Abstract

The saturation and the lightness of hematite layers are quantitatively related to the size distribution of the pigments, when applied on skin or a perfect white substrate. The optical properties, reflectance spectra, and colorimetric coordinates are calculated from the complex refractive index of hematite and by use of the radiative transfer equation. Monodisperse pigments are investigated first and the maximum of saturation is calculated as a function of the pigment radius. Polydisperse pigments are then investigated with a log-normal size distribution. The maximum of saturation is then calculated as a function of the width of the pigment distribution, for different mean radii. This modeling can be extended to any mineral pigments.

© 2011 Optical Society of America

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References

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  1. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. Leipzig 25, 377–445 (1908).
    [CrossRef]
  2. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles(Cambridge University, 2002).
  3. V. E. Cachorro and A. M. De Frutos, “A revised study of the validity of the general junge relationship at solar wavelengths: Application to vertical atmospheric aerosol layer studies,” Atmos. Res. 39, 113–126 (1995).
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  4. O. Ulloa, S. Sathyendranath, and T. Platt, “Effect of the particle-size distribution on the backscattering ratio in seawater,” Appl. Opt. 33, 7070–7077 (1994).
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    [CrossRef]
  7. A. P. Pdshivalko, “Optimization of the covering power and coloristic properties of pigmented coatings on the basis of a four-flux approximation of radiative transfer theory,” J. Appl. Spectrosc. 63, 675–683 (1996).
    [CrossRef]
  8. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  9. M. Elias and G. Elias, “Radiative transfer in inhomogeneous stratified media using the auxiliary function method,” J. Opt. Soc. Am. A 21, 580–589 (2004).
    [CrossRef]
  10. V. Goossens, J. Wielant, S. Van Gils, R. Finsy, H. Terryn, “Optical properties of thin oxide films on steel,” Surf. Interface Anal. 38, 489–493 (2006).
  11. M. Elias, C. Chartier, G. Prévot, and H. Garay, “Relationship between color and composition of ochres,” Mater. Sci. Engng. B 127, 70–80 (2006).
    [CrossRef]
  12. C. Magnain, M. Elias, and J. M. Frigerio, “Skin color modeling using the radiative transfer equation solved by the auxiliary function method,” J. Opt. Soc. Am. A 24, 2196–2205 (2007).
    [CrossRef]
  13. URL:http://www.atm.ox.ac.uk/code/mie/ (for the code Mie_single).
  14. M. Elias and M. Menu, “Characterization of surface states on patrimonial works of art,” Surf. Eng. 17, 225–229(2001).
    [CrossRef]
  15. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982).
  16. G. Latour, M. Elias, and J. M. Frigerio, “Determination of the absorption and scattering coefficients of pigments,” Appl. Spectrosc. 63, 604–610 (2009).
    [CrossRef] [PubMed]
  17. D. Duncan, “The colour of pigment mixtures,” J. Oil Colour Chem. Assoc. 32, 296 (1949).

2009 (1)

2008 (1)

P. Günthert, P. Hauser, and V. Radtke, “Effect of pigment particle size on application properties,” Review of Progress in Coloration and Related Topics 19, 41–48 (2008).
[CrossRef]

2007 (1)

2006 (2)

V. Goossens, J. Wielant, S. Van Gils, R. Finsy, H. Terryn, “Optical properties of thin oxide films on steel,” Surf. Interface Anal. 38, 489–493 (2006).

M. Elias, C. Chartier, G. Prévot, and H. Garay, “Relationship between color and composition of ochres,” Mater. Sci. Engng. B 127, 70–80 (2006).
[CrossRef]

2004 (1)

2002 (1)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles(Cambridge University, 2002).

2001 (1)

M. Elias and M. Menu, “Characterization of surface states on patrimonial works of art,” Surf. Eng. 17, 225–229(2001).
[CrossRef]

1996 (1)

A. P. Pdshivalko, “Optimization of the covering power and coloristic properties of pigmented coatings on the basis of a four-flux approximation of radiative transfer theory,” J. Appl. Spectrosc. 63, 675–683 (1996).
[CrossRef]

1995 (1)

V. E. Cachorro and A. M. De Frutos, “A revised study of the validity of the general junge relationship at solar wavelengths: Application to vertical atmospheric aerosol layer studies,” Atmos. Res. 39, 113–126 (1995).
[CrossRef]

1994 (1)

1982 (1)

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982).

1960 (1)

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

1949 (1)

D. Duncan, “The colour of pigment mixtures,” J. Oil Colour Chem. Assoc. 32, 296 (1949).

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. Leipzig 25, 377–445 (1908).
[CrossRef]

Cachorro, V. E.

V. E. Cachorro and A. M. De Frutos, “A revised study of the validity of the general junge relationship at solar wavelengths: Application to vertical atmospheric aerosol layer studies,” Atmos. Res. 39, 113–126 (1995).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Chartier, C.

M. Elias, C. Chartier, G. Prévot, and H. Garay, “Relationship between color and composition of ochres,” Mater. Sci. Engng. B 127, 70–80 (2006).
[CrossRef]

De Frutos, A. M.

V. E. Cachorro and A. M. De Frutos, “A revised study of the validity of the general junge relationship at solar wavelengths: Application to vertical atmospheric aerosol layer studies,” Atmos. Res. 39, 113–126 (1995).
[CrossRef]

Duncan, D.

D. Duncan, “The colour of pigment mixtures,” J. Oil Colour Chem. Assoc. 32, 296 (1949).

Elias, G.

Elias, M.

Finsy, R.

V. Goossens, J. Wielant, S. Van Gils, R. Finsy, H. Terryn, “Optical properties of thin oxide films on steel,” Surf. Interface Anal. 38, 489–493 (2006).

Frigerio, J. M.

Garay, H.

M. Elias, C. Chartier, G. Prévot, and H. Garay, “Relationship between color and composition of ochres,” Mater. Sci. Engng. B 127, 70–80 (2006).
[CrossRef]

Goossens, V.

V. Goossens, J. Wielant, S. Van Gils, R. Finsy, H. Terryn, “Optical properties of thin oxide films on steel,” Surf. Interface Anal. 38, 489–493 (2006).

Günthert, P.

P. Günthert, P. Hauser, and V. Radtke, “Effect of pigment particle size on application properties,” Review of Progress in Coloration and Related Topics 19, 41–48 (2008).
[CrossRef]

Hauser, P.

P. Günthert, P. Hauser, and V. Radtke, “Effect of pigment particle size on application properties,” Review of Progress in Coloration and Related Topics 19, 41–48 (2008).
[CrossRef]

Kippax, P.

P. Kippax, “Why particle sizing?,” Paint & Coatings Industry 0884-3848 (March 2005).

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles(Cambridge University, 2002).

Latour, G.

Magnain, C.

Menu, M.

M. Elias and M. Menu, “Characterization of surface states on patrimonial works of art,” Surf. Eng. 17, 225–229(2001).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. Leipzig 25, 377–445 (1908).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles(Cambridge University, 2002).

Pdshivalko, A. P.

A. P. Pdshivalko, “Optimization of the covering power and coloristic properties of pigmented coatings on the basis of a four-flux approximation of radiative transfer theory,” J. Appl. Spectrosc. 63, 675–683 (1996).
[CrossRef]

Platt, T.

Prévot, G.

M. Elias, C. Chartier, G. Prévot, and H. Garay, “Relationship between color and composition of ochres,” Mater. Sci. Engng. B 127, 70–80 (2006).
[CrossRef]

Radtke, V.

P. Günthert, P. Hauser, and V. Radtke, “Effect of pigment particle size on application properties,” Review of Progress in Coloration and Related Topics 19, 41–48 (2008).
[CrossRef]

Sathyendranath, S.

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982).

Terryn, H.

V. Goossens, J. Wielant, S. Van Gils, R. Finsy, H. Terryn, “Optical properties of thin oxide films on steel,” Surf. Interface Anal. 38, 489–493 (2006).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles(Cambridge University, 2002).

Ulloa, O.

Van Gils, S.

V. Goossens, J. Wielant, S. Van Gils, R. Finsy, H. Terryn, “Optical properties of thin oxide films on steel,” Surf. Interface Anal. 38, 489–493 (2006).

Wielant, J.

V. Goossens, J. Wielant, S. Van Gils, R. Finsy, H. Terryn, “Optical properties of thin oxide films on steel,” Surf. Interface Anal. 38, 489–493 (2006).

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982).

Ann. Phys. Leipzig (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. Leipzig 25, 377–445 (1908).
[CrossRef]

Appl. Opt. (1)

Appl. Spectrosc. (1)

Atmos. Res. (1)

V. E. Cachorro and A. M. De Frutos, “A revised study of the validity of the general junge relationship at solar wavelengths: Application to vertical atmospheric aerosol layer studies,” Atmos. Res. 39, 113–126 (1995).
[CrossRef]

J. Appl. Spectrosc. (1)

A. P. Pdshivalko, “Optimization of the covering power and coloristic properties of pigmented coatings on the basis of a four-flux approximation of radiative transfer theory,” J. Appl. Spectrosc. 63, 675–683 (1996).
[CrossRef]

J. Oil Colour Chem. Assoc. (1)

D. Duncan, “The colour of pigment mixtures,” J. Oil Colour Chem. Assoc. 32, 296 (1949).

J. Opt. Soc. Am. A (2)

Mater. Sci. Engng. B (1)

M. Elias, C. Chartier, G. Prévot, and H. Garay, “Relationship between color and composition of ochres,” Mater. Sci. Engng. B 127, 70–80 (2006).
[CrossRef]

Review of Progress in Coloration and Related Topics (1)

P. Günthert, P. Hauser, and V. Radtke, “Effect of pigment particle size on application properties,” Review of Progress in Coloration and Related Topics 19, 41–48 (2008).
[CrossRef]

Surf. Eng. (1)

M. Elias and M. Menu, “Characterization of surface states on patrimonial works of art,” Surf. Eng. 17, 225–229(2001).
[CrossRef]

Other (6)

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982).

V. Goossens, J. Wielant, S. Van Gils, R. Finsy, H. Terryn, “Optical properties of thin oxide films on steel,” Surf. Interface Anal. 38, 489–493 (2006).

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

URL:http://www.atm.ox.ac.uk/code/mie/ (for the code Mie_single).

P. Kippax, “Why particle sizing?,” Paint & Coatings Industry 0884-3848 (March 2005).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles(Cambridge University, 2002).

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Figures (17)

Fig. 1
Fig. 1

Complex refractive index n ˜ = n + i α of hematite according to [10].

Fig. 2
Fig. 2

Relative size distribution of hematite in different natural ochers from Madras, India (M), Italy (I), and Sweden (S) according to [11] and the corresponding log-normal distributions as solid curves.

Fig. 3
Fig. 3

Simulated reflectance spectrum of a Caucasian skin, according to [12].

Fig. 4
Fig. 4

Variation of the absorption and scattering coefficients with the wavelength for different mean radii of hematite.

Fig. 5
Fig. 5

Variation of the optical thickness and of the albedo with the wavelength for different mean radii of hematite.

Fig. 6
Fig. 6

Reflectance spectra for different mean radii of hematite and two different substrates.

Fig. 7
Fig. 7

Reflectance factor at 680 nm as a function of 1 / r for a skin substrate.

Fig. 8
Fig. 8

Saturation and lightness as a function of the mean radius of hematite and variation of lightness as a function of saturation for skin and white substrates. The colorimetric coordinates of the substrates are plotted as dashed curves.

Fig. 9
Fig. 9

Variation in the absorption coefficient with the wavelength for different widths of size distribution and two mean radii of hematite.

Fig. 10
Fig. 10

Variation in the scattering coefficient with the wavelength for different widths of size distribution and two mean radii of hematite.

Fig. 11
Fig. 11

Variation in the optical thickness with the wavelength for different widths of size distribution and two mean radius of hematite.

Fig. 12
Fig. 12

Variation in the albedo with the wavelength for different widths of size distribution and two mean radii of hematite.

Fig. 13
Fig. 13

Variation in the reflectance spectra for different widths of size distribution and two mean radii of hematite and a skin substrate.

Fig. 14
Fig. 14

Variation in the reflectance spectra for different widths of size distribution and two mean radii of hematite and a white substrate.

Fig. 15
Fig. 15

Variation in color saturation for different widths of size distribution and different mean radii of hematite and a skin substrate.

Fig. 16
Fig. 16

Variation in color saturation for different widths of size distribution and two mean radii of hematite and a white substrate.

Fig. 17
Fig. 17

Variation in lightness for different widths of size distribution and different mean radii of hematite on a skin and a white substrate.

Equations (6)

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k = 3 4 c Q abs r = 3 4 c Q ext Q sca r ,
s = 3 4 c Q sca r and s = ( 1 g ) 3 4 c Q sca r .
ω = s k + s = ( 1 g ) Q sca Q abs + ( 1 g ) Q sca .
d f ( u , τ ) d τ = f ( u , τ ) μ + ω 4 π μ | μ | [ F k ( u k , τ ) | μ k | + f ( u 1 , τ ) | μ 1 | d Ω 1 ] .
P z ( z ) = 1 2 π σ z exp [ 1 2 ( z z o σ z ) 2 ] or P r ( r ) = 1 2 π r σ z exp [ 1 2 ( log r log r o σ z ) 2 ] .
k = 3 4 c 0 Q abs r P r ( r ) d r s = 3 4 c 0 ( 1 g ) Q sca r P r ( r ) d r .

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