1. INTRODUCTION

Porous Vycor glass with nanopores is transparent in the visible region and is used in colorimetric chemical sensing when impregnated with selectively reacting regents [1–3]. The pores in Vycor have an average radius 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. Optical methods [4] 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 changes in the humidity of ambient air strongly affect its transmission. Although completely dried or wetted nanoporous Vycor glass is transparent for visible light even though there exist innumerable pores within it, Vycor during the drying process after it has been filled with a wetting fluid such as water vapor exhibits a transitory whitish turbidity phenomenon. The same phenomenon has been observed by several authors [5–9]. Page and co-workers [5,6] have observed that Vycor samples partially filled with hexane prepared by adsorption are transparent, but for samples prepared by desorption, the transmission is lower by several orders of magnitude. They attributed this lowering of transmission to “pore-space correlations” and explained the phenomenon qualitatively by the existence of the long-range correlations of empty pores in the pore space of draining Vycor with a quite long correlation length of about 10 μm. Furthermore, this turbidity phenomenon has more general character in nanoporous media, and as pointed out recently by Gruener et al. [10], the turbidity problem also appears at the interface during an imbibition process, not restricted to the drying process.

In our previous studies [11,12], we analyzed the wavelength dependence of the transitory whitish turbidity of nanoporous Vycor on draining water vapor, and found that the turbidity is dependent on the inverse fourth power of the wavelength $(1/{\lambda }^4)$, and that, from this dependence, the phenomenon can be consistently interpreted and quantitatively analyzed by a simple Rayleigh scattering model [11,12], which is based on the particle-scattering approach [13]. From the turbidity as a linear function of $1/{\lambda }^4$, we obtained the effective radius of a Rayleigh scatterer, which is regarded as a measure of the extent of inhomogeneities that cause the scattering [11,12]. The effective radius was found to be about 10 nm at most when the maximum of scattering occurs. This value is 1000 times smaller than that estimated by Page and co-workers [5,6]. Furthermore, the slight deviation of the observed turbidity from theoretical linear dependence on $1/{\lambda }^4$ cannot be explained at all by the previous particle-scattering viewpoint.

In general, the problem of light scattering can also be approached from Einstein’s fluctuation viewpoint, apart from the above particle-scattering viewpoint. In the fluctuation approach, the inhomogeneity present in a condensed medium such as porous glasses 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 $\gamma (r)\equiv {〈{\eta }_A{\eta }_B〉}_{\mathrm{AV}}/{〈{\eta }^2〉}_{\mathrm{AV}}$, where ${\eta }_A$ and ${\eta }_B$ are the local fluctuations in dielectric constant from some average value ${〈\epsilon 〉}_{\mathrm{AV}}$ at points A and B a distance $r$ apart and ${〈{\eta }^2〉}_{\mathrm{AV}}$ is the average value of ${\eta }^2$. The problem of characterizing the inhomogeneities in solids and of relating them to the scattered intensity has been treated by Debye and co-workers [14,15]. Their approach has been utilized both for small angle x-ray scattering and conventional light scattering. Their mathematical scheme is summarized as follows: for an unpolarized incident beam of light, the turbidity $(\tau )$ takes the form [14,15]

Several scattering determinations of the inhomogeneities of Vycor during drainage of a wetting fluid [5,6,16] can be found in the literature. By means of light scattering, which is a definitive probe of the long-range spatial correlations of the empty pores, Page and co-workers [5,6] reported that the draining pores, which are surrounded by filled pores, exhibit long-range correlations of a fractal dimension of about 2.6, and their length is larger than 10 μm. However, they did not report the correlation length as a function of the amount of wetting fluid in the pore space.

In this paper, we reexamine our results for the transitory whitish turbidity of nanoporous Vycor during the drying process 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 θ, for which the theoretical scheme was originally proposed by Debye and co-workers [14,15]. The slight deviation from the ${\lambda }^{-4}$ Rayleigh wavelength dependence directly provides the correlation length of an interconnected network of drained pores.

2. THEORY—TURBIDITY ANALYSIS TO OBTAIN THE CORRELATION LENGTH

Below we outline some equations we have developed for analyzing scattering data in terms of fractal scattering and the corresponding turbidity. As mentioned above [5,6,16], when porous Vycor glass is partially saturated with a wetting fluid, a fractal-like percolation network may be formed, although the pores themselves are not distributed with fractal geometry. As the draining proceeds, the transmission becomes small due to strong scattering of light. This scattering may be characterized by the fractal-like percolation network with a fractal dimension of 2.6 [5,6] using the following fractal correlation function [16]:

Here, the parameter $\xi$, the correlation length, characterizes the average distance between fluctuations in dielectric constant. Inserting the above correlation function into Eq. (2), with analytically treatable fractal dimension ($D$) of 2.5, the correlation volume is given as $\omega (q)={\omega }_0·f_{5/2}(q)$, with a constant ${\omega }_0(D)=4\pi {\xi }^D\mathrm{\Gamma }(D)=4\pi {\xi }^{\frac{5}{2}}(3\sqrt{\pi }/4)$ and the dissymmetry factor

Integrating the correlation volume over all angles, the turbidity is given as

Equation (5) implies that the wavelength dependence of the turbidity $\tau$ is determined mainly by $1/{\lambda }_0^4$ (which comes from the factor $k_0^4$) and the correlation volume, ${\omega }_0\cdot {\mathrm{\Omega }}_{5/2}$, which is a function of $\xi$ only and determines the slight deviation from the above ${\lambda }^{-4}$ Rayleigh wavelength dependence.

3. RESULTS AND DISCUSSION

We have already reported the time-dependent change in the transmission spectrum of a porous glass chip in the UV–vis–near-IR region (300–2500 nm) during the drying process (see Fig. 1 in Ref. [11]). Here, we will replot the same data in a different way, since the original transmission of the porous Vycor glass showed several inherent characteristics, such as an ultraviolet absorption edge, absorption due to adsorbed organic contaminants, and transmission loss due to reflection at interfaces between the glass slab and air. The wavelength dependence of the turbidity derived from the original transmission reveals not only the ${\lambda }_0^{-4}$ dependence but also another type of wavelength dependence, characterized by the convex-downward curve of the turbidity versus $1/{\lambda }_0^4$. This convex-downward dependence can be explained by the spinodal decomposition process that takes place in Vycor as part of the manufacturing process, which will be discussed elsewhere. Here, in order to focus on the change in turbidity upon draining a wetting fluid, we separate the transmission change due to the drainage only from the inherent properties of the porous glass matrix by dividing all transmission spectra measured during the drying process by the final transmission spectrum measured at low relative humidity condition. That is, we analyzed the change in the relative transmission, defined as $T_r({\lambda }_0)\equiv T({\lambda }_0,t)/T({\lambda }_0,t_e)$. Here, as a reference transmission, we used the final transmission data, $T({\lambda }_0,t_e)$, which were measured at the last moment, $t_e=165\min$ after the removal from the immersion chamber. In this case, the turbidity change in the ratio, $T_r({\lambda }_0)$, is considered to be purely caused by the drainage of a wetting fluid from pores deep inside the porous glass to the surface.

 

Fig. 1. (a) Change in UV–vis light transmission spectra of a porous Vycor glass chip relative to that in the dry state at 165 min, after drying for 0, 15, 30, 45, 60, 75, and 90 min immediately after removal from ultrapure water immersion for 2 h at room temperature. (b) Changes in the turbidity (estimated from the logarithm of the relative transmission) as a function of the inverse fourth power of wavelength in air $(1/{\lambda }_0^4)$. The slight deviation of the turbidity from the ${\lambda }^{-4}$ Rayleigh wavelength dependence is well fitted by the one-parameter theoretical curves based on the fractal scattering with fractal dimension of 2.5. In both (a) and (b), the previous results (Fig. 1 in Ref. [11]) are reexamined on the basis of the theory of dielectric constant fluctuation, and a comparison between the measured and fitted data is shown by dotted and solid lines, respectively.

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The time-dependent change in the relative transmission spectrum of a porous glass chip in the UV–vis region (300–850 nm) during the drying process is shown in Fig. 1(a). Measured data are shown by dotted lines and fitted ones by solid lines. The transmission curves were measured every 15 min immediately after the sample removal from the immersion from time 0 to 165 min. In the figure, only the first seven (0–90 min) curves are depicted. As shown in Fig. 1(a), the relative transmission of Vycor approaches the maximum of 1.0 at long wavelength because the transmission loss due to the reflection at the interfaces inherent to the original transmission data is canceled out when each transmission at time $t$ is divided by the reference transmission at $t_e=165\min$. However, only the initial relative transmission measured immediately after the removal from the immersion container (i.e., $t=0\min$, dotted line 1) deviates considerably from the ideal transmission. This might be due to the light scattering by surface roughening caused by intense evaporation of a wetting fluid for the surface.

In the absence of absorption and reflection at interfaces, from Eqs. (1) and (5), the turbidity $\tau$ can be expressed as

To determine the parameter $\xi$ by fitting the experimentally observed turbidity to the theoretical curve, we need to estimate the proportional coefficient ${〈{\eta }^2〉}_{\mathrm{AV}}$. A dry Vycor glass can be considered to consist of two homogeneous phases. One of these may be pore regions of varying and undetermined shape distributed within a homogeneous matrix. Only two dielectric constants need to be considered: ${\epsilon }_s$ in the matrix and ${\epsilon }_p$ in the pores. If the pore is filled with air or water, then ${\epsilon }_p$ is equal to that of air or water (${\epsilon }_{\text{air}}$ or ${\epsilon }_{\text{water}}$). The mean square variation of the dielectric constant is expressed as [17]

On the basis of the above consideration, the transmission profile between 300 and 850 nm is replotted as a function of $1/{\lambda }_0^4$ in Fig. 1(b), which shows the convex-upward curves as a function of the inverse fourth power of the wavelength in air. These convex-upward curves are predicted by the dielectric fluctuation theory, since the correlation volume, ${\omega }_0{\mathrm{\Omega }}_{5/2}$, behaves as a convex-upward function of $1/{\lambda }_0^4$.

With a specified value of the pore-filling fraction $f$, each turbidity curve is fitted by the theoretical curve, which contains only one fitting parameter, $\xi$. Here, the theoretical curve is determined by combining Eqs. (8) and (11). In Figs. 1(a) and 1(b), a comparison between the measured and fitted data is shown by dotted and solid lines, respectively. The agreement between the theory and experiment in both figures is excellent.

Figure 2 shows the obtained correlation length $\xi$ as a function of the pore-filling fraction $f$. The longest correlation length is estimated to be 18 nm at most, which is still one-tenth of the wavelength of the incident light.

 

Fig. 2. Correlation length $\xi$ as a function of pore-filling fraction $f$, extracted from the peak absorbance at around 1900 nm, normalized by the initial maximum value measured immediately after removal from the immersion container [11]. The maximum correlation length reaches a value of about 18 nm at most at about $f=0.6$. The large error bar at $f=0.97$ is due to the deviation of measured curve 1 from the corresponding theoretical curve, as shown in Figs. 1(a) and 1(b). For comparison, the scatterer’s effective radius $r_{\text{sca}}$, based on the previous Rayleigh scattering viewpoint [11,12], as a function of $f$ is also included in the figure.

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Some authors [5,6,16] reported that the correlation length might extend to the order of several micrometers, especially Page and co-workers [5,6], who estimated that the correlation length of the fractal region was about $\xi \sim 10\mathrm{\mu m}$, which was determined from the crossover at the scattering vector $q\sim 0.1{\mathrm{\mu m}}^{-1}$ of the $I(q)$ curve. This value seems to be long compared to the optical path length and the incident monochromatic wavelength (514.5 nm). All of these authors guessed such a large value of $\xi$ on the basis of the consideration of evacuating a wetting fluid deep inside the pore space of Vycor to the surface. However, from the optical point of view, such a large extent of inhomogeneity leads to no light transmission from the sample, which is not the case. Thus, the correlation length on the order of micrometers might be an overestimation. Furthermore, if the correlation length becomes more than the incident light wavelength or an order of micrometers, the wavelength dependence of the scattering will behave as Mie scattering. However, this is also not the case. The experimental results [dotted lines in Fig. 1(b)] show that the deviation from the ${\lambda }^{-4}$ Rayleigh scattering dependence is quite small, which implies that the possibility of Mie scattering is negligibly small.

Figure 2 also shows that the maximum of the correlation length is observed at the filling fraction of about 0.6. In this $f$ range approximately the capillary condensation takes place, which is just a crossover state between the entire filling $(f=1)$ and adsorbed layer film $(f\le 0.4)$ regimes. The maximum $\xi$ observed at about $f\sim 0.6$ corresponds to the appearance of long-range capillary bridges inside the pores.

Our analysis of the wavelength dependence of turbidity of forward-scattered light neglects the short-range atomic structure in Vycor and then provides the long-range correlations in pore space, whereas the scattering at a large angle is determined primarily by the short-range order correlations, which permit resolution of structures with atomic dimensions.

4. CONCLUSIONS

From the slight deviation from the ${\lambda }^{-4}$ wavelength dependence of the photospectroscopically observed turbidity of nanoporous Vycor glass, we estimated the correlation length, $\xi$, of the fractal-like network of drained pores on the basis of the theory of dielectric constant fluctuation. The correlation length is considered to be the extent of optical inhomogeneity that causes scattering. The correlation length ranged from 0.5 to 18 nm at most, which is almost the same order as that indirectly estimated from our previous Rayleigh scatterer’s model. Both pictures, i.e., the particle-scattering viewpoint and the fluctuation viewpoint, can reasonably explain the transient white turbidity phenomenon during the drying process in nanoporous Vycor.