SPIE Proceedings [SPIE SPIE Optical Metrology - Munich, Germany (Monday 23 May 2011)] O3A: Optics...

Determination of the complex optical index of a red pigment, Vermillon

Raphaëlle Jarrigea, Christine Andrauda, Jacques Lafaita, Myriam Evenob, Michel Menub, Nuno Dinizc

aInstitut des NanoSciences de Paris, Case 840, 4 Place Jussieu, 75252 Paris bC2RMF, Palais du Louvre, 14 quai François Mitterrand, 75001 Paris

cGlaizer Group, 43 rue Pierre Valette, 92240 Malakoff

ABSTRACT

The non-destructive analysis of works of art and more specifically the paintings with the aim of a non-ambiguous identification of their components and the understanding of the techniques of the artists still remains a challenge. The aim of our research is to elaborate a purely optical way for this identification, based on the exclusive use of the intrinsic characteristic optical parameters of the components, instead the derived parameters presently commonly used, depending on several other parameters (morphology, environment…). The approach we propose is based on the resolution of the RTE using the 4-Flux approximation, combined with the Mie theory, allowing the identification of the pigments via the spectrum of their complex optical index entered into the model via a database. The key point of this approach is the index data bank. We report in this communication one the method’s crucial steps: the determination of the intrinsic optical index of pigments under the form of grains of micrometric size. This step is far from trivial and presents many difficulties that are not completely solved. This is one of the reasons why a more rigorous analysis of the paintings has not been up to now developed. We illustrate this problem with a red pigment: vermillion randomly dispersed at low concentration in a transparent polymer. The morphology of the sample is well characterized (thickness, concentration, size and dispersion of the pigments, surface roughness) as well as the index of the matrix. We use the same approach and model as presented above, applied this time to the calculation of the complex index of the pigments. The model is supposed to account for the diffuse flux and the specular flux, both measured on our samples, by spectrophotometry with an integrating sphere in the visible spectral range 400-800 nm. This resolution allows determining independently the coefficients of scattering and absorption of the pigment, which are finally related to the complex index of refraction through Mie’s Theory. Keywords: Complex optical index, pigments, 4-flux model, RTE, Mie theory, Vermillon, Malachite

1. INTRODUCTION

All the knowledge behind a work of art or an archeological object, such as the structure or composition of materials, is a prerequisite for the research in art history and in any conservation or restoration intervention. Concerning the study of painted artwork, optical methods are being developed more and more. In fact, the wavelengths used (in the visible and near infrared ranges) and the intensities that are commonly applied are non-destructive to the materials studied. Moreover, one can observe in recent years a rise in the interest for optical methods in reflection due to the possibility to perform direct measurements on the object without extracting it from its own environment. From a general point of view, the identification of the optical spectra of unknown materials involves a comparison of the reflection spectrum by means of a reference database. This method has its limitations nonetheless : the reflection spectrum of a paint layer depends of many parameters (such as size and concentration of pigments, binder, presence of varnish, number of layers…) and is therefore not characteristic of each of its constituent materials. An alternative method for such identification is to define the materials of paint layers by the ratio of the absorption coefficient to the scattering coefficient (K/S). This model is widely used in the industry because it can be more easily

O3A: Optics for Arts, Architecture, and Archaeology III, edited by Luca Pezzati, Renzo Salimbeni, Proc. of SPIE Vol. 8084, 80840D · © 2011 SPIE · CCC code: 0277-786X/11/$18 · doi: 10.1117/12.893264

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implemented to describe a bath of paint or ink, and appears increasingly in the world of analysis of art objects and heritage [2]. In this case, Kubelka and Munk theory, or the 2-flux model, is used to resolve the Radiative Transfert Equation (RTE) [3]. Its validity is restricted to layers with strong scattering and weak absorbing, which is not the case with pigmented layers. Additionally, the K/S ratio is dependent on the format and size of pigments, and is therefore unsuitable for the analysis of painted works. Moreover, in recent years critical articles are emerging about choices made for the usual identification of pigments. The works which are interesting for our purpose are those that challenge the attempt to determine pigments by use of K/S, rather than by their fundamental optical property, which is a refractive index [9][10]. Other interesting articles describe the resolution of the RTE by the 4-flux model and try to determine scattering and absorption coefficients in terms of quantities measured by spectrophotometry. Some papers describe the desire to define the K and S coefficients independently of one another [11][12]. The aim of our research is to identify a pigment by its complex optical index, dependant on the wavelength, the real part, n, the refractive index and an imaginary part, k, linked to the absorption. This index is a fundamental optical property because it is intrinsic to the material studied, since it is directly related to its electronic structure. It will allow us to define unambiguously the optical signature of each pigment, and to predict visual effects caused by shaping this pigment in any given medium The theories involved in calculating the optical properties (reflectance, possibly transmittance) of a paint, starting from the complex optical index of its components (pigments, binder…) are the following:

- The Mie theory for calculating the scattering and absorption coefficients of an electromagnetic wave by a single pigment.

- The Radiative Transfer Equation and its resolution by the 4-flux model. This equation models the interaction of the electromagnetic wave with a set of pigments randomly dispersed in a matrix. The sample can be thin layer with parallel interfaces or possibly multilayer.

- The Kirchhoff theory for calculating the scattering of the electromagnetic wave at the interfaces of each layer due to the surface roughness.

The present work is still ongoing. In this paper we will demonstrate the impact of a morphological parameter, such as the size of the pigment, on the color observed. Malachite is the pigment chosen to illustrate this effect. We will then focus on the results obtained on a red pigment, Vermillon, and develop the methodology required to solve the problem of pigment identification. We will give the first results concerning the calculation of the complex optical index, starting from the calculation of the absorption and scattering coefficients and then of the cross sections of Vermillon samples. The cross sections thus determined are independent of the thickness of the layer and of the pigment concentration, but still dependent on the pigment radius and the refractive index of the matrix. It is again the only complex optical index of the pigments that comes completely from the morphological parameters. We show some cross section results, using abacus, concerning the Vermillon.

2. THE CHANGE OF THE MALACHITE COLOR AS A FUNCTION OF ITS GRAIN SIZE

We have justified our approach of the problem by the fact that the only intrinsic parameter of a pigment is its complex optical index. Other derived parameters involve the morphology of the pigment and its environment and therefore may characterize different colors related to the same pigment. We illustrate this statement with malachite, by studding experimentally the variation of its color as a function of the size of the pigments grains. The choice of malachite is not trivial. We observed on three batches of powders containing malachite pigments of different sizes that the color of the pigment varies with size. One can observe a variation of the hue from green to blue-gray when the size decreases [Fig.1].



Figure 1: 3 batches of powdered Kremer Malachit - 3 grain diameter [5, 14, 24] µm

To quantify the change in color by colorimetry three samples were elaborated. The pigments were put by cold mixing in a transparent polyester Sody 33, made by ESCIL. This method allows a random dispersion of the pigments in a transparent medium without any agglomeration at low concentration. Moreover, one gets a solid sample, which can be reused using optical techniques. The samples were finely polished in order to limit the roughness, and to try to eliminate surface scattering. By using this technique, we elaborated optically polished samples of thickness, 1,3 mm ; with a low mass concentration of pigments, of 0,22%. The only variable parameter is the grain size, respectively 5 microns, 14 microns and 27 microns. The total reflectance (specular + diffuse) , Rs, has been measured using an UV-VIS-NIR spectrometer, Varian , Cary 5000 in the wavelength interval 350-800 nm [Fig.2]. The spectrometer is coupled with a 150 mm diameter integrating sphere inside the spectrometer. The three curves represent the variation of Rs for the three samples. When the grain size decreases there is an increase of the total intensity reflected with a maximum intensity in the short wavelengths of the visible spectrum. For malachite 5 µm the maximum is at 476 nm with a reflectance of 10.8% and for malachite 25 µm the maximum is at 511 nm with a reflectance of 8,1 %. Below 420 nm, the drop in reflectance is due to the absorption in the U-V of the polymer used as a host medium.



Malachit Matsuba-Rookusyou From Kremer

4

5

6

7

8

9

10

11

12

350 400 450 500 550 600 650 700 750 800

wavelength ( nm )

Rs

%

malachite-5malachite-14malachite-25

Figure 2: Influence of grain diameter of malachit on the total reflection according to wavelength

The color calculation of the three samples has been performed from these curves, in the range of 420-800nm within the framework of the CIE Lab system. This system characterizes the color using a parameter of intensity corresponding to the luminance, L, and two chrominance parameters, a * and b *. In the table 1, two colorimetric parameters a * and b * are presented. The component a* is the range of the axis red to green, and b * represents that of the axis yellow to blue through the gray (0). Table 1: Colorimetric parameters (La*b* system) of three samples composed of Malachite pigments of different sizes: 5µm, 14µm and 27µm.

Samples containing the smallest grains exhibit a high chroma value. The color is more saturated, and the tint tends to the blue. At increasing pigment size, the tint slowly tends towards the green. Thus, by simply considering its reflectance spectrum or its scattering and absorption coefficients (which depend on the grain size), one is unable to identify unambiguously this type of pigment.

3. STUDY OF VERMILLON

The necessity of identifying pigments using only their intrinsic characteristic (the complex optical index) being now established, we illustrate the next step of our approach, the determination of this index. For that purpose we use another

Sample a* b* c Hue (-a.O.-b)

Malachite 5 µm -9.17 -4.59 10,25 0,55 rad

Malachite 14 µm -4.42 -0.06 4,42 0,01 rad

Malachite 27 µm -3.05 0.77 3,15 0,26 rad



pigment, red vermillon. The reason for working in the red is that we have had some difficulties on the lower part of the visible spectrum. This is due to the absorption in the blue of the polymer Sody 33 that we use presently as a matrix. We summarize below the method selected for determining this index. We have chosen to solve step by step the inverse problem of the following direct calculation of the optical properties of a dispersion of the pigment under study in a transparent host:

- Input: complex index of refraction ñ = n + ik, volume fraction p (or N number of scatterers per unit volume), mean radius of the pigment, r, optical index (usually real) of the matrix, nmat, thickness of the sample, Z ;

- Calculation of the scattering and absorbing cross-sections: Csca and Cabs, by using the Mie theory ; - Calculation of the scattering and absorption coefficients S and K: S=NCsca , K=NCabs ; - Calculation of the specular and diffuse flux reflected and transmitted by the sample, using the Radiative

Transfer Equation solved within the 4-flux model. To be within the range of validity of the RTE, the following conditions have to be fulfilled:

- The medium must have a low concentration of scattering elements. This condition prevents phase relations, which can give rise to interference phenomena between fields scattered by the various elements (dependent scattering) ;

- Scattering elements must be randomly dispersed to again avoid coherent phenomena ; - The thickness of the medium must be larger than both the diameter of the pigments and their spacing in

order to get an average behavior of the flux propagating through the medium. 3.1 Sample Vermillon is a mercury sulfide, HgS, red pigment produced by inorganic synthesis of sulfur and mercury, obtained from Kremer-Pigmente. The mean radius of the pigment is 1.5 µm with a standard deviation of 1.25 µm. In order to fulfill the conditions of a valid use of the RTE, recalled above, the samples were elaborated with the same protocol as the Malachite samples, with values of the main parameters summarized in Table 2. Table 2:characteristics parameters of the samples

Sample name

average radius of pigment, r

Pigment volume

Mass concentration

Volume concentration

number of particles per unit volume,

N

sample thickness, Z

µm µm 3 % % Ptcles/ mm3 mm Ver-thick-C1 1,5 14 0,151 0,021 1,5E+4 1,3 Ver-thin-C1 1,5 14 0,151 0,021 1,5E+4 0,7 Ver-thick-C2 1,5 14 0,057 0,008 5,67E+3 1,3 Ver-thin-C2 1,5 14 0,057 0,008 5,67E+3 0,7

Matrix - - - - - 1,3 Two volume concentrations were taken into account C1=0,021 % and C2=0,008% ; and two thicknesses, 0,7 and 1,3 mm in order to roughly double the thickness for each concentration. The thicknesses were chosen with the aim of being both small enough to limit the light losses at each lateral side of the sample, and thick enough to avoid interference phenomena in the layer. We said previously that the values of the cross sections are independent of the pigment concentration and the thickness of the medium where they are dispersed. Our aim was to test the ability of inverting the direct calculation presented above. We did this by matching the spectral variations of the specular and diffuse reflectance and transmittance measured with the Cary 5000, under the same conditions as the previous study of Malachite samples.



Besides the morphological parameters entered as inputs, the refractive index of the matrix is an important parameter of the model. We recall that the matrix is a (almost, see further) transparent polyester Sody 33, made by ESCIL. With the aim of the determination of its optical index, a first series of optical measurements has been made on the polished matrix, without any embedded pigment. Despite a careful polishing, we highlighted the presence of scattering at the interfaces, due to a residual roughness. The amount of scattered light thus measured with the integrating sphere, is about 2% at 900 nm and increases up to 5.7% around 430 nm. The origin of this scattering may be twofold. Either the presence of roughness at the interfaces (surface scattering) or the presence of air micro-bubbles within the matrix (volume scattering). Observation by optical microscopy dismissed the second assumption and allows us to consider the diffusion to be due essentially due to surface scattering. Instead of taking this effect into account by modeling, we preferred to solve experimentally the problem. Both interfaces of the matrix were coated with a liquid of optical index as close as possible to the refractive index of the polymer. The refractive index liquid used is from Cargille Labs, formulated for a specific refractive index n=1.5120 at the wavelength 589.3 nm, at 25° C. By using this procedure, we got maximum decrease of the scattered intensity from 7.7% to 0.9% [Fig.3].

0

0,01

0,02

0,03

0,04

0,05

0,06

350 450 550 650 750 850

wavelength (nm)

R,T

(/

1)

matrix-Rd matrix-Td

matrix-Rd-RIL matrix-Td-RIL

Figure 3: diffused reflectance and transmittance of the polymer matrix, before and after impregnation with the refractive index liquid.

In the following of the study, we will work systematically by impregnating the two interfaces of the samples with this refractive index liquid before performing spectrophotometric measurements. The specular interface reflectance and transmittance, r and t, and the refractive index of the matrix, ñmat=nmat+ikmat are deduced from the measurement of the specular transmission, T, and reflection, R, specular. The interface coefficients are deduced from the Fresnel equations :

!

r =R

2 " R (1)

3.2 Determination of the refractive index of the matrix



We can then estimate the refractive index of the matrix, assumed at this first step, to be real :

!

n = "r +1r "1

+r +1r "1#

$ %

&

' ( 2

"1 (2)

It is expected that a small absorption in the matrix is present. This is characterized by a small imaginary part of the index, kmat . This imaginary part can be deduced from the transmittance as follows :

!

k = "#4$Z

.Ln 12"t 2

r2T+

t 2

r2T%

& '

(

) * +

4r2

%

& ' '

(

) * *

+

,

- -

.

/

0 0

(3)

assuming, in a first approximation, that the values of nmat and therefore r are not affected by this small absorption. Taking into account the matrix absorption, the values of nmat and r are then recalculated.

1,4

1,45

1,5

1,55

1,6

350 400 450 500 550 600 650 700 750 800

wavelength (nm)

n

0

0,00005

0,0001

0,00015

0,0002

0,00025

0,0003

0,00035

k

matrix nmatrix k

Figure 4:Complex optical index of matrix

In the visible range beyond 420 nm the matrix can be considered as transparent (kmat is smaller than 10-7). Bellow 420 nm the absorption of the polymer matrix increases up to 10-5 in the lowest part of the visible spectrum. The relative uncertainties in the calculation, of the optical index of the matrix are 0,13% on n and 5,74% on k.



3.3.1. Optical measurements The measurements of the specular and diffuse flux have been realized, as before, with the Cary 5000 spectrophotometer coupled with an integrating sphere. The illumination is realized with a collimated beam (first index in the determination of the measured quantities). The quantities thus measured are: the specular transmittance, Tcc, the specular reflectance, Rcc, the hemispherical diffuse transmittance, Tcd and the hemispherical reflectance Rcd. The results were only exploited in the spectral range 420nm–800nm in order to avoid the absorption of the matrix.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

420 470 520 570 620 670 720 770

wavelength (nm)

R,T

(/

1)

Ver-thick-C1 Rcd Ver-thick-C1 Rcc

Ver-thick-C1 Tcd Ver-thick-C1 Tcc

Vermillon C= 0,151 % by massZ= 1,265 mm

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

420 470 520 570 620 670 720 770

wavelength (nm)

R,T

(/

1)

Ver-thin-C1 Rcd Ver-thin-C1 Rcc

Ver-thin-C1 Tcd Ver-thin-C1 Tcc


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

420 470 520 570 620 670 720 770

wavelength (nm)

R,T

(/

1)

Ver-thick-C2 Rcd Ver-thick-C2 Rcc

Ver-thick-C2 Tcd Ver-thick-C2 Tcc


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

420 470 520 570 620 670 720 770

wavelength (nm)

R,T

(/

1)

Ver-thin-C2 RcdVer-thin-C2 RccVer-thin-C2 TcdVer-thin-C2 Tcc


Figure 5: Diffuse and specular, reflectance and transmittance, spectra of the 4 samples of Vermillon. Wavelength range 420-800nm

The spectra of the collimated transmission and reflection are almost constant at all wavelengths. It means that the specular flux carry very few information about pigment coloration. This information is therefore essentially held by the diffuse flux. An absorption edge is observed at 576 nm and the diffuse flux increases at long wavelengths. The value of the relative uncertainty on these measurements is 4.84% on the diffuse flux and 0.38% on the collimated flux. 3.3.2. Determination of the extinction coefficient [K+S] The extinction coefficient measures the energy loss of the electromagnetic radiation when passing through the matrix containing the pigments. It takes into account the absorbance of the medium and the effects related to scattering.

!

Extinction = Absorption + scattering

3.3 Study of theVermillon samples:



The extinction [K+S] can be deduced directly from the expression of the specular transmittance, Tcc, of the sample:

ZkSK

ZkSK

eretTcc )(22

)(2

1 ++!

++!

!= (4)

It depends on the absorption of the matrix (taken into account via the parameter k=2πkmat/λ), the thickness Z, and the interfaces coefficients, r and t. The numerical expression of extinction is therefore:

krTccr

TccrLn

ZSK !

""#

$

%%&

'

((

)

*

++

,

-+

!+

!=+ 2

2

42

4)1()1(211)( (5)

A measurement of quality for our method will be to get the same value of the extinction for two samples with same pigment concentration but with different thicknesses [Fig.6].

0

50

100

150

200

250

300

350

400

420 470 520 570 620 670 720 770

wavelength (nm)

Exti

nct

ion

[K

+S

] (m

-1)

Ver-thick-C1 Ver-thin-C1

Ver-thick-C2 Ver-thin-C2

Figure 6: Extinction [K + S] versus the wavelength – visible [420-800] nm

The extinction of the two samples of concentrations C2, but of different thicknesses, is almost identical in the upper part of the visible spectrum. It is roughly true for the samples of concentration C1, but with a larger discrepancy about 5%. We attribute this discrepancy to small flatness defects, due to the polishing, that we observed on the thin sample with concentration C1. The spectrum of [K+S] is relatively flat. Therefore, like Tcc and Rcc the extinction that is connected to the specular flux, does not hold information on the color of the pigment. Consequently, one has to go further in the separate calculation of K and S.



3.3.3.Determination of the scattering coefficient, S, and absorption coefficient, K The value of the coefficients K and S, independently of one an other cannot be obtained analytically. The Radiative Transfer Equation (RTE)[1], with the due changes will be a relevant approach for this calculation. The 4-flux model that we have chosen is one of the several methods that allow the resolution of the RTE in stratified media. In this case, the scattered and collimated fluxes are taken into account [3][6][7]. The formulation of numerical expressions has been given by Maheu et al. [4] in the 1980s, and then generalized by Rozé et al. in 2001 [5] to a medium with an arbitrary number of layers. This method takes into account a collimated or diffuse illumination, the thickness of the layer and the interfaces discontinuities. In addition, the relationship between its characteristic magnitudes and the Mie cross sections allows taking into account the size, shape and volume concentration of the layers’ inclusions. Knowing that the intensity of the transmitted flux is higher than that of the reflected flux and that the color information carried by Tcd is higher than that of Rcd, K and S are extracted from Tcc and Tcd. The collimated (c)-collimated (c) transmittance, Tcc:

!

Tcc =(1" rc )

2

e(K +S )Z " rc2e"(K +S )Z (6)

where

!

Z is the thickness of the layer ;

!

rc the reflection coefficient of the collimated beam at the interface layer-air. K and S are respectively the absorption coefficient and the scattering coefficient. They are independent of the thickness of the layer. The collimated-diffuse transmittance is [4][8] :

!

Tcd =(1" rd )(1" rc )e

"(K +S )Z

[A1 " (K + S)2][1" rcA6e"2(K +S )Z ]

#NT

D (7)

With the coefficients:

!

A1 = " 2K[K + 2S(1#$d )]A2 = S["K$c +$c (K + S) +"S(1#$d )]A3 = S[(1#$c )(" #1)K + [(1#$d )" # (1#$c )]S]A4 = "[K + S(1#$d )]A5 = "S(1#$d )

(8)

!

" is the forward scattering ratio and is equal to the energy scattered by a particle in the forward hemisphere over the total scattered energy. Due, the occurrence of collimated (c) and diffuse (d) flux at the interfaces, we consider two forward scattering ratio

!

"d and

!

"c , with the relation:

!

"d =12

+2"c #1$

(9)

!

" is the average crossing parameter defined by considering that in an elementary layer of thickness dZ,

!

dZ , the average path length actually experienced by the scattered light is

!

".dZ . For collimated beams,

!

" is obviously equal to 1, and for an isotropic diffuse flux it is equal to 2 [4,8].



As part of the study of pigment with a size higher than the visible wavelength range, the Mie regime is best suited with hypothetically

!

" =1 and

!

"c =1. Then,

!

D = A1 (rd2 "1)cosh( A1Z) + [rd (A5 " rd A4 ) + rd A5 " A4 ]sinh( A1Z) (10)

!

NT= A1[rd A3 " A2 + rc (rd A2 " A3)]cosh( A1Z)

+[(A5 " rd A4 )(A3 + A2rc ) " (A4 " rd A5)(A2 + A3rc )]sinh( A1Z)

+ A1[(A2 " rd A3)e(K +S )Z + rc (A3 " rd A2)e

"(K +S )Z ]

(11)

The reflection coefficients rc and rd are calculated from the Fresnel formulae applied respectively to the collimated and scattered flux at the interface layer-air. The Mie theory has no size limitation and therefore may be used to describe most spherical (and even non spherical) particles, including those with very small sizes (Rayleigh scattering). Considering that our pigments are larger than the wavelength, we decided to treat the problem within the Mie model. In these conditions, the scattering occurs essentially in the forward direction and it becomes less and less diffuse and more collimated. We therefore cannot consider the flux falling at the interfaces as isotropic. It is the reason why we decided to consider the coefficient of diffuse reflectance rd equal to that of the collimated one rc. The value of the parameter ε is also directly related to the degree of isotropy of the diffuse flux propagating in the medium. In the case of a propagation tending towards a collimated flux, its value tends to 1. Following this procedure, the spectral variations of K and S have been calculated for the four samples containing Vermillon pigments and are presented in Fig.7.

Figure 7: Scattering and absorption coefficients of the 4 samples of Vermillon. Wavelength range 420-800nm



A kind of duality is observed between absorption scattering and the crossing of the two coefficients around the absorption edge. Scattering is predominant at large wavelengths. By contrast, the absorption coefficient takes over below 576nm. This qualitative behavior is consistent with the red coloration of Vermillon. The variations of K and S for samples of the same concentration but with different thickness are almost identical. This result is in agreement with the theoretical independence of these coefficients with the thickness Z. The difference lies within the uncertainties on the parameters (Z and N notably). The algebraic sum of K and S at each wavelengths is equal to the value of [K+S] showed in Fig.5. Moreover, these values of K and S, reinserted in the RTE-4 flux model, lead to calculated values of Tcd and Tcc equal to the measured values. The calculated values of Rcc are also very close to the measured ones. The values of Rcd differ from the measured ones by less than 4%. The increase of K below 500nm can be interpreted as an effect of the absorption in the matrix which is too approximately taken into account in our model. It is in fact not a trivial problem that has already been treated in a thesis [14]. The best way for eliminating this problem is to use a matrix totally transparent in the visible spectrum. 3.3.4 Scattering cross section, Csca and absorption cross section, Cabs The scattering cross sections, Csca and absorption cross section, Cabs can be directly deduced from, respectively, S and K values determined in 3.3.3, as soon as the number of particles per unit volume, N, is known [Fig.8].

!

K = N.Cabs (12)

!

S = N.Csca (13)

Figure 8: Scattering and absorption cross sections of the 4 samples of Vermillon. Wavelength range [420-800]nm



The observation already made on S and K curves applies of course here: duality scattering/absorption, absorption edge around 576nm, non-intrinsic increase of the absorption below 500nm due to the absorption of the matrix. The scattering cross section in the scattering zone is about 13 µm2, almost equal to twice the geometric section of the pigment. 3.3.5.Optical index of pigments The previous step enabled us to determine, over cross sections, the values of absorption and scattering, independently of one another. Now we want to implement a program to reverse the resolution of the Mie theory and calculate the value of the complex optical index functions of the cross sections determined from experimental data. This step is actually ongoing. Here we will give an idea of the complex optical index of Vermillon by establishing a comparison between cross section function on experimental data and cross section data obtained from abacus. From abacus we obtained variations of the scattering cross section and absorption as a function of the real part of refractive index, n, and its imaginary part, k. The input data of the testing program are the wavelength, λ0 , the refractive index of the host matrix, nmat, the radius of the pigments, r, and its size dispersion, σ. Two wavelengths were selected, one at 520nm, corresponding to the spectral range of absorption of the pigment, and the second at 670nm corresponding to the domain where the diffusion predominates. The index values of the matrix are calculated for these two wavelengths. Finally, the mean radius of the pigment was estimated at 1.5µm and its dispersion at 1.25 microns.

Figure 9: Absorption and scattering on cross sections (Abacus).



Due to the uncertainties on the "experimental" values of Cabs and Csca, one can define bounds on these values, which in turn will lead to bounds in the expected values of n and k, provided there is a close relation between experimental and simulated values on the cross sections. It is in fact the case in our example. It confirms the ability of this procedure to evaluate the complex index of the pigment. It is noticeable that Csca governs the determination of n while Cabs allows the refinement of the k value. We thus obtain:

- at 670 nm: n values equal to 1.6 ± 0.05 and k equal to 0.01 ± 0.005 - at 520 nm : n values equal to 1.52 ± 0.05 and k equal to 0.02 ± 0.005

One can conclude from these first results that the determination of n and k is possible following this procedure. By improving the procedure itself as well as the knowledge concerning the morphological parameters will, in time, lead to a reduction of the number of uncertainties in determining the cross sections.

4. CONCLUSION We have demonstrated in this paper, the necessity of using the only intrinsic optical parameters of the pigments, i.e. their complex optical index, for identifying unambiguously the components of paint by non-invasive optical techniques. We have proposed a protocol for this identification that we have tested and validated, step by step. It is based on the elaboration of a data bank of optical index of pigments, presently inexistent. We have illustrated this last point, with the Vermillon, which complex optical index has been determined on samples specially prepared for this purpose. They consisted on a random dispersion of the pigments at low concentration, in a polymer matrix supposed to be transparent in the visible spectrum. The chain of the calculation has been detailed, starting from the optical measurements of the transmitted/reflected specular and diffuse flux, calculating then the extinction coefficient (K+S), then K and S separately, then the cross sections and eventually the complex index n+ik. We have proved, by estimating the n and k values from abacus that following this procedure will lead to the final determination of the index. This last step has yet to be improved by solving correctly the inverse problem of the Mie calculation. The procedure itself merits some improvement in order to enhance the accuracy of the determination of the optical index of the pigments. The morphological parameters, for instance, might be more correctly taken into account as input in the problem, fitting the experimental values within their range of uncertainty. Eventually, the samples have to be improved by finding a polymer matrix effectively totally transparent in the visible spectrum.

5. BIBLIOGRAPHIE [1]Chandrasekhar, S, [Radiative Transfer], Dover Publications (1960). [2] Vargas, W, Niklasson, N, « Applicability conditions of the Kubelka-Munk theory », Applied Optics, Vol.36 (1997). [3] Mudgett, P.S, Richards, L.W, « Multiple scattering calculations for technology », Applied Optics, Vol.10 (1971). [4] Maheu, B, Letoulouzan, J.N, Gouesbet, G, « Four-flux models to solve the scattering transfer equation in term of lorenz-Mie parameters », Applied Optics, Vol. 23 (1984). [5] Rozé, C, Girasole, T, Tafforin, A.G, « Multilayer four-flux model of scattering, emitting and absorbing media », Atmospheric environment, Vol. 35 (2001). [6] Vargas, W.E, « Generalized four-flux radiative transfer model », Applied Optics, Vol.37 (1998). [7] Aden, M, Roesner, A, Olowinsky, A, « Optical caracterization of polycarbonate : Influence of additives on optical properties », Journal of polymer science (2009). [8] Witz, C, [Etude des propriétés optiques des sulfures de terres rares et de leur utilisation en tant que pigments dans les milieux hétérogènes diffusants], thesis of Université Pierre et Marie Curie, Paris 6 (1995). [9] Juuti, M, Koivunen, K, Silvennoinen, M, Paulapuro, H, Peiponen, K.E, « Light scattering study from nanoparticle-coated pigments of paper », Colloids and surfaces A: Physicochemical and engineering aspects, Vol. 352, 94-98 (2009). [10] Niskanen, I, Räty, J, Peiponen, K.E, « A method for the detection of the refractive index of irregular shape solid pigments in light absorbing liquid matrix », Talanta, Vol.81, 1322-1324 (2010). [11] Levinson, R, Berdahl, P, Akbari, H, « Solar spectral optical properties of pigments. Part I : model for deriving scattering and absorption coefficients from transmittance and reflectance measurements », Solar energy materials and solar cells (2004).



[12] Aden, M, Roesner, A, Olowinsky, A, « Optical caracterization of polycarbonate : Influence of additives on optical properties », Journal of polymer science: Part B: Polymer Physics (2009). [13] Caron, J, [Diffusion de la lumière dans les milieux stratifies: prise en compte des interfaces rugueuses et des effets de polarization], thesis of Université Pierre et Marie Curie, Paris 6 (2003). [14] Fardella, G, [Modélisation de l’émission thermique du rayonnement infrarouge par les milieux inhomogènes], thesis of Université Pierre et Marie Curie, Paris 6 (1995).



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SPIE Proceedings [SPIE SPIE Optical Metrology - Munich, Germany (Monday 23 May 2011)] O3A: Optics...

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