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Optical properties of random heterogeneous composite materials: from binary to ternary compounds
ELSEVIER
Physica A 207 (1994) 92-99
Optical properties of random heterogeneous composite materials: from binary to ternary compounds Laurent
ServanP'r
, Eric
Tinetb,
Franqois
Carmonaa,
Sigrid Avrillierb
“Centre de Recherche Paul Pascal-CNRS, Av. A. Schweitzer, 33600 Pessac, France bLaboratoire de Physique des Lasers, Universitt Paris XIII, Av. J.B. CMment, 94430 Villetaneuse, France
Abstract
We report an experimental study on the optical properties in the visible and near infrared of three-dimensional random composite materials, made of pellets of KBr filled either with Ge or TiO, particles (binary composite materials), or with both Ge and TiO, particles (ternary composite materials). Hemispherical factors (reflectance and transmittance) of binary and ternary composites have been measured on pellets with various concentrations, in a wavelength range extending between 0.3 and 1.8 km. The transport of light through the materials has been modelled with a Monte Carlo simulation, used together with the Mie theory. In the case of ternary compounds, we propose two approaches to describe multiple scattering in the samples, which prove to be equivalent in the wavelength and concentration ranges considered. One approach is based on the definition of a unique equivalent scatterer and provide the definition of an effective binary medium (whose characteristics are explicitly given) optically equivalent to the ternary compound.
1. Introduction
Composite materials constitute an interesting way for obtaining materials with tailored optical characteristics, due to the manoeuvrability of their properties which can be obtained through the control of size, shape and relative composition of their constituents [l-3]. The dielectric behaviour and optical behaviour of such composite systems have been studied extensively, but there is still no general theory that can relate their properties to the geometry and microstructure of the composite material. For granular systems, a great amount of research has been devoted to the case where the particle size is much smaller than the incident ’ Now at Laboratoire Bordeaux I, F-33405
de Spectroscopic Molkculaire Talence Cedex, France.
0378-4371/94/$07.00 0 1994 Elsevier SSDI 0378-4371(93)E0539-Q
Science
et Cristalline
B.V. All rights
(CNRS
reserved
URA
124), UniversitC
de
L. Servant et al. I Physica
A 207 (1994) 92-99
93
wavelength, but large enough so that it can be characterized by a well behaved dielectric function. However, in general, the propagation of an electromagnetic wave in a inhomogeneous medium involves both scattering and absorption of the waves by the inhomogeneities, which complicate tremendously any prediction or treatment of the optical properties of such complex systems. In this work, we study the optical properties of two- and three-component scattering model composite materials (whose particle size is not small in comparison to the incident wavelength), and we propose a way to model their optical properties with a Monte Carlo simulation.
2. Experimental
results
The samples are made of particles randomly dispersed in a nonabsorbing KBr matrix. We used two kinds of particles: germanium (Ge) of mean radius a = 5 (*rn and titanium dioxide (TiO,) of mean radius a = 0.2 pm. We prepared three series of samples: Ge-KBr, TiO,-KBr (binary compounds) and Ge-TiO,-KBr (ternary compounds) with thickness d ranging between 0.8 mm and 1 mm. For all the samples, the powders were mixed up in appropriate proportions, and pelletized under 5 ton/cm*. The composition of the binary compounds is specified by the volume concentration @ in Ge or TiO, particles dispersed in the samples [4]. In the case of ternary compounds, we define @ as being the total volume concentration in particles [@ = (V,, + VTio2)/(VG, + VT,,, + VKBr), V, being the volume of species A dispersed in the KBr matrix]. We introduce C as the relative volume concentration of TiO, particles in the samples: C = VTio, /(VT,,, + V,,). It is then possible to express easily, for ternary compounds, the volumic concentration of each component: GTio, = C@ and @oe = (1 - C)@. The optical constants of pure KBr (mKBr = ltKBr - ikKBr) have been measured on compressed pellets, and are reported in Fig. 1; the extinction coefficient kKBr has been found to be 0 within experimental uncertainty. For Ge and TiO, particles, we used the optical constants as given in refs. [5-61. Fig. 1 shows that both TiO, and Ge exhibit two different types of behavior in the wavelength range under investigation; TiO, and Ge particles can be considered as dielectric when h > 0.2 p,m and A > 1.5 pm respectively, and they are absorbing elsewhere. Hemispherically integrated reflectance (R) and transmittance (T) spectra for samples of various compositions (Ge-KBr, TiO,-KBr and Ge-TiO,-KBr) have been recorded between 300 and 1800nm, using a Perkin-Elmer model 330 spectrophotometer (giving a collimated incident beam) with an integrated sphere attachment. Typical measured hemispherical factors, for each of the three systems under investigation, are given in Fig. 2 as a function of the wavelength. In the case of binary compounds, the experimental results agree with the general trends exhibited by the optical constants in Fig. 1: a higher reflectivity is found in the spectral region where the particles are dielectric (i.e., when A > 1.5 pm for Ge
L. Servant et al. I Physica A 207 (1994) 92-99
94
0.5
Fig. (n -rq
1 x en pm
1. Optical constants of KBr - ikTI02) particles as functions
1.5
2
matrix (n,,,), of wavelength.
germanium
(n,,
- ik,,)
and
titanium
dioxide
particles and when A > 2 km for TiO, particles) and the TiO,-KBr samples exhibit a maximum for the hemispherical reflectance at approximately A= 0.2 pm. For ternary compounds, when A < 1.5 pm, both R and T seem dominated by the optical properties of the TiO, particles, whereas when A > 1.5 pm, they are clearly greatly influenced by the optical properties of Ge particles.
3. Discussion Due to the size of the particles used in our materials and the wavelength domain investigated, effective medium theories cannot be used and scattering has to be considered. The difficulties that arise in treating multiple scattering of light in inhomogeneous media can be avoided in the strong-scattering limit, in which photons execute a random walk [7] through the sample. In this limit, it is possible to define a transport mean free path of the photons (or average step size of the random walk), and the spatial variation of the photon density can be described by a diffusion equation. Solving analytically the diffusion equation can be complicated by the boundary conditions, i.e. the nature of the conversion of collimated incident photons to diffused photons, the consequences of reflectivity at the sample interfaces, etc. For these reasons we used a Monte Carlo procedure in order to evaluate both R and T. We will suppose that the scattering material is homogeneous and characterized by its geometric thickness d and by the absorption and scattering coefficients per unit length ( K, CT), equal to the inverse of the scattering and absorption mean free paths (respectively Esea, labs). The angular scattering pattern for a single scatterer is known as the phase function p(B), 0 being the scattering angle measured from the forward direction. In order to simplify the simulation, we did not take into account the polarization; p(8) is a scalar. The asymmetry of
95
L. Servant et al. I Physica A 207 (1994) 92-99 1
‘t s o.a(a) 2
0.6
El Q33 . (a) i E 0.6 .
Y
0.4
i
0.2
t
E B
0.2 0.4
.w
0
l-l u 1
0.5 h
0.5
1.5
2
0.5
h (elm>
1.5
2
1.5
2
X @m)
(Pm)
1
1
1.5
2
0.5
1
A bm)
Fig. 2. Hemispherical reflectance and transmittance of Ge-KBr, TiO,-KBr and Ge-TiO,-KBr samples (d - 1 mm), as a function of wavelength (thick solid lines: experimental; thin solid lines: simulation): (a) Ge-KBr, experimental @ = 0.9%; simulation @ = 0.8%; (b) TiO,-KBr, experimental @ = 0.3%; simulation @ = 0.2%; (c) TiO,-Ge-KBr, experimental @ = 3.4%, C = 0.43; simulation @=3.2%. C=O.43.
this function can be expressed in terms of a single number g, the asymmetry parameter (- 1 s g s 1) defined as the cosine of the scattering angle averaged over many single scattering events. This number is positive (negative) for forward (backward) scatterers and g = 0 for isotropic scatterers. If we assume that each material contains a monodispersed collection of spherical scatterers, then it is possible to evaluate from the Mie theory [8,9] p(8), g, C,,, and Cabs (the scattering and absorption cross sections respectively), which can give estimates of
L. Servant et al. I Physica A 207 (1994) 92-99
96
K and U: K = l/l,,, = 3@C,,,/4ra3 and u = Ill,,, = 3@C,,,/47ra3, as functions of the wavelength and of the composition of the samples. In the case of anisotropic scatterers
(g # 0), it is convenient
to introduce
E* = Z,,,/(l - g), which is the length is randomized The main
the transport
over which the direction
mean
free
path
of light propagation
[7]. steps of our Monte
Carlo
simulation
can be found
in refs.
[lO,ll],
and we just summarize here the basic steps of the simulation. (i) A group of incident photons, of initial weight 1, is generated with an incident direction perpendicular to the sample surface. (ii) The path length between successive interactions is calculated as I= (-ln rand a)/(~ + K), where rand CYis a random number uniformly distributed between 0 and 1. (iii) At each interaction point, a fraction of the incident group of photons equal to w, = (T/(u + K) is scattered according to p(8), and a fraction (1 - 0”) is absorbed. (iv) Nonrandom events occur when the photon trajectory crosses one of the slab surfaces: part of the light is reflected toward the medium and in the case of specular reflection, it is easy to calculate its weight and its direction (through the unpolarized reflection coefficient corresponding to the angle of incidence 8: r(0) = +(]Y~(I~)]’ + ]ril(0)12), where rL (0) and r,, (0) are the reflection coefficients for perpendicular and parallel polarization, respectively, obtained from Fresnel’s equations). The refracted part of the light participates in R or T, depending on the surface. 3.1.
Case of binary compounds
In the wavelength range of interest, K, CT and g were obtained from the particle characteristics, the Mie theory and the composition of the samples, and r(0) was evaluated using the values of the refractive index of KBr. In the spectral range considered, g > 0.5 for both Ge and TiO,, typical orders of magnitude of K + u for Ge and TiO, particles, respectively. In order are about 0.4 ym-’ and 20 km-’ to simplify the calculations, we used the classical scalar Henyey-Greenstein phase function [12] (polarization is not taken into account) instead of the one given by the Mie theory. This phase function depends on g and fits fairly well various phase
functions.
Two
typical
results
for
both
binary
systems
Ge-KBr
and
TiO,-KBr are given in Fig. 2. The calculated R and T factors were fitted to the experimental data by varying the volume fraction @ occupied by the particles in the sample with the mean thickness d being kept constant and equal to 1 mm. In the data-fitting procedure, we sought the best fits for R, and in all cases good agreement was found with the experimental data for both R and T. The value of @ used in the calculations compared well to the experimental ones, within experimental uncertainty. The approach described here is based upon a simplified model. Better agreement could certainly be achieved by considering a size distribution of the particles and using average cross sections. Furthermore, the actual shape of the particles is not rigorously spherical, and nonspherical particles are stronger side-scatters and weaker back-scatterers than equivalent spheres [13].
L. Servant et al.
I Physica A 207 (1994) 92-99
97
This limitation could be taken into account, but at this stage, it would complicate the physical picture. 3.2. Case of ternary compounds We consider here the propagation in a medium comprising two types of scatterers (Ge, TiO,), whose scattering parameters are respectively defined to a first approximation by (g,, K~, ~~1) and (g,, K*, u2), where subscript 1 and 2 denotes Ge and TiO, respectively. Restricting our attention to very dilute samples, we suppose that the scattering and absorption coefficients per unit length for the two scatterers can be simply obtained by substituting Q1 = @de and QZ = QTio2 for @ in the relations giving u and K. In order to perform a direct simulation of the transport of the photons through the scattering media, we have to define a criterion, which can be easily incorporated into our Monte Carlo scheme, stating which kind of scatterer (scatterer 1 or 2) a photon is most likely to meet in a sample of a given composition. Such a criteria can be obtained by considering the ratio X = r1 /(TV+ T*), where pi = o1 + K~ and T*= a, + K~. The numbers TVand TVcontain complete information relative to the sample composition and to its scattering properties. Then, in step (ii) of the Monte Carlo procedure, we generate a random number rand p (between 0 and 1). If rand /3 < X, then the photon is supposed to meet a particle of type (1) and is scattered according to the phase function of particle 1 (asymmetry parameter g,); if not, the photon will be scattered according to the phase function of particle 2 (asymmetry parameter g2). We present in Fig. 2 the results obtained with this procedure. The qualitative shape of the R and T curves is well reproduced by the calculations and reasonable agreement is obtained simultaneously for both R and T for (@, C) values quite close to the experimental ones, although the hypotheses adopted for the model are very rough, e.g., we suppose that the scattering properties of one scatterer (e.g., type 1) are not modified by the presence of the other one (e.g., type 2). However, let us emphasize that the approach we choose involves no free parameters and shows clearly that a key parameter in determining the optical properties a ternary compound is the dimensionless number X, which is both a function of the composition of the samples and of the optical properties of the scatterers. In order to simplify the analysis or predictions one may wish to make concerning ternary compounds, we have studied how to define an effective binary medium, which would have the same optical properties (hemispherical factors) as the ones of a given ternary compound. This approach is made possible due to the simple hypothesis needed to account for the results by the direct simulation. Assuming no interactions between scatterers, the equivalent scattering (a,,) and absorption (Key) coefficients per unit length will be taken as a,, = ~i + oZ and K = K1 + K2. The second step is to characterize the phase function of the eq
L. Servant et al. I Physica A 207 (1994) 92-99
98
effective scatterer. In order to obtain the asymmetry parameter ge_, of the effective scatterer, we assume that the inverses of the transport mean free paths are additive, then: (1 - g,,)a,, = (1 - g,)a, + (1 - g,)a,, giving geq = 1 - [( 1 gr)~r + (1 - g,)a,]/(or + c2). Thus, the effective scattering medium is completely characterized by the three numbers (geq, K,~, ueq), which depend on the wavelength and the concentration through cl,* and K~,~. One can notice that for ternary compounds, ge4 depends on the composition, which is not the case for binary compounds. We used this procedure to analyse our experimental data, and it turned out that we found exactly the same results as those obtained with the direct simulation. We then conclude that the two approaches are strictly equivalent for all of the configurations we investigated (range of values of (T, K). The interest of the latter approach is that it deals with a unique (virtual) type of scatterer, and thus provides a much more convenient way for making a prediction or analysis on such complex systems.
4. Summary
and conclusions
We have presented an experimental study of the optical properties (hemispherical factors) of random heterogeneous materials made of particles randomly dispersed in a nonabsorbing matrix. The analysis of the experimental data accounts fairly well for the properties of the binary and ternary compounds, both in the wavelength and concentration ranges. For ternary compounds, we introduced a dimensionless parameter (X) used in the direct Monte Carlo simulation, whose order of magnitude indicates roughly which scatterer will play a prevailing role in the average optical properties of the samples, and we discuss the properties of a possible equivalent effective scatterer. This work shows that the simplified assumptions we chose are sufficient to account well for the experimental data, and do not involve any free parameters. In particular, we did not take into account the possible existence of clusters within the materials; the loss of accuracy by doing so, in this case, is not that significant, and largely compensated by the gain in simplicity. These results pave the way for further studies, especially in material science, where there is a need for simple methods and criteria in order to design systems with selective optical properties.
References [l] [2] [3] [4] [5] [6]
J.C. Maxwell-Garnett, Philos. Trans. R. Sot. London 203 (1904) 385. J.C. Maxwell-Garnett, Philos. Trans. R. Sot. London 205 (1906) 237. Th. Robin and B. Souillard, Europhys. Lett. 21 (1993) 273. L. Servant and F. Carmona, Appl. Opt. 32 (1993) 2789. E.D. Palik, ed., in: Handbook of Optical Constants of Solids (Academic Press, New York, M. Herzberg and C.D. Salzberg, J. Opt. Sot. Am. 52 (1962) 420.
1985).
99
L. Servant et al. I Physica A 207 (1994) 92-99 [7] P. Sheng, [8] [9] [lo] [ll] [12] [13]
in: Scattering
and Localization
of Classical
Waves in Random
Media
(World
Scientific,
New York, 1991). A. Ishimaru, in: Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978). CF. Bohren and D.R. Huffman, in: Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983). S. Avrillier, E. Tinet and E. Delettre, J. Phys. (Paris) 51 (1990) 2.521. E. Tinet, Ph.D. Thesis, University of Paris XIII (1992). L.G. Henyey and J.L. Greenstein, Astrophys. J. 93 (1941) 70. M.I. Mischenko, Appl. Opt. 32 (1993) 4652.
Physica A 207 (1994) 92-99
Optical properties of random heterogeneous composite materials: from binary to ternary compounds Laurent
ServanP'r
, Eric
Tinetb,
Franqois
Carmonaa,
Sigrid Avrillierb
“Centre de Recherche Paul Pascal-CNRS, Av. A. Schweitzer, 33600 Pessac, France bLaboratoire de Physique des Lasers, Universitt Paris XIII, Av. J.B. CMment, 94430 Villetaneuse, France
Abstract
We report an experimental study on the optical properties in the visible and near infrared of three-dimensional random composite materials, made of pellets of KBr filled either with Ge or TiO, particles (binary composite materials), or with both Ge and TiO, particles (ternary composite materials). Hemispherical factors (reflectance and transmittance) of binary and ternary composites have been measured on pellets with various concentrations, in a wavelength range extending between 0.3 and 1.8 km. The transport of light through the materials has been modelled with a Monte Carlo simulation, used together with the Mie theory. In the case of ternary compounds, we propose two approaches to describe multiple scattering in the samples, which prove to be equivalent in the wavelength and concentration ranges considered. One approach is based on the definition of a unique equivalent scatterer and provide the definition of an effective binary medium (whose characteristics are explicitly given) optically equivalent to the ternary compound.
1. Introduction
Composite materials constitute an interesting way for obtaining materials with tailored optical characteristics, due to the manoeuvrability of their properties which can be obtained through the control of size, shape and relative composition of their constituents [l-3]. The dielectric behaviour and optical behaviour of such composite systems have been studied extensively, but there is still no general theory that can relate their properties to the geometry and microstructure of the composite material. For granular systems, a great amount of research has been devoted to the case where the particle size is much smaller than the incident ’ Now at Laboratoire Bordeaux I, F-33405
de Spectroscopic Molkculaire Talence Cedex, France.
0378-4371/94/$07.00 0 1994 Elsevier SSDI 0378-4371(93)E0539-Q
Science
et Cristalline
B.V. All rights
(CNRS
reserved
URA
124), UniversitC
de
L. Servant et al. I Physica
A 207 (1994) 92-99
93
wavelength, but large enough so that it can be characterized by a well behaved dielectric function. However, in general, the propagation of an electromagnetic wave in a inhomogeneous medium involves both scattering and absorption of the waves by the inhomogeneities, which complicate tremendously any prediction or treatment of the optical properties of such complex systems. In this work, we study the optical properties of two- and three-component scattering model composite materials (whose particle size is not small in comparison to the incident wavelength), and we propose a way to model their optical properties with a Monte Carlo simulation.
2. Experimental
results
The samples are made of particles randomly dispersed in a nonabsorbing KBr matrix. We used two kinds of particles: germanium (Ge) of mean radius a = 5 (*rn and titanium dioxide (TiO,) of mean radius a = 0.2 pm. We prepared three series of samples: Ge-KBr, TiO,-KBr (binary compounds) and Ge-TiO,-KBr (ternary compounds) with thickness d ranging between 0.8 mm and 1 mm. For all the samples, the powders were mixed up in appropriate proportions, and pelletized under 5 ton/cm*. The composition of the binary compounds is specified by the volume concentration @ in Ge or TiO, particles dispersed in the samples [4]. In the case of ternary compounds, we define @ as being the total volume concentration in particles [@ = (V,, + VTio2)/(VG, + VT,,, + VKBr), V, being the volume of species A dispersed in the KBr matrix]. We introduce C as the relative volume concentration of TiO, particles in the samples: C = VTio, /(VT,,, + V,,). It is then possible to express easily, for ternary compounds, the volumic concentration of each component: GTio, = C@ and @oe = (1 - C)@. The optical constants of pure KBr (mKBr = ltKBr - ikKBr) have been measured on compressed pellets, and are reported in Fig. 1; the extinction coefficient kKBr has been found to be 0 within experimental uncertainty. For Ge and TiO, particles, we used the optical constants as given in refs. [5-61. Fig. 1 shows that both TiO, and Ge exhibit two different types of behavior in the wavelength range under investigation; TiO, and Ge particles can be considered as dielectric when h > 0.2 p,m and A > 1.5 pm respectively, and they are absorbing elsewhere. Hemispherically integrated reflectance (R) and transmittance (T) spectra for samples of various compositions (Ge-KBr, TiO,-KBr and Ge-TiO,-KBr) have been recorded between 300 and 1800nm, using a Perkin-Elmer model 330 spectrophotometer (giving a collimated incident beam) with an integrated sphere attachment. Typical measured hemispherical factors, for each of the three systems under investigation, are given in Fig. 2 as a function of the wavelength. In the case of binary compounds, the experimental results agree with the general trends exhibited by the optical constants in Fig. 1: a higher reflectivity is found in the spectral region where the particles are dielectric (i.e., when A > 1.5 pm for Ge
L. Servant et al. I Physica A 207 (1994) 92-99
94
0.5
Fig. (n -rq
1 x en pm
1. Optical constants of KBr - ikTI02) particles as functions
1.5
2
matrix (n,,,), of wavelength.
germanium
(n,,
- ik,,)
and
titanium
dioxide
particles and when A > 2 km for TiO, particles) and the TiO,-KBr samples exhibit a maximum for the hemispherical reflectance at approximately A= 0.2 pm. For ternary compounds, when A < 1.5 pm, both R and T seem dominated by the optical properties of the TiO, particles, whereas when A > 1.5 pm, they are clearly greatly influenced by the optical properties of Ge particles.
3. Discussion Due to the size of the particles used in our materials and the wavelength domain investigated, effective medium theories cannot be used and scattering has to be considered. The difficulties that arise in treating multiple scattering of light in inhomogeneous media can be avoided in the strong-scattering limit, in which photons execute a random walk [7] through the sample. In this limit, it is possible to define a transport mean free path of the photons (or average step size of the random walk), and the spatial variation of the photon density can be described by a diffusion equation. Solving analytically the diffusion equation can be complicated by the boundary conditions, i.e. the nature of the conversion of collimated incident photons to diffused photons, the consequences of reflectivity at the sample interfaces, etc. For these reasons we used a Monte Carlo procedure in order to evaluate both R and T. We will suppose that the scattering material is homogeneous and characterized by its geometric thickness d and by the absorption and scattering coefficients per unit length ( K, CT), equal to the inverse of the scattering and absorption mean free paths (respectively Esea, labs). The angular scattering pattern for a single scatterer is known as the phase function p(B), 0 being the scattering angle measured from the forward direction. In order to simplify the simulation, we did not take into account the polarization; p(8) is a scalar. The asymmetry of
95
L. Servant et al. I Physica A 207 (1994) 92-99 1
‘t s o.a(a) 2
0.6
El Q33 . (a) i E 0.6 .
Y
0.4
i
0.2
t
E B
0.2 0.4
.w
0
l-l u 1
0.5 h
0.5
1.5
2
0.5
h (elm>
1.5
2
1.5
2
X @m)
(Pm)
1
1
1.5
2
0.5
1
A bm)
Fig. 2. Hemispherical reflectance and transmittance of Ge-KBr, TiO,-KBr and Ge-TiO,-KBr samples (d - 1 mm), as a function of wavelength (thick solid lines: experimental; thin solid lines: simulation): (a) Ge-KBr, experimental @ = 0.9%; simulation @ = 0.8%; (b) TiO,-KBr, experimental @ = 0.3%; simulation @ = 0.2%; (c) TiO,-Ge-KBr, experimental @ = 3.4%, C = 0.43; simulation @=3.2%. C=O.43.
this function can be expressed in terms of a single number g, the asymmetry parameter (- 1 s g s 1) defined as the cosine of the scattering angle averaged over many single scattering events. This number is positive (negative) for forward (backward) scatterers and g = 0 for isotropic scatterers. If we assume that each material contains a monodispersed collection of spherical scatterers, then it is possible to evaluate from the Mie theory [8,9] p(8), g, C,,, and Cabs (the scattering and absorption cross sections respectively), which can give estimates of
L. Servant et al. I Physica A 207 (1994) 92-99
96
K and U: K = l/l,,, = 3@C,,,/4ra3 and u = Ill,,, = 3@C,,,/47ra3, as functions of the wavelength and of the composition of the samples. In the case of anisotropic scatterers
(g # 0), it is convenient
to introduce
E* = Z,,,/(l - g), which is the length is randomized The main
the transport
over which the direction
mean
free
path
of light propagation
[7]. steps of our Monte
Carlo
simulation
can be found
in refs.
[lO,ll],
and we just summarize here the basic steps of the simulation. (i) A group of incident photons, of initial weight 1, is generated with an incident direction perpendicular to the sample surface. (ii) The path length between successive interactions is calculated as I= (-ln rand a)/(~ + K), where rand CYis a random number uniformly distributed between 0 and 1. (iii) At each interaction point, a fraction of the incident group of photons equal to w, = (T/(u + K) is scattered according to p(8), and a fraction (1 - 0”) is absorbed. (iv) Nonrandom events occur when the photon trajectory crosses one of the slab surfaces: part of the light is reflected toward the medium and in the case of specular reflection, it is easy to calculate its weight and its direction (through the unpolarized reflection coefficient corresponding to the angle of incidence 8: r(0) = +(]Y~(I~)]’ + ]ril(0)12), where rL (0) and r,, (0) are the reflection coefficients for perpendicular and parallel polarization, respectively, obtained from Fresnel’s equations). The refracted part of the light participates in R or T, depending on the surface. 3.1.
Case of binary compounds
In the wavelength range of interest, K, CT and g were obtained from the particle characteristics, the Mie theory and the composition of the samples, and r(0) was evaluated using the values of the refractive index of KBr. In the spectral range considered, g > 0.5 for both Ge and TiO,, typical orders of magnitude of K + u for Ge and TiO, particles, respectively. In order are about 0.4 ym-’ and 20 km-’ to simplify the calculations, we used the classical scalar Henyey-Greenstein phase function [12] (polarization is not taken into account) instead of the one given by the Mie theory. This phase function depends on g and fits fairly well various phase
functions.
Two
typical
results
for
both
binary
systems
Ge-KBr
and
TiO,-KBr are given in Fig. 2. The calculated R and T factors were fitted to the experimental data by varying the volume fraction @ occupied by the particles in the sample with the mean thickness d being kept constant and equal to 1 mm. In the data-fitting procedure, we sought the best fits for R, and in all cases good agreement was found with the experimental data for both R and T. The value of @ used in the calculations compared well to the experimental ones, within experimental uncertainty. The approach described here is based upon a simplified model. Better agreement could certainly be achieved by considering a size distribution of the particles and using average cross sections. Furthermore, the actual shape of the particles is not rigorously spherical, and nonspherical particles are stronger side-scatters and weaker back-scatterers than equivalent spheres [13].
L. Servant et al.
I Physica A 207 (1994) 92-99
97
This limitation could be taken into account, but at this stage, it would complicate the physical picture. 3.2. Case of ternary compounds We consider here the propagation in a medium comprising two types of scatterers (Ge, TiO,), whose scattering parameters are respectively defined to a first approximation by (g,, K~, ~~1) and (g,, K*, u2), where subscript 1 and 2 denotes Ge and TiO, respectively. Restricting our attention to very dilute samples, we suppose that the scattering and absorption coefficients per unit length for the two scatterers can be simply obtained by substituting Q1 = @de and QZ = QTio2 for @ in the relations giving u and K. In order to perform a direct simulation of the transport of the photons through the scattering media, we have to define a criterion, which can be easily incorporated into our Monte Carlo scheme, stating which kind of scatterer (scatterer 1 or 2) a photon is most likely to meet in a sample of a given composition. Such a criteria can be obtained by considering the ratio X = r1 /(TV+ T*), where pi = o1 + K~ and T*= a, + K~. The numbers TVand TVcontain complete information relative to the sample composition and to its scattering properties. Then, in step (ii) of the Monte Carlo procedure, we generate a random number rand p (between 0 and 1). If rand /3 < X, then the photon is supposed to meet a particle of type (1) and is scattered according to the phase function of particle 1 (asymmetry parameter g,); if not, the photon will be scattered according to the phase function of particle 2 (asymmetry parameter g2). We present in Fig. 2 the results obtained with this procedure. The qualitative shape of the R and T curves is well reproduced by the calculations and reasonable agreement is obtained simultaneously for both R and T for (@, C) values quite close to the experimental ones, although the hypotheses adopted for the model are very rough, e.g., we suppose that the scattering properties of one scatterer (e.g., type 1) are not modified by the presence of the other one (e.g., type 2). However, let us emphasize that the approach we choose involves no free parameters and shows clearly that a key parameter in determining the optical properties a ternary compound is the dimensionless number X, which is both a function of the composition of the samples and of the optical properties of the scatterers. In order to simplify the analysis or predictions one may wish to make concerning ternary compounds, we have studied how to define an effective binary medium, which would have the same optical properties (hemispherical factors) as the ones of a given ternary compound. This approach is made possible due to the simple hypothesis needed to account for the results by the direct simulation. Assuming no interactions between scatterers, the equivalent scattering (a,,) and absorption (Key) coefficients per unit length will be taken as a,, = ~i + oZ and K = K1 + K2. The second step is to characterize the phase function of the eq
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effective scatterer. In order to obtain the asymmetry parameter ge_, of the effective scatterer, we assume that the inverses of the transport mean free paths are additive, then: (1 - g,,)a,, = (1 - g,)a, + (1 - g,)a,, giving geq = 1 - [( 1 gr)~r + (1 - g,)a,]/(or + c2). Thus, the effective scattering medium is completely characterized by the three numbers (geq, K,~, ueq), which depend on the wavelength and the concentration through cl,* and K~,~. One can notice that for ternary compounds, ge4 depends on the composition, which is not the case for binary compounds. We used this procedure to analyse our experimental data, and it turned out that we found exactly the same results as those obtained with the direct simulation. We then conclude that the two approaches are strictly equivalent for all of the configurations we investigated (range of values of (T, K). The interest of the latter approach is that it deals with a unique (virtual) type of scatterer, and thus provides a much more convenient way for making a prediction or analysis on such complex systems.
4. Summary
and conclusions
We have presented an experimental study of the optical properties (hemispherical factors) of random heterogeneous materials made of particles randomly dispersed in a nonabsorbing matrix. The analysis of the experimental data accounts fairly well for the properties of the binary and ternary compounds, both in the wavelength and concentration ranges. For ternary compounds, we introduced a dimensionless parameter (X) used in the direct Monte Carlo simulation, whose order of magnitude indicates roughly which scatterer will play a prevailing role in the average optical properties of the samples, and we discuss the properties of a possible equivalent effective scatterer. This work shows that the simplified assumptions we chose are sufficient to account well for the experimental data, and do not involve any free parameters. In particular, we did not take into account the possible existence of clusters within the materials; the loss of accuracy by doing so, in this case, is not that significant, and largely compensated by the gain in simplicity. These results pave the way for further studies, especially in material science, where there is a need for simple methods and criteria in order to design systems with selective optical properties.
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