Abstract

A ray-optics scattering model has been developed to determine if multiple reflections between the sphere and the plate could alter the exponential relationship between the scattering intensity and the separation distance as contact is approached. Results indicate that the effect of multiple reflections is dependent on sphere size, refractive indices, and the penetration depth of the evanescent wave. An experimental validation of the model was performed with polystyrene spheres (diameters 7–30 μm) immersed in an alcohol mixture and resting on an MgF2 film that had the same refractive index. Film thicknesses varied between 0 and 300 nm. No significant effect of multiple reflections was measured at an incident angle approximately 2° above the critical angle, which was in agreement with the predictions of the ray-optics model. By contrast, the scattering intensity from a 300-μm sphere was predicted to be much more sensitive to the separation distance at separations below one penetration depth when the incident angle was increased to over 6° above the critical angle.

© 1993 Optical Society of America

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References

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  1. D. C. Prieve, F. Luo, F. Lanni, “Brownian motion of a hydrosol particle in colloidal force field,” Faraday Discuss. Chem. Soc. 83, 297–307 (1987).
    [CrossRef]
  2. S. G. Bike, D. C. Prieve, “Measurement of double-layer repulsion for slightly overlapping counterion clouds,” Int. J. Multiphase Flow 16, 727–740 (1990).
    [CrossRef]
  3. D. C. Prieve, N. A. Frej, “Total internal reflection microscopy: a quantitative tool for the measurement of colloidal forces,” Langmuir 6, 396–403 (1990).
    [CrossRef]
  4. D. C. Prieve, S. G. Bike, N. A. Frej, “Brownian motion of a single microscopic sphere in a colloidal force field,” Faraday Discuss. Chem. Soc. 90, 209–222 (1990).
    [CrossRef]
  5. N. A. Frej, D. C. Prieve, “Measurement of the hindered diffusion coefficient of a single sphere near a wall in a non-uniform force field,” J. Chem. Phys. (to be published).
  6. H. Chew, D. S. Wang, M. Kerker, “Elastic scattering of evanescent electromagnetic waves,” Appl. Opt. 18, 2679–2687 (1979).
    [CrossRef] [PubMed]
  7. I. N. Court, F. K. von Willisen, “Frustrated total internal reflection and application of its princple to laser cavity design,” Appl. Opt. 3, 719–726 (1964).
    [CrossRef]
  8. S. G. Lipson, H. Lipson, Optical Physics, 2nd ed. (Cambridge U. Press, New York, 1981), Chap. 4, p. 69.
  9. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), Chap. 12, pp. 200–215.
  10. R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer-Verlag, New York, 1982), Chap. 3, p. 76.
  11. Optics Guide 4 (Melles Griot, Irvine, Calif., 1988).

1990 (3)

S. G. Bike, D. C. Prieve, “Measurement of double-layer repulsion for slightly overlapping counterion clouds,” Int. J. Multiphase Flow 16, 727–740 (1990).
[CrossRef]

D. C. Prieve, N. A. Frej, “Total internal reflection microscopy: a quantitative tool for the measurement of colloidal forces,” Langmuir 6, 396–403 (1990).
[CrossRef]

D. C. Prieve, S. G. Bike, N. A. Frej, “Brownian motion of a single microscopic sphere in a colloidal force field,” Faraday Discuss. Chem. Soc. 90, 209–222 (1990).
[CrossRef]

1987 (1)

D. C. Prieve, F. Luo, F. Lanni, “Brownian motion of a hydrosol particle in colloidal force field,” Faraday Discuss. Chem. Soc. 83, 297–307 (1987).
[CrossRef]

1979 (1)

1964 (1)

Bike, S. G.

S. G. Bike, D. C. Prieve, “Measurement of double-layer repulsion for slightly overlapping counterion clouds,” Int. J. Multiphase Flow 16, 727–740 (1990).
[CrossRef]

D. C. Prieve, S. G. Bike, N. A. Frej, “Brownian motion of a single microscopic sphere in a colloidal force field,” Faraday Discuss. Chem. Soc. 90, 209–222 (1990).
[CrossRef]

Chew, H.

Court, I. N.

Frej, N. A.

D. C. Prieve, N. A. Frej, “Total internal reflection microscopy: a quantitative tool for the measurement of colloidal forces,” Langmuir 6, 396–403 (1990).
[CrossRef]

D. C. Prieve, S. G. Bike, N. A. Frej, “Brownian motion of a single microscopic sphere in a colloidal force field,” Faraday Discuss. Chem. Soc. 90, 209–222 (1990).
[CrossRef]

N. A. Frej, D. C. Prieve, “Measurement of the hindered diffusion coefficient of a single sphere near a wall in a non-uniform force field,” J. Chem. Phys. (to be published).

Kerker, M.

Lanni, F.

D. C. Prieve, F. Luo, F. Lanni, “Brownian motion of a hydrosol particle in colloidal force field,” Faraday Discuss. Chem. Soc. 83, 297–307 (1987).
[CrossRef]

Lipson, H.

S. G. Lipson, H. Lipson, Optical Physics, 2nd ed. (Cambridge U. Press, New York, 1981), Chap. 4, p. 69.

Lipson, S. G.

S. G. Lipson, H. Lipson, Optical Physics, 2nd ed. (Cambridge U. Press, New York, 1981), Chap. 4, p. 69.

Luo, F.

D. C. Prieve, F. Luo, F. Lanni, “Brownian motion of a hydrosol particle in colloidal force field,” Faraday Discuss. Chem. Soc. 83, 297–307 (1987).
[CrossRef]

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer-Verlag, New York, 1982), Chap. 3, p. 76.

Prieve, D. C.

S. G. Bike, D. C. Prieve, “Measurement of double-layer repulsion for slightly overlapping counterion clouds,” Int. J. Multiphase Flow 16, 727–740 (1990).
[CrossRef]

D. C. Prieve, N. A. Frej, “Total internal reflection microscopy: a quantitative tool for the measurement of colloidal forces,” Langmuir 6, 396–403 (1990).
[CrossRef]

D. C. Prieve, S. G. Bike, N. A. Frej, “Brownian motion of a single microscopic sphere in a colloidal force field,” Faraday Discuss. Chem. Soc. 90, 209–222 (1990).
[CrossRef]

D. C. Prieve, F. Luo, F. Lanni, “Brownian motion of a hydrosol particle in colloidal force field,” Faraday Discuss. Chem. Soc. 83, 297–307 (1987).
[CrossRef]

N. A. Frej, D. C. Prieve, “Measurement of the hindered diffusion coefficient of a single sphere near a wall in a non-uniform force field,” J. Chem. Phys. (to be published).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), Chap. 12, pp. 200–215.

von Willisen, F. K.

Wang, D. S.

Appl. Opt. (2)

Faraday Discuss. Chem. Soc. (2)

D. C. Prieve, F. Luo, F. Lanni, “Brownian motion of a hydrosol particle in colloidal force field,” Faraday Discuss. Chem. Soc. 83, 297–307 (1987).
[CrossRef]

D. C. Prieve, S. G. Bike, N. A. Frej, “Brownian motion of a single microscopic sphere in a colloidal force field,” Faraday Discuss. Chem. Soc. 90, 209–222 (1990).
[CrossRef]

Int. J. Multiphase Flow (1)

S. G. Bike, D. C. Prieve, “Measurement of double-layer repulsion for slightly overlapping counterion clouds,” Int. J. Multiphase Flow 16, 727–740 (1990).
[CrossRef]

Langmuir (1)

D. C. Prieve, N. A. Frej, “Total internal reflection microscopy: a quantitative tool for the measurement of colloidal forces,” Langmuir 6, 396–403 (1990).
[CrossRef]

Other (5)

N. A. Frej, D. C. Prieve, “Measurement of the hindered diffusion coefficient of a single sphere near a wall in a non-uniform force field,” J. Chem. Phys. (to be published).

S. G. Lipson, H. Lipson, Optical Physics, 2nd ed. (Cambridge U. Press, New York, 1981), Chap. 4, p. 69.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), Chap. 12, pp. 200–215.

R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer-Verlag, New York, 1982), Chap. 3, p. 76.

Optics Guide 4 (Melles Griot, Irvine, Calif., 1988).

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Figures (14)

Fig. 1
Fig. 1

Simple schematic illustrating use of MgF2-coated glass slides for measuring scattering intensity versus separation distance. (Note the drawing is not to scale). The laser beam strikes the glass prism at normal incidence and reflects off the glass–MgF2 interface, forming an evanescent in the film. The alcohol solution has the same refractive index as the MgF2 coating, so the particle appears to be resting on an optically invisible support.

Fig. 2
Fig. 2

Scattering of an evanescent wave formed at a glass–water interface by a polystyrene sphere; θ1 is the incident angle.

Fig. 3
Fig. 3

Comparison of the simple exponential decay with the tunneling of an evanescent wave through a thin planar film. The calculations are for the glass–alcohol–polystyrene system with an incident angle of 68°. The penetration depth is 313 nm.

Fig. 4
Fig. 4

Coordinate system and angles utilized in the geometric-optics model. The origin is at the center of the sphere.

Fig. 5
Fig. 5

Typical ray paths through the sphere. The integers p and w denote the number of contacts made with the inner surface of the sphere and the plate, respectively.

Fig. 6
Fig. 6

Typical two-dimensional scattering profile from a ray-optics model. This profile was calculated for a 7-μm polystyrene sphere suspended in an alcohol solution 10 nm above a glass plate; ϕsca = 0°

Fig. 7
Fig. 7

Results from a ray-optics model for polystyrene spheres immersed in a 1-mM NaCl aqueous solution. The incident angle is 68°. The model results are compared with the exponential decay that ignores multiple reflections. Note that the values of I0 have been arbitrarily adjusted for ease of viewing.

Fig. 8
Fig. 8

Raw scattering data collected from 7-, 15-, and 30-μm polystyrene spheres in an alcohol mixture. The data for the different particle sizes have been scaled differently for ease of viewing.

Fig. 9
Fig. 9

Experimental results from experiments with MgF2-coated slides and an alcohol solution. The error bars are the standard errors of the mean. The incident angle is 68°. The exponential decay lines have been fit to match the measured values at zero separation distance, which have been adjusted for ease of viewing.

Fig. 10
Fig. 10

Results from a ray-optics model for polystyrene spheres immersed in a 91.74% wt. 1-propanol, 8.26% wt. ethanol solution. The incident angle is 68°.

Fig. 11
Fig. 11

Effect of sphere size on the intensity of light scattered from an evanescent wave striking a polystyrene sphere in alcohol. The measured values are the experimental results from the 296-nm MgF2 slide shown in Fig. 9. Sphere diameter is in micrometers.

Fig. 12
Fig. 12

Transmittance and reflectance of an incident ray in alcohol striking a polystyrene sphere. A contact angle of 90° indicates normal incidence at the center of the sphere, whereas 0° is the grazing incidence. The incident ray is polarized parallel to the plane of incidence.

Fig. 13
Fig. 13

Fraction of total scattered energy contributed by the initial contact of the evanescent wave (no multiple reflections). Results are shown for a sphere in both an alcohol solution and a 1-mM NaCl aqueous solution. The incident angle is 68° in both solutions, and the sphere is located at approximately zero separation distance. Sphere diameter is in micrometers.

Fig. 14
Fig. 14

Results from a ray-optics model for a 300-μm polystyrene sphere immersed in a 91.74% wt. 1-propanol, 8.26% wt. ethanol solution. The incident angle is 72°. The model results are compared with the simple exponential decay (solid curve), which ignores multiple reflections.

Tables (2)

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Table 1 Standard Deviation of Experimental Results as a Fraction of Mean

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Table 2 Summary of Experimental Results

Equations (12)

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E ( y ) = E 0 exp ( k 2 β y ) ,
β = [ ( n 1 / n 2 ) sin 2 θ 1 1 ] 1 / 2 ,
I sca ( h ) = I 0 exp ( 2 k 2 β h ) ,
T ( l ) = T 0 γ sinh 2 ( k 2 β l / 2 ) + γ ,
T ( l ) 4 T 0 exp ( 2 k 2 β l ) , as l ,
I 0 a 2 cos τ sin τ d τ d ϕ inc .
r 2 sin θ d θ d ϕ sca .
I sca = 2 I 0 a 2 cos τ sin τ d τ d ϕ inc r 2 sin θ d θ d ϕ sca ,
D = cos τ sin τ d τ d ϕ inc sin θ d θ d ϕ sca .
I sca = a 2 I 0 2 D r 2 .
δ = 2 π λ 0 ( reference path length actual path length ) ,
E sca = E inc D 1 / 2 exp ( i σ ) ,

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