1. INTRODUCTION

Porous Vycor glass with nanopores is transparent in the visible region and is used in colorimetric chemical gas sensing when impregnated with selectively reacting regents [1–4]. The pores in Vycor have an average diameter of about 4.2 nm and fill about 0.3 of the total volume of the material. Vycor’s large internal surface area of about $200{\mathrm{m}}^2/\mathrm{g}$ and its transparency are very attractive for gas sensing by adsorption. This is because the larger the area the higher the adsorption, resulting in higher sensitivity. In addition, optical methods [5] have an advantage over conventional ones in that they can distinguish gas species by chemically detecting the inherent optical absorption.

However, they have some disadvantages in sensing, since it turns out that changes in the humidity of ambient air strongly affect the transmittance of porous glass-based sensor elements [6,7]. In the transmittance measurement, we measure the attenuation or extinction of the light emerging from the sensor element impregnated with a proper chemical reagent. Unfortunately, this extinction is the result of both absorption and scattering. The cumulative absorbance changes at a specified wavelength in the visible region certainly originate from the absorption due to selective reactions of impregnated reagent with the target gas. But the scattering is originated from the optical inhomogeneities occurring in the pore space of the porous glass matrix [8]. The transitory turbidity is found to occur even for the elements without reagents and is a problem common to all the elements that use porous glass as a base material. In our previous studies [8–10], we have shown that this transitory whitish turbidity phenomenon can be attributed to Rayleigh scattering caused by the spatial fluctuation (inhomogeneity) of constituent materials within the glass, i.e., the distribution of wetting fluid such as water vapor in the pore space.

Given this situation, there is considerable value in modeling and understanding the cause of turbidity in inhomogeneous nanoporous media such as Vycor. The study of the optical properties of porous glasses, in particular light attenuation in scattering and absorption, is important for correcting the unnecessary contribution from the transitory white turbidity from the porous matrix. A numerical model to account for the turbidity is required to precisely estimate the concentrations from the cumulative absorbance change in the porous glass-based sensor element.

For a long time, porous Vycor glass had been used an intermediate product for obtaining high-silica glass (96–98%) [11], which implies that its silica skeleton is highly pure in composition. Since silica glass is a dielectric insulator, it is transparent in the near ultraviolet and the visible regions because the electrons of the atomic shells are bound to the individual atom. Its optical properties are determined by its complex refractive index, $\tilde{n}(\omega )=n(\omega )-ik(\omega )$, where $n$ is the refractive index and $k$ is the extinction coefficient. For wavelengths in the range $3.70\mathrm{\mu m}>\lambda >0.21\mathrm{\mu m}$, the refractive index $n$ of silicon dioxide (${\mathrm{SiO}}_2$) (glass) is almost constant around the value of 1.45, and no $k$ values are given in this wavelength range [12], which implies that no absorption is observed for pure ${\mathrm{SiO}}_2$ glass [12,13]. Its electronic structure [14] indicates that the energy gap of about 8.9 eV between the conduction and valence bands suppresses the electronic excitations by absorbing visible light photons. Thus, the classical electromagnetic theory is sufficient for our purpose.

In general, attenuation (extinction) or power loss, as light travels a distance $d_{\text{opt}}$ through any optical materials, can be measured in terms of turbidity ${\tau }_{\text{ext}}=d_{\text{opt}}^{-1}\ln [I(0)/I(d_{\text{opt}})]$ with the dimension of reciprocal length (usually ${\mathrm{\mu m}}^{-1}$), where $I(0)$ is the incident light intensity and $I(d_{\text{opt}})$ is the intensity of the light beam emerging from the sample. As the sources of attenuation in optically transparent materials, loss mechanisms fall into two categories, one involving the slight absorption of radiation ${\tau }_{\text{abs}}$ (e.g., by reagent molecules in the listed sensor elements with light energy causing electronic transitions in the visible region), and the other mainly concerned with scattering ${\tau }_{\text{sca}}$, in which radiation is diverted (with or without change of frequency) from its primary propagating path. The law of conservation of energy then requires that ${\tau }_{\text{ext}}={\tau }_{\text{abs}}+{\tau }_{\text{sca}}$.

The sources of optical loss in glass are of three types: (i) intrinsic absorption in the ultraviolet region and molecular vibrations in the infrared region [15], (ii) impurities such as transition elements which causes colors [15], and (iii) scattering caused by bubbles, phase separation, and/or density fluctuations [16,17]. Among these, the optical properties of porous glasses in the visible region are mainly determined by scattering [18,19] because they are structurally inhomogeneous due to the presence of the innumerable pores and surrounding silica skeleton, and compositionally inhomogeneous due to the nonuniform distribution of the imbibed or drained wetting fluid such as water in the pore space during imbibition or drainage of fluid.

In this paper, we will treat the optical properties of porous Vycor glass in the view of scattering caused by structural and compositional inhomogeneities, both of which might scatter light strongly.

Structural studies by means of small-angle x-ray and neutron scattering (SAXS and SANS) on Vycor [20–36] have shown that nanoporous Vycor glass has a pronounced inhomogeneous structure. The inhomogeneities of the refractive index observed in nanoporous Vycor glass can be conveniently divided into two classes: Inhomogeneities of type I and those of type II. The former are due to the inhomogeneity of the original leached borosilicate glass structure itself and arise in the process of glass manufacturing—the so-called spinodal phase separation and decomposition process. Several SAXS and SANS measurements [21–23,25,26–36] have clearly shown that the structure of the pore space of Vycor is reminiscent of spinodal decomposition, with a broad peak centered around $q=0.25{\mathrm{nm}}^{-1}$ (where $q$ is the scattering vector) in the structure factor corresponding to a length of about 20–30 nm. This kind of inhomogeneity should be called “physical inhomogeneity,” as distinguished from the type-II inhomogeneity.

Inhomogeneities of type II are due to local deviations from the mean glass composition with long-range correlation in the pore space [8–10,37,38], that is, “chemical inhomogeneity.” The porous matrix medium itself is static and rigid, but the distribution of wetting fluid imbibed into pores randomly varies in the pore space, so that the amplitude and phase of the light waves traveling through the whole medium might fluctuate randomly in space, which, subsequently, would change the transparency of the whole medium. Several SANS studies [27–29] have shown that, upon the filling or draining of pores, incomplete, partially saturated Vycor can exhibit fractal properties via the formation of a fractal-like percolation network of imbibed or drained pores in the pore space. In this sense, to clarify the turbidity problem in nanocomposite materials, it is very important to characterize properties of soft materials in hard confinement, being rather general in the characterization of the nanomaterials.

In our previous studies [8–10], we have shown that the transitory whitish turbidity phenomenon occurs in nanoporous Vycor glass that was partially saturated with a wetting fluid such as water, that is, the light scattering caused by the inhomogeneities of type II. The phenomenon can be consistently interpreted and quantitatively analyzed by a simple Rayleigh scattering model [8–10], which is based on the particle-scattering approach [39]. In our subsequent study [40], we attributed the phenomenon to the fractal-like percolation network of drained pores, on the basis of the theory of dielectric constant fluctuations [41,42] with the previously reported experimental observations [27–29,37,38]. In the study, it was very important to explain the slight deviation of the observed turbidity from the ${\lambda }^{-4}$ Rayleigh wavelength dependence, which cannot be explained at all from the previous particle-scattering viewpoint [39].

This paper is organized as follows. The next section briefly describes the theoretical scheme to analyze the transmittance spectrum in terms of turbidities on the basis of the theory of dielectric constant fluctuations. In Section 2.A, we develop mathematical expressions for a correlation volume with experimentally observed value of the fractal dimension $D=5/2$ (for details for values other than 5/2, see Appendix B). In Section 2.B, the same fluctuation analysis scheme is applied to the case for the structural inhomogeneity caused by the spinodal decomposition. For both cases, we derived the deviation from the ${\lambda }^{-4}$ wavelength dependence of corresponding turbidities explicitly. In the subsequent sections, these derived expressions are compared with the previously obtained experimental results, and then we offer a possible explanation of the transmittance spectra of dried and completely saturated Vycor glass and that of incompletely saturated Vycor glass in terms of the previously mentioned spinodal-decomposed and fractal turbidities, respectively. Finally, Section 5 concludes the paper. The detailed deviation of the basic equations for the turbidity analysis based on the classical electromagnetic theory is available in Appendix A, and the detailed mathematical expressions for correlation volumes with half-integer and integral values of the fractal dimension $D$ are available in Appendix B.

2. THEORETICAL EXPRESSIONS

As is well known, the problem of light scattering can be approached from Einstein’s fluctuation viewpoint [41,42], apart from the previously mentioned particle-scattering one. This approach can treat the problem more systematically and comprehensively by considering the correlation function that characterizes both classes of inhomogeneities. That is, in the fluctuation approach, the inhomogeneity present in a condensed medium is generally characterized by introducing the concept of a spatial correlation function, in which the inhomogeneity of the medium is considered to be due to a continuous variation of the dielectric constant or refractive index rather than to the presence of discrete particles. The spatial correlation function is defined as $\gamma (r)\equiv ⟨{\eta }_A{\eta }_B⟩/{⟨{\eta }^2⟩}_{AV}$, where ${\eta }_A$ and ${\eta }_B$ are the local fluctuations in dielectric constant from some average value ${\epsilon }_{AV}$ at points $A$ and $B$ a distance $r$ apart, and ${⟨{\eta }^2⟩}_{AV}$ is the average value of ${\eta }^2$. The problem of characterizing the inhomogeneities in solids and relating them to the scattered intensity has been treated by Debye et al. [43,44]. Their approach has been utilized both for SAXS and conventional light scattering.

Their mathematical scheme is summarized as follows: for an unpolarized incident beam of light, the scattered intensity for an isotropic system is simply proportional to $\omega (q)$, that is, the Fourier transform of $\gamma (r)$, namely,

In all scattering determinations of the inhomogeneities of Vycor [20–36], Eq. (1) implies that, to obtain the correlation function via Eq. (2), the angular dependence of the scattered intensity [which is proportional to the correlation volume $\omega (q)$] from the sample with a monochromatic incident radiation beam is measured as a function of the scattering vector $q$.

On the other hand, in this paper, we re-examine our transmittance spectrum of nanoporous Vycor glass from the fluctuation viewpoint and show that the photospectroscopically observed forward-scattered intensity from Vycor glass provides the same information as that obtained from the scattering intensity of monochromatic incident light at small angle $\theta$, for which the theoretical scheme was originally proposed by Debye et al. [43,44]. To discuss the slight deviation of the observed turbidity from the ${\lambda }^{-4}$ Rayleigh wavelength dependence, we derive the explicit formula for the turbidity as a function of the incident wavelength corresponding to the assumed correlation functions. In the following section we outline some equations we have developed for analyzing scattering data in terms of fractal scattering and the corresponding turbidity.

A. Fractal Correlation Function and Corresponding Turbidity

As mentioned previously, when porous Vycor glass is partially saturated with a wetting fluid, a fractal-like percolation network might form [10,27–29,37,38], although the pores themselves are not distributed with fractal geometry. As the draining proceeds, the transmittance becomes small due to strong scattering of light. Taking into account both the finite size and the fractal-like percolation network structure of drained pores, Sinha, Freltoft, and Kjems [45–47] proposed the fractal correlation function ${\gamma }_f(r)$ of the form

On the other hand, in the case of a spectrophotometer with a varying wavelength of the incident light, we have to analyze the turbidity of the forward-scattered light to derive any structural information about the wavelength dependence of the transmittance spectra [40]. Unfortunately, there is no closed formula for the integral of the correlation volume with any noninteger value of $D$ in Eq. (5) over the entire solid angle $d\mathrm{\Omega }$. As a consequence, we have to integrate the correlation volume in Eq. (5) only for specified values of $D$, for which the analytical expression of the corresponding turbidity is obtainable.

Several scattering determinations of the inhomogeneities of Vycor during drainage of a wetting fluid [37,38] can be found in the literature. By means of light scattering, which is a definite probe of the long-range spatial correlations of the empty pores, Page et al. [37,38] reported that the draining pores, which are surrounded by filled pores, exhibits long-range correlations of a fractal dimension of about 2.6.

Inserting the correlation function of Eq. (4) for $d=3$ with analytically treatable fractal dimension ($D$) of 2.5 into Eq. (2), the correlation volume of Eq. (5) is given as ${\omega }_f(q)={\omega }_0(5/2)·f_{5/2}(q)$, with a constant ${\omega }_0(5/2)=4\pi {\xi }^{5/2}\mathrm{\Gamma }(5/2)=3{\pi }^{3/2}{\xi }^{5/2}$ and the dissymmetry factor

Substituting this expression into Eq. (3) and integrating over all angles, the corresponding turbidity ${\tau }_f(D=5/2)$ is given as

Equation (7) implies that the wavelength dependence of the turbidity ${\tau }_f$ due to the fractal-like percolation network of drained pores is determined mainly by $1/{\lambda }_0^4$ [which comes from the factor $k^4/(16{\pi }^2{\epsilon }_{AV}^2)$] and the integrated correlation volume ${\omega }_0(5/2)·{\mathrm{\Omega }}_{5/2}$, which is a function of $\xi$ only and determines the slight deviation from the ${\lambda }^{-4}$ Rayleigh wavelength dependence. We hereafer refer to ${\tau }_f$ as “fractal turbidity.”

Figure 1(a) shows the dependence of the anisotropic correlation volume ${\omega }_f(q)$ on the scattering wave vector $q$, illustrating the fractal correlations of half-integer and integer dimensions $D$. For the sake of comparison with the experimental data represented by open circles, which are reproduced from [37,38], the correlation length $\xi$ is temporally set equal to 10 μm. It turns out that the curve with the fractal dimension of 5/2 (solid line) well approximates the slope of the reported data. Figure 1(b) shows that the corresponding integrated dissymmetry factors ${\mathrm{\Omega }}_D$ of the fractal dimension of $D$ are convex-upward functions of $1/{\lambda }_0^4$. Here, the correlation length $\xi$ is set to about 18.5 nm. This value of $\xi$ is chosen to fit our observed data, as discussed in the next section. This ${\lambda }^{-4}$ wavelength dependence makes the turbidity ${\tau }_f$ due to a fractal-like percolation network of drained pores to also be a convex-upward function of $1/{\lambda }_0^4$, since ${\tau }_f$ is proportional to the integrated factor ${\mathrm{\Omega }}_D$ as in Eq. (7).

 

Fig. 1. (a) Dependence of anisotropic correlation volume ${\omega }_f(q)$ on scattering wave vector $q$, illustrating the fractal correlations of half and integral numbers of fractal dimension $D$. The curve with $D=5/2$ well approximates the slope of experimental data depicted by open circles, which represent scattered light intensity as a function of $q$ with a monochromatic incident light from a laser source (514.2 nm) [37,38]. This comparison is justified by considering the fact that ${\omega }_f(q)$ is proportional to the scattered intensity. For this comparison with data in [37,38], we have to intentionally set the correlation length ($\xi$) equal to 10 μm, which is about ${10}^3$ times longer than our estimated value [40]. (b) Dependence of integrated dissymmetry factors (${\mathrm{\Omega }}_D$) for various values of $D$, where ${\mathrm{\Omega }}_D$ is proportional to the fractal turbidity ${\tau }_f$ on $1/{\lambda }_0^4$, indicating both ${\tau }_f$ and ${\mathrm{\Omega }}_D$ are convex-upward functions of $1/{\lambda }_0^4$.

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A wetting fluid such as water in the pores of Vycor glass is reported to have a fractal structure with a fractal dimension of 2.6 [37,38], which will be well approximated by the given expression with $D=5/2$, while the matrix skeleton of silica glass does not have a fractal structure [21,22]. Next, we will discuss the turbidity inherent to the structural inhomogeneities, such as spindal decomposition, in the following subsection.

B. Spinodal-Decomposition Correlation Function and Corresponding Turbidity

Leached porous glass such as Vycor is produced by a spinodal phase separation and leaching process [11] in which sodium borosilicate glass is heat-treated below the liquidus temperature to induce separation into continuous phases. The ${\mathrm{Na}}_2\mathrm{O}\text{-}{\mathrm{B}}_2{\mathrm{O}}_3$-rich phase is then leached out with hydrochloric acid. Nanoporous Vycor glass has a narrow pore-size distribution, an average nominal pore radius of 2.0 nm, a porosity $\phi$ of 0.28–0.3, and an internal surface area of $200{\mathrm{m}}^2/\mathrm{g}$. Several SAXS and SANS measurements have shown that the structure of this pore space is reminiscent of spinodal decomposition, with a broad peak centered around 0.3 [21], 0.23 [22,36], 0.22 [25], 0.20 [23,32], and 0.25 [26–29,30,31,33–35] ${\mathrm{nm}}^{-1}$ in the structure factor, corresponding to a periodicity of 20–30 nm.

The reported pair correlation function inherent to the spinodal decomposition is [28]

Now, we assume that the local dielectric fluctuation is driven by a local density fluctuation and that the dielectric correlation function $⟨\eta ({\overrightarrow{r}}_1)\eta ({\overrightarrow{r}}_2)⟩$ is proportional to the density–density correlation function $g(r)-1$ [16,17]. Therefore, we can express the correlation function, which characterizes the dielectric fluctuation due to the spinodal decomposition, in the form

By fitting our observed data with the given model, a number of parameters can be obtained from Eq. (13). Figure 2(a) shows the dependence of the integrated spinodal correlation volume ${\mathrm{\Omega }}_{sp}$ on $1/{\lambda }_0^4$, which indicates that ${\mathrm{\Omega }}_{sp}$ is a convex-downward function of $1/{\lambda }_0^4$, with the parameter $\beta =0.25{\mathrm{nm}}^{-1}$ [26–29,30,31,33–35] and the correlation length $\zeta =19.95\mathrm{nm}$. The value of $\beta$—the widely accepted one in the literature [26–29,30,31,33–35]—corresponds to the density fluctuation wavelength ${\lambda }_m=25\mathrm{nm}$ (where ${\lambda }_m=2\pi /\beta$), while the value of $\zeta$ is determined to fit best our measured turbidity for the dry sample, as will be discussed later (see Fig. 4). Its value is about twice as large as that in the literature [28]. Figure 2(b) shows the dependence of spinodal-decomposed turbidity ${\tau }_{sp}$ on $1/{\lambda }_0^4$, which indicates that ${\tau }_{sp}$ is also a convex-downward function of $1/{\lambda }_0^4$.

 

Fig. 2. (a) Dependence of integrated spinodal correlation volume ${\mathrm{\Omega }}_{sp}$ on $1/{\lambda }_0^4$, indicating ${\mathrm{\Omega }}_{sp}$ is a convex-downward function of $1/{\lambda }_0^4$, with the parameter $\beta =0.251{\mathrm{nm}}^{-1}$ [27–29] and the correlation length $\zeta =19.95\mathrm{nm}$, which was determined to fit best our data described in the text. (b) Dependence of corresponding turbidity ${\tau }_{sp}$ on $1/{\lambda }_0^4$, indicating ${\tau }_{sp}$ is also a convex-downward function of $1/{\lambda }_0^4$.

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The parameters for our dry sample fitted using this model provide both the estimated spinodal correlation function ${\gamma }_{sp}(r)$ and the corresponding estimated correlation volume ${\omega }_{sp}(q)$, as shown in Figs. 3(a) and 3(b), respectively. In Fig. 3(a), below 60 nm a clear oscillation can be observed with a period of around 25 nm. Figure 3(b) shows a strong correlation peak around $0.25{\mathrm{nm}}^{-1}$, as expected.

 

Fig. 3. (a) Dependence of spinodal correlation function ${\gamma }_{sp}$ on the radius $r$ in nanometers. (b) Dependence of the corresponding correlation volume ${\omega }_{sp}(q)$ on the scattering vector $q$. Both curves are estimated from the respective models, namely, Eq. (11) for (a) and Eq. (12) for (b), with parameters $\beta =0.251{\mathrm{nm}}^{-1}$ and $\zeta =19.95\mathrm{nm}$, both of which were determined to fit best the measured ${\tau }_{sp}$ in the range of 350 to 850 nm.

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In the given developments both for fractal and spinodal-decomposed trubidities, we assumed the dielectric correlation function $⟨\eta (0)\eta (r)⟩$ can be expressed as in Eq. (4) or (11). However, strictly speaking, the expressions in Eqs. (4) and (11) are those for the density–density correlation, which originates from the angular dependence of the scattered intensities observed by SAXS or SANS. The dielectric constant is mainly a function of local density, and then the dielectric constant fluctuation can be related to the density fluctuation around its thermodynamic equilibrium value. Therefore, the correlation function for $\eta$ is related to that for $\rho$ in the following manner [16,17]:

Hence, the constants $C_f$ in Eq. (4) and $C_{sp}$ in Eq. (11), both of which include the coefficient ${(\partial {\epsilon }_{AV}/\partial \rho )}^2$, have been assumed to be constant in this derivation.

3. EXPERIMENTAL VERIFICATION

We have reported the time-dependent change in the transmittance spectrum of a porous glass chip in the UV-Vis-near-IR region (300–2500 nm) during the drying process (see Fig. 1 in [8]). Subsequently, we have also reported the effect of water desorption on the transparency change of the nanoporous Vycor glass from the fluctuation viewpoint [40]. In that report, the effect of water desorption could be clearly separated from the intrinsic optical properties of the Vycor glass itself, such as ultraviolet absorption, structural inhomogeneities due to spinodal decomposition, and so on, by dividing the observed transmission by the transmission observed in a very low humidity state. Here, let us consider how this treatment is justified.

The turbidities derived in the previous section, ${\tau }_f$ and ${\tau }_{sp}$, are considered to be additive. Thus, for nonabsorbing glasses, the observed turbidity ${\tau }_{\text{ext}}$ can be expressed as ${\tau }_{\text{ext}}={\tau }_{\text{sca}}={\tau }_f+{\tau }_{sp}$. Since the transient white turbidity is the scattering caused by drainage of a wetting fluid from the pore space of porous Vycor glass in the process of forming a fractal-like percolation network of drained pores, no fractal turbidity ${\tau }_f$ is expected to be observed in samples for which drainage has been completed. For such a dry sample, only the turbidity inherent to structural inhomogeneity, which was formed during the process of spinodal decomposition, ${\tau }_{sp}$, might be observed. Thus, the sample during the drying process will exhibit both the turbidity due to the fractal-like percolation network of drained pores and that due to the structural inhomogeneity caused by the spinodal decomposition, whereas the dried sample will exhibit only that due to the spinodal decomposition.

In terms of the incident light intensity ($I_{\text{in}}$) and the transmitted intensity through the scattering medium ($I_{\text{out}}$), the observed transmittance is expressed as

Using the additive property of the turbidities, we can factorize the transmittance as follows:

For the time-dependent change in the transmittance of a porous glass during the drying process, if we assign the time-dependent and final transmittances, $T({\lambda }_0,t)$ and $T({\lambda }_0,t_e)$, to the term involving the factor $\exp (-{\tau }_f·d_{\text{opt}})$ and that involving the factor ${(1-r)}^2\exp (-{\tau }_{sp}·d_{\text{opt}})$, respectively, then the transmittance due to the fractal-like percolation network of drained pores can be obtained from the relative transmittance, defined as

In the absence of absorption and reflection at interfaces, the fractal turbidity ${\tau }_f$ can be simply expressed as

On the other hand, the spinodal-decomposed turbidity ${\tau }_{sp}$ is expressed in a slightly complicated form as

To determine parameters $\xi$ or $\zeta$ by fitting the experimentally observed turbidity to the respective theoretical curve, we need to estimate the respective proportional coefficient ${⟨{\eta }_f^2⟩}_{AV}$ or ${⟨{\eta }_{sp}^2⟩}_{AV}$.

A completely dry or completely water-saturated Vycor glass can be considered to consist of two homogeneous phases. One of them is a pore region of varying and undetermined shape distributed within a homogeneous matrix. The other is a homogeneous matrix in which pores are formed. Only two dielectric constants need to be considered: ${\epsilon }_p$ in the pores and ${\epsilon }_s$ in the matrix. If the pores are filled with air or water, then ${\epsilon }_p$ is equal to that of air or water (${\epsilon }_{\text{air}}$ or ${\epsilon }_{\text{water}}$). In both the completely dry or completely water-saturated cases, it should be considered that there remain only the inhomogeneities of type I, that is, those reminiscent of spinodal decomposition. Therefore, the mean square variation of the dielectric constant ${⟨{\eta }_{sp}^2⟩}_{AV}$ is expressed as [49]

On the other hand, for the partially water-saturated case, we have to consider the change in the dielectric constant of porous glass, ${\epsilon }_{\mathrm{pG}}$, as a function of the amount of water in the pores (precisely speaking, in terms of the pore-filling fraction, as explained subsequently) because of the imbibition/drainage of water into/from pores. In this case, the dielectric constant of porous glass can be estimated by using effective-medium models in which a pore is regarded as a superposition of its components. The most frequently used model is that of Bruggeman [50]. It is assumed in this model that the following formula is valid for pore clusters, which are cavities that connect with each other:

During the drying process, as the pore-filling fraction is reduced, the turbidity due to the fractal-like percolation network of drained pores shows characteristic transient behavior: it is initially small, rapidly increases to its maximum value, and then decreases and approaches the saturated value. This time dependence exactly corresponds to the time sequence of the phenomenological appearance of the sample, that is, initially transparent, then transitory whitish when it scatters light, and finally transparent again. This behavior parallels that of a binary alloy, where the intensity of the diffuse scattering varies as $c(1-c)$, $c$ being the proportion of one of the components, such as the empty pore concentration. Instead of $c$, we will use the pore-filling fraction $f$ to specify the transient mean square variation of the dielectric constant as follows:

Here, $f(1-f)$-type dependence of ${⟨{\eta }_f^2⟩}_{AV}$ on the filling fraction $f$ has already been proposed by Li et al. [27–29].

On the basis of this consideration, we replot the turbidities derived from the measured transmittance profiles between 350 and 850 nm as a function of $1/{\lambda }_0^4$ in Fig. 4(a), when the sample exhibits white turbidity (2) and when the completely saturated or dry sample exhibits its transparency (1,3). The figure shows the convex-upward curve for 2 or convex-downward curves for 1 and 3, respectively, as a function of the inverse fourth power of the wavelength in air. Convex-upward curve 3 is predicted by the fractal turbidity model, namely, ${\omega }_0{\mathrm{\Omega }}_{5/2}$, whereas convex-downward curves 1 and 3 are predicted by the spinodal-decomposed turbidity model, namely, ${\mathrm{\Omega }}_{sp}$, both of which are derived by the dielectric fluctuation theory. Correspondingly, Fig. 4(b) shows the respective transmittance between 350 and 850 as a function of wavelength.

 

Fig. 4. (a) Changes in the turbidity (estimated from the logarithm of the observed transmittance) as a function of the inverse fourth power of wavelength in air ($1/{\lambda }_0^4$). The slight deviations of the turbidities for 1 and 3 from the ${\lambda }^{-4}$ Rayleigh wavelength dependence are well fitted by the spinodal-decomposed turbidity curves, while that for 2 is well fitted by the one-parameter theoretical curve based on the fractal scattering with fractal dimension of 2.5. (b) Corresponding change in UV-Vis-light transmittance spectra of a porous Vycor glass after drying for 0, 45, and 165 min immediately after removal from ultrapure water immersion for 2 h at room temperature. In both (a) and (b), the previous results (Fig. 1 in [8]) are re-examined on the basis of the theory of dielectric constant fluctuations, and a comparison between the measured and fitted data is shown by solid and dotted lines, respectively.

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In particular, with a specified value of the pore-filling fraction $f$, the measured fractal turbidity curve 2 is fitted by the theoretical curve, which contains only one fitting parameter, $\xi$. In Figs. 4(a) and 4(b), a comparison between the measured and fitted data is shown by solid and dotted lines, respectively. The agreement between the theory and experiment in both figures is excellent. From this fitting, the correlation length $\xi$ can be obtained as a function of the pore-filling fraction $f$, whose functional form has been already reported in [40].

4. DISCUSSION

So far we have shown that how the microscopic structures described by the correlation functions—which were experimentally observed by light scattering, SAXS, and SANS measurements—affect the optical properties of nanoporous Vycor glass in the visible range via scattering. In our analysis, the key assumption is based on the form of dielectric functions, which are taken to be equivalent to the correlation functions. We will start to discuss this point first.

Our approach might be justified by the following reasons: (i) the optical properties of isotropic transparent nanoporous materials are reported to be mainly determined by their microscopic structure consisting of innumerable nanopores and almost pure silica skeleton [18,19]; (ii) their microscopic structure has been extensively studied by the scattering techniques such as SAXS and SANS measurements [20–36] and has been shown to have a pronounced inhomogeneous structure which is well characterized by the relevant correlation function; (iii) optical inhomogeneities are known to scatter light strongly [16,17]; nevertheless (iv) porous Vycor glasses are transparent in the visible range and predominantly exhibit insignificant light attenuation, determined by the relatively low level of scattering by inhomogeneities, even they are structurally inhomogeneous due to the presence of the innumerable nanopores. This implies that inhomogeneities are almost uniform within the glass and much smaller than the wavelength of the visible light scale. In this sense, a scheme is desirable to account for two kinds of turbidities, that arising due to fractal percolation of pores with fluid inside and that arising due to the structural inhomogeneity in terms of the correlation function, which contains a parameter called the correlation length; when the correlation length becomes comparable to the incident wavelength, strong scattering will be expected.

In applications of scattering techniques, interaction of radiation with matter is used for studying structural properties of the system of interest. In that case, the probing radiation has been chosen to be weakly coupled to the system and to match the condition for a required special resolution, which is inversely proportional to the momentum transfer, $q$. Measurements of the time-averaged scattered intensity $I_s(q)$ can be related to structural inhomogeneity, from which the spatial correlation functions such as the pair correlation function $g(r)$, that is, Eq. (10), are obtained. In this sense, our approach is just the reverse of the scattering measurements; that is, from the assumed correlation functions, we have shown what optical responses are expected in the visible region on the basis of the fluctuation theory of dielectric constants.

All transparent matter scatters light due to thermal fluctuations which, in turn, generate fluctuations in dielectric constants or refractive index [16,17]. Glass differs in that these fluctuations are frozen in when the material is cooled through the annealing range of temperature [16]. As mentioned in the last part of Section 2, Eq. (15) directly relates the dielectric function to the density-density correlation function $g(r)-1$, observed by small-angle scattering (SAS) measurements. In the liquid state, single component systems are physically inhomogeneous, the average density ${\rho }_0$ being broken up by thermal fluctuations of mean square density $⟨{(\mathrm{\Delta }\rho )}^2{⟩}_{AV}$ given by [52]

As shown in Section 2, the structural information such as the fractal-like percolation network or spinodal-decomposed structure is incorporated into the functional form of the correlation function, Eq. (4) or Eq. (11), respectively. As discussed in [28], the functional form of the density–density correlation function, Eq. (10), is derived on the basis of Cahn’s linear theory for spinodal decomposition [56]. On the other hand, the compositional information is incorporated into the mean square variation of the dielectric constant, ${⟨{\eta }_{sp}^2⟩}_{AV}$ or ${⟨{\eta }_f^2⟩}_{AV}$, that is, Eq. (22) or Eq. (25) [49]. The form given by Eq. (23) to Eq. (24) is based on the effective medium model [50,51,57,58].

The functional form of the dielectric function due to spinodal decomposition, Eq. (12), contains two adjustable parameters, $\beta$ and $\zeta$, while its proportional constant ${⟨{\eta }_{sp}^2⟩}_{AV}$, Eq. (22), is uniquely determined by material parameters such as the porosity and dielectric constants of constituent materials.

To see the effect of these adjustable parameters in the correlation function on the optical properties of Vycor in the visible range, a series of estimations with various combinations of $\beta$ and $\zeta$ is demonstrated in Fig. 5. The effects of parameter changes of $\beta$ and $\zeta$ on the correlation volume ${\omega }_{sp}(q)$ and on the transmittance are demonstrated in Figs. 5(a) and 5(b), respectively. As depicted by curves denoted as E, A, and D, the value of $\beta$ becomes smaller from 0.3 to $0.2{\mathrm{nm}}^{-1}$ while the correlation length $\zeta$ is fixed to the value of 19.95 nm. As shown in Fig. 5(a), the value of $\beta$ determines the position of the so-called Vycor peak while the value of $\zeta$ determines the width of half-height of the correlation volume. The corresponding transmittance curves in Fig. 5(b) demonstrate that the transmittance in the short wavelength range is remarkably improved as the value of $\beta$ becomes short. Because of the relation $\beta =2\pi /{\lambda }_m$, the smaller value of $\beta$ implies the longer quasi-periodicity ${\lambda }_m$ of the structural inhomogeneity existing even in the inhomogeneous porous structure, and subsequently the scattering due to inhomogeneities in the short wavelength range becomes less. As depicted by curves denoted by B, A, and C, the correlation length $\zeta$ becomes long from 10.2 to 29.0 nm, while the value of $\beta$ is fixed to $0.251{\mathrm{nm}}^{-1}$, the most frequently reported value. Correspondingly, the transmittances in the short wavelength range are improved as the correlation length $\zeta$ becomes longer. This tendency of transparency improvement is the same as that of the effect of smaller $\beta$, but in this case it is less effective. In this case, the correlation length $\zeta$ determines the extent of the local quasi-periodic range caused by the spinodal decomposition. Thus, the longer the correlation length $\zeta$, the more transparent the porous glass in the short wavelength range. However, this effect is considered to be local as compared to that caused by the small value of $\beta$. That is to say, the value of $\beta$ seems to determine the quasi-periodicity of the entire structural inhomogeneity.

 

Fig. 5. (a) Effects of changes in parameters $(\beta ,\zeta )$ on the correlation volume ${\omega }_{sp}(q)$ as a function of $q$. Each ${\omega }_{sp}(q)$ curve corresponds to the parameter pair denoted by A: ($\beta$ [${\mathrm{nm}}^{-1}$], $\zeta$ [nm]) = (0.251, 19.95), B: (0.251, 10.2), C: (0.251, 29.0), D: (0.200, 19.95), and E: (0.300, 19.95), respectively. (b) Corresponding changes in the estimated UV-Vis-light transmittance spectra of a porous Vycor glass.

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Next, let us discuss the SAS data on Vycor 7930 in literatures. There are numerous contradictions and discussions in the literature regarding the fractal structures in Vycor glasses [21–38] (see Table 1). The fractal dimensions of solids are mainly determined by SAXS and SANS. For example, an analysis of the SAS pattern from dry Vycor [23,25,32] suggests that the glass possesses a rough surface with a fractal dimension $D\sim 2.5$ and that the carriers of the fractal surface properties are interconnected units (cluster, drained pores) with an average diameter of 35 nm for Vycor glass [22]. However, several other investigations have shown no evidence of surface fractality in ${\mathrm{H}}_2\mathrm{O}$-saturated samples [26,30].

Table 1. Structural Information on Vycor 7930 Obtained by SAS Measurements^a

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Mildner and Hall [20] provided a unified scheme to interpret SAS data in terms of the concept of fractal structure. Its essence is that from the power law dependence of the scattered intensity $I_s(Q)\propto Q^{-n}$ at the high-$Q$ region within the SAS regime, the fractal dimensionality $D_s$ is defined as $n=6-D_s$ in three-dimensional objects. However, in the case of Vycor, the so-called Vycor peak around $0.25{\mathrm{nm}}^{-1}$ had prevented the accurate determination of the gradient of $I_s(Q)$. If the slope of $-4$ at large $Q$ is observed, it is the signature of scattering from a smooth surface, that is, nonfractal.

In subsequent investigations [29–33], it has been shown that a fractal surface can be defractalized upon deposition of an adsorbed film of water [24]. On the other hand, a fractal geometry was indicated in Vycor glass on a length scale larger than 100 nm with the percolation network formed by the empty pores and the value of the fractal dimension of 1.75 by SANS studies [26–29]. This geometry was also indicated on a length scale larger than 10 μm with a pore-space correlation of the fractal dimension of 2.6 by laser light scattering (LLS) measurements [37,38]. Contrary to these results, it has been noted [23] that the SANS and SAXS data leave little room for the idea that Vycor glass has a fractal pore network on a scale above 4 nm.

We will discuss this issue in terms of the turbidities derived in the previous section. The slight deviation from the ${\lambda }^{-4}$-wavelength dependence of the observed turbidity, which is quantitatively expressed as in Eq. (7) or (13), is expected to shed light on this problem.

As shown in Fig. 4(a), the observed turbidities for the completely water-saturated Vycor (1) and dry Vycor (3) exhibit the convex-downward curves, while the turbidity observed for the partially saturated Vycor (2) exhibits the convex-upward curve. The convex-downward curve can be well reproduced by our derived spinodal-decomposed turbidity ${\tau }_{sp}$, while the convex-upward curve can be reproduced by our derived fractal turbidity ${\tau }_f$. Since we have adopted the $f(1-f)$-type dependence of $⟨{\eta }_f^2⟩$ on the pore-filling fraction $f$ in Eq. (25) as the coefficient for Eq. (7), it is natural to observe that there is no fractal scattering for the completely water-saturated Vycor and dry Vycor, as previously reported in [21,22,26,30]. However, this functional choice for $⟨{\eta }_f^2⟩$ conflicts with the observations in [23,36].

Our simple interpretation of these contradictions is that porous Vycor glasses are not fractal in themselves (although they are very porous), but the filling of their pores, if incomplete, can exhibit fractal properties. Our observations [8–10] on the transitory whitish turbidity phenomenon strongly support this interpretation that the transient phenomenon is due to a fractal-like percolation network of drained/imbibed pores in the pore space.

Our third issue concerns the effective optical constants for porous dielectric structures proposed by Vereshchagin et al. [57]. In their derivation of the effective refractive index for a porous nonabsorbing layer, they started from the following expression:

Their approach seems to be based on the particle-scattering viewpoint with absorption, originally from the famous text on light scattering by Van de Hulst [59]. However, within the given framework, this formulation will lead to a simple ${\lambda }^{-4}$-wavelength dependence, because the field amplitude of a wave traveling in the $z$-direction in the medium specified by Eq. (29) is proportional to

In contrast, our approach based on the fluctuation viewpoint has the following advantage: by introducing two types of turbidity—one caused by structural inhomogeneity such as that reminiscent of spinodal decompition and the other by compositional inhomogeneity such as inhomogeneous distribution of wetting fluid in the pore space—we can describe exactly the transmittance spectrum of porous Vycor glass in the visible region without introducing the complex refractive indices, namely, $\tilde{n}=n+i\kappa$, as had been done by Kuchinskii et al. [51] and other groups [18,19,50,57].

Next, we will briefly comment on the absorption peak at around 1900 nm in the NIR region, although this is not the main subject of this paper. As described in the previous section, we used the absorbance peak at around 1900 nm to estimate the pore-filling fraction, which is related to the amount of water within the pore space. This treatment is based on the result reported by Wood et al. [60], who performed infrared spectroscopic studies on dried alkoxide silica gels. They found that the broad absorption at $5292{\mathrm{cm}}^{-1}$ ($\cong 1890\mathrm{nm}$) is attributed to hydrogen-bonded ${\mathrm{H}}_2\mathrm{O}$ molecules only and that its isolation from other absorption bands makes it possible to study this molecular species independently of other OH-containing groups. Therefore, a convenient quantitative determination of ${\mathrm{H}}_2\mathrm{O}$ content in the pore space can be made from the intensity of the $5292{\mathrm{cm}}^{-1}$ infrared absorption band by using the appropriate value of the extinction coefficient. Subsequent infrared studies on porous Vycor glasses [61,62] verified their result, and now the absorption band near $5260{\mathrm{cm}}^{-1}$ ($\cong 1900\mathrm{nm}$) is used for the characterization of molecular water within the pore space.

Now, we will discuss the difference between our fitted and measured transmittances in the shorter range of less than 350 nm. As shown in Fig. 6(a), the measured transmittance for the dry Vycor drops off more steeply than the estimated transmittance, which was best fitted in the range between 350 and 850 nm. If the transmittance theoretically estimated by the spinodal-decomposed turbidity is considered to provide a theoretical limit of optical transmittance in the ultraviolet region, this observed steep drop of the transmittance implies that, for the following two reasons, there must exist some absorbing species within this energy range: First, as shown in Fig. 4(b), both observed transmittances for completely water-saturated and dry Vycor glass start to drop off steeply at about 297 nm, where the two curves intersect and below which both curves drop off along the same line. This strongly indicates the existence of an absorbing species around that wavelength. Second, since the optical bandgap and the Urbach tail for vitreous silica are reported to be about 9 eV [13] and 7.4–8.8 eV [63,64], which are much larger than our predicted limit of 155 nm (= 8 eV), the porous structure of Vycor with innumerable pores, which is characterized by the spinodal correlation function, Eq. (10), does not cause the observed steep drop of transmittance.

 

Fig. 6. (a) Measured and fitted UV-Vis light transmittance spectra of a porous Vycor in the dry state at 165 min, on the basis of the theory of dielectric constant fluctuations. The fitted curve is drawn with the parameters derived from the best fitting of ${\tau }_{sp}$ in the range of 350 to 850 nm. (b) Difference between measured and fitted UV-Vis light absorbances, estimated from $\mathrm{Abs}({\lambda }_0)={\log }_{10}\{T_{\text{fit}}({\lambda }_0)/T_{\text{meas}}({\lambda }_0)\}$, as a function of the photon energy. The observed optical absorption spectrum is best fitted by three Gaussian curves, denoted as A, B, and C.

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Two possible candidates for the absorption in this region are as follows: one is the absorption caused by some organic or inorganic contaminants, which are unintentionally adsorbed on the pore walls; the other might be the optical absorption due to structural defects inherent to the amorphous network of vitreous silica itself, such as peroxy linkage ($\equiv \mathrm{Si}─\mathrm{O}─\mathrm{O}─\mathrm{Si}\equiv$), $E^{\prime }$ centers ($\equiv \mathrm{Si}·$), nonbridging oxygen hole centers (NBOHCs, $\equiv \mathrm{Si}─\mathrm{O}·$), and oxygen-deficiency centers ($\equiv \mathrm{Si}─\mathrm{Si}\equiv$) [65].

In order to identify these absorption peaks, as shown in Fig. 6(b), we have tried to separate them by fitting the absorbance versus photon energy plot with three Gaussians of the form

Table 2. Optical Absorption Bands Separated by Three-Gaussian Fitting

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$E^{\prime }$ centers around 5.8 eV [65–67] and Si-Si bonds (oxygen vacancy) around 7.6 eV [65,66,68,69] are the major reported optical absorption bands of defects in vitreous silica. $\mathrm{Si}─\mathrm{Si}$ bonds seem to be easy to exclude because of their higher peak positions. The observed largest peak, denoted as “C,” might be from $E^{\prime }$ centers because our fitting with the peak energy of 5.29 eV has some arbitrariness of peak positioning. On the other hand, three oxygen-excess defects—NBOHC [65,67], peroxy radicals (PORs) [67], and interstitial ${\mathrm{O}}_3$ (ozone)—are not so easy to exclude from our consideration because those defects are observed around 4.8 eV. For example, the POR absorption band is found at 4.8 eV in bulk samples [67]. Furthermore, even the $\equiv \mathrm{Si}─\mathrm{Si}\equiv$ oxygen vacancy defect has also been attributed to the optical absorption band near 5.0 eV [68]. Unfortunately, the most controversial region in the literature is also that around 4.8 eV. This fact makes the assignment of these peaks to oxygen-excess defects very difficult. Thus, the attribution of the observed peak to one of these defects is beyond the scope of our consideration in the present circumstances.

Another possible reason for the strong absorption peak at around 5.29 eV might be related to the extrinsic charge transfer and $s\to p$ absorption bands due to trace impurities of metal ions such as ${\mathrm{Fe}}^{3+}/{\mathrm{Fe}}^{2+}$ [70]. In the course of an investigation focused on the transparency improvement in the vacuum-ultraviolet region of high-purity silica glass, reducing the trace impurities of transition metals turned out to be the most effective way to improve the ultraviolet transmittance [71]. In glasses melted under normal atmospheric conditions, El-Batal et al. [70] reported that the iron impurity exists mainly as ${\mathrm{Fe}}^{3+}$ species with a UV band at about 230–250 nm. This absorption band seems to be coincidental with our observed absorbance peaks. However, we cannot draw any definitive conclusion about this issue because of a lack of related evidence.

Finally, we will discuss the applicability of our scheme to other transparent nanoporous materials. In this paper, we are mainly concerned with isotropic transparent nanoporous materials such as Vycor glasses, which are characterized by a three-dimensional random pore network. However, there are other types of nanoporous materials, which particularly are characterized by the network of parallel cylindrical nanochannels, that is, considerably different from the three-dimensional pore network of Vycor. It is reported that such materials also exhibit turbidities and that the light scattering in such materials drastically rises in the capillary condensate regime, quite similar to Vycor glass [72,73]. At this moment, it seems quite difficult to account for such an anisotropy in our scheme because the integrated correlation volume is obtained by assuming the fundamental spatial isotropy so that the expressions such as Eqs. (8) and (14) are possible. If the correlation function is not isotropic but possesses rotational symmetry around a unique axis, as claimed by these experimental observations, one would obtain a scattering law that has an elliptically symmetric dependence on the azimuthal orientation of the scattering vector $q$. This elliptical dependence necessarily might be removed by averaging the scattering law in terms of a reduced scattering vector to obtain the integrated correlation volume $\mathrm{\Omega }$ in Eqs. (8) and (14). In this sense, our analysis will encounter difficulty at this point. Further work is required to address this point.

5. CONCLUSION

We have studied the applicability of the turbidity analysis of the photospectroscopically measured data as a method to study the correlation functions that characterize the pore space and the structural features of isotropic transparent porous media, on the presupposition that there exists no light attenuation other than the scattering. This kind of structural analysis has been conventionally carried out by measuring the small-angle scattered intensity with a monochromatic probing radiation as a function of scattering vector.

In this paper, we have shown that the light propagation and scattering in monolithic nanoporous Vycor glass exhibit two optical turbidities in the visible (Vis) region: one is a transient white turbidity, characterized by the convex-upward dependence to the inverse fourth power of wavelength ($1/{\lambda }_0^4$), and the other is the turbidity inherent to the structural inhomogeneity, characterized by the convex-downward dependence. The former is assumed to be caused by a fractal-like percolation network of imbibed/drained pores as a consequence of imbibition/drainage of wetting fluid into/from pore space. The latter is considered to be caused by the structural inhomogeneities inherent to the Vycor glass itself, which are produced by spinodal decomposition.

To explain and verify the slight deviation from the ${\lambda }^{-4}$ wavelength dependence of photospectroscopically observed turbidities, we have derived respective analytical expressions for them on the basis of the theory of dielectric constant fluctuation. For the fractal-like percolation network, we obtained the general expressions for the turbidity due to a fractal-like percolation network, ${\tau }_f$, with half and integer values of fractal dimensions from 1/2 to 3, and verified that all of them are the convex-upward functions of $1/{\lambda }_0^4$. On the other hand, the derived expression for the spinodal-decomposed turbidity ${\tau }_{sp}$ turned out to be a convex-downward function of $1/{\lambda }_0^4$.

In summary, we have developed a general scheme to estimate the transmittance spectra of isotropically transparent nanoporous glass such as Vycor through the turbidity in the visible region. The fluctuation approach can more systematically and comprehensively treat the problem of concern with a slight deviation from the ${\lambda }^{-4}$ Rayleigh wavelength dependence of the observed turbidity.

APPENDIX A: DERIVATION OF THE BASIC EQUATIONS

Scattering theory based on the Maxwell equations for a nonconducting, nonmagnetic medium can be used to derive Eqs. (1) to (3), the basic equations for the turbidity analysis. We follow the treatment of Landau et al. [41] and Chu [42].

In the fluctuation approach, the inhomogeneity present in a condensed medium such as porous glass is generally characterized by introducing the concept of a spatial correlation function, in which the inhomogeneity of the medium is considered to be due to a continuous variation of the dielectric constant or refractive index rather than to the presence of discrete particles. Let $\epsilon (\overrightarrow{r})$ denote a local dielectric constant at position $\overrightarrow{r}$ and ${\epsilon }_{AV}$ be its average (which is related to the refractive index $n=\sqrt{{\epsilon }_{AV}}$). The function $\eta (\overrightarrow{r})=\epsilon (\overrightarrow{r})-{\epsilon }_{AV}$ represents the local fluctuation of the dielectric constant, and

(A1)$$\gamma ({\overrightarrow{r}}_1,{\overrightarrow{r}}_2)\equiv ⟨\eta ({\overrightarrow{r}}_1)\eta ({\overrightarrow{r}}_2)⟩/{⟨{\eta }^2⟩}_{AV}$$

defines a dimensionless spatial correlation function, where ${⟨{\eta }^2⟩}_{AV}$ is the average value of ${\eta }^2$. Here $⟨\dots ⟩$ represents an ensemble average. For a plane-wave incident electric field,

(A2)$${\overrightarrow{E}}_{\text{in}}(\overrightarrow{R},t)={\widehat{n}}_i{E}_0\exp [i({\overrightarrow{k}}_i·\overrightarrow{R}-{\omega }_{i}t)],$$

where ${\overrightarrow{E}}_{\text{in}}$ is assumed to be a simple sinusoidal traveling wave of frequency ${\omega }_i$ in the direction ${\overrightarrow{k}}_i$ with phase velocity $c_m(={\omega }_i/|{\overrightarrow{k}}_i|)$. Here, ${\overrightarrow{k}}_i$ is the wave vector in the propagating direction with magnitude of $|{\overrightarrow{k}}_i|=2\pi n/{\lambda }_0$ (${\lambda }_0$ is the wavelength in vacuo), ${\widehat{n}}_i$ is a unit vector in the direction of polarization of the incident field, and $E_0$ is the field amplitude. According to the classical theory of electromagnetic waves, the component of the scattered electric field at a large distance $R$ ($=|\overrightarrow{R}|$) from the scattering volume $V$ with the dielectric fluctuation $\eta (\overrightarrow{r})$ can be expressed in vector form as [41,42]

(A3)$$\begin{gathered} {\overrightarrow{E}}_s(\overrightarrow{R},t)={\overrightarrow{k}}_s\times ({\overrightarrow{k}}_s\times {\widehat{n}}_i)\frac{E_0}{{\epsilon }_{AV}}\frac{\exp [i(k_{s}R-{\omega }_{i}t)]}{4\pi R} \\ \times {\int }_V{\mathrm{d}}^3\overrightarrow{r}\eta (\overrightarrow{r})\exp [i({\overrightarrow{k}}_i-{\overrightarrow{k}}_s)·\overrightarrow{r}], \end{gathered}$$

where the wave vector ${\overrightarrow{k}}_s$ of the scattered field is in the same direction as $\overrightarrow{R}$ and the subscript $V$ indicates that the integral is over the scattering volume $V$.

Assuming that the scattering is weak and the Born approximation is valid, the time-averaged scattered intensity is expressed as [41,42]

(A4)$$\begin{gathered} I_s({\overrightarrow{k}}_i-{\overrightarrow{k}}_s)=\frac{1}{2}\sqrt{\frac{{\epsilon }_{AV}}{\mu }}{|E_0|}^2\frac{{|{\overrightarrow{k}}_s\times ({\overrightarrow{k}}_s\times {\widehat{n}}_i)|}^2}{{(4\pi R)}^2{\epsilon }_{AV}^2} \\ \times {\iint }_v{\mathrm{d}}^3{\overrightarrow{r}}_1{\mathrm{d}}^3{\overrightarrow{r}}_2\exp [i({\overrightarrow{k}}_i-{\overrightarrow{k}}_s)·({\overrightarrow{r}}_1-{\overrightarrow{r}}_2)]⟨\eta ({\overrightarrow{r}}_1)\eta ({\overrightarrow{r}}_2)⟩, \end{gathered}$$

where $\mu$ is the permeability, which is unity for a nonmagnetic dielectric medium. Now, we introduce the vector defined by $\overrightarrow{q}={\overrightarrow{k}}_i-{\overrightarrow{k}}_s$ as the scattering vector and the angle between ${\overrightarrow{k}}_i$ and ${\overrightarrow{k}}_s$ as the scattering angle $\theta$. For an elastic scattering such as Rayleigh scattering, the scattered wave has essentially the same wavelength as that of the incident wave: $|{\overrightarrow{k}}_s|\cong |{\overrightarrow{k}}_i|\equiv k=2\pi n/{\lambda }_0$. Thus it follows that $q=2k\sin (\theta /2)$.

For an isotropic medium, the spatial correlation function $\gamma ({\overrightarrow{r}}_1,{\overrightarrow{r}}_2)$ depends only on the relative distance $r=|{\overrightarrow{r}}_1-{\overrightarrow{r}}_2|$ and not on the direction in space. Then $\gamma (r)$ becomes a spherically symmetric function that has a value of unity at $r=0$ and decays to zero as $r\to \infty$. The steepness with which this function drops off from unity to zero is a measure of the average extension of the inhomogeneities. If the dielectric medium is isotropic, the double integral in Eq. (A4) can be replaced by the product of the scattering volume $V$ and the integral over $d^3\overrightarrow{r}$ whose angular integration can be performed, yielding

(A5)$$I_s(q)=I_{\text{in}}\frac{k_s^4{\sin }^2\chi }{{(4\pi R)}^2}·\frac{{⟨{\eta }^2⟩}_{AV}}{{\epsilon }_{AV}^2}V{\int }_0^{\infty }4\pi r^2\gamma (r)\frac{\sin (qr)}{qr}\mathrm{d}r,$$

where we use the facts that the incident intensity of the form $I_{\text{in}}=\frac{1}{2}\sqrt{\frac{{\epsilon }_{AV}}{\mu }}{|E_0|}^2$ is the flow of power per unit area of the incoming light beam and that the magnitude of the vector cross product ${\overrightarrow{k}}_s\times ({\overrightarrow{k}}_s\times {\overrightarrow{n}}_i)$ is equal to $k_s^2\sin \chi$, where $\chi$ is the angle measured from the scattering direction to the dipole. The ${\sin }^2\chi$ angular dependence in Eq. (A5) is characteristic of the dipole radiation, so its dependence can also be expressed in terms of the scattering angle $\theta$ as $(1+{\cos }^2\theta )/2$ with respect to the scattering plane.

The differential cross section is defined as

(A6)$$\frac{dP}{d\mathrm{\Omega }}=\frac{1}{2}\mathrm{Re}[R^2\widehat{R}·{\overrightarrow{E}}_s\times {\overrightarrow{H}}_s^{*}]=\frac{I_s(q)R^2}{I_{\text{in}}V},$$

where $V$ is the scattering volume and $\widehat{R}$ is the unit vector in the same direction as $\overrightarrow{R}$, that is, $\widehat{R}\equiv \overrightarrow{R}/|\overrightarrow{R}|$, and $*$ indicates the complex conjugate. The differential cross section has the dimensions of reciprocal length. The total scattering cross section, known as the turbidity $\tau$, is related to the differential cross section by integrating over the whole solid angle, namely, $\tau ={\int }_{4\pi }\mathrm{d}\mathrm{\Omega }(\theta ,\phi )·\frac{dP}{d\mathrm{\Omega }}$. Substitution of Eqs. (A5) and (A6) into this expression gives

(A7)$$\begin{gathered} \tau ={⟨{\eta }^2⟩}_{AV}\frac{k_s^4}{{(4\pi )}^2{\epsilon }_{AV}^2}{\int }_{4\pi }\mathrm{d}\mathrm{\Omega }\frac{1+{\cos }^2\theta }{2}\omega (q) \\ ={⟨{\eta }^2⟩}_{AV}\frac{{\pi }^2}{{\lambda }_0^4}{\int }_{4\pi }\mathrm{d}\mathrm{\Omega }\frac{1+{\cos }^2\theta }{2}\omega (q), \end{gathered}$$

with the correlation volume, which is a function both $\gamma (r)$ and the scattering angle $\theta$, defined as

(A8)$$\omega (q)={\int }_0^{\infty }4\pi r^2\gamma (r)\frac{\sin (qr)}{qr}\mathrm{d}r.$$

A simple physical interpretation of Eq. (A7) is that the turbidity is proportional to the product of the mean square fluctuation of the dielectric constant, ${⟨{\eta }^2⟩}_{AV}$, and an associated correlation volume $\omega (q)$.

A measure of the extent of the inhomogeneities is mathematically expressed by correlation length $\xi$, with the help of the correlation function involving the length scale in the form $r/\xi$ as the exponential argument. Conventionally, the angular dependence of the scattered intensity [which is proportional to correlation volume $\omega (q)$] from the sample with a monochromatic incident light beam [or the dependence of the scattered intensity on scattering vector $q$, i.e., $I(q)$] is measured to obtain the correlation function.

APPENDIX B: DERIVATION OF ${\mathsf{\Omega }}_{\mathit{D}}$

Detailed mathematical expressions for correlation volumes and related quantities with half-integer and integral values of the fractal dimension $D$ for the fractal correlation function of Eq. (4) with $d=3$ are summarized here. The integrated dissymmetry factor ${\mathrm{\Omega }}_D$ is required to obtain the turbidity ${\tau }_f$ due to a fractal-like percolation network of drained pores, as explained in Section 2.A.

For the integer and half-integer dimensions from 1/2 to 3, the explicit expressions for the anisotropic correlation volume ${\omega }_f(q)$ are tabulated in Table 3 in terms of the dissymmetric factor of $D$th order $f_D(q)$, defined as

(B1)$$\begin{gathered} f_D(q)=\frac{{\omega }_f(q)}{{\omega }_0(D)} \\ =\frac{\mathrm{\Gamma }(D-1)}{\mathrm{\Gamma }(D)}{\left[\frac{1+q^2{\xi }^2}{{(q\xi )}^2}\right]}^{1/2}\frac{\sin [(D-1){\tan }^{-1}(q\xi )]}{{[1+q^2{\xi }^2]}^{D/2}}, \end{gathered}$$

with the symmetrical correlation volume ${\omega }_0(D)$, defined as

(B2)$${\omega }_0(D)=4\pi {\xi }^D·\mathrm{\Gamma }(D).$$

Table 3. Correlation Volumes and Dissymmetrical Factors of Fractal Correlation Functions

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Thus, the correlation volume ${\omega }_f(q)$ is provided by the product of Eqs. (B1) and (B2).

As shown in Table 3, Eqs. (4) and (5) for $d=3$ provide two particular cases: $D=3$, ${\gamma }_f(r)\sim \exp (-r/\xi )$, which gives the Debye law [43], ${\omega }_f(q)\sim {\xi }^3/{(1+q^2{\xi }^2)}^2$, and $D=2$, ${\gamma }_f(r)\sim r^{-1}\exp (-r/\xi )$, which gives the Ornstein–Zernike law, ${\omega }_f(q)\sim {\xi }^2/(1+q^2{\xi }^2)$.

The corresponding turbidity [${\tau }_f(D)$] is given by integrating the correlation volume over the entire solid angle $d\mathrm{\Omega }$, as follows:

(B9)$${\tau }_f(D)={⟨{\eta }_f^2⟩}_{AV}\frac{k^4}{{(4\pi )}^2{\epsilon }_{AV}^2}{\omega }_0(D)·{\mathrm{\Omega }}_D,$$

where ${\mathrm{\Omega }}_D$ is the integrated dissymmetrical factor, defined as

(B10)$${\mathrm{\Omega }}_D=2\pi {\int }_0^{\pi }\mathrm{d}\theta \sin \theta \frac{1+{\cos }^2\theta }{2}{f}_D\left(2k\sin \right(\frac{\theta }{2}\left)\right).$$

The product of Eq. (B10) with the symmetrical correlation volume ${\omega }_0(D)$ provides the integrated correlation volume for the fractal scattering, that is, the integrated expression in Eq. (3): ${\int }_{4\pi }\mathrm{d}\mathrm{\Omega }\frac{1+{\cos }^2\theta }{2}\omega (q)={\omega }_0(D)·{\mathrm{\Omega }}_D$.

Equation (B9) implies that the wavelength dependence of the turbidity ${\tau }_f$ due to the fractal-like percolation network of fractal dimensionality $D$ is determined mainly by $1/{\lambda }_0^4$ [which comes from the factor $k^4/(16{\pi }^2{\epsilon }_{AV}^2)$] and the integrated correlation volume ${\omega }_0(D)·{\mathrm{\Omega }}_D$, which is a function of $\xi$ only and determines the slight deviation from the ${\lambda }^{-4}$ Rayleigh wavelength dependence.

For half-integer and integer values of $D$, the explicit expressions for ${\mathrm{\Omega }}_D$ are as follows:

For $D=1/2$, in the difference form with $b\equiv 4k^2{\xi }^2$,

(B11a)$${\mathrm{\Omega }}_{1/2}=\frac{16\sqrt{2}\pi }{b^3}\{f(1+\sqrt{1+b},b)-f(2,b)\},$$

where

(B11b)$$f(z,b)=2\sqrt{z}\{\frac{z^5}{11}-\frac{5z^4}{9}+\frac{8z^3}{7}-\frac{4z^2}{5}-b(\frac{z^3}{7}-\frac{3z^2}{5}+\frac{2z}{3})+\frac{b^2}{2}(\frac{z}{3}-1)\}.$$

In the explicit form,

(B11c)$$\begin{gathered} {\mathrm{\Omega }}_{1/2}=\frac{16\sqrt{2}\pi }{3465b^3}[\sqrt{2}\{256+33b(16+35b)\} \\ -\sqrt{1+\sqrt{1+b}}\{128(1+\sqrt{1+b}) \\ +8b(31+23\sqrt{1+b})+b^2(1822-795\sqrt{1+b})\}]. \end{gathered}$$

For $D=3/2$, in the difference form with $b\equiv 4k^2{\xi }^2$,

(B12a)$${\mathrm{\Omega }}_{3/2}=\frac{16\sqrt{2}\pi }{b^3}\{g(1+\sqrt{1+b},b)-g(2,b)\},$$

where

(B12b)$$g(z,b)=2\sqrt{z}\{\frac{z^4}{9}-\frac{4z^3}{7}+\frac{4z^2}{5}-b(\frac{z^2}{5}-\frac{2z}{3})+\frac{b^2}{2}\}.$$

In the explicit form,

(B12c)$$\begin{gathered} {\mathrm{\Omega }}_{3/2}=\frac{16\sqrt{2}\pi }{315b^3}[-\sqrt{2}\{256+21b(16+15b)\} \\ +\sqrt{1+\sqrt{1+b}}\{128(1+\sqrt{1+b}) \\ +8b(19+11\sqrt{1+b})+259b^2\}]. \end{gathered}$$

For $D=5/2$, in the difference form with $b\equiv 4k^2{\xi }^2$,

(B13a)$${\mathrm{\Omega }}_{5/2}=\frac{16\sqrt{2}\pi }{3b^3}\{h(1+\sqrt{1+b},b)-h(2,b)\},$$

where

(B13b)$$h(z,b)=2\sqrt{z}\{\frac{z^3}{7}-\frac{z^2}{5}-\frac{z}{3}-(2+b)(\frac{z}{3}+1)-\frac{1+b+b^2/2}{z-1}\}.$$

In the explicit form,

(B13c)$$\begin{gathered} {\mathrm{\Omega }}_{5/2}=\frac{16\sqrt{2}\pi }{315b^3}[\sqrt{2}(768+560b+105b^2) \\ -\frac{\sqrt{1+\sqrt{1+b}}}{\sqrt{1+b}}\{384(1+\sqrt{1+b}) \\ +8b(53+29\sqrt{1+b})+145b^2\}]. \end{gathered}$$

For $D=1$, with $b=4k^2{\xi }^2$,

(B14)$${\mathrm{\Omega }}_1=\frac{4\pi }{15}[\frac{22}{\sqrt{b}}{\tan }^{-1}\left(\sqrt{b}\right)+\frac{6+7b}{b^2}-\frac{(6+10b+15b^2)}{b^3}\ln (1+b)]$$

For $D=2$, with $b=4k^2{\xi }^2$,

(B15)$${\mathrm{\Omega }}_2=\frac{4\pi }{b^2}\{-(2+b)+\frac{2+2b+b^2}{b}\ln (1+b)\}.$$

For $D=3$, with $b=4k^2{\xi }^2$,

(B16)$${\mathrm{\Omega }}_3=\frac{4\pi }{b^2}\{\frac{{(2+b)}^2}{1+b}-\frac{2(2+b)}{b}\ln (1+b)\},$$

which is the same as that of Eq. (6′′) of Debye [43].

Acknowledgment

The authors are sincerely grateful to Drs. Osamu Kagami and Akinori Furuya of NTT for their encouragement and support throughout this work.

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