Abstract

In this paper we outline a new all-optical non-contact technique for measurement of the surface tension of a Newtonian fluid. It is based on the accurate measurement of the spacing of the supernumerary fringes produced by the diffraction pattern of a laser beam transmitted through or reflected by a thin vertically-draining film of the liquid. We discuss the basic theory and application of this technique, and several issues which must be addressed before it can be used commercially.

© 2008 Optical Society of America

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References

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  1. H. Jeffreys, "The drainage of a vertical plate," Proc. Cambridge Philos. Soc. 26, 204-205 (1930).
    [CrossRef]
  2. P. C. Hiemenz, Principles of Colloid and Surface Chemistry, 2nd. ed. (Marcel Dekker, New York, 1986), pp. 287-299.
  3. L. H. Tanner, "The surface tension effect on the flow of liquid down vertical or inclined surfaces," J. Phys. D 13, 1633-1641 (1980).
    [CrossRef]
  4. M. Abromowitz and I. A. Stegun, "Handbook of Mathematical Functions," (Dover Publications, New York, 1970), pp. 16 (formula 3.6.25).
  5. G. B. Airy, "On the intensity of light in the neighbourhood of a caustic," Trans. Cambridge Philos. Soc. 6, 397-403 (1836), reprinted in P. L. Marston (ed.), Geometrical Aspects of Scattering, SPIE Milestone Series, vol. MS87 (SPIE, Bellingham, WA, 1994) pp. 298-309.
  6. T. Poston and I. Stewart, Catastrophe theory and its applications (Dover Publications, New York, 1978) pp. 252-255.
  7. J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, Bristol, 1999) pp. 123-136.
  8. R. Greenler, Rainbows, Halos and Glories (Cambridge University Press, Cambridge, 1980), pp. 8-10.
  9. M. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993), pp. 195-197.
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 57-74.
  11. M. Abromowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970), pp. 470-478 (formula 3.6.25).
  12. A good online resource for Airy functions is: http://mathworld.wolfram.com/AiryFunctions.html.
  13. M. Abromowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970), pp. 478 (Table 10.13).
  14. C. L. Adler, J. A. Lock, J. K. Nash, and K. W. Saunders, "Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference," Appl. Opt. 40, 1548-1558 (2001).
    [CrossRef]
  15. C. L. Adler, J. A. Lock, I. P. Rafferty, W. A. Hickok, and B. R. Keating, "Twin-Rainbow Metrology I: Measurement of the Thickness of a Thin Liquid Film Draining Under Gravity," Appl. Opt. 42, 6584-6594 (2003) .
    [CrossRef]
  16. L. H. Tanner, "A study of the optics and motion of draining oil drops," Opt. Technol. 125-129 (1978) .
    [CrossRef]

2003 (1)

C. L. Adler, J. A. Lock, I. P. Rafferty, W. A. Hickok, and B. R. Keating, "Twin-Rainbow Metrology I: Measurement of the Thickness of a Thin Liquid Film Draining Under Gravity," Appl. Opt. 42, 6584-6594 (2003) .
[CrossRef]

2001 (1)

1980 (1)

L. H. Tanner, "The surface tension effect on the flow of liquid down vertical or inclined surfaces," J. Phys. D 13, 1633-1641 (1980).
[CrossRef]

1978 (1)

L. H. Tanner, "A study of the optics and motion of draining oil drops," Opt. Technol. 125-129 (1978) .
[CrossRef]

1930 (1)

H. Jeffreys, "The drainage of a vertical plate," Proc. Cambridge Philos. Soc. 26, 204-205 (1930).
[CrossRef]

Adler, C. L.

C. L. Adler, J. A. Lock, I. P. Rafferty, W. A. Hickok, and B. R. Keating, "Twin-Rainbow Metrology I: Measurement of the Thickness of a Thin Liquid Film Draining Under Gravity," Appl. Opt. 42, 6584-6594 (2003) .
[CrossRef]

C. L. Adler, J. A. Lock, J. K. Nash, and K. W. Saunders, "Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference," Appl. Opt. 40, 1548-1558 (2001).
[CrossRef]

Hickok, W. A.

C. L. Adler, J. A. Lock, I. P. Rafferty, W. A. Hickok, and B. R. Keating, "Twin-Rainbow Metrology I: Measurement of the Thickness of a Thin Liquid Film Draining Under Gravity," Appl. Opt. 42, 6584-6594 (2003) .
[CrossRef]

Jeffreys, H.

H. Jeffreys, "The drainage of a vertical plate," Proc. Cambridge Philos. Soc. 26, 204-205 (1930).
[CrossRef]

Keating, B. R.

C. L. Adler, J. A. Lock, I. P. Rafferty, W. A. Hickok, and B. R. Keating, "Twin-Rainbow Metrology I: Measurement of the Thickness of a Thin Liquid Film Draining Under Gravity," Appl. Opt. 42, 6584-6594 (2003) .
[CrossRef]

Lock, J. A.

C. L. Adler, J. A. Lock, I. P. Rafferty, W. A. Hickok, and B. R. Keating, "Twin-Rainbow Metrology I: Measurement of the Thickness of a Thin Liquid Film Draining Under Gravity," Appl. Opt. 42, 6584-6594 (2003) .
[CrossRef]

C. L. Adler, J. A. Lock, J. K. Nash, and K. W. Saunders, "Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference," Appl. Opt. 40, 1548-1558 (2001).
[CrossRef]

Nash, J. K.

Rafferty, I. P.

C. L. Adler, J. A. Lock, I. P. Rafferty, W. A. Hickok, and B. R. Keating, "Twin-Rainbow Metrology I: Measurement of the Thickness of a Thin Liquid Film Draining Under Gravity," Appl. Opt. 42, 6584-6594 (2003) .
[CrossRef]

Saunders, K. W.

Tanner, L. H.

L. H. Tanner, "The surface tension effect on the flow of liquid down vertical or inclined surfaces," J. Phys. D 13, 1633-1641 (1980).
[CrossRef]

L. H. Tanner, "A study of the optics and motion of draining oil drops," Opt. Technol. 125-129 (1978) .
[CrossRef]

App. Opt. (1)

C. L. Adler, J. A. Lock, I. P. Rafferty, W. A. Hickok, and B. R. Keating, "Twin-Rainbow Metrology I: Measurement of the Thickness of a Thin Liquid Film Draining Under Gravity," Appl. Opt. 42, 6584-6594 (2003) .
[CrossRef]

Appl. Opt. (1)

J. Phys. D (1)

L. H. Tanner, "The surface tension effect on the flow of liquid down vertical or inclined surfaces," J. Phys. D 13, 1633-1641 (1980).
[CrossRef]

Opt.Technol. (1)

L. H. Tanner, "A study of the optics and motion of draining oil drops," Opt. Technol. 125-129 (1978) .
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

H. Jeffreys, "The drainage of a vertical plate," Proc. Cambridge Philos. Soc. 26, 204-205 (1930).
[CrossRef]

Other (11)

P. C. Hiemenz, Principles of Colloid and Surface Chemistry, 2nd. ed. (Marcel Dekker, New York, 1986), pp. 287-299.

M. Abromowitz and I. A. Stegun, "Handbook of Mathematical Functions," (Dover Publications, New York, 1970), pp. 16 (formula 3.6.25).

G. B. Airy, "On the intensity of light in the neighbourhood of a caustic," Trans. Cambridge Philos. Soc. 6, 397-403 (1836), reprinted in P. L. Marston (ed.), Geometrical Aspects of Scattering, SPIE Milestone Series, vol. MS87 (SPIE, Bellingham, WA, 1994) pp. 298-309.

T. Poston and I. Stewart, Catastrophe theory and its applications (Dover Publications, New York, 1978) pp. 252-255.

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, Bristol, 1999) pp. 123-136.

R. Greenler, Rainbows, Halos and Glories (Cambridge University Press, Cambridge, 1980), pp. 8-10.

M. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993), pp. 195-197.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 57-74.

M. Abromowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970), pp. 470-478 (formula 3.6.25).

A good online resource for Airy functions is: http://mathworld.wolfram.com/AiryFunctions.html.

M. Abromowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970), pp. 478 (Table 10.13).

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Figures (9)

Fig. 1.
Fig. 1.

Profile of a draining fluid layer as a function of time. Fluid parameters: Surface tension, σ=0.02 N/m (20 dyne/cm); specific viscosity, µ/ρ=5×10-5 m2/s (50 cS); density, ρ=740 kg/m3 (0.74 g/cm3). A) t=0.001 s; B) t=0.01 s; C) t=0.1 s.

Fig. 2.
Fig. 2.

Refraction of a parallel bundle of light rays by the fluid layer near the contact line. Note the point of inflection leading to the maximum deviation of the light ray (indicated by a dashed line.) Units are normalized.

Fig. 3.
Fig. 3.

Graph of the function v=(Ai(z))2. Note that the function decays without oscillation for z>0, while it oscillates (i.e., displays interference fringes) for z<0.

Fig. 4.
Fig. 4.

Experimental setup to measure interference fringes from the draining liquid layer. A) Experimental layout as seen from above. B) Profile of slide and draining layer.

Fig. 5.
Fig. 5.

Image of rainbow fringes due to scattering from the fluid layer. The different supernumerary peaks are labelled on the figure.

Fig. 6.
Fig. 6.

Digitized intensity of the interference fringes of fig. 5. Pixel number increases toward the bottom of the image (see Fig. 5). Supernumerary peaks are indicated on the figure.

Fig. 7.
Fig. 7.

Computer simulation of the intensity of light scattered by a draining fluid layer.

Fig. 8.
Fig. 8.

Position of peak of “zeroth” interference fringe (θ0) as a function of time.

Fig. 9.
Fig. 9.

Angular separation of interference fringe maxima (θ0i)Dots: Data from computer simulation Solid lines: Separation determined using the analytic approximation (Eq. (12)).

Tables (1)

Tables Icon

Table 1. Positions of Airy Function Maxima and Fringe Maxima

Equations (15)

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y = ( ρ g μ ) x 2 t + k ( σ ρ g ) 1 3 x 1 3 ( = A x 2 + B x 1 3 )
x 0 ( t ) = ( 1 9 B A ) 3 5 = 0.167 σ 3 5 μ 3 5 ρ 6 5 g 6 5 t 3 5
y 0 ( t ) = 0.27904 μ 1 5 σ 6 5 ρ 7 5 g 7 5 t 1 5
x x 0 ( t ) = β ( y y 0 ( t ) ) + 1 3 ! δ ( y y 0 ( t ) ) 3
x x 0 t 3 5 ; y y 0 t 1 5
d 2 z d y 2 = d 2 y d z 2 ( d y d z ) 3 = 0 at the point of inflection ,
d 3 z d y 3 = ( d y d z ) ( d 3 y dz 3 ) 3 ( d 2 y dz 2 ) 2 ( d y d z ) 5
β = 1.197 μ 2 5 ρ 1 5 g 1 5 σ 1 5 t 2 5
δ = 41.0 ρ g σ
ϕ ( y ) = k ( n 1 ) x ( y ) = 2 π ( n 1 ) λ [ β ( y y 0 ) + δ 3 ! ( y y 0 ) 3 ]
E ( Y ) = i λ L E ( y ) e i ( 2 π y Y λ L ϕ ( y ) ) d y
I ( θ ) = K ( A i ( ( 4 π 2 3 ) 1 3 ( σ ( n 1 ) ρ g λ 2 ) 1 3 ( θ θ 0 ) ) ) 2
σ = 1.0395 ( n 1 ) ρ g λ 2 ( z i z 0 θ i θ 0 ) 3
τ = ( μ 2 ρ g σ ) 1 2
θ 0 = ( n 1 ) β = 1.197 ( n 1 ) μ 2 5 ρ 1 5 g 1 5 σ 1 5 t 2 5

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