Abstract

In this paper we study the application of photon correlation spectroscopy to a system of randomly diffusing particles suspended in a fluid undergoing uniform translational motion relative to the optical scattering volume. To do so we derive theoretical expressions for both the homodyne and heterodyne correlation functions in both the dilute and nondilute particle limits. We then test these results with experiments on a flowing system and find good agreement. We discuss a useful method of analysis and define limits to particle sizing in such a system using this light-scattering technique.

© 1984 Optical Society of America

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References

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  1. G. B. King, C. M. Sorensen, T. W. Lester, J. F. Merklin, “Photon Correlation Spectroscopy Used as a Particle Size Diagnostic in Sooting Flames,” Appl. Opt. 21, 976 (1982).
    [Crossref] [PubMed]
  2. W. L. Flower, “Optical Measurements of Soot Formation in Pre-mixed Flames,” Combust. Sci. Technol. 33, 17 (1983).
    [Crossref]
  3. W. Hinds, P. C. Reist, “Aerosol Measurement by Laser Doppler Spectroscopy‐I. Theory and Experimental Results for Aerosol Homogeneous,” Aerosol Sci. 3, 501 (1972).
    [Crossref]
  4. R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Doppler Flow meter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
    [Crossref]
  5. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  6. N. A. Clark, J. H. Lunacek, G. B. Benedek, “A Study of Brownian Motion Using Light Scattering,” Am. J. Phys. 38, 575 (1970).
    [Crossref]
  7. C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple Scattering from a System of Brownian Particles,” Phys. Rev. 17, 2030 (1978).
    [Crossref]

1983 (1)

W. L. Flower, “Optical Measurements of Soot Formation in Pre-mixed Flames,” Combust. Sci. Technol. 33, 17 (1983).
[Crossref]

1982 (1)

1978 (1)

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple Scattering from a System of Brownian Particles,” Phys. Rev. 17, 2030 (1978).
[Crossref]

1972 (2)

W. Hinds, P. C. Reist, “Aerosol Measurement by Laser Doppler Spectroscopy‐I. Theory and Experimental Results for Aerosol Homogeneous,” Aerosol Sci. 3, 501 (1972).
[Crossref]

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Doppler Flow meter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[Crossref]

1970 (1)

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A Study of Brownian Motion Using Light Scattering,” Am. J. Phys. 38, 575 (1970).
[Crossref]

Angus, J. C.

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Doppler Flow meter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[Crossref]

Benedek, G. B.

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A Study of Brownian Motion Using Light Scattering,” Am. J. Phys. 38, 575 (1970).
[Crossref]

Berne, B. J.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Clark, N. A.

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A Study of Brownian Motion Using Light Scattering,” Am. J. Phys. 38, 575 (1970).
[Crossref]

Dunning, J. W.

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Doppler Flow meter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[Crossref]

Edwards, R. V.

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Doppler Flow meter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[Crossref]

Flower, W. L.

W. L. Flower, “Optical Measurements of Soot Formation in Pre-mixed Flames,” Combust. Sci. Technol. 33, 17 (1983).
[Crossref]

French, M. J.

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Doppler Flow meter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[Crossref]

Hinds, W.

W. Hinds, P. C. Reist, “Aerosol Measurement by Laser Doppler Spectroscopy‐I. Theory and Experimental Results for Aerosol Homogeneous,” Aerosol Sci. 3, 501 (1972).
[Crossref]

King, G. B.

Lester, T. W.

Lunacek, J. H.

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A Study of Brownian Motion Using Light Scattering,” Am. J. Phys. 38, 575 (1970).
[Crossref]

Merklin, J. F.

Mockler, R. C.

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple Scattering from a System of Brownian Particles,” Phys. Rev. 17, 2030 (1978).
[Crossref]

O’Sullivan, W. J.

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple Scattering from a System of Brownian Particles,” Phys. Rev. 17, 2030 (1978).
[Crossref]

Pecora, R.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Reist, P. C.

W. Hinds, P. C. Reist, “Aerosol Measurement by Laser Doppler Spectroscopy‐I. Theory and Experimental Results for Aerosol Homogeneous,” Aerosol Sci. 3, 501 (1972).
[Crossref]

Sorensen, C. M.

G. B. King, C. M. Sorensen, T. W. Lester, J. F. Merklin, “Photon Correlation Spectroscopy Used as a Particle Size Diagnostic in Sooting Flames,” Appl. Opt. 21, 976 (1982).
[Crossref] [PubMed]

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple Scattering from a System of Brownian Particles,” Phys. Rev. 17, 2030 (1978).
[Crossref]

Aerosol Sci. (1)

W. Hinds, P. C. Reist, “Aerosol Measurement by Laser Doppler Spectroscopy‐I. Theory and Experimental Results for Aerosol Homogeneous,” Aerosol Sci. 3, 501 (1972).
[Crossref]

Am. J. Phys. (1)

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A Study of Brownian Motion Using Light Scattering,” Am. J. Phys. 38, 575 (1970).
[Crossref]

Appl. Opt. (1)

Combust. Sci. Technol. (1)

W. L. Flower, “Optical Measurements of Soot Formation in Pre-mixed Flames,” Combust. Sci. Technol. 33, 17 (1983).
[Crossref]

J. Appl. Phys. (1)

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Doppler Flow meter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[Crossref]

Phys. Rev. (1)

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple Scattering from a System of Brownian Particles,” Phys. Rev. 17, 2030 (1978).
[Crossref]

Other (1)

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

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Figures (3)

Fig. 1
Fig. 1

Plot of Y(nΔt) vs time = nt for the 0.091-μm system at a variety of flow rates. Incident lens focal length was 30.0 cm.

Fig. 2
Fig. 2

Square root of the slope of the curves in Fig. 1 vs flow rate for F = 30.0 cm and two other incident lens focal lengths. The beam transit time τ2 is the inverse of the square root of the slope.

Fig. 3
Fig. 3

Correlation time τ1 or the beam transit time τ2 vs flow rate for both sizes of microspheres. The flow velocity in the center of the tube is given by v(cm/sec) = 4.68 × 10−2 × flow rate (mliter/min).

Equations (26)

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E s ( t ) = j = 1 N T exp ( i q · r j ) P ( r j ) .
q = 4 π n λ sin θ / 2 ,
P ( r ) = E 0 exp ( - r 2 / w 2 ) ,
r j ( t ) = v t + r j ( t ) .
I 1 ( t ) E s ( t ) E s * ( 0 ) ,
I 2 ( t ) E s ( t ) 2 E s ( 0 ) 2 .
I ( t ) I ( 0 ) het = [ I lo 2 + 2 I lo Re I 1 ( t ) ] ,
I ( t ) I ( 0 ) hom = I 2 ( t ) .
I 1 = exp ( i q · v t ) j , k exp [ i q · δ r j , k ( t ) ] δ r × P [ r j ( 0 ) ] P [ r k ( t ) + v t ] r ( 0 ) .
exp [ i q · δ r ( t ) ] = exp ( - D q 2 t ) ,
E 0 2 V T - 1 d 3 r exp ( - r 2 w 2 ) exp [ - ( r + v t ) 2 w 2 ] E 0 2 V T - 1 ( π 2 ) 3 / 2 × w 2 exp ( - v 2 t 2 / 2 w 2 ) .
P P ( t ) r ( 0 ) = E 0 2 V s V T - 1 exp ( - v 2 t 2 / 2 w 2 ) .
I 1 ( t ) = I 0 N exp ( i q · v t ) exp ( - D q 2 t ) exp ( - v 2 t 2 / 2 w 2 ) ,
I 2 ( t ) = j , k , l , m exp [ i q · r j ( 0 ) ] P [ r j ( 0 ) ] × exp [ - i q · r k ( 0 ) ] P [ r k ( 0 ) ] × exp ( i q · v t ) exp [ i q · r l ( t ) ] P [ r l ( t ) + v t ] × exp ( - i q · v t ) exp [ i q · r m ( t ) ] P [ r m ( t ) + v t ] .
I 2 , a ( t ) = j , l P 2 [ r j ( 0 ) ] P 2 [ r l ( t ) + v t ] r ( 0 ) δ r .
I 2 , a ( t ) = E 0 4 j l V T - 2 d 3 r j d 3 r l exp ( - 2 r j 2 w 2 ) exp [ - 2 ( r l + v t ) 2 w 2 ] = I 0 2 N ( N - 1 ) .
I 2 , a ( t ) = E 0 4 j V T - 1 d 3 r j exp ( - 2 r j 2 w 2 ) exp [ - 2 ( r j + v t ) 2 w 2 ] = E 0 4 N T V T - 1 exp ( - v 2 t 2 w 2 ) 2 - 3 / 2 ( π / 2 ) 3 / 2 w 3 = I 0 2 N 2 - 3 / 2 exp ( - v 2 t 2 w 2 ) .
I 2 , b ( t ) = j l P [ r j ( 0 ) ] P [ r j ( t ) + v t ] × P [ r l ( 0 ) P [ r l ( t ) + v t ] r ( 0 ) × exp ( i q · δ r j , m ) exp ( i q · δ r l , k ) δ r .
I 2 , b ( t ) = j l P P ( t ) 2 exp ( i q · δ r ) 2 = I 0 2 N ( N - 1 ) exp ( - v 2 t 2 w 2 ) exp ( - 2 D q 2 t ) .
I 2 ( t ) N 2 [ 1 + exp ( 2 D q 2 t ) exp ( - v 2 t 2 w 2 ) ] + 2 - 3 / 2 N exp ( - v 2 t 2 w 2 ) .
I 2 ( t ) = N 2 I 0 2 + I 1 ( t ) 2 .
I I ( n Δ t ) = B + C ( 0 ) exp [ - n Δ t τ 1 - ( n Δ t 2 ) 2 τ 2 2 ] .
Y ( n Δ t ) = 1 n Δ t ln C ( n Δ t ) C ( 0 ) = 1 τ 1 + ( 1 τ 2 ) 2 n Δ t .
Y m ( n Δ t ) = - 1 ( n - m ) Δ t ln C ( n ) C ( m ) = 1 τ 1 + ( 1 τ 2 ) 2 ( n + m ) Δ t .
w 1 = λ π w 0 f ,
D = K T 6 π η r ,

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