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OPTICAL MATERIAL SCIENCE AND TECHNOLOGY Features of the anomalous scattering of light in two-phase sodium borosilicate glass M. P. Shepilov,a) O. S. Dymshits, and A. A. Zhilin OAO Scientific Research and Technological Institute of Optical Material Science, S. I. Vavilov State Optical Institute All-Russia Science Center, St. Petersburg, Russia

A. E. Kalmykov A. F. Ioffe Physics-and-Engineering Institute, Russian Academy of Sciences, St. Petersburg, Russia

G. A. Sycheva I. V. Grebenshchikov Institute of Silicate Chemistry, Russian Academy of Sciences, St. Petersburg, Russia

(Submitted June 4, 2013) Opticheskiı˘ Zhurnal 80, 83–93 (November 2013) This paper is devoted to the study of light scattering in inhomogeneous glasses. The spectral behavior of the extinction coefficient of the initial sodium borosilicate glass and two glasses obtained by heat treatment is experimentally studied. It is established that the extinction coefficient of heat-treated glasses in the visible region is determined by the scattering of light. Experimental data on the anomalous spectral dependence of the extinction coefficient of the glass subjected to more prolonged heat treatment are compared with the results of calculations carried out in terms of various scattering models based on literature data on the structure of this glass, which consists of liquation spherical particles in a matrix. It is shown that the ordering effects observed earlier in the relative position of the particles play an important role in the light scattering. It is pointed out that, to theoretically describe the scattering properties of a system of polydisperse particles, it is necessary to know the pairwise correlation function of the particles, which depends on the size of the particles in the pair. © 2014 Optical Society of America. OCIS codes: (290.0290) Scattering; (290.2200) Extinction; (290.5820) Scattering measurements; (290.5850) Scattering, particles; (290.7050) Turbid media. http://dx.doi.org/10.1364/JOT.80.000706

INTRODUCTION

The study of the scattering of light in glasses and in other optical materials is an important task of optical material science. If the optical material is intended for the fabrication of transparent optical elements, light scattering is a negative factor and must be minimized. In other cases (the fabrication of diffuse-reflection standards, diffuse reflectors, etc.), the scattering is a characteristic property that determines the use of the material. Data on light scattering can be useful in discussing the question of the structure of the scattering material. The cause of light scattering is the inhomogeneity of the material’s structure. Typical representatives of inhomogeneous glasses are liquated (phase-separated) glasses. Liquated glasses are characterized by anomalous light scattering, the study of which has a long history.1 We should point out that the early studies, begun at the S. I. Vavilov State Optical Institute by D. I. Levin2 and M. M. Gurevich3 and continued by A. I. Kolyadin4 and N. A. Voı˘shvillo,5,6 were carried out on silicon borosilicate glasses. 706

J. Opt. Technol. 80 (11), November 2013

The phenomenon of anomalous scattering consists of predominant backscattering (i.e., at angles that exceed 90°) and of a spectral dependence of the scattering coefficient on wavelength of the type λ−p , where p > 4,1 whereas, in the case of Rayleigh scattering (incoherent scattering by particles significantly smaller than the wavelength and a small relative refractive index), p 4, and the scattering index is symmetric relative to the scattering angle θ 90°.7 The anomalous scattering by glass undergoing liquation of nucleation type (binodal phase separation) and consisting of particles of a separated phase in a matrix, was interpreted as the result of the interference of waves scattered by different particles (interparticle interference).8 This interpretation was based on an idea expressed earlier by A. I. Kolyadin.4 Later research concerned the applicability of such an approach, called the interference approximation, to the calculation of the extinction (scattering) index of a turbid medium consisting of a system of particles in a homogeneous matrix.9 Homogeneous monodisperse spherical particles, identical to

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© 2014 Optical Society of America

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each other, were considered, and the pairwise correlation function (PCF), calculated in the Percus–Yevick approximation,10 was used to describe their relative spatial position. The use of the PCF makes it possible to take into account correlation in the mutual position of the particles. Dik and Ivanov9 found the boundaries of the applicability of the interference approximation in calculating the extinction coefficient of the medium and showed that, when the applicability conditions are met, the use of the interference approximation gives results that agree well with those of more rigorous calculations and with the experimental data obtained for a suspension of latex particles in water. In what follows, we restrict ourselves to a consideration of situations in which the absorption of light can be neglected in comparison with scattering. The light attenuation is determined in these cases by the scattering, and the extinction coefficient is equal to the scattering coefficient. We shall therefore speak of extinction coefficient ε, which can be determined from experiments on transmission. The interference approximation was used in Refs. 11 and 12 in the theoretical discussion of the question of the extinction coefficient of a system of identical spherical particles when the scattering of an individual particle is by Rayleigh scattering. It was shown that the fact that the spectral behavior of the system’s extinction coefficient differs from Rayleigh scattering (ε ∝ λ−4 ) may be associated with the character of the spectral behavior of the structure factor, which is determined by the PCF of the particles—i.e., by the structure of the system. It was pointed out that the structure factor used in Ref. 8 is based on a PCF that contradicts the experimental data. Several examples of systems with different PCFs were considered. It was shown that the use of the Debye PCF in the calculations,13 which describes a random position of mutually nonpenetrating spherical particles in the limit of small volume fraction, does not give appreciable differences of the behavior of the extinction coefficient ε ∝ λ−p from Rayleigh behavior (p ≈ 4). When the Percus–Yevick PCF is used,10 small differences from the Rayleigh behavior can occur in the spectral behavior of the extinction coefficient—for example, in one of the versions discussed there, a value of p 4.55 was obtained. In other words, there are signs of anomalous scattering behavior in this case. Finally, the use in the calculations of the PCF obtained for sodium borosilicate glass on the basis of the experimental data of Refs. 14 and 15, which provides evidence of the presence of substantial correlations (ordering elements) in the placement of the particles, resulted in a wavelength dependence of the calculated extinction coefficient of the form ε ∝ λ−p , where p 5.5 6.0; i.e., it resulted in pronounced anomalous behavior. Thus, the value of p in the examples considered in Refs. 11 and 12 falls into the interval from 4 to 6. The paper raises questions the answers to which would make it possible to explain why the values of p obtained in the calculation do not reach the upper limit of pmax ≈ 8 9 of the values recorded experimentally for scattering in sodium borosilicate glasses.3,5 It follows from the formulas in Refs. 11 and 12 that the scattering coefficient that takes into account interference effects is less than the coefficient for independent Rayleigh 707


scatterers, even when its spectral dependence is close to Rayleigh scattering.16 A theoretical treatment of how the polydisperse nature of a system of Rayleigh spherical particles affects the extinction coefficient was carried out in terms of the interference approximation.17 On the assumption that the PCF of the particles is independent of their size, it was shown that the interference effects are reduced, and the scattering is increased as the degree of polydisperseness of the particles increases. A large amount of experimental material (see the review in Ref. 1) was accumulated in years of intense study of the anomalous scattering of light in glasses. However, it is impossible from our viewpoint to unambiguously interpret the experimental data, since the scattering data were obtained for glasses whose structure was not experimentally studied. Data on anomalous scattering in glasses are discussed for the first time in this paper on the basis of results of a detailed experimental study of the structure of these glasses. EXPERIMENTAL TECHNIQUE

The initial G0 glass, of composition 13.9Na2 O · 36.0B2 O3 · 50.1SiO2 (mol. % by analysis), was prepared from a melt with volume 180 L at a temperature of 1250°C, cooling it for 70 h to 500°C, and holding for 8 h at that temperature, after which it was allowed to cool to room temperature at a rate of 4°C∕h. The samples of the initial glass were heat-treated at 610°C for 5 h (G1 glass) and 10 h (G2 glass). The structure of the G1 and G2 glasses was studied by the methods of stereology on the basis of data obtained by mathematically processing electron-microscope images of carbon replicas of microsections of the samples. The details of obtaining the replicas and their images, the image-processing procedure, the stereological methods used, and the results of the structural study of the glasses are described in detail in Ref. 18. The results of the structural study will be used below to the extent necessary. Here we mention two qualitative conclusions concerning the structure of the glasses that were studied. First, the structure of each of the two glasses is a system of polydisperse spherical particles in a matrix. Second, for each of the glasses, there are ordering effects in the relative position of the particles. As far as we know, ordering effects in the relative position of the particles in the glasses, based on a direct (stereological) study of the structure, were first derived in Ref. 14 and were confirmed in Ref. 15 for G2 glass. The results of an investigation of the ordering effects in G1 and G2 glasses are presented in Refs. 19 and 20. The transmission spectra of the glass samples were measured on a Shimadzu UV 3600 spectrophotometer in the wavelength range 300–1200 nm. The samples were polished plane-parallel plates. Two samples each of different thicknesses were fabricated for the initial and heat-treated glasses: 1.01 and 20.00 mm (G0), 0.99 and 15.97 mm (G1), and 0.48 and 10.00 mm (G2). After the transmittance of the samples of two thicknesses was measured at a given wavelength, it was possible to compute the extinction coefficient ε and refractive index (RI) n for this wavelength using Eq. (2.76) from Ref. 21. Shepilov et al.

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EXPERIMENTAL RESULTS AND DISCUSSION

The initial G0 glass is virtually transparent in the wavelength region λ > 400 nm. The extinction coefficient is estimated as ε ≈ 0.015 cm−1 at λ 400 nm and decreases with increasing wavelength (ϵ ≈ 0.005 cm−1 at λ 550 nm). Because of the large relative error in measuring small optical densities, the wavelength dependence of the extinction coefficient obtained from the results of one measurement for each thickness is noticeably noisy. For example, with a measurement step of 2 nm, the dependence is nonmonotonic and becomes monotonic only with a step of ≈20 nm. The approximation of this dependence, averaged over several opticaldensity measurements by the expression ελ aλ−p

(1)

in the wavelength region 420–540 nm, results in an estimate of the exponent of p 4.1 0.4;

(2)

which is close to the value of p 4 for Rayleigh scattering. The large error in determining p is associated not only with the large relative error in measuring small optical densities but also with the inhomogeneity of the glass samples. Since there are no data on the presence of inhomogeneities in G0 glass, it is not possible to discuss the extinction behavior. Glass G1, which contains spherical particles having a mean radius of 20.7 nm and volume fraction 0.047,18 scatters light a factor of 4–5 more strongly than the initial G0 glass. The extinction coefficient of G1 glass is ε ≈ 0.08 cm−1 at λ 400 nm and decreases with increasing wavelength to a value of ε ≈ 0.019 cm−1 at λ 550 nm. The approximation of the ελ dependence by Eq. (1) in the wavelength interval 395–535 nm gives p 4.8 0.2;

(3)

which appreciably exceeds the value given by Eq. (2), obtained for G0 glass. Thus the scattering of light in G1 glass has an anomalous character. This conclusion qualitatively agrees with that concerning the presence of ordering elements

in the relative position of the scattering particles in G1 glass,18 and this must involve anomalous light scattering.11,12 A quantitative interpretation of the ελ dependence for G1 glass is not currently possible, since this would require data on the structure factor of the system of particles in G1 glass, which do not appear in the literature. Glass G2 contains spherical particles with a mean radius of 33.8 nm and volume fraction v 0.065.18 The optical densities of the thin sample (thickness h1 0.48 mm) and the thick sample (h2 10.00 mm) of G2 glass are shown in Fig. 1. The results of measurements obtained from different parts of it (three spectra for the thin sample and six for the thick sample) are shown in Fig. 1(a) for each sample, while the averaged results are shown in Fig. 1(b). The averaged results were used to calculate the spectral dependence of the extinction coefficient shown in Fig. 2(a) on a log–log scale, and the spectral dependence of the RI [Fig. 2(b)] of G2 glass. The extinction coefficient is estimated as ε ≈ 0.25, 0.029, and 0.005 cm−1 at λ 400, 550, and 700 nm, respectively. The extinction coefficient of G2 glass at λ 400 nm is a factor of 17 greater than in G0 glass and a factor of 3 larger than in G1 glass, with this difference decreasing as the wavelength increases. The increase of the extinction coefficient as one goes from G1 glass to G2 glass is associated with the increase of the light scattering, and this corresponds to the observed strengthening of the opalescence and is explained by the increase of the size and volume fraction of the particles. When λ > 700 nm, this ελ dependence strongly fluctuates, and this is associated with the large relative measurement error of small optical densities. In the wavelength interval 360–700 nm, the ελ dependence plotted on a log–log scale [Fig. 2(a)] is approximated by a straight line with slope p 6.9 0.1.

(4)

Light scattering in G2 glass thus has a clearly expressed anomalous character. The value of p is significantly greater for G2 glass than for G1 glass, and this corresponds to the fact that the ordering effects of the particles are expressed more strongly in G2 glass than in G1 glass.18 The RI is more sensitive to small measurement errors of the optical density, and this results in appreciable fluctuations

(a)

(b)

nm

nm

FIG. 1. Experimental data on the optical density D of a thin sample (thickness h1 0.48 mm) (1) and a thick sample (h2 10.00 nm) (2) of G2 glass. The results of measurements taken from its different parts (a) and the averaged results (b) are presented for each sample. 708


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cm–1

(b)

(a)

nm

nm

FIG. 2. Extinction coefficient ελ imaged on a log–log scale (a, curve 1) and RI nλ (b, curve 1) for G2 glass. The linear section of the log ε versus log λ dependence is approximated by a straight line with slope p ≈ 6.9 (a, curve 2), while the nλ dependence is represented by a fourth-order polynomial (b, curve 2).

of the wavelength dependence of the RI in the entire wavelength interval studied here [Fig. 2(b), curve 1]. We shall ignore these fluctuations in what follows, approximating the experimental dependence with a smooth nλ curve in the 360–700 nm wavelength interval [Fig. 2(b), curve 2]. The resulting RI values can be compared with the values shown in the literature (Ref. 22, pp. 277–84) for homogeneous glasses of similar compositions. As a rule, Ref. 22 shows RI values nD at the wavelength 589.29 nm and values of Δ nF − nC × 105 (where nF and nC are the RIs at wavelengths 479.99 and 656.27 nm, respectively), which characterize the mean RI dispersion (nF − nC ). For example, for the glass 13.3Na2 O · 33.3B2 O3 · 53.4SiO2 (mol. % as synthesized), nD 1.5014, Δ 770 and, for the glass 15Na2 O · 40B2 O3 · 45SiO2 (mol. % as synthesized), nD 1.505, Δ 783 (Ref. 22, p. 284). For G2 glass, the RI nD ≈ 1.482 is somewhat lower, while the dispersion Δ ≈ 2400 is significantly greater than in the homogeneous glasses mentioned above. The observed RI of G2 glass cannot be explained on the basis of simple considerations. Actually, two-phase G2 glass consists of particles enriched with silicon dioxide,18 and accordingly of a matrix depleted from silicon dioxide with respect to the original glass; this depletion is insignificant because of the smallness of the volume fraction of particles v 0.065. It can be concluded on the basis of handbook data (Ref. 22, p. 277–92), that the RI of the particles enriched with silicon dioxide cannot be less than that of quartz glass nD 1.458, whereas the RI of a matrix close in composition to the initial glass is estimated as nD > 1.5. Using these RI values of the matrix and of particles with a volume fraction of v 0.065 in the Maxwell–Garnett formula [Ref. 21, Eq. (8.50)] or in the Bruggeman formula [Ref. 21, Eq. (8.51)], it is possible to obtain an estimate of the RI of the two-phase G2 glass of nD > 1.497, which is appreciably greater than the experimental value of nD ≈ 1.482. The difference of the RI of the matrix from the handbook RI value for the glass of the same composition should be regarded as the most probable cause of such a discrepancy; this may be caused by the difference in the thermal history and by the presence of stresses in the glass of the matrix. In the subsequent calculations, we shall use the nλ dependence obtained in the approximation of the experimental data that we obtained from the smooth curve. 709


Let us proceed to a comparison of the experimental ελ dependence obtained for G2 glass with the results of calculations carried out in terms of one model or another of light scattering by a substance using the available data concerning the structure of G2 glass. In all versions of the calculation, nλ will be used as the RI of the matrix, nm . This is a good approximation, because the volume fraction of particles is small, whereas their RI np , as will be shown below, is close to the RI of the matrix. At the same time, the exact value of the ratio of the RIs of the particles and the matrix, m np ∕nm ≈ np ∕nλ [or of the difference Δn np − nm ≈ np − nλ] is unknown. We shall choose a value of m (or Δn) individually for each of the models, starting from the fact that the extinction coefficient calculated at a wavelength of 450 nm for the value nm n450 nm 1.5094 coincides with the experimental ε450 nm 0.115 cm−1 , and we assume that the value of m (or Δn) is independent of wavelength. We should point out that such a method of determining m (or Δn) does not make it possible to answer the question of whether the value of m is larger or smaller than unity (or whether Δn is positive or negative). Without knowing the exact phase compositions, it is impossible to answer that question on the basis of handbook data,22 the more so that the RI of the matrix differs from the handbook value, as mentioned in the preceding paragraph. Nevertheless, the handbook data22 show that, for glasses of similar compositions, the RI decreases with increasing silica concentration (see, for example, the RI given two paragraphs previously for the two compositions). If so, the RI of the particles must be less than that of the matrix—i.e., Δn < 0;

m 0, m > 1) does not change the conclusions made in this paper). Let us begin with the simplest case—a model of light scattering by a system of independent polydisperse scatterers in the Rayleigh approximation (model 1). In this case, the extinction coefficient determined by the scattering has the form (see, for example, Ref. 17) Shepilov et al.

709

εex;1 λ

27 π 5 m21 − 1 2 nλ 4 N hR6 i; 3 λ m21 2

(6)

where N is the number of particles in unit volume, hR6 i is the mean value of the sixth power of the radius of the particles, and subscript 1 denotes model 1. It follows from the results of a structural study of G2 glass18 that N 3.8 · 1014 cm−3 ; hR i 3

1.096 R3a ;

Ra 33.8 nm;

hR i 1.518 R6a ; 6

(7)

which also gives the mean value of the cube of the radius of the particles hR3 i, used below, and the quantities hR3 i and hR6 i are expressed in terms of the mean radius Ra of the particles when the probability density of the distribution of the particle radii is used. By varying m1 , it is possible to fit the theoretical value given by Eq. (6) to the experimental extinction coefficient for λ 450 nm and then to construct the theoretical εex;1 λ dependence with the value found for m1 const (model 1, version I). The εex;1 λ dependence thus obtained is well approximated by Eq. (1) in the wavelength interval 360–700 nm (Fig. 3, curve 1), while the values found for the parameters are m1 0.99572;

p1;I 4.28.

(8)

The difference of the value of parameter p from the Rayleigh value p 4 and the insignificant nonlinearity of the εex;1 λ dependence plotted on a log–log scale is associated with the presence of nλ dispersion. A second version of the calculation (II) was carried out for the constant value Δn1 −0.00646, which ensures that the theory and experiment will coincide at 450 nm [Δn1 m1 − 1n450 nm]. In the second version, p1;II 4.14. It should be pointed out that the particles are not small enough to justify using the Rayleigh approximation. The application of Mie theory to independent polydisperse scatterers [Ref. 23, Eq. (80)] reduced the values of p given above by ≈0.4. The values of p obtained by using the model of independent Rayleigh scatterers are significantly less than those observed experimentally. This can be expected, since substantial correlations (ordering elements) in the relative position of

. cm–1

.

nm FIG. 3. Results of calculations of the extinction coefficient in models 1–4 (curves 1–4) in comparison with the experimental results (curve 5) for G2 glass on a log–log scale. 710


the particles are observed for G2 glass14,18 that are neglected in the independent-scatterer model. As pointed out in the introduction, the correlations in the position of the particles when the extinction coefficient is calculated are taken into account by using the PCF in the interference approximation.9 For a system of identical spherical Rayleigh particles, the interference approximation results in the expression11,12 (model 2) 2 m − 1 2 nλ 4 ¯ Sλ: (9) εex;2 λ 4π 4 vD3 22 λ m2 2 Here subscript 2 denotes model 2, v and D are the volume fraction and diameter of the particles, and Z π ¯Sλ 3 dθ sin θ1 cos2 θS4πnλD sinθ∕2∕λ 8 0 Z π dθ sin θ1 cos2 θS4πnλD sinθ∕2∕λ∕ 0 Z π dθ sin θ1 cos2 θ; (10) 0

where the integration is carried out over the scattering angle θ, while the structure factor of the system of particles is Z ∞ sin μρ 2 ρ dρ; gρ − 1 Sμ 1 24ν μρ 0 ρ ≡ r∕D;

(11)

as a function of the modulus of the dimensionless scattering vector μ is expressed in terms of the PCF gρ of the particles, which depends on the dimensionless distance ρ between the centers of the particles. As pointed out in Ref. 17, quantity Sλ has the meaning of the structure factor averaged over scattering angle θ with weighting factor (1 cos2 θ), which corresponds to the scattering index of natural light (or the scattering index of plane-polarized light averaged over angle φ). For randomly placed particles, gρ 1, Sμ 1, and Eq. (9) after integration reduces to an expression that characterizes Rayleigh scattering [see, for example, Eq. (6) for the particular case of a monodisperse system—i.e., when hR6 i D∕26 ]. The PCF gρ for G2 glass, calculated by the stereological method in the approximation of a monodisperse system and based on electron-microscope data obtained from a flat section of the sample, was first presented in Ref. 14 and was discussed in more detail in Ref. 18. This PCF substantially differs from unity and demonstrates ordering elements in the relative position of the particles. The structure factor Sμ based on it for G2 glass is shown in Refs. 11 and 12 and is reproduced in Fig. 4. Using this structure factor and the experimental values18 of the volume fraction v 0.065 and using the mean particle diameter 2Ra 67.6 nm from Eq. (7) as diameter D, we varied m2 in order to fit the theoretical value of Eq. (9) to the experimental extinction coefficient for λ 450 nm and then constructed the theoretical dependence εex;2 λ with the resulting value of m2 const (Fig. 3, curve 2). The resulting εex;2 λ dependence is well approximated by Eq. (1) in the wavelength interval 360–700 nm, and the values found for the parameters are Shepilov et al.

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of m3 and p3;I , obtained as a result of approximating curve 3 by a straight line using the method of least squares in the wavelength interval 360–700 nm, are m3 0.99204;

FIG. 4. Structure factor Sμ from Eq. (11), calculated for a system of particles in G2 glass in the approximation of a monodisperse system (according to Ref. 11) and used in calculating the extinction coefficient in models 2–4.

m2 0.99306;

p2;I 5.78.

(12)

The second version of the calculation for model 2 was carried out for the constant value Δn2 −0.01048, which ensures that theory and experiment agree at a wavelength of 450 nm [Δn2 m2 − 1n450 nm]. In the second version, p2;II 5.64. As already pointed out, the individual scattering of light by particles of the sizes under consideration as given by Eqs. (7) can be considered Rayleigh scattering only in first approximation. The use of Mie theory and the interference approximation in calculating the extinction coefficient of an ensemble of monodisperse spherical particles (model 3) results in the formula9,1) εex;3 λ

Z π 3ν Qex − Qsc dθ sin θpθ 2D 0 × 1 − S4πnλD sin θ∕2∕λ ;

(13)

where Qex , Qsc , and pθ are the attenuation efficiency factor, the R scattering efficiency factor, and the scattering index [ 0π dθ sin θpθ 1] of a separate particle.21 In the case of nonabsorbing particles that we are considering, the attenuation and scattering efficiency factors are equal, while their value is determined by the particle diameter, the RI of the matrix, and the RI of the particles (or by the relative RI), Qex D; nλ; m3 Qsc D; nλ; m3 :

(14)

Based on Eqs. (13) and (14), calculations were carried out similar to those that were done using Eq. (9). The εex;3 λ dependence obtained in the first version of the calculation is represented by curve 3 in Fig. 3. Curve 3 differs from a straight line more noticeably than do curves 1, 2, and 4. Namely, the absolute value of the slope of curve 3 decreases from p 5.43 at λ 680 nm to p 4.70 at λ 380 nm. The slope decreases because, as the wavelength decreases, there is a transition from the Rayleigh-scattering regime to Mie scattering, which is characterized by a lower value of p (see, for example, the “Introduction” in Ref. 11). The values 711


p3;I 5.33.

(15)

In the second version of the calculation using Eqs. (13) and (14), Δn3 −0.01201 and p3;II 5.19. A system of monodisperse particles is considered in the versions of the interference approximation used above [Eqs. (9) and (13)]. At the same time, the system of particles in G2 glass is substantially polydisperse.18 For example, the standard deviation of the particle radius from the mean value, given by Eq. (7), is 6 nm, i.e., 18% (Ref. 18, Table 1). The interference approximation was used in Ref. 17 to calculate the extinction coefficient of a system of polydisperse Rayleigh spherical particles. On the assumption that the PCF gr is independent of the size of the particles in a pair, the following formula was obtained (model 4):

m24 − 1 2 nλ 4 λ m24 2 6 hR i ¯ × vhR3 i Sλ − 1 ; hR3 i2

εex;4 λ 4π 4

(16)

where the averaged structure factor Sλ is determined by Eqs. (10) and (11). hR3 i and hR6 i are the mean values of the cube and the sixth power of the radius of the particles. The averaged structure factor used in model 2 and calculated on the basis of the PCF found for G2 glass by the stereological method in the approximation of a monodisperse system was used as Sλ in model 4 and is independent of the particle size. The values of hR3 i and hR6 i for a system of particles in G2 glass are given by Eqs. (7). On this basis, calculations for model 4 were carried out using Eq. (16), similar to those that were carried out using Eq. (9). The calculation of the εex;4 λ dependence obtained in the first version (curve 4 in Fig. 3) is well approximated by Eq. (1) in the wavelength interval 360–700 nm, while the values found for the parameters are m4 0.99455;

p4;I 5.22.

(17)

In the second version of the calculation using Eq. (16), Δn4 −0.00822 and p4;II 5.08. We should point out that it does not seem possible at present to calculate the extinction coefficient of a polydisperse system in the interference approximation using Mie theory, because the necessary theoretical basis is lacking. The results of the calculation of the spectral dependence of the extinction coefficient (Fig. 3, curves 1–4) can be compared with the experimental results (Fig. 3, curve 5). In accordance with the procedure for choosing the relative RI value mi in model i, all the dependences intersect at one point—at a wavelength of 450 nm. At the same time, each model is characterized by its own type of spectral dependence of the extinction coefficient (in terms of Fig. 3, by its own slope p). The model of independent scatterers (model 1, curve 1) demonstrates the greatest differences from the experimental results. Shepilov et al.

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Taking interference effects in the scattering into account (models 2–4, curves 2–4) substantially improves the agreement of theory and experiment. We should point out that the use of rigorous Mie theory (model 3, curve 3 in Fig. 3) results in the somewhat smaller value of p from Eq. (15) than does the use of the Rayleigh approximation [model 2, curve 2 in Fig. 3, Eq. (12)]. This fact is especially important if it is impossible or difficult to use Mie theory for one reason or another. A comparison of the results of the calculations carried out in the interference approximation for systems of monodisperse [model 2, curve 2 in Fig. 3, Eq. (12)] and polydisperse [model 4, curve 4 in Fig. 3, Eq. (17)] Rayleigh particles shows that taking polydisperseness into account reduces the value of p—i.e., it decreases the interference effects in the scattering. The agreement of the calculated results with experiment is degraded in this case. It should be kept in mind that the conclusion that the interference effects decrease when polydisperseness is taken into account is not unconditional. In fact, this conclusion is based on assumption made in model 4 that the PCF of a polydisperse system of particles is independent of their sizes. Such an assumption is not justified for a system of particles formed as a result of nucleation phase separation. Actually, the particles “collect” the material by depleting the surrounding matrix of the segregated component; therefore, it is significantly less probable that two large particles will lie a short distance apart than that small particles will be closely spaced. In other words, the PCF must depend on the size of the particles in a pair. The particle-size-dependent PCF of a polydisperse system in principle can be found by the stereological method on the basis of data obtained from a flat cross section of a sample by solving the integral equation derived in Ref. 25. However, as pointed out in Ref. 25, there are two reasons that it is currently impossible to put such a procedure into practice. First, this would require a method to be developed for numerically solving an integral equation. Second, a significantly greater volume of experimental data is needed in this case than in the case of a size-independent PCF of a system of monodisperse particles. Since it is not realistic at present to experimentally determine the particle-size-dependent PCF of a polydisperse system, there is only one way to theoretically analyze how the polydisperseness of the particles affects the extinction coefficient—to use the interference approximation with some model PCFs or the other, dependent on the size of the particles in a pair. To carry out this procedure, it is necessary to develop new models or to elucidate models of nucleation phase separation existing in the literature that can be used to calculate the size-dependent PCFs of polydisperse particles, to calculate these PCFs, and to use them in implementing the interference approximation for a polydisperse system.17 It is desirable in this case to start not only from Rayleigh scattering, as was done in Ref. 17, but also from Mie scattering. If one speaks of models of nucleation phase separation represented in the literature that make it possible to calculate PCFs dependent on the size of the particles in a pair, one can refer to models 712


with instantaneous26 and continuous27 nucleation of the particles of a new phase. Thus, the question remains open of how the degree of polydisperseness of the particles affects the extinction coefficient of a material in the presence of interference effects. As pointed out above, taking interference effects into account in scattering (models 2–4, curves 2–4 in Fig. 3) substantially improves the agreement of theory and experiment. Nevertheless, there is an appreciable difference in the spectral dependence of the calculated and experimental extinction coefficient. Possible causes of such a discrepancy can include: (1) inadequately taking into account the polydisperseness of the particles, considered above; (2) the spectral dependence of the relative RI of the particles m, which was assumed in the calculations to be wavelength independent; and (3) the presence of concentration gradients and accordingly of RI gradients around the particles in the G2 glass, for which a recondensation stage (Ostwald ripening) occurs.18 CONCLUSION

The spectral behavior of the extinction coefficient of the original glass of composition 13.9Na2 O · 36.0B2 O3 · 50.1SiO2 (mol. %) (G0 glass) and glasses obtained by heattreating the original glass at 610°C for 5 h (G1 glass) and 10 h (G2 glass) has been experimentally studied. The original G0 glass is virtually transparent in the visible region, and its extinction coefficient is characterized by a power dependence on the inverse wavelength with an exponent p close to 4. This is characteristic of losses due to Rayleigh scattering. A comparison of the results obtained for glasses G1 and G2 shows that the extinction coefficient in the visible region increases as the heat-treatment time increases, while anomalies in its spectral dependence grow stronger: p ≈ 4.8 for G1 glass and p ≈ 6.9 for G2 glass. Such behavior of the extinction coefficient is associated with light scattering and qualitatively corresponds to the results presented in the literature for experimental studies of the structure of G1 and G2 glasses. According to these studies, the structure of both glasses is represented by spherical particles of a new phase in the matrix (binodaltype liquation structures). The simplicity of the structure of these glasses makes them model objects for studying the connection of the structure and the light scattering. The volume fraction and mean particle radius are greater in G2 glass than in G1 glass, and therefore the extinction coefficient is greater in G2 glass. The ordering effects experimentally observed earlier in the relative position of the particles are greater for G2 glass than for G1 glass, and accordingly the interference effects in scattering and the deviation of p from the value of 4 are greater for G2 glass (the value of 4 occurs when there is random position of Rayleigh scatterers). An attempt to qualitatively explain the spectral behavior of the extinction coefficient was made for G2 glass, for which the structure factor needed in such calculations for a system of particles is presented in the literature. The calculations showed that the independent-scatterer model cannot describe the observed spectral behavior of the extinction coefficient, whereas Shepilov et al.

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taking into account the interference effects in the scattering, caused by the ordering elements in the relative position of the particles, substantially improves the agreement of theory and experiment. It was pointed out that the question of how the degree of polydisperseness of the particles affects the extinction coefficient of the material in the presence of interference effects remains open. To solve this question, it is necessary to know the pairwise correlation function of the particles, which depends on the size of the particles in a pair. An attempt was made for the first time in this paper to explain the anomalous scattering of light in glass, based on quantitative data on the actual structure of the material. ACKNOWLEDGMENTS

This work was carried out with the support of the Russian Foundation for Basic Research (Grant No. 13-03-01289-a). a)

Email: [email protected] We should point out that this formula in Ref. 9 contains an obvious misprint: Instead of a plus sign in front of the term that contains the integral over the scattering angle, there should be a minus sign [see, for example, Eq. (3) in the later paper24 by the same author].

1)

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