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Light scattering in monodisperse systems – from suspensions to transparent ceramics
Journal Pre-proof Light Scattering in Monodisperse Systems – from Suspensions to Transparent Ceramics ˇ Hˇr´ıbalova, ´ Willi Pabst Sona
PII:
S0955-2219(19)30798-8
DOI:
https://doi.org/10.1016/j.jeurceramsoc.2019.11.053
Reference:
JECS 12878
To appear in:
Journal of the European Ceramic Society
Received Date:
13 June 2019
Revised Date:
29 October 2019
Accepted Date:
15 November 2019
Please cite this article as: Hˇr´ıbalova´ S, Pabst W, Light Scattering in Monodisperse Systems – from Suspensions to Transparent Ceramics, Journal of the European Ceramic Society (2019), doi: https://doi.org/10.1016/j.jeurceramsoc.2019.11.053
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Light Scattering in Monodisperse Systems – from Suspensions to Transparent Ceramics Soňa Hříbalová, Willi Pabst Department of Glass and Ceramics University of Chemistry and Technology, Prague Technická 5, 16628 Prague 6, Czech Republic Email: [email protected]
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Abstract: Predicting the in-line transmittance of transparent ceramics via model calculations is a useful guide for materials design and optimization of preparation process. However, the absence of reliable input information (volume fractions and size) usually precludes the direct verification of these calculations. On the other hand, suspensions, which can be prepared with well controlled volume fractions of the solids selected, may serve as model systems that are amenable to verification. This paper describes the procedure to perform these calculations, and compares calculated data for suspensions of monodisperse spheres of amorphous silica with spectrophotometric measurements of silica monosphere suspensions with four different concentrations. Moreover, completely analogous model calculations are performed for bubbles in silica glass and pores in spinel ceramics. The results are discussed on the basis on 3D graphs, the use of which is highly recommended. The results may be considered as a benchmark for future model calculations for polydisperse systems.
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1. Introduction
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Keywords: Transmittance; Mie theory; transparent ceramics (spinel); monodisperse silica spheres; light scattering.
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Transparent ceramics are an important topic of current research due to their specific properties and wide range of uses, such as impact resistant windows [1], lamp envelopes [2], solar energy devices [3], missile domes [4], solid state lasers [5]. A major challenge in its preparation is the elimination of residual porosity and other phases (second-phase inclusions) in order to achieve high transparency. Light scattering (and absorption when absorbing scatterers are present) on residual pores and/or other phases is the dominant phenomenon responsible for the decrease of the in-line transmittance in such ceramics. Even though highly transparent ceramics have been successfully prepared in the past [6-9], the elimination of light scattering still remains a challenge. It is, therefore, necessary to have an idea about the impact of the amount and size of pores (or other second phase inclusions) on the resulting transparency (in-line transmittance). The Mie theory of light scattering is widely used in many fields of scientific research, e.g. meteorology [10], space research [11], characterization of droplet size in emulsions [12] and particle sizing (laser diffraction) [13]. Mie’s theory of light scattering, derived for homogeneous and optically isotropic 1
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spheres, is based on the solution of Maxwell equations under suitable boundary conditions. Since this solution contains two infinite series, the practical application of Mie theory requires the application of numerical calculations, which can be an obstacle for simple use; therefore analytical approximations are very popular, e.g. the Fraunhofer approximation for sufficiently large or highcontrast scatterers [14] or the Rayleigh approximation for sufficiently small or low-contrast scatterers [15]. However, the range of wavelengths, scatterer sizes and refractive indices often exceeds the range of validity of any these approximations alone. Therefore, wherever possible, Mie theory should be used. The measurement of porosity and pore size in almost fully densified (i.e. almost transparent) ceramics is a complicated task, because the resolution of common 3D characterization techniques currently available is often insufficient for capturing relevant pore sizes, especially when the pore volume fractions are low. Only few authors have used suitable approaches – FIB-SEM with tomography [16] or confocal laser microscopy [17, 18], but their work shows how difficult this task is. The choice of a representative size value from a measured size distribution is not simple either. It is common practice today to choose some kind of mean, often the arithmetic one, without considering possible alternatives. This practice is indirectly reflected in the frequent use of transmission spectra for a certain mean size of scatterer as a representation of light scattering in a system. In reality, however, most systems are polydisperse, so that the choice of the appropriate mean size is actually a crucial one. The work of Pecharromán et al. [19] has shown that indeed the arithmetic mean of the volume-weighted size distribution (in the original paper called “medium” value of the “volume grain distribution”), which may in practice be often close to the maximum diameter of the number-weighted size distribution, is indeed appropriate for size distributions of low-contrast inclusions when the RayleighGans-Debye approximation is used. However, according to Stuer et al. [16], another quantity, viz. the third general moment, which is closely related to a cubic mean of a volume-weighted distribution [20], would be appropriate for pores within the Rayleigh approximation. Therefore it is clear that for broad pore size distributions, which may contain both small pores for which the Rayleigh approximation can be used and larger ones for which it cannot be used, the choice of these mean values becomes debatable. On the other hand, before attacking the intricate problem of polydisperse system it is necessary to fully understand light scattering in monodisperse systems in the first place. The behavior of light scattering in monodisperse systems can be conveniently displayed in three-dimensional (3D) graphs (transmittance versus scatterer diameter versus wavelength of incident light), which contain much more information than the commonly used two-dimensional (2D) graphs, which are actually just sections of the former. To the best of our knowledge, no attempt has been made in the ceramic community to compare the model calculations obtained via Mie theory with experimental measurements of monodisperse systems. Doubtless this is caused by the fact that there are very few systems that can be considered as really monodisperse. To the best of our knowledge one of the few systems that fulfill this desideratum to at least a very good degree are the almost monodisperse amorphous silica spheres made via the Stöber process [21,22]. Suspensions of these “silica monospheres” exhibit optical properties that are in many respects 2
analogous to transparent ceramics and can be considered as model systems for experimentally verifying transmittance calculations. It is the aim of this paper to recall the procedure of transmittance calculation based on Mie theory (taking pores in spinel and bubbles in silica glass as typical examples) and to compare calculated data with results of spectrophotometric measurements. 2. Theory
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(1)
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𝑇 = 𝑇max 𝐼 .
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The basic quantity describing the optical properties of a material is the refractive index (n), which is dependent on the wavelength of light. This dependence is often described by a so-called Sellmeier equation [23]. In a heterogeneous system, scattering of light occurs at optical heterogeneities, i.e. scatterers, the refractive index of which is different from the medium (at least in a certain range of wavelengths). The (in-line) transmittance of a system, more precisely a slab of material with given thickness, is defined as a ratio of light intensity passing through the system (I) in the direction of the normally incident light intensity (I0), multiplied by the theoretical limit of transmittance (𝑇max = 2𝑛/(1 + 𝑛2 ), where n is the refractive index of the medium without any optical heterogeneities): 0
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When the light passes through the heterogeneous system and multiple scattering is negligible, then the relative transmittance (Trel) describing the light intensity loss in the direction of incident light can be expressed using LambertBeer extinction law: 𝑇 𝐼 𝑇rel = 𝑇 = 𝐼 = 𝑒 −𝛾extℎ , (2)
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where γext is the extinction coefficient and h the light path length in the system (i.e. the thickness of a slab of material or the inner width of a cuvette). Assuming that all of the scatterers are identical, the extinction coefficient is expressed as 𝛾ext = 𝑁𝑉 𝐶ext ,
(3)
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where NV is the number of scatterers per unit volume (a quantity that principally cannot be obtained from 2D sections) and Cext is the so-called extinction cross section. In general, the extinction cross section is the combination (sum) scattering and absorption cross sections. In this paper, however, all scatterers are non-absorbing, therefore the absorption cross section can be considered to be zero. Negligible multiple scattering is one of the assumptions that have to be made when predicting the transparency (in-line transmittance) based on light scattering in heterogeneous system. This assumption amounts to assuming that the light is scattered only once in its path through the system. This assumption is generally fulfilled when the concentration of optical heterogeneities (scatterers) is low enough. According to van de Hulst [24], multiple scattering is negligible when the turbidity (defined as 𝜏 = 𝛾ext ℎ) is ≪ 1, which means of course that the relative transmittance is close to 1 (see Equation 2), a very strong condition 3
indeed. According to other authors [12, 25, 26, 27], multiple scattering occurs when the transmittance decreases below 70–90 % of its maximum value. Moreover, statements of Bohren and Huffman [23] can be interpreted in the sense that only a deviation from the simple exponential relation (Lambert-Beer extinction law) is indicative of multiple scattering. That means, when the experimental data are well fitted by a simple exponential, multiple scattering can be assumed to be negligible. This is the interpretation adhered to in this paper. 3. Experimental
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For the Mie theory calculations, Prahl’s Mie scattering calculator [28] was used to calculate the extinction cross sections for the given input data (refractive indices for the wavelength in question). The wavelength dependences of the refractive indices are displayed in Figure 1, the resulting in-line transmittance was calculated using Equations 1-3. Three systems are described in this paper: porous spinel (ceramics example), bubbles in amorphous silica (as an analogous system to porous spinel) and amorphous silica particles in water, for which a comparison of predicted and measured data is presented. For the predictions, the thickness of the systems (solid slabs and suspensions in the cuvette) and the volume fraction of scatterers in the systems is h = 10 mm and ϕ = 0.00001, respectively for the comparison of measured and predicted data, different volume fractions (denoted in each graph) were used to calculate the transmittance versus wavelength dependence; the thickness of the systems is also in this case h = 10 mm. The number of particle systems that corresponds directly to the assumptions of Mie theory (optical isotropy, spherical shape) is very limited. Therefore, amorphous silica (SiO2) monodisperse powder (Monosphere 1000, Merck, Germany) synthetized by the Stöber process (which is based on hydrolysis of alkyl silicates and condensation of silicic acid in alcohol solutions with the use of ammonia as a morphological catalyst) [21,22] was selected for the comparison of predicted and measured data. This powder, characterized via electron microscopy (Lyra 3, TESCAN), X-ray fluorescence (XRF, ARL 9400 XP, Thermo ARL) and X-ray diffraction (XRD, PANanalytical X’Pert3 Powder) consists of amorphous silica particles of spherical shape with a diameter of 1 μm. For the purpose of spectrophotometric measurements and a comparison of measured and predicted data, aqueous suspensions of this powder were prepared. The powder was precisely weighed and mixed with an accurate amount of distilled water. Then the suspension (50 ml) was intensely ultrasonicated for approximately 30 s (Dr. Hielscher UP 200S, amplitude 60 % of 200 W and cycle 0.5, titanium sonotrode S14 with diameter 14 mm). The spectrophotometric measurements (Shimadzu UV-2400PC) were done right after the preparation of each suspension. The same new (i.e. not previously used) cuvette (Hellma Analytics, Quartz Suprasil® High Precision Cell, with inner width or light path 10 mm) was used for all measurements. The transmittance spectrum for each suspension was measured three times in a row, but the results were completely reproducible, which allows us to exclude sedimentation effects.
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Figure 1: Wavelength dependences of refractive indices used as the input data for Mie theory calculations, top left – amorphous silica [29], top right – spinel [30], bottom left – water [31] and bottom right – vacuum or gas. 4. Results
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In the context of transparent ceramics the transmittance is usually displayed as a dependence on the wavelength of light (transmittance spectrum) or the diameter of the scatterer. However, using either of these approaches, some information is suppressed. The complete information is displayed only in the three-dimensional plots (3D graphs), where both wavelength and scatterer diameter dependences are presented simultaneously. On the other hand, 3D graphs are not as straightforward as their two-dimensional sections (2D graphs), and the interpretation of the results is more intricate. Therefore, as a guide for the reader, cross sections for the wavelength of 600 nm of the 3D graphs (Figures 35) are presented in Figure 2 and side projections are shown throughout.
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Figure 2: Predicted dependence of transmittance on scatterer diameter of three heterogeneous systems: amorphous silica in water, bubbles in amorphous silica and pores in spinel; thickness of the system is h = 10 mm and the volume fraction of the scatterers is ϕ = 0.00001; wavelength of light 600 nm. As expected, in all of the three systems similar trends were found, see Figures 2-5. In the resulting dependences of the transmittance on the scatterer diameter and the wavelength of incident light a valley occurs in the system, which becomes deeper with increasing difference of refractive indices. The valley shifts with the wavelength and, as is clearly visible in Figure 2, its position is dependent on the optical properties of the systems. For scatterers sufficiently 6
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small or sufficiently large, the transmittance regains its theoretical level (i.e. maximum transmittance).
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Figure 3: Mie theory predictions of the transmittance of a heterogeneous system consisting of amorphous silica spheres in water in the dependence on the wavelength of incident light and the scatterer diameter; the thickness of the system is h = 10 mm and the volume fraction of scatterers is ϕ = 0.00001.
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Figure 4: Mie theory predictions of the transmittance of a heterogeneous system consisting of bubbles in amorphous silica in the dependence on the wavelength of incident light and the scatterer diameter; the thickness of the system is h = 10 mm and the volume fraction of scatterers is ϕ = 0.00001.
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Figure 5: Mie theory predictions of the transmittance of a heterogeneous system consisting of pores in spinel in the dependence on the wavelength of incident light and the scatterer diameter; the thickness of the system is h = 10 mm and the volume fraction of scatterers is ϕ = 0.00001. In order to compare the predicted transmittance spectra with the spectrophotometry measurements for aqueous suspensions of amorphous silica with monodisperse particles of 1 μm in diameter (see Figure 6) aqueous suspensions were prepared in four concentrations (volume fractions: 0.000 000 773, 0.000 001 500, 0.000 005 909, 0.000 012 273). XRF measurement revealed that the powder is composed of 99.98 wt.% SiO2; the rest being iron oxides. XRD confirmed the amorphous structure (see Figure 7); therefore refractive index data for amorphous silica were used as the input information for modelling. The 9
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SEM micrograph of the powder (Figure 6) reveals the spherical shape and very good monodispersity of the size distribution.
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Figure 6: SEM micrographs of nearly monodisperse amorphous silica powder; the majority of particles is spherical and 1 μm in diameter (median value D50 1.03 µm with a span (D90-D10)/D50 of 0.05 µm, based on number-weighted size distributions obtained by image analysis of 200 particles).
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Figure 7: XRD diffractogram of silica monospheres, indicating their amorphous structure. The measured and predicted transmittance spectra displayed in Figure 8 are generally in good agreement, especially for large wavelengths and lower concentrations. The agreement slightly deteriorates for higher concentrations and shorter wavelengths. Since both the particle size and the wavelength dependence of the refractive index are reliably known, the most probable explanation for this discrepancy is the presence of clusters (agglomerates) of primary particles that may increase the transmittance in the short wavelength region (for a given concentration, the number of large scattering particles is lower than the number of small scattering particles per volume). However, for the visible range of light, the predicted and measured transmittances are very 10
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close. As mentioned above, no significant differences were observed in the repeatedly measured spectra, therefore sedimentation effects can be safely neglected. Figure 9 demonstrates that the experimental results correspond to the exponential Lambert-Beer law of extinction (coefficient of determination R2 is at least 0.9906, i.e. very close to 1). Multiple scattering can therefore be considered to be negligible.
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Figure 8: Comparison of predicted and measured transmittance-wavelength dependences of four aqueous suspensions of monodisperse amorphous silica spheres (thickness h = 10 mm, volume fractions are displayed in each graph).
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Figure 9: Demonstration of validity of the Lambert-Beer law for predicted transmittances and simple exponential fit of the measured transmittances of suspensions, indicating negligible multiple scattering.
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5. Discussion
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It is well known that the optical transmittance of transparent ceramics is critically dependent on the volume fractions, size distribution and shape of inclusions and pores. However, as mentioned above, even using cutting edge imaging technologies the determination of these microstructural parameters is not trivial, as shown in the few relevant experimental papers on this topic [16– 18]. Moreover, there is no agreement in the literature about the type of mean size that would have to be used for the calculation of transmittance in the case of a polydisperse system of scatterers, when Mie theory is applied. Therefore the prediction of transmittance based on microstructural information from asprepared real-world transparent ceramics is elusive so far, although without doubt the work of Pecharromán et al. [19] gives valuable hints for practical use in the Rayleigh and Rayleigh-Gans-Debye approximation, which have been successfully connected to independently determined microstructural information (pore size distributions obtained via 3D-FIB tomography) by Stuer et al. [16]. In this context we fully agree with the finding of Stuer et al. [16] that pores smaller than 50 nm do not significantly affect the in-line transmittance. We are convinced, however, that the full treatment of the influence of pore size distributions on transmittance via Mie theory requires our concept of generalized mean values [20], when the calculation is to be based on a characteristic size instead of the whole distribution. On the other hand, material and process design requires certain knowledge of the influence of volume fractions, size distribution and shape of inclusions and pores on the transmittance. It is important to know, for example, what concentration or size of inclusions or pores can be tolerated in a 12
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transparent ceramic or whether a finer or coarser raw powder would have to be used to achieve transparency. For this reason theoretical calculations of transmittance by Mie theory are indispensable, and checking the results of these calculations with easily controllable real-world systems (e.g. suspensions) surely adds to their credibility. For this purpose aqueous suspensions can be used, the optical properties of which are in many respects similar to transparent solids with inclusions or pores, can serve as model systems for transparent ceramics, because the same physics apply to both systems and because it is the phase contrast (i.e. the ratio of refractive indices) that determines the transmittance in both cases. Suspensions can be readily and accurately prepared with solid particles of a well-defined shape, size distribution and volume fraction, including very small volume fractions, whereas the determination of these quantities in transparent ceramics is a highly non-trivial task. However, before attacking the intricate problem of systems with polydisperse size distributions of real-shaped scatterers, it was necessary to obtain benchmark results for the case of monodisperse spheres. This was the purpose of the present paper. In order to allow for a comparison of theoretical predictions and experimental results, silica monospheres were used. Although not perfectly monodisperse and even though a small degree of agglomeration could probably not be completely excluded, this particle system comes indeed very close to the realization of a monodisperse system. The results of the Mie calculations in this paper were presented in the form of 3D graphs, which we consider to be an essential step towards a future deeper understanding of the scattering of polydisperse systems and transmittance in realistic situations that may range from the ultraviolet to the infrared. 3D graphs naturally contain more information than the commonly presented transmittance spectra (commonly presented in experimental papers) or scatterer size dependences of transmittance for a given wavelength (commonly presented in numerical papers), and are harder to capture for the reader. Therefore 2D sections and projections have been used in this paper in order to facilitate the interpretation of the 3D plots and to emphasize some of their specific features. For the suspensions the predicted and measured transmittance spectra are in reasonably good agreement, which gives credibility also to the results for the transparent solids, for which microstructural input data are not available. Of course, detailed in-depth comparison of the results indicates a slight discrepancy between measured and predicted data, probably due to the presence of clusters in the suspensions, but this effect is significant only in suspensions with higher concentrations of scattering particles and for short wavelengths. Multiple scattering does not occur to any significant degree (as was demonstrated by fitting the concentration dependence of transmittance via simple exponentials). In conclusion, a concise overview of the light scattering in (nonabsorbing) heterogeneous systems has been given, in-line transmittances have been calculated via Mie theory and for suspensions of silica monospheres compared with experimental results (spectrophotometric measurements). The procedure may be used as a guideline for investigating light scattering in other heterogeneous systems, including transparent ceramics in all cases where the aforementioned microstructural parameters are available, at least the volume 13
fraction. Three-dimensional graphs, which include not only the wavelength dependence (spectrum) but also the scatterer size dependence of transmittance, seem to be indispensable tools for understanding the transmittance of polydisperse heterogeneous systems. 6. Conclusion and outlook
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Declaration of interests
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Predicting the in-line transmittance of transparent ceramics via model calculations is a useful guide for materials design and optimization of preparation process. However, the absence of reliable input information (volume fractions and size) usually precludes the direct verification of these calculations. On the other hand, suspensions, which can be prepared with well controlled volume fractions of the solids selected, may serve as model systems that are amenable to verification. It is clear, however, that even in this case both the calculation and the verification of transmittance predictions are prone to many pitfalls. In particular, before attacking the intricate problem of systems with broad (polydisperse) size distributions, it is necessary to obtain benchmark results for the case of monodisperse spheres, a problem for which Mie theory provides a numerical solution. Indeed, the simplest thinkable case is a monodisperse system of spherical particles, which can be realized in a very good approximation with so-called silica monospheres, i.e. a nearly monodisperse system of nearly spherical spheres of amorphous silica. The results of the Mie calculations in this paper were presented in the form of 3D graphs, which is an essential step towards a future deeper understanding of the scattering of polydisperse systems and transmittance in realistic situations that may range from the ultraviolet to the infrared. For the suspensions the predicted and measured transmittance spectra are in reasonably good agreement, which gives credibility also to the results for the transparent solids, for which microstructural input data are not available.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Acknowledgement: This work is part of the project “Partially and fully sintered ceramics - processing, microstructure, properties, modeling and sintering theory” (GA18-17899S), supported by the Czech Science Foundation (GAČR). References
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PII:
S0955-2219(19)30798-8
DOI:
https://doi.org/10.1016/j.jeurceramsoc.2019.11.053
Reference:
JECS 12878
To appear in:
Journal of the European Ceramic Society
Received Date:
13 June 2019
Revised Date:
29 October 2019
Accepted Date:
15 November 2019
Please cite this article as: Hˇr´ıbalova´ S, Pabst W, Light Scattering in Monodisperse Systems – from Suspensions to Transparent Ceramics, Journal of the European Ceramic Society (2019), doi: https://doi.org/10.1016/j.jeurceramsoc.2019.11.053
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Light Scattering in Monodisperse Systems – from Suspensions to Transparent Ceramics Soňa Hříbalová, Willi Pabst Department of Glass and Ceramics University of Chemistry and Technology, Prague Technická 5, 16628 Prague 6, Czech Republic Email: [email protected]
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Abstract: Predicting the in-line transmittance of transparent ceramics via model calculations is a useful guide for materials design and optimization of preparation process. However, the absence of reliable input information (volume fractions and size) usually precludes the direct verification of these calculations. On the other hand, suspensions, which can be prepared with well controlled volume fractions of the solids selected, may serve as model systems that are amenable to verification. This paper describes the procedure to perform these calculations, and compares calculated data for suspensions of monodisperse spheres of amorphous silica with spectrophotometric measurements of silica monosphere suspensions with four different concentrations. Moreover, completely analogous model calculations are performed for bubbles in silica glass and pores in spinel ceramics. The results are discussed on the basis on 3D graphs, the use of which is highly recommended. The results may be considered as a benchmark for future model calculations for polydisperse systems.
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1. Introduction
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Keywords: Transmittance; Mie theory; transparent ceramics (spinel); monodisperse silica spheres; light scattering.
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Transparent ceramics are an important topic of current research due to their specific properties and wide range of uses, such as impact resistant windows [1], lamp envelopes [2], solar energy devices [3], missile domes [4], solid state lasers [5]. A major challenge in its preparation is the elimination of residual porosity and other phases (second-phase inclusions) in order to achieve high transparency. Light scattering (and absorption when absorbing scatterers are present) on residual pores and/or other phases is the dominant phenomenon responsible for the decrease of the in-line transmittance in such ceramics. Even though highly transparent ceramics have been successfully prepared in the past [6-9], the elimination of light scattering still remains a challenge. It is, therefore, necessary to have an idea about the impact of the amount and size of pores (or other second phase inclusions) on the resulting transparency (in-line transmittance). The Mie theory of light scattering is widely used in many fields of scientific research, e.g. meteorology [10], space research [11], characterization of droplet size in emulsions [12] and particle sizing (laser diffraction) [13]. Mie’s theory of light scattering, derived for homogeneous and optically isotropic 1
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spheres, is based on the solution of Maxwell equations under suitable boundary conditions. Since this solution contains two infinite series, the practical application of Mie theory requires the application of numerical calculations, which can be an obstacle for simple use; therefore analytical approximations are very popular, e.g. the Fraunhofer approximation for sufficiently large or highcontrast scatterers [14] or the Rayleigh approximation for sufficiently small or low-contrast scatterers [15]. However, the range of wavelengths, scatterer sizes and refractive indices often exceeds the range of validity of any these approximations alone. Therefore, wherever possible, Mie theory should be used. The measurement of porosity and pore size in almost fully densified (i.e. almost transparent) ceramics is a complicated task, because the resolution of common 3D characterization techniques currently available is often insufficient for capturing relevant pore sizes, especially when the pore volume fractions are low. Only few authors have used suitable approaches – FIB-SEM with tomography [16] or confocal laser microscopy [17, 18], but their work shows how difficult this task is. The choice of a representative size value from a measured size distribution is not simple either. It is common practice today to choose some kind of mean, often the arithmetic one, without considering possible alternatives. This practice is indirectly reflected in the frequent use of transmission spectra for a certain mean size of scatterer as a representation of light scattering in a system. In reality, however, most systems are polydisperse, so that the choice of the appropriate mean size is actually a crucial one. The work of Pecharromán et al. [19] has shown that indeed the arithmetic mean of the volume-weighted size distribution (in the original paper called “medium” value of the “volume grain distribution”), which may in practice be often close to the maximum diameter of the number-weighted size distribution, is indeed appropriate for size distributions of low-contrast inclusions when the RayleighGans-Debye approximation is used. However, according to Stuer et al. [16], another quantity, viz. the third general moment, which is closely related to a cubic mean of a volume-weighted distribution [20], would be appropriate for pores within the Rayleigh approximation. Therefore it is clear that for broad pore size distributions, which may contain both small pores for which the Rayleigh approximation can be used and larger ones for which it cannot be used, the choice of these mean values becomes debatable. On the other hand, before attacking the intricate problem of polydisperse system it is necessary to fully understand light scattering in monodisperse systems in the first place. The behavior of light scattering in monodisperse systems can be conveniently displayed in three-dimensional (3D) graphs (transmittance versus scatterer diameter versus wavelength of incident light), which contain much more information than the commonly used two-dimensional (2D) graphs, which are actually just sections of the former. To the best of our knowledge, no attempt has been made in the ceramic community to compare the model calculations obtained via Mie theory with experimental measurements of monodisperse systems. Doubtless this is caused by the fact that there are very few systems that can be considered as really monodisperse. To the best of our knowledge one of the few systems that fulfill this desideratum to at least a very good degree are the almost monodisperse amorphous silica spheres made via the Stöber process [21,22]. Suspensions of these “silica monospheres” exhibit optical properties that are in many respects 2
analogous to transparent ceramics and can be considered as model systems for experimentally verifying transmittance calculations. It is the aim of this paper to recall the procedure of transmittance calculation based on Mie theory (taking pores in spinel and bubbles in silica glass as typical examples) and to compare calculated data with results of spectrophotometric measurements. 2. Theory
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𝑇 = 𝑇max 𝐼 .
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The basic quantity describing the optical properties of a material is the refractive index (n), which is dependent on the wavelength of light. This dependence is often described by a so-called Sellmeier equation [23]. In a heterogeneous system, scattering of light occurs at optical heterogeneities, i.e. scatterers, the refractive index of which is different from the medium (at least in a certain range of wavelengths). The (in-line) transmittance of a system, more precisely a slab of material with given thickness, is defined as a ratio of light intensity passing through the system (I) in the direction of the normally incident light intensity (I0), multiplied by the theoretical limit of transmittance (𝑇max = 2𝑛/(1 + 𝑛2 ), where n is the refractive index of the medium without any optical heterogeneities): 0
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When the light passes through the heterogeneous system and multiple scattering is negligible, then the relative transmittance (Trel) describing the light intensity loss in the direction of incident light can be expressed using LambertBeer extinction law: 𝑇 𝐼 𝑇rel = 𝑇 = 𝐼 = 𝑒 −𝛾extℎ , (2)
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where γext is the extinction coefficient and h the light path length in the system (i.e. the thickness of a slab of material or the inner width of a cuvette). Assuming that all of the scatterers are identical, the extinction coefficient is expressed as 𝛾ext = 𝑁𝑉 𝐶ext ,
(3)
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where NV is the number of scatterers per unit volume (a quantity that principally cannot be obtained from 2D sections) and Cext is the so-called extinction cross section. In general, the extinction cross section is the combination (sum) scattering and absorption cross sections. In this paper, however, all scatterers are non-absorbing, therefore the absorption cross section can be considered to be zero. Negligible multiple scattering is one of the assumptions that have to be made when predicting the transparency (in-line transmittance) based on light scattering in heterogeneous system. This assumption amounts to assuming that the light is scattered only once in its path through the system. This assumption is generally fulfilled when the concentration of optical heterogeneities (scatterers) is low enough. According to van de Hulst [24], multiple scattering is negligible when the turbidity (defined as 𝜏 = 𝛾ext ℎ) is ≪ 1, which means of course that the relative transmittance is close to 1 (see Equation 2), a very strong condition 3
indeed. According to other authors [12, 25, 26, 27], multiple scattering occurs when the transmittance decreases below 70–90 % of its maximum value. Moreover, statements of Bohren and Huffman [23] can be interpreted in the sense that only a deviation from the simple exponential relation (Lambert-Beer extinction law) is indicative of multiple scattering. That means, when the experimental data are well fitted by a simple exponential, multiple scattering can be assumed to be negligible. This is the interpretation adhered to in this paper. 3. Experimental
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For the Mie theory calculations, Prahl’s Mie scattering calculator [28] was used to calculate the extinction cross sections for the given input data (refractive indices for the wavelength in question). The wavelength dependences of the refractive indices are displayed in Figure 1, the resulting in-line transmittance was calculated using Equations 1-3. Three systems are described in this paper: porous spinel (ceramics example), bubbles in amorphous silica (as an analogous system to porous spinel) and amorphous silica particles in water, for which a comparison of predicted and measured data is presented. For the predictions, the thickness of the systems (solid slabs and suspensions in the cuvette) and the volume fraction of scatterers in the systems is h = 10 mm and ϕ = 0.00001, respectively for the comparison of measured and predicted data, different volume fractions (denoted in each graph) were used to calculate the transmittance versus wavelength dependence; the thickness of the systems is also in this case h = 10 mm. The number of particle systems that corresponds directly to the assumptions of Mie theory (optical isotropy, spherical shape) is very limited. Therefore, amorphous silica (SiO2) monodisperse powder (Monosphere 1000, Merck, Germany) synthetized by the Stöber process (which is based on hydrolysis of alkyl silicates and condensation of silicic acid in alcohol solutions with the use of ammonia as a morphological catalyst) [21,22] was selected for the comparison of predicted and measured data. This powder, characterized via electron microscopy (Lyra 3, TESCAN), X-ray fluorescence (XRF, ARL 9400 XP, Thermo ARL) and X-ray diffraction (XRD, PANanalytical X’Pert3 Powder) consists of amorphous silica particles of spherical shape with a diameter of 1 μm. For the purpose of spectrophotometric measurements and a comparison of measured and predicted data, aqueous suspensions of this powder were prepared. The powder was precisely weighed and mixed with an accurate amount of distilled water. Then the suspension (50 ml) was intensely ultrasonicated for approximately 30 s (Dr. Hielscher UP 200S, amplitude 60 % of 200 W and cycle 0.5, titanium sonotrode S14 with diameter 14 mm). The spectrophotometric measurements (Shimadzu UV-2400PC) were done right after the preparation of each suspension. The same new (i.e. not previously used) cuvette (Hellma Analytics, Quartz Suprasil® High Precision Cell, with inner width or light path 10 mm) was used for all measurements. The transmittance spectrum for each suspension was measured three times in a row, but the results were completely reproducible, which allows us to exclude sedimentation effects.
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Figure 1: Wavelength dependences of refractive indices used as the input data for Mie theory calculations, top left – amorphous silica [29], top right – spinel [30], bottom left – water [31] and bottom right – vacuum or gas. 4. Results
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In the context of transparent ceramics the transmittance is usually displayed as a dependence on the wavelength of light (transmittance spectrum) or the diameter of the scatterer. However, using either of these approaches, some information is suppressed. The complete information is displayed only in the three-dimensional plots (3D graphs), where both wavelength and scatterer diameter dependences are presented simultaneously. On the other hand, 3D graphs are not as straightforward as their two-dimensional sections (2D graphs), and the interpretation of the results is more intricate. Therefore, as a guide for the reader, cross sections for the wavelength of 600 nm of the 3D graphs (Figures 35) are presented in Figure 2 and side projections are shown throughout.
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Figure 2: Predicted dependence of transmittance on scatterer diameter of three heterogeneous systems: amorphous silica in water, bubbles in amorphous silica and pores in spinel; thickness of the system is h = 10 mm and the volume fraction of the scatterers is ϕ = 0.00001; wavelength of light 600 nm. As expected, in all of the three systems similar trends were found, see Figures 2-5. In the resulting dependences of the transmittance on the scatterer diameter and the wavelength of incident light a valley occurs in the system, which becomes deeper with increasing difference of refractive indices. The valley shifts with the wavelength and, as is clearly visible in Figure 2, its position is dependent on the optical properties of the systems. For scatterers sufficiently 6
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small or sufficiently large, the transmittance regains its theoretical level (i.e. maximum transmittance).
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Figure 3: Mie theory predictions of the transmittance of a heterogeneous system consisting of amorphous silica spheres in water in the dependence on the wavelength of incident light and the scatterer diameter; the thickness of the system is h = 10 mm and the volume fraction of scatterers is ϕ = 0.00001.
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Figure 4: Mie theory predictions of the transmittance of a heterogeneous system consisting of bubbles in amorphous silica in the dependence on the wavelength of incident light and the scatterer diameter; the thickness of the system is h = 10 mm and the volume fraction of scatterers is ϕ = 0.00001.
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Figure 5: Mie theory predictions of the transmittance of a heterogeneous system consisting of pores in spinel in the dependence on the wavelength of incident light and the scatterer diameter; the thickness of the system is h = 10 mm and the volume fraction of scatterers is ϕ = 0.00001. In order to compare the predicted transmittance spectra with the spectrophotometry measurements for aqueous suspensions of amorphous silica with monodisperse particles of 1 μm in diameter (see Figure 6) aqueous suspensions were prepared in four concentrations (volume fractions: 0.000 000 773, 0.000 001 500, 0.000 005 909, 0.000 012 273). XRF measurement revealed that the powder is composed of 99.98 wt.% SiO2; the rest being iron oxides. XRD confirmed the amorphous structure (see Figure 7); therefore refractive index data for amorphous silica were used as the input information for modelling. The 9
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SEM micrograph of the powder (Figure 6) reveals the spherical shape and very good monodispersity of the size distribution.
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Figure 6: SEM micrographs of nearly monodisperse amorphous silica powder; the majority of particles is spherical and 1 μm in diameter (median value D50 1.03 µm with a span (D90-D10)/D50 of 0.05 µm, based on number-weighted size distributions obtained by image analysis of 200 particles).
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Figure 7: XRD diffractogram of silica monospheres, indicating their amorphous structure. The measured and predicted transmittance spectra displayed in Figure 8 are generally in good agreement, especially for large wavelengths and lower concentrations. The agreement slightly deteriorates for higher concentrations and shorter wavelengths. Since both the particle size and the wavelength dependence of the refractive index are reliably known, the most probable explanation for this discrepancy is the presence of clusters (agglomerates) of primary particles that may increase the transmittance in the short wavelength region (for a given concentration, the number of large scattering particles is lower than the number of small scattering particles per volume). However, for the visible range of light, the predicted and measured transmittances are very 10
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close. As mentioned above, no significant differences were observed in the repeatedly measured spectra, therefore sedimentation effects can be safely neglected. Figure 9 demonstrates that the experimental results correspond to the exponential Lambert-Beer law of extinction (coefficient of determination R2 is at least 0.9906, i.e. very close to 1). Multiple scattering can therefore be considered to be negligible.
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Figure 8: Comparison of predicted and measured transmittance-wavelength dependences of four aqueous suspensions of monodisperse amorphous silica spheres (thickness h = 10 mm, volume fractions are displayed in each graph).
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Figure 9: Demonstration of validity of the Lambert-Beer law for predicted transmittances and simple exponential fit of the measured transmittances of suspensions, indicating negligible multiple scattering.
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5. Discussion
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It is well known that the optical transmittance of transparent ceramics is critically dependent on the volume fractions, size distribution and shape of inclusions and pores. However, as mentioned above, even using cutting edge imaging technologies the determination of these microstructural parameters is not trivial, as shown in the few relevant experimental papers on this topic [16– 18]. Moreover, there is no agreement in the literature about the type of mean size that would have to be used for the calculation of transmittance in the case of a polydisperse system of scatterers, when Mie theory is applied. Therefore the prediction of transmittance based on microstructural information from asprepared real-world transparent ceramics is elusive so far, although without doubt the work of Pecharromán et al. [19] gives valuable hints for practical use in the Rayleigh and Rayleigh-Gans-Debye approximation, which have been successfully connected to independently determined microstructural information (pore size distributions obtained via 3D-FIB tomography) by Stuer et al. [16]. In this context we fully agree with the finding of Stuer et al. [16] that pores smaller than 50 nm do not significantly affect the in-line transmittance. We are convinced, however, that the full treatment of the influence of pore size distributions on transmittance via Mie theory requires our concept of generalized mean values [20], when the calculation is to be based on a characteristic size instead of the whole distribution. On the other hand, material and process design requires certain knowledge of the influence of volume fractions, size distribution and shape of inclusions and pores on the transmittance. It is important to know, for example, what concentration or size of inclusions or pores can be tolerated in a 12
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transparent ceramic or whether a finer or coarser raw powder would have to be used to achieve transparency. For this reason theoretical calculations of transmittance by Mie theory are indispensable, and checking the results of these calculations with easily controllable real-world systems (e.g. suspensions) surely adds to their credibility. For this purpose aqueous suspensions can be used, the optical properties of which are in many respects similar to transparent solids with inclusions or pores, can serve as model systems for transparent ceramics, because the same physics apply to both systems and because it is the phase contrast (i.e. the ratio of refractive indices) that determines the transmittance in both cases. Suspensions can be readily and accurately prepared with solid particles of a well-defined shape, size distribution and volume fraction, including very small volume fractions, whereas the determination of these quantities in transparent ceramics is a highly non-trivial task. However, before attacking the intricate problem of systems with polydisperse size distributions of real-shaped scatterers, it was necessary to obtain benchmark results for the case of monodisperse spheres. This was the purpose of the present paper. In order to allow for a comparison of theoretical predictions and experimental results, silica monospheres were used. Although not perfectly monodisperse and even though a small degree of agglomeration could probably not be completely excluded, this particle system comes indeed very close to the realization of a monodisperse system. The results of the Mie calculations in this paper were presented in the form of 3D graphs, which we consider to be an essential step towards a future deeper understanding of the scattering of polydisperse systems and transmittance in realistic situations that may range from the ultraviolet to the infrared. 3D graphs naturally contain more information than the commonly presented transmittance spectra (commonly presented in experimental papers) or scatterer size dependences of transmittance for a given wavelength (commonly presented in numerical papers), and are harder to capture for the reader. Therefore 2D sections and projections have been used in this paper in order to facilitate the interpretation of the 3D plots and to emphasize some of their specific features. For the suspensions the predicted and measured transmittance spectra are in reasonably good agreement, which gives credibility also to the results for the transparent solids, for which microstructural input data are not available. Of course, detailed in-depth comparison of the results indicates a slight discrepancy between measured and predicted data, probably due to the presence of clusters in the suspensions, but this effect is significant only in suspensions with higher concentrations of scattering particles and for short wavelengths. Multiple scattering does not occur to any significant degree (as was demonstrated by fitting the concentration dependence of transmittance via simple exponentials). In conclusion, a concise overview of the light scattering in (nonabsorbing) heterogeneous systems has been given, in-line transmittances have been calculated via Mie theory and for suspensions of silica monospheres compared with experimental results (spectrophotometric measurements). The procedure may be used as a guideline for investigating light scattering in other heterogeneous systems, including transparent ceramics in all cases where the aforementioned microstructural parameters are available, at least the volume 13
fraction. Three-dimensional graphs, which include not only the wavelength dependence (spectrum) but also the scatterer size dependence of transmittance, seem to be indispensable tools for understanding the transmittance of polydisperse heterogeneous systems. 6. Conclusion and outlook
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Declaration of interests
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Predicting the in-line transmittance of transparent ceramics via model calculations is a useful guide for materials design and optimization of preparation process. However, the absence of reliable input information (volume fractions and size) usually precludes the direct verification of these calculations. On the other hand, suspensions, which can be prepared with well controlled volume fractions of the solids selected, may serve as model systems that are amenable to verification. It is clear, however, that even in this case both the calculation and the verification of transmittance predictions are prone to many pitfalls. In particular, before attacking the intricate problem of systems with broad (polydisperse) size distributions, it is necessary to obtain benchmark results for the case of monodisperse spheres, a problem for which Mie theory provides a numerical solution. Indeed, the simplest thinkable case is a monodisperse system of spherical particles, which can be realized in a very good approximation with so-called silica monospheres, i.e. a nearly monodisperse system of nearly spherical spheres of amorphous silica. The results of the Mie calculations in this paper were presented in the form of 3D graphs, which is an essential step towards a future deeper understanding of the scattering of polydisperse systems and transmittance in realistic situations that may range from the ultraviolet to the infrared. For the suspensions the predicted and measured transmittance spectra are in reasonably good agreement, which gives credibility also to the results for the transparent solids, for which microstructural input data are not available.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Acknowledgement: This work is part of the project “Partially and fully sintered ceramics - processing, microstructure, properties, modeling and sintering theory” (GA18-17899S), supported by the Czech Science Foundation (GAČR). References
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