Abstract

Correlation spectroscopy is an analytical technique that can identify the residence time of reflective or fluorescent particles in a measurement spot, allowing particle velocity or diffusion to be inferred. We show that the technique can be applied to data measured with a time-domain terahertz sensor. The speed of reflectors such as silica ballotini or bubbles can thus be measured in fluid samples. Time-domain terahertz sensors can therefore be used, for the first time, to measure rheological properties of optically opaque fluids that contain entrained reflectors, such as polyethylene beads.

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References

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  1. W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325 (2007).
    [Crossref]
  2. J. Obradovic, J. Collins, O. Hirsch, M. Mantle, M. Johns, and L. F. Gladden, Polymer 48, 3494 (2007).
    [Crossref]
  3. D. Madge, W. W. Webb, and E. L. Elson, Biopolymers 17, 361 (1978).
    [Crossref]
  4. M. Gosch, H. Blom, J. Holm, T. Heino, and R. Rigler, Anal. Chem. 72, 3260 (2000).
    [Crossref]
  5. S. T. Hess, S. Huang, A. A. Heikal, and W. W. Webb, Biochemistry 41, 697 (2002).
    [Crossref]
  6. Y.-C. Shen and P. F. Taday, IEEE J. Sel. Top. Quantum Electron. 14, 407 (2008).
    [Crossref]
  7. D. M. Mittleman, R. H. Lacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
    [Crossref]
  8. www.npl.co.uk/upload/pdf/091217_terahertz_naftaly.pdf (accessed May2016).
  9. E. Rees, Sample Dataset and Matlab Analysis Code for THz Correlation Velocimetry (University of Cambridge, 2016), https://www.repository.cam.ac.uk/handle/1810/254844 .
  10. T. E. Faber, Fluid Dynamics for Physicists (Cambridge University, 1995).
  11. J. Y. Huang, Y. M. J. Chew, and D. I. Wilson, J. Food Eng. 109, 49 (2012).
    [Crossref]
  12. J. F. Davidson and D. Harrison, Fluidised Particles (Cambridge University, 1963).

2012 (1)

J. Y. Huang, Y. M. J. Chew, and D. I. Wilson, J. Food Eng. 109, 49 (2012).
[Crossref]

2008 (1)

Y.-C. Shen and P. F. Taday, IEEE J. Sel. Top. Quantum Electron. 14, 407 (2008).
[Crossref]

2007 (2)

W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325 (2007).
[Crossref]

J. Obradovic, J. Collins, O. Hirsch, M. Mantle, M. Johns, and L. F. Gladden, Polymer 48, 3494 (2007).
[Crossref]

2002 (1)

S. T. Hess, S. Huang, A. A. Heikal, and W. W. Webb, Biochemistry 41, 697 (2002).
[Crossref]

2000 (1)

M. Gosch, H. Blom, J. Holm, T. Heino, and R. Rigler, Anal. Chem. 72, 3260 (2000).
[Crossref]

1996 (1)

D. M. Mittleman, R. H. Lacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
[Crossref]

1978 (1)

D. Madge, W. W. Webb, and E. L. Elson, Biopolymers 17, 361 (1978).
[Crossref]

Blom, H.

M. Gosch, H. Blom, J. Holm, T. Heino, and R. Rigler, Anal. Chem. 72, 3260 (2000).
[Crossref]

Chan, W. L.

W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325 (2007).
[Crossref]

Chew, Y. M. J.

J. Y. Huang, Y. M. J. Chew, and D. I. Wilson, J. Food Eng. 109, 49 (2012).
[Crossref]

Collins, J.

J. Obradovic, J. Collins, O. Hirsch, M. Mantle, M. Johns, and L. F. Gladden, Polymer 48, 3494 (2007).
[Crossref]

Davidson, J. F.

J. F. Davidson and D. Harrison, Fluidised Particles (Cambridge University, 1963).

Deibel, J.

W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325 (2007).
[Crossref]

Elson, E. L.

D. Madge, W. W. Webb, and E. L. Elson, Biopolymers 17, 361 (1978).
[Crossref]

Faber, T. E.

T. E. Faber, Fluid Dynamics for Physicists (Cambridge University, 1995).

Gladden, L. F.

J. Obradovic, J. Collins, O. Hirsch, M. Mantle, M. Johns, and L. F. Gladden, Polymer 48, 3494 (2007).
[Crossref]

Gosch, M.

M. Gosch, H. Blom, J. Holm, T. Heino, and R. Rigler, Anal. Chem. 72, 3260 (2000).
[Crossref]

Harrison, D.

J. F. Davidson and D. Harrison, Fluidised Particles (Cambridge University, 1963).

Heikal, A. A.

S. T. Hess, S. Huang, A. A. Heikal, and W. W. Webb, Biochemistry 41, 697 (2002).
[Crossref]

Heino, T.

M. Gosch, H. Blom, J. Holm, T. Heino, and R. Rigler, Anal. Chem. 72, 3260 (2000).
[Crossref]

Hess, S. T.

S. T. Hess, S. Huang, A. A. Heikal, and W. W. Webb, Biochemistry 41, 697 (2002).
[Crossref]

Hirsch, O.

J. Obradovic, J. Collins, O. Hirsch, M. Mantle, M. Johns, and L. F. Gladden, Polymer 48, 3494 (2007).
[Crossref]

Holm, J.

M. Gosch, H. Blom, J. Holm, T. Heino, and R. Rigler, Anal. Chem. 72, 3260 (2000).
[Crossref]

Huang, J. Y.

J. Y. Huang, Y. M. J. Chew, and D. I. Wilson, J. Food Eng. 109, 49 (2012).
[Crossref]

Huang, S.

S. T. Hess, S. Huang, A. A. Heikal, and W. W. Webb, Biochemistry 41, 697 (2002).
[Crossref]

Johns, M.

J. Obradovic, J. Collins, O. Hirsch, M. Mantle, M. Johns, and L. F. Gladden, Polymer 48, 3494 (2007).
[Crossref]

Lacobsen, R. H.

D. M. Mittleman, R. H. Lacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
[Crossref]

Madge, D.

D. Madge, W. W. Webb, and E. L. Elson, Biopolymers 17, 361 (1978).
[Crossref]

Mantle, M.

J. Obradovic, J. Collins, O. Hirsch, M. Mantle, M. Johns, and L. F. Gladden, Polymer 48, 3494 (2007).
[Crossref]

Mittleman, D. M.

W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325 (2007).
[Crossref]

D. M. Mittleman, R. H. Lacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
[Crossref]

Nuss, M. C.

D. M. Mittleman, R. H. Lacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
[Crossref]

Obradovic, J.

J. Obradovic, J. Collins, O. Hirsch, M. Mantle, M. Johns, and L. F. Gladden, Polymer 48, 3494 (2007).
[Crossref]

Rees, E.

E. Rees, Sample Dataset and Matlab Analysis Code for THz Correlation Velocimetry (University of Cambridge, 2016), https://www.repository.cam.ac.uk/handle/1810/254844 .

Rigler, R.

M. Gosch, H. Blom, J. Holm, T. Heino, and R. Rigler, Anal. Chem. 72, 3260 (2000).
[Crossref]

Shen, Y.-C.

Y.-C. Shen and P. F. Taday, IEEE J. Sel. Top. Quantum Electron. 14, 407 (2008).
[Crossref]

Taday, P. F.

Y.-C. Shen and P. F. Taday, IEEE J. Sel. Top. Quantum Electron. 14, 407 (2008).
[Crossref]

Webb, W. W.

S. T. Hess, S. Huang, A. A. Heikal, and W. W. Webb, Biochemistry 41, 697 (2002).
[Crossref]

D. Madge, W. W. Webb, and E. L. Elson, Biopolymers 17, 361 (1978).
[Crossref]

Wilson, D. I.

J. Y. Huang, Y. M. J. Chew, and D. I. Wilson, J. Food Eng. 109, 49 (2012).
[Crossref]

Anal. Chem. (1)

M. Gosch, H. Blom, J. Holm, T. Heino, and R. Rigler, Anal. Chem. 72, 3260 (2000).
[Crossref]

Biochemistry (1)

S. T. Hess, S. Huang, A. A. Heikal, and W. W. Webb, Biochemistry 41, 697 (2002).
[Crossref]

Biopolymers (1)

D. Madge, W. W. Webb, and E. L. Elson, Biopolymers 17, 361 (1978).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

Y.-C. Shen and P. F. Taday, IEEE J. Sel. Top. Quantum Electron. 14, 407 (2008).
[Crossref]

D. M. Mittleman, R. H. Lacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
[Crossref]

J. Food Eng. (1)

J. Y. Huang, Y. M. J. Chew, and D. I. Wilson, J. Food Eng. 109, 49 (2012).
[Crossref]

Polymer (1)

J. Obradovic, J. Collins, O. Hirsch, M. Mantle, M. Johns, and L. F. Gladden, Polymer 48, 3494 (2007).
[Crossref]

Rep. Prog. Phys. (1)

W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325 (2007).
[Crossref]

Other (4)

J. F. Davidson and D. Harrison, Fluidised Particles (Cambridge University, 1963).

www.npl.co.uk/upload/pdf/091217_terahertz_naftaly.pdf (accessed May2016).

E. Rees, Sample Dataset and Matlab Analysis Code for THz Correlation Velocimetry (University of Cambridge, 2016), https://www.repository.cam.ac.uk/handle/1810/254844 .

T. E. Faber, Fluid Dynamics for Physicists (Cambridge University, 1995).

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Figures (4)

Fig. 1.
Fig. 1.

Principle of terahertz correlation spectroscopy. Reflective silica ballotini (yellow) sink in a terahertz-transparent paraffin oil. Successive time-domain scans of terahertz reflection intensity contain time-varying signals from which the average residence time of particles in the imaging volume can be evaluated using correlation spectroscopy. Therefore the mean particle velocity can be inferred. The waterfall plot shows a typical reflection measurement of 425 μm diameter ballotini.

Fig. 2.
Fig. 2.

(a) Time-resolved terahertz reflection intensity (inset) due to silica ballotini falling through a static observation volume in paraffin oil. The autocorrelations of these data are approximately Gaussian, and by fitting Eq. (3) to these values the average particle residence time can be inferred. (b) The velocities corresponding to the fitted residence times are consistent with the calculated terminal Stokes velocities of spherical particles, and with drop-time measurements of velocity.

Fig. 3.
Fig. 3.

(a) The terahertz sensor setup for measuring the speed of entrained particles in the Taylor–Couette viscometer. (b) Time- and depth-resolved terahertz reflections show that reflective particles move faster with increasing proximity to the rotating bob. (c) The radially resolved velocity profiles obtained by analyzing the reflection data in B with correlation spectroscopy (circles) are broadly consistent with the velocity profiles that would be set up in ideal Newtonian fluids at the applied rotation rates (solid lines of matching color correspond to rotation rates of 10, 13.5, 15, 19, and 24 rpm of the 4 mm radius bob).

Fig. 4.
Fig. 4.

Characteristic double echoes due to terahertz reflection from both sides of air bubbles in paraffin oil. The separation ( d ) indicates bubble diameter (here, about 100 μm). Fitting the residence times of individual echoes ( δ t ) indicates the bubble rise velocity.

Equations (3)

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I ( x , y , z ) = I 0 exp ( 2 ( x 2 + y 2 + ( z ξ ) 2 ) w 2 ) ,
G ( τ ) = δ I ( t ) δ I ( t + τ ) d t δ I ( t ) δ I ( t ) d t .
G ( τ ) = 1 N exp ( ( τ τ flow ) 2 ) .

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