A 37, 2125 (1988).
    [CrossRef] [PubMed]
  3. P. M. Chaikin, J. M. di Meglio, W. D. Dozier, H. M. Lindsay, D. A. Weitz, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 65.
  4. P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25 (1983).
    [CrossRef]
  5. R. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097 (1981).
    [CrossRef] [PubMed]
  6. A. Ishimaru, Wave Propagation in Random Media (Academic, New York, 1978).
  7. G. Maret, P. E. Wolf, “Multiple light scattering from disordered media: the effect of Brownian motion of scatterers,” Z. Phys. B 65, 409 (1987).
    [CrossRef]
  8. D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134 (1988).
    [CrossRef] [PubMed]
  9. F. C. MacKintosh, S. John, Phys. Rev. B 40, 2383 (1989).
    [CrossRef]
  10. The correction to Brownian motion that is due to the convective flow is called Taylor dispersion. Taylor dispersion modifies particle diffusion in the direction of velocity gradient. In the entire range of shear rate in this experiment the correction (Γτ)2/3 is much smaller than 1. This justifies our approximation that the particle diffusion and convective shear are decoupled.
  11. For some scattering geometries there will be an additional term in the sum corresponding to the difference between the input and output wave vectors. This term is proportional to the velocity (rather than to the velocity gradient) and does not contribute to the homodyne correlation function. We also note that for the common case that the flow direction is perpendicular to the input and output wave vectors, this term is identically zero.
  12. G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge U. Press, Cambridge, 1977), p. 83.
  13. More generally, to include the effects of particle interactions, one must replace F(q) by the full scattering function S(q)F(q), where S(q) is the structure factor. We note that in these experiments, however, the volume fraction of PSS’s is low (ϕ= 0.02) and the Coulomb interaction between spheres is highly screened. Under these conditions, S(q) ≃ 1.
  14. B. Chu, Laser Light Scattering (Academic, New York, 1974), pp. 101–104.
  15. L. D. Landau, E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1984), p. 55.
  16. Carboxylated polystyrene spheres were purchased from Duke Scientific, Palo Alto, California.
  17. D. J. Tritton, Physical Fluid Dynamics, 2nd ed. (Clarendon, Oxford, 1988), p. 20.
  18. P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (Paris) 49, 63 (1988).
    [CrossRef]
  19. E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77 (1988).
    [CrossRef]
  20. We note that the average intensity of transmitted light, 〈I〉, through the sample did not vary with Γ¯. Since 〈I〉 ∼ l*/L, we conclude that l*, and hence P(s), does not vary with Γ¯ for our samples (see Refs. 6 and 8). This also suggests that S(q) is essentially independent of Γ for the weakly interacting samples used in this study.
  21. J. S. Huang, M. W. Kim, “Critical behavior of a microemulsion,” Phys. Rev. Lett. 47, 1462 (1981).
    [CrossRef]
  22. S. H. Chen, T. L. Lin, J. S. Huang, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 285.

1989 (1)

F. C. MacKintosh, S. John, Phys. Rev. B 40, 2383 (1989).
[CrossRef]

1988 (4)

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon-correlation spectroscopy,” Phys. Rev. A 37, 2125 (1988).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef] [PubMed]

1987 (1)

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

1983 (1)

P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25 (1983).
[CrossRef]

1981 (2)

R. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097 (1981).
[CrossRef] [PubMed]

J. S. Huang, M. W. Kim, “Critical behavior of a microemulsion,” Phys. Rev. Lett. 47, 1462 (1981).
[CrossRef]

1980 (1)

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555 (1980).
[CrossRef]

Akkermans, E.

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

Batchelor, G. K.

G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge U. Press, Cambridge, 1977), p. 83.

Bonner, R.

Chaikin, P. M.

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef] [PubMed]

P. M. Chaikin, J. M. di Meglio, W. D. Dozier, H. M. Lindsay, D. A. Weitz, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 65.

Chan, C. K.

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon-correlation spectroscopy,” Phys. Rev. A 37, 2125 (1988).
[CrossRef] [PubMed]

Chen, S. H.

S. H. Chen, T. L. Lin, J. S. Huang, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 285.

Chu, B.

B. Chu, Laser Light Scattering (Academic, New York, 1974), pp. 101–104.

di Meglio, J. M.

P. M. Chaikin, J. M. di Meglio, W. D. Dozier, H. M. Lindsay, D. A. Weitz, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 65.

Dozier, W. D.

P. M. Chaikin, J. M. di Meglio, W. D. Dozier, H. M. Lindsay, D. A. Weitz, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 65.

Fuller, G. G.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555 (1980).
[CrossRef]

Goldburg, W. I.

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon-correlation spectroscopy,” Phys. Rev. A 37, 2125 (1988).
[CrossRef] [PubMed]

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef] [PubMed]

Huang, J. S.

J. S. Huang, M. W. Kim, “Critical behavior of a microemulsion,” Phys. Rev. Lett. 47, 1462 (1981).
[CrossRef]

S. H. Chen, T. L. Lin, J. S. Huang, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 285.

Ishimaru, A.

A. Ishimaru, Wave Propagation in Random Media (Academic, New York, 1978).

John, S.

F. C. MacKintosh, S. John, Phys. Rev. B 40, 2383 (1989).
[CrossRef]

Kim, M. W.

J. S. Huang, M. W. Kim, “Critical behavior of a microemulsion,” Phys. Rev. Lett. 47, 1462 (1981).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1984), p. 55.

Leal, L. G.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555 (1980).
[CrossRef]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1984), p. 55.

Lin, T. L.

S. H. Chen, T. L. Lin, J. S. Huang, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 285.

Lindsay, H. M.

P. M. Chaikin, J. M. di Meglio, W. D. Dozier, H. M. Lindsay, D. A. Weitz, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 65.

MacKintosh, F. C.

F. C. MacKintosh, S. John, Phys. Rev. B 40, 2383 (1989).
[CrossRef]

Maret, G.

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

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

Maynard, R.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

Nossal, R.

Pieranski, P.

P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25 (1983).
[CrossRef]

Pine, D. J.

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef] [PubMed]

Rallison, J. M.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555 (1980).
[CrossRef]

Schmidt, R. L.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555 (1980).
[CrossRef]

Sirivat, A.

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon-correlation spectroscopy,” Phys. Rev. A 37, 2125 (1988).
[CrossRef] [PubMed]

Tong, P.

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon-correlation spectroscopy,” Phys. Rev. A 37, 2125 (1988).
[CrossRef] [PubMed]

Tritton, D. J.

D. J. Tritton, Physical Fluid Dynamics, 2nd ed. (Clarendon, Oxford, 1988), p. 20.

Weitz, D. A.

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef] [PubMed]

P. M. Chaikin, J. M. di Meglio, W. D. Dozier, H. M. Lindsay, D. A. Weitz, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 65.

Wolf, P. E.

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

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

Appl. Opt. (1)

Contemp. Phys. (1)

P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25 (1983).
[CrossRef]

J. Fluid Mech. (1)

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555 (1980).
[CrossRef]

J. Phys. (Paris) (2)

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

Phys. Rev. A (1)

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon-correlation spectroscopy,” Phys. Rev. A 37, 2125 (1988).
[CrossRef] [PubMed]

Phys. Rev. B (1)

F. C. MacKintosh, S. John, Phys. Rev. B 40, 2383 (1989).
[CrossRef]

Phys. Rev. Lett. (2)

J. S. Huang, M. W. Kim, “Critical behavior of a microemulsion,” Phys. Rev. Lett. 47, 1462 (1981).
[CrossRef]

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef] [PubMed]

Z. Phys. B (1)

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

Other (12)

We note that the average intensity of transmitted light, 〈I〉, through the sample did not vary with Γ¯. Since 〈I〉 ∼ l*/L, we conclude that l*, and hence P(s), does not vary with Γ¯ for our samples (see Refs. 6 and 8). This also suggests that S(q) is essentially independent of Γ for the weakly interacting samples used in this study.

P. M. Chaikin, J. M. di Meglio, W. D. Dozier, H. M. Lindsay, D. A. Weitz, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 65.

A. Ishimaru, Wave Propagation in Random Media (Academic, New York, 1978).

The correction to Brownian motion that is due to the convective flow is called Taylor dispersion. Taylor dispersion modifies particle diffusion in the direction of velocity gradient. In the entire range of shear rate in this experiment the correction (Γτ)2/3 is much smaller than 1. This justifies our approximation that the particle diffusion and convective shear are decoupled.

For some scattering geometries there will be an additional term in the sum corresponding to the difference between the input and output wave vectors. This term is proportional to the velocity (rather than to the velocity gradient) and does not contribute to the homodyne correlation function. We also note that for the common case that the flow direction is perpendicular to the input and output wave vectors, this term is identically zero.

G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge U. Press, Cambridge, 1977), p. 83.

More generally, to include the effects of particle interactions, one must replace F(q) by the full scattering function S(q)F(q), where S(q) is the structure factor. We note that in these experiments, however, the volume fraction of PSS’s is low (ϕ= 0.02) and the Coulomb interaction between spheres is highly screened. Under these conditions, S(q) ≃ 1.

B. Chu, Laser Light Scattering (Academic, New York, 1974), pp. 101–104.

L. D. Landau, E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1984), p. 55.

Carboxylated polystyrene spheres were purchased from Duke Scientific, Palo Alto, California.

D. J. Tritton, Physical Fluid Dynamics, 2nd ed. (Clarendon, Oxford, 1988), p. 20.

S. H. Chen, T. L. Lin, J. S. Huang, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 285.

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Figures (6)

Fig. 1
Fig. 1

Geometry for the calculation of the phase shift due to shear for the ith scattering event: The flow is in the x direction, and the velocity gradient, Γ, is in the z direction. The distance between successive events is Λi = |ri+1(0) − ri(0)|, and the change in position of ith scatterer in the time interval t owing to convective shear is ΔriS(τ).

Fig. 2
Fig. 2

Sample geometry: The cell was made of quartz, which has the following dimensions: a = 0.1 cm, b = 1.2 cm, and c = 50 cm. A suspension of 0.415-μm-diam PSS’s at a volume fraction of 2% was pumped through the cell with a 50-cm3 syringe. The resulting velocity profiles with Vmax at the front are shown. The incident laser beam, directed along the z axis and polarized at 45 deg with respect to the x axis, passed through the center of the cell, where the only velocity gradient was in z direction.

Fig. 3
Fig. 3

Log[G2(τ)] versus τ without flow. The circles (upper curve) and squares (lower curve) are experimental data for backscattering and forward scattering, respectively. The curves were fitted to the data using Eqs. (8) and (9) with Eq. (10) and letting τS → ∞. With γ = 2.2, we found l B * = 83 μ m and l F * = 105 μ m for backscattering and forward scattering, respectively.

Fig. 4
Fig. 4

Log[G2(τ)] versus τ at a pumping rate of 0.165 cm3/sec. The circles (upper curve) and squares (lower curve) are experimental data for backscattering and forward scattering, respectively. The curves were fitted to the data using Eqs. (9) and (10) with Eq. (10). The values of γ and l* were taken from quiescent measurements, so that the only fitting parameter is τS.

Fig. 5
Fig. 5

1/τSk0l* versus average shear rate Γ ¯. The meanings of the symbols are the same as in Figs. 2 and 3. For the entire range of experimentally accessible pumping rates, 0.5 sec 1 < Γ ¯ < 200 sec 1, our measurements agree with the theory, which is plotted as a straight line. The inset is an expanded view for low Γ ¯.

Fig. 6
Fig. 6

Log[G2(τ)] versus τ for measurement at the same pumping rate as in Fig. 5 but with different PSS concentrations. The measurements were made in the forward-scattering geometry with a pumping rate of 0.15 cm3/sec. The PSS’s at volume fractions of 2% (squares), 1% (triangles), 0.5% (circles), and 0.25% (crosses) were used. Fits to Eq. (9) gave l* = 105, 215, 420, and 780 μm, respectively for the same concentrations given above. As is indicated, G2(τ) was independent of l*, in agreement with the theory.

Equations (13)

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Φ ( n ) ( τ ) = i = 1 n { q i Δ r i B ( τ ) + k i [ Δ r i + 1 S ( τ ) Δ r i S ( τ ) ] } .
k i [ Δ r i + 1 S ( τ ) Δ r i S ( τ ) ] = Γ τ k 0 Λ i cos θ i sin θ i cos ϕ i ,
G 1 ( n ) ( τ ) = E ( n ) ( 0 ) E * ( n ) ( τ ) = | E ( n ) ( 0 ) | 2 i = 1 n exp [ i Φ i ( τ ) ] .
G 1 ( n ) ( τ ) = I 0 P ( n ) i = 1 n exp [ i Φ i ( τ ) ] ,
G 1 ( τ ) = I 0 i = 1 P ( n ) i = 1 n exp [ i Φ i ( τ ) ] .
G 1 ( τ ) I 0 n = 1 P ( n ) exp [ i Φ i ( τ ) ] n .
G 1 ( τ ) = I 0 P ( s ) exp { 2 [ τ / τ B + ( τ / τ S ) 2 ] s / l } d s ,
l * / l = F ( q ) d q / F ( q ) ( 1 cos θ ) d q .
G 1 ( τ ) = I 0 P ( s ) exp { 2 [ τ / τ B + ( τ / τ S ) 2 ] s / l * } d s ,
G 1 ( τ ) = L γ l * ζ ( 3 ) L l * { 6 [ τ / τ B + ( τ / τ S ) 2 ] } 1 / 2 z sinh ( γ l * z / L ) sinh ( z ) d z ,
G 1 ( τ ) = 1 1 γ l * / L sinh ( L l * { 6 [ τ / τ B + ( τ / τ S ) 2 ] } 1 / 2 ( 1 γ l * / L ) ) sinh ( L l * { 6 [ τ / τ B + ( τ / τ S ) 2 ] } 1 / 2 ) .
g 2 ( τ ) = G 2 ( τ ) / G 2 ( 0 ) = | G 1 ( τ ) / G 1 ( 0 ) | 2 .
G 1 ( τ ) 1 ζ ( 3 ) 6 L l * τ τ S z 2 sinh ( z ) d z ,

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