Abstract

Despite serious nonsphericity of the particulate entities involved, observed extinction spectra for dilute carbon black sols are fitted precisely by Mie theory for ensembles of spheres which it is shown must be quite nearly volume-equivalent to the actual colloidal carbon units. Hence volume distribution statistics are obtainable for carbon blacks by inversion of spectrophotometric data on suspensions.

© 1980 Optical Society of America

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References

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  1. P. W. Barber, presented at the Fifty-Third Colloid and Surface Science Symposium, Rolla, Mo. (June 1979);Adv. Colloid Interface Sci. (to be published).
  2. P. Chýlek, J. Opt. Soc. Am. 67, 1348 (1977).
    [CrossRef]
  3. J. McK. Ellison, Proc. Phys. Soc. B 70, 102 (1957).
    [CrossRef]
  4. C. R. Berry, J. Opt. Soc. Am. 52, 888 (1962).
    [CrossRef]
  5. J. R. Hodkinson, in Electromagnetic Scattering, M. Kerker, Ed. (Pergamon, New York, 1963), p. 87.
  6. D. H. Napper, R. H. Ottewill, J. Colloid Sci. 18, 262 (1963).
    [CrossRef]
  7. B. Donn, R. S. Powell, in Electromagnetic Scattering, M. Kerker, Ed. (Pergamon, New York, 1963), p. 151.
  8. A. C. Holland, G. Gagne, Appl. Opt. 9, 1113 (1970).
    [CrossRef] [PubMed]
  9. T. D. Proctor, D. Barker, Aerosol Sci. 5, 91 (1974).
    [CrossRef]
  10. T. D. Proctor, G. W. Harris, Aerosol Sci. 5, 81 (1974).
    [CrossRef]
  11. R. G. Pinnick, D. E. Carroll, D. J. Hoffmann, Appl. Opt. 15, 384 (1976).
    [CrossRef] [PubMed]
  12. R. J. Perry, A. J. Hunt, D. R. Hufman, Appl. Opt. 17, 2700 (1978).
    [CrossRef] [PubMed]
  13. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  14. J. R. Hodkinson, Am. Ind. Hyg. Assoc. J. 26, 64 (1965).
    [CrossRef]
  15. Logically the interpretation of optical data for an irregular-particle suspension is exactly analogous to the interpretation of sedimentation coefficients. Stokes’s law ṙ/ω2r = D3(Δρ)/(18ηD) relates a sphere’s centrifugal rate of motion ṙ per unit field ω2r when sedimenting in a fluid at a low Reynolds number to its diameter D. (The sphere’s density is ρ and the fluid’s viscosity and density are η and ρ − Δρ, respectively.) This is readily solved for D: D = [(18η/Δρ)(ṙ/ω2r)]1/2, which is taken as the definition of phenomenological sedimentation-equivalent sphere diameters DStokes, when measured coefficients ṙ/ω2r for nonspherical particles are inserted into the right-hand side. The inversion step (solving for D) happens to be less difficult with Stokes’s law than with Mie’s equations, but subsequently connecting DStokes to intrinsic dimensions of irregular particles is as problematic as in the analogous optical case because of the difficulty of calculating fluid drag on irregular shapes.
  16. P. Chýlek, G. W. Grams, R. G. Pinnick, Science 193, 480 (1976).
    [CrossRef]
  17. A. I. Medalia, L. W. Richards, J. Colloid Interface Sci. 40, 233 (1972).
    [CrossRef]
  18. T. S. Ng, Angew. Makromol. Chem. 44, 165 (1975).
    [CrossRef]
  19. T. S. Ng, Prog. Colloid Polym. Sci. 65, 271 (1978).
    [CrossRef]
  20. I. N. Tang, H. R. Munkelwitz, J. Colloid Interface Sci. 63, 297 (1978).
    [CrossRef]
  21. G. N. Plass, G. W. Kattawar, Appl. Opt. 10, 1172 (1971).
    [CrossRef]
  22. A. C. Holland, G. Gagne, Appl. Opt. 10, 1173 (1971).
    [CrossRef] [PubMed]
  23. E. E. Underwood, Quantitative Stereology (Addison-Wesley, Reading, Mass., 1970), Sec. 6.4.3.
  24. A. C. Holland, J. S. Draper, Appl. Opt. 6, 511 (1967).
    [CrossRef] [PubMed]
  25. R. G. Pinnick, D. E. Carroll, D. J. Hofmann, Appl. Opt. 15, 384 (1976).
    [CrossRef] [PubMed]
  26. The degree of separation obtainable can be appreciated by noting that the ratio DS/Ds for a floc of K monodisperse spheres is K1/2, and it has been shown [J. Janzen, I. Goodarz-Nia, J. Colloid Interface Sci. 69, 476 (1979)] reasonably accurate to model carbon blacks in terms of such flocs, with values of K1/2 at least as large as 4 not at all uncommon. The ratio DS/Ds for cubes, in contrast, is only (6/π)1/2 or 1.38. (See Table V.)
    [CrossRef]
  27. P. Chýlek, J. Opt. Soc. Am. 66, 285 (1976).
    [CrossRef]
  28. C. Acquista, Appl. Opt. 17, 3851 (1978).
    [CrossRef] [PubMed]
  29. P. Chýlek, R. G. Pinnick, Appl. Opt. 18, 1123 (1979).
    [CrossRef] [PubMed]
  30. J. Janzen, J. Colloid Interface Sci. 69, 436 (1979).
    [CrossRef]
  31. J. Janzen, E. E. Rush, U.S. Patent3,807,704 (1974), licensed to Lako Manufacturing Co.
  32. These ultrasonification times, totaling 3 h, have been found necessary and sufficient to reach stationary plateaus in optical properties of the carbon black sols.
  33. Sodev Inc., model 02D.
  34. Ref. 13, p. 324. Note that here λ is in nanometers.
  35. P. Latimer, J. Colloid Interface Sci. 39, 497 (1972).
    [CrossRef]
  36. i≡−1.
  37. E. A. Taft, H. R. Philipp, Phys. Rev. A: 138, 197 (1965).
  38. R. P. DiNardo, A. N. Goland, J. Opt. Soc. Am. 61, 1321 (1971).
    [CrossRef]
  39. M. W. Williams, E. T. Arakawa, J. Appl. Phys. 43, 3460 (1972).
    [CrossRef]
  40. This quantity multiplied by ln10 is the same as what Ng19 denotes by λs.
  41. C. deBoor, in Mathematical Software, J. R. Rice, Ed. (Academic, New York, 1971), Chap. 7.
  42. G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]
  43. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Eq. (10.1.31).
  44. A. Brockes, Optik 21, 550 (1964).
  45. J. Janzen, I. Goodarz-Nia, J. Colloid Interface Sci. 69, 476 (1979).
    [CrossRef]
  46. The tails of some distributions extend somewhat beyond.
  47. R. M. Welch, S. K. Cox [Appl. Opt. 17, 3159 (1978)] misquote Chýlek when they say “the extinction cross section of a randomly oriented nonspherical particle is always larger than the extinction cross section of a spherical particle of equal volume,” without prescribing large x.
    [CrossRef] [PubMed]
  48. A. Fiorenza, Rubber Age 80, 69 (1956).
  49. A. Voet, Rubber Age 82, 657 (1958).
  50. A. Fiorenza, Rubber Age 84, 945 (1959).
  51. H. C. Donoian, A. I. Medalia, J. Paint Technol. 39, 716 (1967).
  52. J. C. Giddings, M. N. Myers, F. J. F. Yang, L. K. Smith, in Colloid and Interface Science, Vol. 4, M. Kerker, Ed. (Academic, New York, 1976), p. 381.
  53. These difficulties were mentioned briefly in Ref. 45; obtaining quantitative microscopic data of sufficient precision and accuracy to allow critical comparison with precise macroscopic measurements places a heavy burden upon the crucial microscope specimen preparation step as well as upon the image analysis data collection and reduction.
  54. A. I. Medalia, J. Colloid Interface Sci. 32, 115 (1970).
    [CrossRef]

1979

The degree of separation obtainable can be appreciated by noting that the ratio DS/Ds for a floc of K monodisperse spheres is K1/2, and it has been shown [J. Janzen, I. Goodarz-Nia, J. Colloid Interface Sci. 69, 476 (1979)] reasonably accurate to model carbon blacks in terms of such flocs, with values of K1/2 at least as large as 4 not at all uncommon. The ratio DS/Ds for cubes, in contrast, is only (6/π)1/2 or 1.38. (See Table V.)
[CrossRef]

J. Janzen, J. Colloid Interface Sci. 69, 436 (1979).
[CrossRef]

J. Janzen, I. Goodarz-Nia, J. Colloid Interface Sci. 69, 476 (1979).
[CrossRef]

P. Chýlek, R. G. Pinnick, Appl. Opt. 18, 1123 (1979).
[CrossRef] [PubMed]

1978

1977

1976

1975

T. S. Ng, Angew. Makromol. Chem. 44, 165 (1975).
[CrossRef]

1974

T. D. Proctor, D. Barker, Aerosol Sci. 5, 91 (1974).
[CrossRef]

T. D. Proctor, G. W. Harris, Aerosol Sci. 5, 81 (1974).
[CrossRef]

1972

A. I. Medalia, L. W. Richards, J. Colloid Interface Sci. 40, 233 (1972).
[CrossRef]

M. W. Williams, E. T. Arakawa, J. Appl. Phys. 43, 3460 (1972).
[CrossRef]

P. Latimer, J. Colloid Interface Sci. 39, 497 (1972).
[CrossRef]

1971

1970

A. C. Holland, G. Gagne, Appl. Opt. 9, 1113 (1970).
[CrossRef] [PubMed]

A. I. Medalia, J. Colloid Interface Sci. 32, 115 (1970).
[CrossRef]

1967

1965

E. A. Taft, H. R. Philipp, Phys. Rev. A: 138, 197 (1965).

J. R. Hodkinson, Am. Ind. Hyg. Assoc. J. 26, 64 (1965).
[CrossRef]

1964

A. Brockes, Optik 21, 550 (1964).

1963

D. H. Napper, R. H. Ottewill, J. Colloid Sci. 18, 262 (1963).
[CrossRef]

1962

1959

A. Fiorenza, Rubber Age 84, 945 (1959).

1958

A. Voet, Rubber Age 82, 657 (1958).

1957

J. McK. Ellison, Proc. Phys. Soc. B 70, 102 (1957).
[CrossRef]

1956

A. Fiorenza, Rubber Age 80, 69 (1956).

Acquista, C.

Arakawa, E. T.

M. W. Williams, E. T. Arakawa, J. Appl. Phys. 43, 3460 (1972).
[CrossRef]

Barber, P. W.

P. W. Barber, presented at the Fifty-Third Colloid and Surface Science Symposium, Rolla, Mo. (June 1979);Adv. Colloid Interface Sci. (to be published).

Barker, D.

T. D. Proctor, D. Barker, Aerosol Sci. 5, 91 (1974).
[CrossRef]

Berry, C. R.

Brockes, A.

A. Brockes, Optik 21, 550 (1964).

Carroll, D. E.

Chýlek, P.

Cox, S. K.

deBoor, C.

C. deBoor, in Mathematical Software, J. R. Rice, Ed. (Academic, New York, 1971), Chap. 7.

DiNardo, R. P.

Donn, B.

B. Donn, R. S. Powell, in Electromagnetic Scattering, M. Kerker, Ed. (Pergamon, New York, 1963), p. 151.

Donoian, H. C.

H. C. Donoian, A. I. Medalia, J. Paint Technol. 39, 716 (1967).

Draper, J. S.

Ellison, J. McK.

J. McK. Ellison, Proc. Phys. Soc. B 70, 102 (1957).
[CrossRef]

Fiorenza, A.

A. Fiorenza, Rubber Age 84, 945 (1959).

A. Fiorenza, Rubber Age 80, 69 (1956).

Gagne, G.

Giddings, J. C.

J. C. Giddings, M. N. Myers, F. J. F. Yang, L. K. Smith, in Colloid and Interface Science, Vol. 4, M. Kerker, Ed. (Academic, New York, 1976), p. 381.

Goland, A. N.

Goodarz-Nia, I.

The degree of separation obtainable can be appreciated by noting that the ratio DS/Ds for a floc of K monodisperse spheres is K1/2, and it has been shown [J. Janzen, I. Goodarz-Nia, J. Colloid Interface Sci. 69, 476 (1979)] reasonably accurate to model carbon blacks in terms of such flocs, with values of K1/2 at least as large as 4 not at all uncommon. The ratio DS/Ds for cubes, in contrast, is only (6/π)1/2 or 1.38. (See Table V.)
[CrossRef]

J. Janzen, I. Goodarz-Nia, J. Colloid Interface Sci. 69, 476 (1979).
[CrossRef]

Grams, G. W.

P. Chýlek, G. W. Grams, R. G. Pinnick, Science 193, 480 (1976).
[CrossRef]

Harris, G. W.

T. D. Proctor, G. W. Harris, Aerosol Sci. 5, 81 (1974).
[CrossRef]

Hodkinson, J. R.

J. R. Hodkinson, Am. Ind. Hyg. Assoc. J. 26, 64 (1965).
[CrossRef]

J. R. Hodkinson, in Electromagnetic Scattering, M. Kerker, Ed. (Pergamon, New York, 1963), p. 87.

Hoffmann, D. J.

Hofmann, D. J.

Holland, A. C.

Hufman, D. R.

Hunt, A. J.

Janzen, J.

The degree of separation obtainable can be appreciated by noting that the ratio DS/Ds for a floc of K monodisperse spheres is K1/2, and it has been shown [J. Janzen, I. Goodarz-Nia, J. Colloid Interface Sci. 69, 476 (1979)] reasonably accurate to model carbon blacks in terms of such flocs, with values of K1/2 at least as large as 4 not at all uncommon. The ratio DS/Ds for cubes, in contrast, is only (6/π)1/2 or 1.38. (See Table V.)
[CrossRef]

J. Janzen, J. Colloid Interface Sci. 69, 436 (1979).
[CrossRef]

J. Janzen, I. Goodarz-Nia, J. Colloid Interface Sci. 69, 476 (1979).
[CrossRef]

J. Janzen, E. E. Rush, U.S. Patent3,807,704 (1974), licensed to Lako Manufacturing Co.

Kattawar, G. W.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Latimer, P.

P. Latimer, J. Colloid Interface Sci. 39, 497 (1972).
[CrossRef]

Medalia, A. I.

A. I. Medalia, L. W. Richards, J. Colloid Interface Sci. 40, 233 (1972).
[CrossRef]

A. I. Medalia, J. Colloid Interface Sci. 32, 115 (1970).
[CrossRef]

H. C. Donoian, A. I. Medalia, J. Paint Technol. 39, 716 (1967).

Munkelwitz, H. R.

I. N. Tang, H. R. Munkelwitz, J. Colloid Interface Sci. 63, 297 (1978).
[CrossRef]

Myers, M. N.

J. C. Giddings, M. N. Myers, F. J. F. Yang, L. K. Smith, in Colloid and Interface Science, Vol. 4, M. Kerker, Ed. (Academic, New York, 1976), p. 381.

Napper, D. H.

D. H. Napper, R. H. Ottewill, J. Colloid Sci. 18, 262 (1963).
[CrossRef]

Ng, T. S.

T. S. Ng, Prog. Colloid Polym. Sci. 65, 271 (1978).
[CrossRef]

T. S. Ng, Angew. Makromol. Chem. 44, 165 (1975).
[CrossRef]

Ottewill, R. H.

D. H. Napper, R. H. Ottewill, J. Colloid Sci. 18, 262 (1963).
[CrossRef]

Perry, R. J.

Philipp, H. R.

E. A. Taft, H. R. Philipp, Phys. Rev. A: 138, 197 (1965).

Pinnick, R. G.

Plass, G. N.

Powell, R. S.

B. Donn, R. S. Powell, in Electromagnetic Scattering, M. Kerker, Ed. (Pergamon, New York, 1963), p. 151.

Proctor, T. D.

T. D. Proctor, D. Barker, Aerosol Sci. 5, 91 (1974).
[CrossRef]

T. D. Proctor, G. W. Harris, Aerosol Sci. 5, 81 (1974).
[CrossRef]

Richards, L. W.

A. I. Medalia, L. W. Richards, J. Colloid Interface Sci. 40, 233 (1972).
[CrossRef]

Rush, E. E.

J. Janzen, E. E. Rush, U.S. Patent3,807,704 (1974), licensed to Lako Manufacturing Co.

Smith, L. K.

J. C. Giddings, M. N. Myers, F. J. F. Yang, L. K. Smith, in Colloid and Interface Science, Vol. 4, M. Kerker, Ed. (Academic, New York, 1976), p. 381.

Taft, E. A.

E. A. Taft, H. R. Philipp, Phys. Rev. A: 138, 197 (1965).

Tang, I. N.

I. N. Tang, H. R. Munkelwitz, J. Colloid Interface Sci. 63, 297 (1978).
[CrossRef]

Underwood, E. E.

E. E. Underwood, Quantitative Stereology (Addison-Wesley, Reading, Mass., 1970), Sec. 6.4.3.

Voet, A.

A. Voet, Rubber Age 82, 657 (1958).

Welch, R. M.

Williams, M. W.

M. W. Williams, E. T. Arakawa, J. Appl. Phys. 43, 3460 (1972).
[CrossRef]

Yang, F. J. F.

J. C. Giddings, M. N. Myers, F. J. F. Yang, L. K. Smith, in Colloid and Interface Science, Vol. 4, M. Kerker, Ed. (Academic, New York, 1976), p. 381.

Aerosol Sci.

T. D. Proctor, D. Barker, Aerosol Sci. 5, 91 (1974).
[CrossRef]

T. D. Proctor, G. W. Harris, Aerosol Sci. 5, 81 (1974).
[CrossRef]

Am. Ind. Hyg. Assoc. J.

J. R. Hodkinson, Am. Ind. Hyg. Assoc. J. 26, 64 (1965).
[CrossRef]

Angew. Makromol. Chem.

T. S. Ng, Angew. Makromol. Chem. 44, 165 (1975).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

M. W. Williams, E. T. Arakawa, J. Appl. Phys. 43, 3460 (1972).
[CrossRef]

J. Colloid Interface Sci.

P. Latimer, J. Colloid Interface Sci. 39, 497 (1972).
[CrossRef]

J. Janzen, I. Goodarz-Nia, J. Colloid Interface Sci. 69, 476 (1979).
[CrossRef]

A. I. Medalia, L. W. Richards, J. Colloid Interface Sci. 40, 233 (1972).
[CrossRef]

I. N. Tang, H. R. Munkelwitz, J. Colloid Interface Sci. 63, 297 (1978).
[CrossRef]

The degree of separation obtainable can be appreciated by noting that the ratio DS/Ds for a floc of K monodisperse spheres is K1/2, and it has been shown [J. Janzen, I. Goodarz-Nia, J. Colloid Interface Sci. 69, 476 (1979)] reasonably accurate to model carbon blacks in terms of such flocs, with values of K1/2 at least as large as 4 not at all uncommon. The ratio DS/Ds for cubes, in contrast, is only (6/π)1/2 or 1.38. (See Table V.)
[CrossRef]

J. Janzen, J. Colloid Interface Sci. 69, 436 (1979).
[CrossRef]

A. I. Medalia, J. Colloid Interface Sci. 32, 115 (1970).
[CrossRef]

J. Colloid Sci.

D. H. Napper, R. H. Ottewill, J. Colloid Sci. 18, 262 (1963).
[CrossRef]

J. Opt. Soc. Am.

J. Paint Technol.

H. C. Donoian, A. I. Medalia, J. Paint Technol. 39, 716 (1967).

Optik

A. Brockes, Optik 21, 550 (1964).

Phys. Rev. A

E. A. Taft, H. R. Philipp, Phys. Rev. A: 138, 197 (1965).

Proc. Phys. Soc. B

J. McK. Ellison, Proc. Phys. Soc. B 70, 102 (1957).
[CrossRef]

Prog. Colloid Polym. Sci.

T. S. Ng, Prog. Colloid Polym. Sci. 65, 271 (1978).
[CrossRef]

Rubber Age

A. Fiorenza, Rubber Age 80, 69 (1956).

A. Voet, Rubber Age 82, 657 (1958).

A. Fiorenza, Rubber Age 84, 945 (1959).

Science

P. Chýlek, G. W. Grams, R. G. Pinnick, Science 193, 480 (1976).
[CrossRef]

Other

P. W. Barber, presented at the Fifty-Third Colloid and Surface Science Symposium, Rolla, Mo. (June 1979);Adv. Colloid Interface Sci. (to be published).

Logically the interpretation of optical data for an irregular-particle suspension is exactly analogous to the interpretation of sedimentation coefficients. Stokes’s law ṙ/ω2r = D3(Δρ)/(18ηD) relates a sphere’s centrifugal rate of motion ṙ per unit field ω2r when sedimenting in a fluid at a low Reynolds number to its diameter D. (The sphere’s density is ρ and the fluid’s viscosity and density are η and ρ − Δρ, respectively.) This is readily solved for D: D = [(18η/Δρ)(ṙ/ω2r)]1/2, which is taken as the definition of phenomenological sedimentation-equivalent sphere diameters DStokes, when measured coefficients ṙ/ω2r for nonspherical particles are inserted into the right-hand side. The inversion step (solving for D) happens to be less difficult with Stokes’s law than with Mie’s equations, but subsequently connecting DStokes to intrinsic dimensions of irregular particles is as problematic as in the analogous optical case because of the difficulty of calculating fluid drag on irregular shapes.

J. R. Hodkinson, in Electromagnetic Scattering, M. Kerker, Ed. (Pergamon, New York, 1963), p. 87.

B. Donn, R. S. Powell, in Electromagnetic Scattering, M. Kerker, Ed. (Pergamon, New York, 1963), p. 151.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

J. Janzen, E. E. Rush, U.S. Patent3,807,704 (1974), licensed to Lako Manufacturing Co.

These ultrasonification times, totaling 3 h, have been found necessary and sufficient to reach stationary plateaus in optical properties of the carbon black sols.

Sodev Inc., model 02D.

Ref. 13, p. 324. Note that here λ is in nanometers.

J. C. Giddings, M. N. Myers, F. J. F. Yang, L. K. Smith, in Colloid and Interface Science, Vol. 4, M. Kerker, Ed. (Academic, New York, 1976), p. 381.

These difficulties were mentioned briefly in Ref. 45; obtaining quantitative microscopic data of sufficient precision and accuracy to allow critical comparison with precise macroscopic measurements places a heavy burden upon the crucial microscope specimen preparation step as well as upon the image analysis data collection and reduction.

The tails of some distributions extend somewhat beyond.

i≡−1.

E. E. Underwood, Quantitative Stereology (Addison-Wesley, Reading, Mass., 1970), Sec. 6.4.3.

This quantity multiplied by ln10 is the same as what Ng19 denotes by λs.

C. deBoor, in Mathematical Software, J. R. Rice, Ed. (Academic, New York, 1971), Chap. 7.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Eq. (10.1.31).

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Figures (5)

Fig. 1
Fig. 1

Transmission electron micrograph of a typical individual carbon black paracrystal.

Fig. 2
Fig. 2

DODSobs (●) and DODScalc (full curve) for carbon black 4 of Table I (N219).

Fig. 3
Fig. 3

DODSobs (●) and DODScalc (full curve) for carbon black 20 of Table I (IRB 4; N330).

Fig. 4
Fig. 4

DODSobs (●) and DODScalc (full curve) for carbon black 27 of Table I (noncommercial; s = 53 m2/g).

Fig. 5
Fig. 5

Location of refractive indices of carbon blacks: a, commercial furnace-process samples; b, special (noncommercial) furnace-process samples; c, channel blacks. The curve is 0.195 normal reflectance isopleth. Cross denotes m0 estimate reported in Ref. 30.

Tables (5)

Tables Icon

Table I Results of DODS Analyses

Tables Icon

Table II Synthetic Data; Least Squares Analyses

Tables Icon

Table III Synthetic Data for XDV; Least Squares Analyses

Tables Icon

Table IV Hierarchy of Intrinsic Dimension-Equivalent Sphere Diameters

Tables Icon

Table V Relative Intrinsic Dimension-Equivalent Sphere Diameters for Flocsa

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

DODS obs ρ λ c 1 b 1 log ( I 0 / I ) ,
R ( 0 ° ) = [ ( 1 n ) 2 + k 2 ] / [ ( 1 + n ) 2 + k 2 ]
[ ( 1 n ) 2 + k 2 ] / [ ( 1 + n ) 2 + k 2 ] = 0.195
DODS calc λ ln 10 C ext V = λ ln 10 C ext π X 3 / 6 π X 3 / 6 V ,
C ext 0 f ( X ) C ext ( X ) d X ,
f ( X ) = 1 X 2 π ln G exp [ 1 2 ( ln X / X g ln G ) 2 ] .
X 606 / X g = exp [ 3 ( ln G ) 2 ] .
D s < D V / a ¯ < D p ¯ / a ¯ < D V < D a ¯ < D S < D p ¯ ,

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