Light scattering materials for energy-related applications: Determination of absorption and scattering coefﬁcients

To facilitate optical design of energy-efﬁcient materials and devices, a detailed knowledge of their basic optical parameters is necessary. In this paper we present a novel method for determining scattering ( S ) and absorption ( K ) coefﬁcients from total transmittance and reﬂectance measurements by inversion of the Kubelka-Munk theory. The reﬂectance parameters appearing in this theory depend on the angular distribution of scattered light inside the material. The versatility of our method is demonstrated by a reanalysis of experimental data for several materials of interest in energy-related applications. Speciﬁcally, we report spectra of S and K for: (a) pigmented polymer foils for radiative cooling applica- tions; (b) suspended particle devices for smart windows; (c) solar reﬂecting TiO 2 -pigmented paints and (d) selective solar absorbing paints for solar collectors. (cid:1) 2019 Elsevier Ltd. All rights reserved. peer-review under scientiﬁc committee E-MRS Meeting, 2019. open access


Introduction
Functional materials with applications in the fields of solar energy and energy-efficient buildings are frequently nano-or microstructured and exhibit significant light scattering. For example, it is possible to accurately tailor functional optical properties by incorporating absorbing and/or scattering particles in a nonabsorbing solid matrix and varying particle concentration, size, shape and separation as well as layer thickness and surface roughness. Such materials have found wide application as pigmented paints [1], translucent sheets for daylighting applications [2], UV-absorbing films [3], radiative cooling devices [4,5], suspended particle devices [6,7] as well as solar absorbing paints for solar collectors [8,9]. An understanding of the optical properties of these materials is crucial in order to evaluate and optimize their performance.
In this paper, we focus on determining the effective optical parameters of composite materials consisting of particles in an embedding matrix. To facilitate optical design of such materials, a detailed knowledge of their basic optical parameters, namely the light scattering (S) and absorption (K) coefficients is necessary. Previous methods to extract these coefficients from experimental data either relied on unverified approximations [9] or required extensive computational effort. The two-background method [10] requires accurate characterization of white and black reference materials. The computationally complex approaches include Monte-Carlo simulations [11,12] and numerical solutions of the general radiative transfer equation [13,14]. A simpler alternative is offered by inversion schemes using the approximative two-flux theory [15,16]. However, it was found that in certain cases the obtained scattering and absorption coefficients could not accurately reproduce experimental spectra [7]. In this paper we present an improved method for inversion of the two-flux theory and demonstrate its versatility by calculations of S and K for various materials of interest for energy applications.

Inverse problem of Kubelka-Munk theory
The optical properties of a composite material depend on the type, size, shape and concentration of the particles together with the type of matrix. If the particle size is much smaller than the wavelength of light, effective medium theory [17,18]  neglected. The effective optical constants of the composite layer can be obtained by inversion of experimental transmittance and reflectance measurements [19]. For particle sizes on the order of, or larger than the wavelength of light, light scattering becomes important and must be taken into account in optical theories. Scattering of light from a single spherical particle is described by Mie theory [20] and the case of multiple scattering in a composite material is usually treated within the framework of radiative transfer theory. The Kubelka-Munk two-flux model [21,22] is an analytical approximation to full radiative transfer, that is frequently used in applied research. It has been extended to the case of partly collimated and partly diffusely scattered light propagating in the material [15], and it constitutes a starting point for efforts to solve the inverse problem of radiative transfer for particulate composite materials. Below we give an overview of Kubelka-Munk theory and a recent improved formulation of the inverse problem, building on a previous paper by some of us [23]. It should be noted that the scattering coefficient S in the Kubelka-Munk model describes only the backscattering, as is clear from Eqs. (1) and (2) below.
A convenient approach to characterizing the optical properties of composite materials is to use a conventional spectrophotometer with an integrating sphere accessory. From spectrophotometric measurements, total transmittance (T tot ) and reflectance (R tot ), as well as the diffuse parts of the transmittance and reflectance (T dif ) and (R dif ) ( Fig. 1a) are readily obtained. The regular transmittance (T reg ) and specular reflectance (R spe ) can subsequently be derived by subtracting the diffuse portions from the total ones [24]. The inverse problem of Kubelka-Munk theory consists in using experimentally measured total transmittance and reflectance in order to calculate the scattering and absorption coefficients of the material under study.
The Kubelka-Munk model [21,22] considers two radiation fluxes moving in the forward and backward direction in a slab of material (Fig. 1a). The total intensities in the forward and backward directions are coupled according to the differential equations: Here the intensity passing towards the back interface is denoted I and the intensity towards the front interface is denoted J. In addition z = d-x, and the coordinates x = 0 and d represent the back and front interfaces, respectively, in conformity with previously used notation [7,[21][22][23]. The solutions of these equations give the transmittance and reflectance for the case of a slab with backside reflectance R g (Fig. 1b), according to, and where d is the thickness, a is equal to (1 + K/S) and b is equal to (a 2 -1) 0.5 . After taking into account reflections at the front interface together with multiple reflections in the slab, we obtain where R c is the reflectance of the incident light on the front interface of the slab while R j is the frontside internal reflectance of the slab (Fig. 1b). It is realized, that in order to invert the equations for R and T to determine S and K, knowledge of the interface reflectance parameters is necessary. The internal interface reflectances R g and R j involve contributions from reflected collimated and diffuse light, at back and front interfaces, respectively (Fig. 1b), where q 0 and q d are defined as the fractions of diffuse light in the film close to the back (x = 0) and front (x = d) interfaces, respectively. Furthermore, R df and R db are diffuse interface reflectances obtained from the forward and backward portions of the angle-dependent scattering profile [23]. Hence they describe diffuse reflection at the back and front interfaces, respectively. It has been shown by Barrios et al [7] that q 0 and q d can be calculated from regular and diffuse transmittance together with specular and diffuse reflectance spectra. After some algebra, the relations for q 0,d were found to be [7] It is seen from Eq. (7) that R df and R db need to be known in order to determine the total interface reflectances R g and R j . A problem here is that the diffusely scattered light exhibits a certain angular distribution and cannot in general be assumed to be isotropic. In recent works by some of us, we developed a method to estimate the diffuse interface reflectances of the composite medium by angle resolved scattering (ARS) measurements [23,25]. However, in many cases, for example in reanalysis of old data as in the present paper, ARS measurements may not be available. Hence, we have to assume an angular scattering profile in order to be able to determine the forward and backward diffuse interface reflectances R df and R db . We have investigated this problem extensively in the previous work [23] and found that strongly peaked forward and/or backward scattering distributions could be well approximated by a constant profile out to a cutoff angle given by the critical angle of total internal reflection. The alternative assumption would be isotropic diffuse scattering and combining these alternatives we proposed three approximations, denoted dif, cri and cri/ dif [23].
In the dif approximation, we assumed an isotropic scattering profile for both forward and backward directions. We obtain the total diffuse interface reflectance with light incident from the sample to air from [26] Here r denotes the reflectance for light incident on the interface from inside the composite material at an angle b.
In the cri approximation, we computed the diffuse interface reflectances in analogy with the dif method, but restricted the scattered light in the forward and backward directions to angles less than the critical angle, hence The critical angle h c is the angle above which total internal reflection occurs so that the reflection coefficients for s-and ppolarized light are equal to unity. This angle exhibits a slight wavelength dependence, which has a negligible influence on the analysis. The critical angle was computed from the refractive indices of the matrix material and the air surrounding the composite slab.
In the cri/dif approximation, the scattering angle was assumed to be restricted by the critical angle in the forward direction and scattering was assumed to be isotropic in the backward direction.

Determination of absorption and scattering coefficients for energy-related materials
In this section we use the approach outlined above to calculate absorption and scattering coefficient spectra for several materials of interest pertaining to energy efficiency and solar energy. In some cases they constitute coatings deposited onto substrates. The substrates have refractive indices similar to those of the coatings and hence their effect on the optical parameters was ignored. We restrict ourselves to a reanalysis of selected data from the scientific literature. However, in certain cases considered below, only the total transmittance and reflectance were measured and little or no information was given on the diffuse and regular/specular parts. For weakly scattering materials, especially those having a low concentration of particles, it is usually a good approximation to put R spe equal to the front surface reflectance, R c , while independent information would be needed in order to approximate T reg and T dif . In the case of strongly scattering materials the situation is different and we may assume dominant diffuse scattering by putting T dif = T tot , and R tot = R dif + R c . In the examples below the front surface reflectance was calculated from the refractive indices of the matrix materials embedding the particles.

Solar absorbing paints: FeMnCu oxide in KBr
High-efficiency solar-thermal energy conversion requires a coating with high solar absorptance and low thermal emittance of the solar collector [27]. Solar selective absorbing paints offer a commercially available low-cost alternative for applications in the low to medium temperature range [8]. These coatings consist of an absorbing pigment, usually iron oxide-based or carbon particles, in an organic resin. The coating is applied on a metallic solar collector surface. In this configuration, only reflectance is measurable, and the method in sect. 2 cannot be applied. Instead, it has been proposed [28,29] to use reflectance measurements on coatings with different thicknesses to obtain values of the Kubelka-Munk coefficients, S and K.
In order to study the optical properties of the efficiently solar absorbing FeMnCu oxide pigment, Tesfamichael et al. [9] dispersed the pigment particles in KBr. Details are given in the original publication [9]; briefly (Fe,Mn) 3 O 4 :CuO (Ferro Corporation, Model F-6331) was mixed with KBr powder, ground in a pearl mill and subsequently pressed into a pellet. The pellets had a diameter of 13 mm and a thickness (d) of 0.4 mm. Pigment particles were aggregated into clusters with a diameter of approximately 0.5 lm [9]. Fig. 2a shows the measured total transmittance and reflectance of a FeMnCu oxide-KBr sample with 0.1 wt% pigment, corresponding to a volume fraction of particles being 0.053%. The total reflectance is mainly caused by the interface reflectance from the front and back sample/air interfaces. The transmittance is lower in the visible than in the near infrared, implying a much higher absorption in the solar region than in the infrared. The diffuse parts of R and T were measured, but not reported in the original publication. However, it was possible to obtain the ratio between regular and total transmittance from the reported simplified evaluation of the S spectrum [9]. The specular reflectance was approximated by the front surface reflectance, which was obtained from the refractive index of KBr. This should be a good approximation for such an extremely low pigment volume fraction. The KBr matrix has a refractive index varying between 1.54 and 1.65 in the studied wavelength range [30], and for simplicity we put the critical angle equal to an average value of 39 . The spectra were analysed with the inversion method described in sect. 2, and it was found that both the cri and cri/dif approximations could be used to accurately fit both the reflectance and transmittance spectra in Fig. 2a. The obtained absorption and scattering coefficients are shown in Fig. 2b and clearly absorption is dominant. The fitted absorption coefficients from the two methods are almost overlapping, but the scattering coefficients exhibit large quantitative differences. Hence, it is critical for the evaluation of correct values of S and K, to use a good description of the angle-dependence of scattering. In the present case, we believe that the cri approximation is the best one, in analogy with our previous studies [23] of low volume fraction nanoparticle composites.

Suspended particle devices
Suspended particle devices (SPDs) can rapidly switch from a dark bluish-black state to a clear grey appearance when a high AC electric field is applied [6,7]. Important applications of these devices occur in smart windows and in switchable optics technology. The optically active layer is made of a polymer matrix containing $1 lm diameter droplets comprising a suspension of polyhalide particles [31]. This layer is sandwiched between glass substrates coated with transparent contacts. The SPD investigated in [6,7] has an active area of 28 Â 22 cm 2 and a thickness of 300 lm. It was operated with a sinusoidal signal at 50 Hz and a peak voltage U between 0 (dark 'off' state) and 100 V (clear 'on' state).
In this case both the diffuse and regular/specular parts of the transmittance and reflectance were measured previously [7], so we could directly apply our improved inversion procedure to the data. Here the critical angle was estimated from the average refractive index of glass. Fig. 3a shows the spectra of T tot and R tot for both 'on' and 'off' states. The modulation on the transmittance curves between 'on' and 'off' states occurs mainly in the visible region. It was found that by using the cri approximation the computed total transmittance and reflectance curves overlap exactly with the experimental ones (Fig. 3a). This is a major improvement on the earlier analysis using the dif approximation [7], which could not accurately fit the reflectance spectrum. The absorption is dominant compared to the scattering (Fig. 3b), and it is mainly the absorption coefficient that is changed between the 'off' state and the 'on' state. It is observed that SPD devices fall into the category of low-scattering materials, which is of course a necessity for any application on windows.

Strongly scattering materials: TiO 2 particles in binder
Diffusely solar reflecting coatings can prevent excessive heating in strong sunlight and TiO 2 is an excellent choice of pigment for this application, due to its high refractive index. Below we report some results from an extensive previous study [32][33][34], in which the solar reflectance of coatings containing TiO 2 particles and TiO 2 coated SiO 2 particles was compared. Pure TiO 2 (rutile) pigment (Flexonyl White R 100 VP, Hoechst), with average particle diameter 230 nm, was mixed with a binder, a copolymer latex of ethylene vinyl acetate (Movolith, Hoechst). Paint coatings were deposited onto 2 mm thick glass plates by the doctor blading technique. Coatings with thickness between 60 and 75 lm, and particle weight fractions from 0.05 to 0.5, were obtained. Film thicknesses were measured over a step edge by a Tencor-Alpha-Step 200 surface profiler instrument. The refractive index of the binder was obtained from measurements on a coating of the pure binder. The average value in the studied wavelength range was found to be 1.36, giving a critical angle of 47°, while the critical angle of the glass substrate was approximately 41°. Fig. 4a depicts the transmittance and reflectance spectra for coatings with TiO 2 particle weight fractions of 0.05, 0.1, 0.2, 0.3, 0.4 and 0.5. The reflectance increases and the transmittance decreases as the pigment concentration increases. The abrupt drop in the near ultraviolet is due to the band gap of TiO 2 , as well as to light absorption in the glass. The average solar reflectance of the high pigment concentration coatings is limited to about 0.8, but would of course be significantly higher when applied on a metal backing, for example on a roof.
The coatings were visually diffuse and preliminary ARS measurements did not show any specular peak in transmittance [35]. Hence it is a very good approximation to neglect the direct transmittance so that T reg = 0 and in addition put R spe = R c . The Kubelka-Munk coefficients were extracted using the three approximations mentioned in section 2. However, the cri approximation failed to converge and is clearly not useful for such strongly scattering materials. The cri/dif approximation converges slightly better than the dif one, but both give qualitatively similar results. The obtained scattering and absorption coefficients S and K are depicted in Fig. 4b, clearly demonstrating that scattering is dominant in these paints. It seems that the scattering coefficient saturates when the weight fraction of TiO 2 reaches 0.4. Absorption bands in the near infrared range are due to the binder and the glass substrate.

Pigmented polymer foils for radiative cooling
Plastic foils containing non-absorbing pigments can display a high reflectance of solar radiation combined with a high transmit- tance in the atmospheric window region in the thermal infrared [4]. They are suitable for keeping the temperature of an underlying thermally emissive material close to the ambient one. Heating by solar radiation is largely prevented and this is compensated, and can even be exceeded, by the infrared radiative cooling effect. Here we consider the optical scattering of ZnS-polyethylene radiative cooling foils [4]. We analyze data only in the visible and near infrared wavelength ranges, since scattering is very low in the thermal infrared. The pigment contained 98% ZnS particles with a mean diameter 0.15 lm, together with 1% BaSO 4 , 0.2% ZnO and 0.8% unspecified materials. The powder was first heated to a temperature above 373 K to evaporate most of the water and was then mixed with polyethylene in an extruder during heating to 430 K in order to split pigment agglomerates. Subsequently, the mixture was pressed at a temperature of 420 K.
The total transmittance and reflectance spectra of ZnS-Polyethylene samples with two different volume fractions 0.0363 and 0.241 are depicted in Fig. 5a. The average solar reflectance is limited to 0.74 for the foil with the higher pigment volume fraction, due to absorption bands of unknown origin [4]. Hence the radiative cooling effect cannot wholly compensate solar heating. Scattering and absorption coefficients were obtained using our formulation of the inverse problem. The appearance of the samples was whitish diffuse and hence we neglect the regular transmittance as in sect. 3.3. Low density polyethylene has a refractive index of 1.48 to 1.52 in the considered spectral range [36], and for simplicity we use an average critical angle of 42 . The spectra of S and K are shown in Fig. 5b. Both the cri/dif and dif approximations converge and give very similar results. Also, in the case of these materials, scattering dominates over absorption.  3. (a) Experimental total transmittance T tot and reflectance R tot , and fitted (cri) total transmittance T t and reflectance R t when the SPD is in the 'on' and 'off' states; (b) K and S derived using the cri approximation for the 'on' and 'off' states. At wavelengths where the transmittance was close to zero, K could not be evaluated.

Conclusion
It is essential to develop reliable and robust methods to analyze the optical parameters of light scattering materials. In this paper, we analyzed weakly and strongly light scattering samples of interest for energy-related applications. The inverse problem of the two-flux Kubelka-Munk theory was solved in order to obtain reliable values of absorption and scattering coefficients. The main difficulty with this method is that experimental data that would be used to estimate the interface reflectances of the scattering layer are often lacking. We tested three approximations, denoted cri, cri/dif, and dif for the angular scattering profiles inside the material. It was found that the cri approximation is the most accurate one for weakly scattering samples. For strongly scattering samples both cri/dif and dif approximations can be used and give reasonable results. However, we hypothesize that cri/dif is most suitable for medium to strong scattering, and that for extremely strongly scattering samples dif should be used. It should be emphasized that all these methods are based on approximations to the angular light scattering profiles. In order to obtain very accurate results, actual measurements of angle resolved scattering are necessary.