Abstract

A two-speed photon diffusion equation is developed for light propagation in a powder bed of high volume fraction or dense particulate suspension, whereby the light speed is impacted by the refractive index difference between particles and the suspending medium. The equation is validated using Monte Carlo simulation of light propagation coupled with dynamic simulation of particle sedimentation for the non-uniform arrangement of powder particles. Frequency domain experiments at 650 nm for a 77-µm-diameter resin-powder and 50-µm-diameter lactose-powder beds as well as resin-water and lactose-ethanol suspensions confirm the scattering and absorption coefficients derived from the two-speed diffusion equation.

© 2005 Optical Society of America

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References

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  1. R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick, �??Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: a novel sensor of blend homogeneity,�?? J. Pharm. Sci. 88, 959-966 (1999).
    [CrossRef] [PubMed]
  2. T. Pan and E. M. Sevick-Muraca, �??Volume of pharmaceutical powders probed by frequency-domain photon migration measurements of multiply scattered light,�?? Anal. Chem. 74, 4228-4234 (2002).
    [CrossRef] [PubMed]
  3. T. Pan, D. Barber, D. Coffin-Beach, Z. Sun, and E. M. Sevick-Muraca, �??Measurement of low dose active pharmaceutical ingredient in a pharmaceutical blend using frequency-domain photon migration,�?? J. Pharm. Sci. 93, 635-645 (2004).
    [CrossRef] [PubMed]
  4. Z. Sun, S. Torrance, F. K. McNeil-Watson, and E. M. Sevick-Muraca, �??Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,�?? Anal. Chem. 75, 1720-1725 (2003).
    [CrossRef] [PubMed]
  5. S. E. Torrance, Z. Sun, and E. M. Sevick-Muraca, �??Impact of excipient particle size on measurement of active pharmaceutical ingredient absorbance in mixtures using frequency domain photon migration,�?? J. Pharm. Sci. 93, 1879-1889 (2004).
    [CrossRef] [PubMed]
  6. V. Venugopalan, J. S. You, and B. J. Tromberg, �??Radiative transport in the diffusion approximation: an extension for highly absorbing medium and small source-detector separations,�?? Phys. Rev. E 58, 2395-2407 (1998).
    [CrossRef]
  7. J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis, (John Wiley & Sons, 1976 ), Chapter 4.
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  10. J. P. Hansen and I. R. McDonald, Theory of Simple Liquid, (Academic Press, 1986), Chapter 3.
  11. Z. Sun and E. M. Sevick-Muraca, �??Investigation of particle interactions in dense colloidal suspensions using frequency domain photon migration: bidisperse systems,�?? Langmuir 18, 1091-1097 (2002).
    [CrossRef]
  12. Y. Huang, Z. Sun, and E. M. Sevick-Muraca, �??Assessment of electrostatic interaction in dense colloidal suspensions with multiply scattered light,�?? Langmuir 18, 2048-2053 (2002).
    [CrossRef]
  13. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, 1983), Chapter 7.
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    [CrossRef]
  16. X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, �??Determination of complex refractive index of polystyrene microsphere from 370 to 1610 nm,�?? Phys. Med. Biol. 48, 4165-4172 (2003).
    [CrossRef]
  17. A. N. Winchell, The Optical Properties of Organic Compounds, (Academic Press, 1954), Chapter 4.
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  22. N. V. Brilliantov, F. Spahn, and J. M. Hertzsch, �??Model for collisions in granular gases,�?? Phys. Rev. E 53, 5382-5392 (1996).
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  23. P. A. Thompson and G. S. Grest, �??Granular flow: friction and the dilatancy transition,�?? Phys. Rev. Lett. 67, 1751-1754 (1991).
    [CrossRef] [PubMed]
  24. L. G. Leal, Laminar Flow and Convective Transport Processes, (Butterworth-Heinemann, 1992), Chapter 4.
  25. F. Podczeck, J. M. Newton, and M. B. James, �??The adhesion force of micronized salmeterol xinafoate particles to pharmaceutically relevant surface materials,�?? J. Phys. D: Appl. Phys. 29, 1878-1884 (1996).
    [CrossRef]
  26. R. J. Roberts, R. C. Rowe, and P. York, �??The relationship between Young�??s modulus of elasticity of organic solids and their molecular structure,�?? Powder Technol. 65, 139-146 (1991).
    [CrossRef]
  27. Y. Shimada, Y. Yonezawa, H. Sunada, R. Nonaka, K. Katou, and H. Morishita, �??Development of an apparatus for measuring adhesive force between fine particles,�?? KONA No. 20, 223-230 (2002).
  28. K. Z. Y. Yen and T. K. Chaki, �??A dynamic simulation of particle rearrangement in powder packings with realistic interactions,�?? J. Appl. Phys. 71, 3164-3173 (1992).
    [CrossRef]
  29. L. M. Zurk, L. Tsang, K. H. Ding, and D. P. Winebrenner, �??Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometrics,�?? J. Opt. Soc. Am A 12, 1772-1781 (1995).
    [CrossRef]

Anal. Chem. (2)

Z. Sun, S. Torrance, F. K. McNeil-Watson, and E. M. Sevick-Muraca, �??Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,�?? Anal. Chem. 75, 1720-1725 (2003).
[CrossRef] [PubMed]

T. Pan and E. M. Sevick-Muraca, �??Volume of pharmaceutical powders probed by frequency-domain photon migration measurements of multiply scattered light,�?? Anal. Chem. 74, 4228-4234 (2002).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

H.-J. Moon, K. An, and J.- H. Lee, �??Single spatial mode selection in a layered square microcavity laser,�?? Appl. Phys. Lett. 82, 2963-2965 (2003).
[CrossRef]

Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques, and L. Zheng, �??MCML �?? Monte Carlo modeling of light transport in multi-layered tissue,�?? Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

J. Appl. Phys. (2)

R. Y. Yang, R. P. Zou, and A. B. Yu, �??Effect of material properties on the packing of fine particles,�?? J. Appl. Phys. 94, 3025-3034 (2003).
[CrossRef]

K. Z. Y. Yen and T. K. Chaki, �??A dynamic simulation of particle rearrangement in powder packings with realistic interactions,�?? J. Appl. Phys. 71, 3164-3173 (1992).
[CrossRef]

J. Opt. Soc. Am A (1)

L. M. Zurk, L. Tsang, K. H. Ding, and D. P. Winebrenner, �??Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometrics,�?? J. Opt. Soc. Am A 12, 1772-1781 (1995).
[CrossRef]

J. Pharm. Sci. (3)

T. Pan, D. Barber, D. Coffin-Beach, Z. Sun, and E. M. Sevick-Muraca, �??Measurement of low dose active pharmaceutical ingredient in a pharmaceutical blend using frequency-domain photon migration,�?? J. Pharm. Sci. 93, 635-645 (2004).
[CrossRef] [PubMed]

S. E. Torrance, Z. Sun, and E. M. Sevick-Muraca, �??Impact of excipient particle size on measurement of active pharmaceutical ingredient absorbance in mixtures using frequency domain photon migration,�?? J. Pharm. Sci. 93, 1879-1889 (2004).
[CrossRef] [PubMed]

R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick, �??Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: a novel sensor of blend homogeneity,�?? J. Pharm. Sci. 88, 959-966 (1999).
[CrossRef] [PubMed]

J. Phys. D: Appl. Phys. (1)

F. Podczeck, J. M. Newton, and M. B. James, �??The adhesion force of micronized salmeterol xinafoate particles to pharmaceutically relevant surface materials,�?? J. Phys. D: Appl. Phys. 29, 1878-1884 (1996).
[CrossRef]

KONA (1)

Y. Shimada, Y. Yonezawa, H. Sunada, R. Nonaka, K. Katou, and H. Morishita, �??Development of an apparatus for measuring adhesive force between fine particles,�?? KONA No. 20, 223-230 (2002).

Langmuir (2)

Z. Sun and E. M. Sevick-Muraca, �??Investigation of particle interactions in dense colloidal suspensions using frequency domain photon migration: bidisperse systems,�?? Langmuir 18, 1091-1097 (2002).
[CrossRef]

Y. Huang, Z. Sun, and E. M. Sevick-Muraca, �??Assessment of electrostatic interaction in dense colloidal suspensions with multiply scattered light,�?? Langmuir 18, 2048-2053 (2002).
[CrossRef]

Phys. Med. Biol. (1)

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, �??Determination of complex refractive index of polystyrene microsphere from 370 to 1610 nm,�?? Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Phys. Rev. E (3)

V. Venugopalan, J. S. You, and B. J. Tromberg, �??Radiative transport in the diffusion approximation: an extension for highly absorbing medium and small source-detector separations,�?? Phys. Rev. E 58, 2395-2407 (1998).
[CrossRef]

R. Y. Yang, R. P. Zou, and A. B. Yu, �??Computer simulation of the packing of fine particles,�?? Phys. Rev. E 62, 3900-3908 (2000).
[CrossRef]

N. V. Brilliantov, F. Spahn, and J. M. Hertzsch, �??Model for collisions in granular gases,�?? Phys. Rev. E 53, 5382-5392 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

P. A. Thompson and G. S. Grest, �??Granular flow: friction and the dilatancy transition,�?? Phys. Rev. Lett. 67, 1751-1754 (1991).
[CrossRef] [PubMed]

Powder Technol. (1)

R. J. Roberts, R. C. Rowe, and P. York, �??The relationship between Young�??s modulus of elasticity of organic solids and their molecular structure,�?? Powder Technol. 65, 139-146 (1991).
[CrossRef]

Review of Modern Physics (1)

S. Chandrasekhar, �??Stochastic problems in physics and astronomy,�?? Review of Modern Physics 15, 1-89 (1943).
[CrossRef]

Other (6)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, 1983), Chapter 7.

A. N. Winchell, The Optical Properties of Organic Compounds, (Academic Press, 1954), Chapter 4.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis, (John Wiley & Sons, 1976 ), Chapter 4.

A. F. Henry, Nuclear-Reactor Analysis, (The MIT Press, 1975), Chapter 9.

J. P. Hansen and I. R. McDonald, Theory of Simple Liquid, (Academic Press, 1986), Chapter 3.

L. G. Leal, Laminar Flow and Convective Transport Processes, (Butterworth-Heinemann, 1992), Chapter 4.

Supplementary Material (1)

» Media 1: AVI (769 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Schematic of light trajectory in ensemble of large dense, scatters (dp≫λ) with significant refractive index difference with surroundings. The trajectory of photons within particles occurs at speed cp=c/np , while outside the particles, photons travel with speed cmed=c/nmed , where c is the light speed in vacuum and np and nmed are refractive indices of particle and surrounding medium, respectively.

Fig. 2.
Fig. 2.

(a) Micrograph of resin particles in packed powder bed. The resin particles have regular spherical shape with a mean diameter of 77µm. In a densely packed powder bed containing such resin particles, the volume fraction, fv, is 0.64; (b) Micrograph of multi-disperse lactose particles with a mean diameter of around 50µm.

Fig. 3.
Fig. 3.

(Movie) Powder structure evolution computed from dynamic simulation for a sedimentation process with 2,000 particles of dp =50µm confined by a plate (consisting of orderly arranged particles and positioned at the top of simulated particles) in a (5×10-4 m).(5×10-4 m).(2×10-3 m) cube with periodic boundary conditions along horizontal directions. [Media 1]

Fig. 4.
Fig. 4.

(a) The time-dependent photon densities computed from time-domain Monte Carlo simulation for ϕp (blue symbols) and ϕmed (red symbols) as a function of time and source-detector separation, r, of 75dp , 165dp , and 255dp in a powder bed with volume fraction of 0.61, np =1.6, and dp =50µm ; (b) The time-dependent photon densities computed from time-domain Monte Carlo simulation for ϕp (blue symbols) and (2.44×ϕmed ) (red symbols) as a function of time and r of 45dp , 75dp , 105dp , 135dp , 165dp , 195dp , 225dp , 255dp , and 285dp in a powder bed with volume fraction of 0.61, np =1.6, and dp =50µm.

Fig. 5.
Fig. 5.

(a) Ratio of mean scattering length in suspending medium to particle diameter versus volume fraction varying from 0.05 to 0.64 at various relative refractive indices, nmed/np , and particle diameters, dp , varying from 20 to 50µm with points representing Monte Carlo simulation results and the curve denoting a fitting relationship; (b) Ratio of mean scattering length within particles to particle diameter versus relative refractive index, nmed/np , with triangles (fv =0.61, dp =50µm), squares (fv =0.30, dp =40µm) and diamonds (fv =0.10, dp =30µm) representing Monte Carlo simulation results and the curve denoting a fitting relationship.

Fig. 6.
Fig. 6.

(a) Y(PS,AC,DC,r) versus the source-detector separation at 65 MHz (diamonds), 95 MHz (triangles), and 125 MHz (squares) in frequency domain photon migration measurements for resin powder bed of dp =77µm and fv =0.64 with the solid lines computed from two-speed diffusion equation using scattering coefficient predicted from Monte Carlo simulation. (b) Y(PS,AC,DC,r) versus the source-detector separation at 40 MHz (diamonds), 80 MHz (triangles), and 120 MHz (squares) in frequency domain photon migration measurements for polydisperse lactose powder bed with fv =0.65 and mean diameter of around of 50 µm with the solid lines denoting fitting curves of the two-speed diffusion equation.

Fig. 7.
Fig. 7.

Absorption coefficients of particulate suspensions, µa,bed , versus volume fraction, fv , for lactose-ethanol (triangles) and resin-water (squares) suspensions with points representing the FDPM measurement results and the lines denoting the fitting curves of two-speed diffusion equation.

Fig. a1.
Fig. a1.

1. (a) The time-dependent photon densities computed from time-domain Monte Carlo simulation for ϕp (blue symbols) and (1.46×ϕmed ) (red symbols) as a function of time and source-detector separation, r, in a powder bed with volume fraction of 0.45 and dp =45µm. (b) The time-dependent photon densities computed from time-domain Monte Carlo simulation for ϕp (blue symbols) and (1.13×ϕmed ) (red symbols) as a function of time and source-detector separation, r, in a powder bed with volume fraction of 0.38 and dp =42µm.

Tables (2)

Tables Icon

Table 1 The parameters are listed in the following table.

Tables Icon

Table 2 The verification of <Nscat,p >=<Nscat,med > has been conducted under various conditions of fv . The Monte Carlo simulation results are shown in the table.

Equations (34)

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ϕ p t + c n p · J p + c n p · μ a , p · ϕ p + r loss , p · ϕ p r loss , med · ϕ med = 0 .
ϕ med t + c n med · J med + c n med · μ a , med · ϕ med r loss , p · ϕ p + r loss , med · ϕ med = 0 .
ϕ p ( r , t ) ϕ med ( r , t ) = K e
J med = D med , med ϕ med + D med , p ϕ p
J p = D p , med ϕ med + D p , p ϕ p
n med c ϕ med t D bed 2 ϕ med + μ a , bed · ϕ med = 0
D bed = ( D med , med + K e · D med , p ) + n med n p ( D p , med + K e · D p , p ) ( 1 + K e )
μ a , bed = μ a , med + n med n p · K e · μ a , p ( 1 + K e ) .
F i = m i g + F i , vdW + F i , normal + F i , friction + F i , drag = m i d v i dt
T i = i = 1 N i , contact T ij , friction + T ij , rolling = I i d ω i dt
r i ( t + h ) = r i ( t ) + h × v i ( t ) + h 2 2 F i ( t ) m i
v i ( t + h ) = v i ( t ) + h m i × [ 3 2 F i ( t ) 1 2 F i ( t h ) ]
ω i ( t + h ) = ω i ( t ) + h I i × [ 3 2 T i ( t ) 1 2 T i ( t h ) ]
( Δ ln w ) i = μ a , j · c n j · t i
< r ( t ) > 2 = 16 · c · D bed π · n med · t
K e = ϕ p ϕ med = < t total , p > < t total , med > .
< t total , j > = < l scat , j > · N scat , j ( c n j ) ( j = p , med )
K e = < t total , p > < t total , med > = n p · < l scat , p > n med · < l scat , med >
μ a , bed = n med ( μ a , med < l scat , med > + μ a , p < l scat , p > ) ( n med < l scat , med > + n p < l scat , p > ) .
Y ( PS , AC , DC , r ) = { [ ( PS PS 0 ) 2 + ln 2 ( r · AC r 0 · AC 0 ) ] 2 ln 4 ( r · DC r 0 · DC 0 ) } 1 4
Y ( PS , AC , DC , r ) = 3 μ s , bed · ω · n med c × ( r r 0 ) .
< l scat , med > d p = 0.395 f v 1.26 ( 0.15 f v 0.64 ) .
μ a , bed = μ a , med + a 1 · f v 1.26 1 + a 2 · f v 1.26 ( 0.15 f v 0.64 )
F i , vdW = j = 1 , j i N m F ij , vdW
F ij , vdW = H vdW 6 · 64 R i 3 R j 3 ( h vdW + R i + R j ) n ij ( h vdW 2 + 2 R i h vdW + 2 R j h vdW ) 2 ( h vdW 2 + 2 R i h vdW + 2 R j h vdW + 4 R i R j ) 2
h vdW = max [ ( R i R j R i R j ) , h vdW 0 ]
F i , normal = j = 1 N i , contact F ij , normal
F ij , normal = [ 2 3 E R ̅ η 3 2 γ n E R ̅ η ( v ij · n ij ) ] n ij
F i , friction = γ s · j = 1 N i , contact F ij , normal t ij
F i , dragging = 6 π μ R i v i
T ij , friction = ( R i n ij ) × F ij , friction
T ij , rolling = γ r R i F ij , normal ω i ω i
P ( α i ) = 1 2 [ sin 2 ( α i α t ) sin 2 ( α i + a t ) + tan 2 ( α i α t ) tan 2 ( α i + a t ) ]
n med sin α i = n p sin α t .

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