1 Introduction

Nanoporous material is an important class of materials that has attracted much attention over the past few years due to their application in many fields and industries [1,2,3,4,5,6,7]. These materials are widely used, for example, in gas capture [1, 2], gas and liquid separation processes [3, 4], catalysis [5] and biomedical applications [6, 7].

One of the most important parameters in determining the suitability of nanoporous materials for a specific application is the average pore size. There is a number of methods to determine the average pore size, such as mercury intrusion porosimetry [8], gas adsorption [9], scanning electron microscopy [10] and nuclear magnetic resonance [11]. These methods have some limitations but complement each other. In addition, they may yield different values, because they use different physical principles to extract the pore size. Therefore, there is always a need for new methods for pore size assessment. One of the relatively new methods to assess the pore size is based on gas in scattering media absorption spectroscopy (GASMAS) [12]. Employing the GASMAS method has opened up new possibilities for monitoring gases embedded in scattering media, such as wood [13], fruits [14], and ceramic materials [15]. Also, it has been utilized in many applications such as studying gas diffusion [15], characterizing pharmaceutical tablets [16, 17], monitoring food packages [18] and diagnosing human sinuses [19].

The principle of GASMAS is based on the different nature of light absorption in gas and in bulk material. It is well known that the width of an absorption dip of a gas is very narrow; typically, it is about 10,000 times narrower than that of a bulk material band [20]. Hence, the gas absorption dip can be easily extracted as it appears as a very narrow signal on an almost flat background from the bulk material absorption [12, 21, 22].

The width of an absorption dip is broadened by collisions of gas molecules. This broadening is inversely proportional to the average time between collisions. Within pores, the gas molecules collide with themselves as well as with the walls of the pores. If the mean free path between intermolecular collisions for O2 at ambient air is comparable with the average pore size, then the pore size will influence the width the absorption dip. At standard conditions for temperature and pressure, the mean free path of the intermolecular collisions is typically of the order of 100 nm [20, 23]. Hence, if the average pore size is of the order of 100 nm or smaller, then the collisions with the pore walls will lead to an extra broadening in the absorption dip. This provides a great opportunity to utilize GASMAS as a nonintrusive and economic technique to assess pore sizes from the broadening of the absorption lines [24].

In this work, GASMAS is used to extract the line broadening of the absorption line at 763.84 nm of oxygen entrapped in porous α-alumina disc samples. The average pore size of these samples is measured independently by Mercury Intrusion Porosimetry (MIP), which is a standard method to measure the pore size distribution. The range of the measured average pore size of the samples is between 70 and 150 nm. A theoretical model is developed to correlate the broadening due to wall collisions with the average pore size of the samples. For the investigated samples, it is found that the line broadening due to wall collisions is inversely proportional to the average pore size.

2 Experiment

A schematic diagram of the experimental setup is shown in Fig. 1. The light source used is a Vertical-Cavity Surface-Emitting Laser (VCSEL) (FU-763-0.3-TO5-01, Frankfurt Laser Company), which has a typical emission wavelength of 763 nm, and a spectral bandwidth of about 100 MHz. The laser has an output power of about 0.3 mW at 25 °C. A temperature controller (TED200C, Thorlabs) and a homemade current controller are used to operate the laser. The current controller is a low-noise voltage controlled current source, which can generate a maximum current of 2 mA with a driving voltage of 3.5 V.

Fig. 1
figure 1

Schematic diagram of the experimental setup

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The temperature of the laser is set such that when the laser is tuned using the current source, it covers three strong oxygen absorption lines of the A-band electric-dipole forbidden transition \({X}^{3}{\sum }_{\mathrm{g}}^{-} \to {b}^{1}{\sum }_{\mathrm{g}}^{+}\), namely, the P9P9 line at 763.84 nm, the P9Q8 line at 763.73 nm, and the P7P7 line at 763.43 nm [25]. Figure 2 shows these lines in open-air over a path length of 5 m. The lines’ wavelengths are checked using a high-resolution spectrometer (SPEX 1403). The absorption line used in this study is the P9P9 line. To convert from injection current to light frequency, a wavemeter (HighFinesse, WS6/200), which has an absolute accuracy of 0.0004 nm, is used to relate \(\lambda \) to \(i\). The laser wavelength in nm as a function of the injection current in mA is found to fit very well to a quadratic equation, and then the frequency is obtained from \(\nu =c/\lambda \), where c is the speed of light.

Fig. 2
figure 2

Molecular oxygen absorption lines P7P7 at 763.43 nm, P9Q8 at 763.73 nm and P9P9 at 763.84 nm measured in open-air of 5 m path length

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The photodetector used is a silicon PIN photodiode (DET36A/M, Thorlabs) with an active area of 13 mm2 and a responsivity of about 0.45 A/W at a wavelength of 760 nm. The current generated by the photodetector is converted to a measurable voltage signal by a transimpedance resistor of 42 MΩ. A 16-bit DAQ board with a sample rate up to 2 MS/s (NI USB-6366, National Instruments) and a LabVIEW code are used to control both the laser diode temperature and injection current and to collect the output signal of the photodetector. The LabVIEW code generates a triangular ramp signal to drive the laser by varying the injection current in a step manner. At each step, the code reads 100 samples and averages them to reduce the noise. Furthermore, the noise is additionally reduced by averaging over 25 ramps.

A dithering lens is employed to reduce optical interference fringes that often limit the sensitivity of laser absorption measurement. The dithering lens convert the interference fringes to noise and then the noise is reduced by averaging. The dithering lens and its two tracking coils are extracted from a CD player. The tracking coils are fed with a low-frequency sinusoidal signal of 200 Hz using a wave generator. The laser, the dithering lens, and the photodetector are vertically stacked, and an alumina disc sample is placed directly on the photodetector. The open-air distance between the laser and the sample is about 1 cm. This distance is much smaller than the effective pathlength within the sample, which is about 1 m. Hence, the absorption from oxygen in the open-air path is negligible compared with the absorption of oxygen entrapped in the sample.

3 Samples and characterization

In this study, seven alumina discs are investigated, of which four are purchased and three are homemade. Three of the purchased discs are from Cobra Technologies B.V., while the fourth disc is from Zircar Ceramics, Inc. The Cobra discs are high-purity alpha alumina (α-Al2O3) (> 99.9%) with identical dimensions of 25.4 mm diameter and 2 mm thickness. The Zircar disc is a multiphase alumina with micrometer pore size distribution with the same diameter as that of Cobra discs but with a thickness of 10 mm. The three homemade discs are prepared by mixing 80 g of aluminum nitrate \({\mathrm{A}\mathrm{l}({\mathrm{N}\mathrm{O}}_{3})}_{3}\bullet {9\mathrm{H}}_{2}\mathrm{O}\) (> 98%) and 350 mL of anhydrous ethanol C2H5OH at room temperature. After the aluminum nitrate is fully dissolved, propylene oxide C3H6O (> 99%) is added drop wise while stirring until the mixture is converted into a transparent gel. The gel is then sealed with a piece of polyethylene film and kept under static conditions for one week for aging. After that, the wet gel is cut into pieces and dried at 100 °C for 2 days. The pieces’ color become transparent yellowish and their size is reduced to about half. Then, the samples are treated at 400 °C for 10 h and their color becomes white yellowish. After that, the samples are further treated at 1100 °C for another 10 h until their color becomes white. Finally, each sample is compressed under a different high pressure and shaped into a disc of a diameter 15 mm. Three different samples are produced using compression pressures of 2-ton, 3-ton and 4-ton with thicknesses of 6.2 mm, 5.8 mm and 5.2 mm, respectively. The discs are made solid and cohesive by heat treatment at 1100 °C for 10 h.

The samples are characterized using MIP, X-Ray powder Diffraction (XRD) and Scanning Electron Microscopy (SEM). Figure 3 shows the results of the MIP for the Cobra discs (\(a\)), and the homemade discs (\(b)\). It is found that the samples have narrow pore distributions with average pore diameters of \(72\) nm (Cobra 1), \(84\) nm (Cobra 2) and \(123\) nm (Cobra 3) for the Cobra discs, and average pore diameters of \(145\) nm (HM 1), \(146\) nm (HM 2) and \(144\) nm (HM 3) for the homemade discs. MIP is not performed for the Zircar disc, because it is so fragile. Its size of about \({10}^{3}\) nm is estimated roughly from its SEM image.

Fig. 3
figure 3

Distribution function of pore dimeter \(D(w)\) as a function of pore diameter \(w\) for the Cobra discs (\(\mathbf{a}\)), and for the homemade discs (\(\mathbf{b}\))

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SEM images of the samples are shown in Fig. 4. The images of Cobra 1, Cobra 2 and Cobra 3 are shown in Figs. 4a, b, and c, respectively, while the image of the Zircar disc is shown in Fig. 4d. The homemade discs have similar SEM images and here only the image of HM 1 disc is shown in Fig. 4e. Although the SEM images cannot give an accurate estimate of the average pore size, they can give a qualitative assessment. It should be noted that the pores in the SEM images have sizes that are consistent with those measured by MIP.

Fig. 4
figure 4

SEM images of a Cobra 1, b Cobra 2, c Cobra 3, and d Zircar alumina e HM 1

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The Cobra and homemade discs have XRD spectra like that of a pure α-alumina. Figure 5 shows an excellent agreement between the XRD spectrum of a reference α-alumina (black) and that of the Cobra 1 (red), and the homemade HM1 (blue). However, the XRD spectrum of the Zircar sample (pink) shows that it is not a pure α-alumina but rather a mixture of α-alumina with other alumina phases.

Fig. 5
figure 5

XRD spectra for the reference \(\alpha \)-alumina, Cobra 1, HM 1 and the Zircar disc

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4 Results and discussion

Although the samples have thicknesses of few millimeters, multiple scattering within the samples makes the effective path length long enough to observe a clear oxygen absorption signal by direct absorption spectroscopy. For example, the absorbance from oxygen in 5 mm open-air is about \({10}^{-4},\) while the observed absorbance in the samples is typically \({10}^{-2}\). Figure 6 shows the detected transmitted signal \(s\left(i\right)\) as a function of the laser injection current \(i\) from Cobra 1 disc. The signal increases almost linearly due to the variation of the laser intensity with the laser injection current, which can be fitted to a polynomial function. The dip at the middle is due to the absorption by the oxygen gas, which is expected to have an approximately Lorentzian shape. Therefore, to obtain the absorbance from the oxygen gas alone, the transmitted signal \(s\left(i\right)\) is fitted to a Lorentzian function summed with a polynomial function of order \(m\).

Fig. 6
figure 6

Raw data of the oxygen absorption line P9P9 (black points) along with a fitting curve (red line) for the Cobra 1 disc

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$$s\left(i\right)=a\frac{({\gamma /2)}^{2}}{{(i-{i}_{0})}^{2}+{(\gamma /2)}^{2}}+\sum_{n=0}^{m}{c}_{n}{\left(i-{i}_{0}\right)}^{n},$$
(1)

where the fitting parameters are \(a, {i}_{0}, \gamma , \mathrm{a}\mathrm{n}\mathrm{d} {c}_{n}.\) Here, \(a\) represents the height of the Lorentzian function, \({i}_{0}\) represents the current at which the Lorentzian function peaks, \(\gamma \) represents the full width at half maximum FWHM of the Lorentzian function measured in mA, \({c}_{n}\)’s represent the coefficients of the polynomial function, and m = 6 is the order of the polynomial function. The oxygen absorbance at the peak of the Lorentzian function can be obtained from \(-a/{c}_{0}\).

Figure 7 is the normalized absorbance as a function of relative frequency obtained from the signal of Fig. 6 using Eq. 1. The maximum absorbance is \(1.66\times {10}^{-2}\), which corresponds to an effective oxygen absorption path length of 69 cm in open-air. The absorbance peak full width at half maximum \(({\Gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}})\) is \(5.12 \mathrm{G}\mathrm{H}\mathrm{z}\). As can be observed, the absorbance fits well to a Lorentzian function.

Fig. 7
figure 7

Normalized absorbance for the P9P9 molecular oxygen line (black points) along with a fitting curve (red line) for the Cobra 1 disc

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Figure 8 shows normalized absorbance spectra for the P9P9 oxygen molecular line in open-air as well as in Cobra 1 and HM 1 discs. The spectrum in open-air has a FWHM of \(3.30\) GHz while the FWHM of the Cobra 1 is \(5.12\) GHz and that of HM 1 is \(4.17\) GHz. The uncertainty in the FWHM measurements is estimated to be 0.05 GHz, which is the standard deviation of ten different linewidth measurements for each sample. For all samples, the uncertainty is found to be approximately the same.

Fig. 8
figure 8

Normalized absorbance of the P9P9 molecular oxygen line from open-air as well as from Cobra 1 and HM 1 discs

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The open-air oxygen absorption linewidth \({\Gamma }_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}-\mathrm{a}\mathrm{i}\mathrm{r} }=3.30\) GHz is a result of convoluting the collision-broadened P9P9 line, the Doppler-broadened P9P9 line, and the laser line. The linewidth of the laser alone is 0.1 GHz, the linewidth due to collision alone is 2.94 GHz according to HITRAN database [26], while the linewidth due to Doppler broadening alone is 0.86 GHz, which is calculated from \(\sqrt{8kT\mathrm{ln}2/m{\lambda }^{2}}\), where \(k\) is the Boltzmann constant, \(T\) is the absolute temperature, \(m\) is the molecular mass of oxygen molecule, and \(\lambda \) is the wavelength. Since the line shapes due to collision broadening and due to the laser have Lorentzian profiles, their linewidths add up to 3.04 GHz. The line shape due to Doppler broadening has a Gaussian profile and its convolution with a Lorentzian of a linewidth of 3.04 GHz results in a linewidth of 3.28 GHz, in agreement with the measured linewidth.

When oxygen is confined in pores with sizes comparable or smaller to the intermolecular mean free path, extra broadening occurs due to the collisions of oxygen molecules with the pore walls. This broadening is commonly known as wall-collision broadening and its contribution to the total linewidth \({\Gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\) will be called \({\Gamma }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}\). Many studies on collision broadening in the microwave region [27], as well as in the near infrared region [13, 22], assume that the broadening contributions from the wall and intermolecular collisions are approximately additive. In our study, the shapes of total absorption line from the sample \({\Gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\) and from open-air \({\Gamma }_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}-\mathrm{a}\mathrm{i}\mathrm{r}}\) are approximately Lorentzian, as in Fig. 6. Therefore, it can be assumed that the contribution from the wall-collision broadening is also Lorentzian. Thus,

$${\Gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\Gamma }_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}-\mathrm{a}\mathrm{i}\mathrm{r}}+{\Gamma }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}.}$$
(2)

The broadening from the wall collisions alone can be estimated using a model based on the kinetic theory of gases [28]. According to the kinetic theory of gases, the average time between two consecutive collisions with the pore walls \({\tau }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}\) is

$$\frac{1}{{\tau }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}}=\frac{A}{V}\frac{\bar{v}}{4},$$
(3)

where \(A\) is the pore surface area, \(V\) is the pore volume, and \(\bar{v}=\sqrt{8kT/\pi m}=445 \mathrm{m}/\mathrm{s}\) is the average speed of oxygen molecules. Hence, the broadening due to wall collisions alone is given by [28, 29]:

$${\Gamma }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}=\frac{1}{\pi {\tau }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}}=\frac{A}{\pi V}\frac{\bar{v}}{4}.$$
(4)

Although the shapes of the pores in our samples are irregular and hence their volumes and areas cannot be determined exactly, the ratio of the pore volume to its area should scale with the average diameter of the pore \(\stackrel{-}{w}\), if the shapes of the pores in different samples have almost the same morphology, as in the case in our samples, see Fig. 4. Therefore, the broadening due to wall collisions alone in the samples should scale with the inverse of the average pore size \(\stackrel{-}{w}\)

$${\Gamma }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}\propto \frac{1}{\stackrel{-}{w}}. $$
(5)

Figure 9 shows the measured line broadening due to wall collisions \({\Gamma }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}\) as a function of the average pore diameter \(\stackrel{-}{w}\) measured by MIP. It is clear, that the linewidth of the P9P9 absorption line increases with decreasing average pore diameter. This trend can be explained by the fact that with smaller average pore diameter, oxygen molecules collide more frequently with the walls of the pores. In the same context, for the case of the Zircar disc, there is almost no broadening due to wall collisions. This is because the Zircar disc has an average pore diameter of about \({10}^{3} \mathrm{n}\mathrm{m}\), which is much bigger than the intermolecular mean free path of oxygen in air of about 100 \(\mathrm{n}\mathrm{m}\).

Fig. 9
figure 9

The line broadening due to wall collisions \({\Gamma }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}\) as a function of the measured average pore diameter \(\stackrel{-}{w}\) by MIP. The solid line is a fit to a function of the form \(b/\stackrel{-}{w}\), where \(b\) is a fitting parameter. The insert is a magnification of the region between 60 and 180 nm

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Figure 9 also shows that \({\Gamma }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}\) fits very well to a function of the form \(b/\stackrel{-}{w}\), as indicated by the solid line in the figure, where \(b\) is a fitting parameter and is found to be 131 \(\mathrm{G}\mathrm{H}\mathrm{z}\bullet \mathrm{n}\mathrm{m}\). The shape of the pores might be modeled as a simple shape like a sphere [30] or a cylinder [8, 31]. If a spherical pore shape is considered, \(A/V=6/\stackrel{-}{w}\) and thus the fitting parameter \(b\) should be \(3\stackrel{-}{v}/2\pi =212 \mathrm{G}\mathrm{H}\mathrm{z}\bullet \mathrm{n}\mathrm{m}.\) If a cylindrical pore shape of diameter \(\stackrel{-}{w}\) and height \(h\) is considered,

$$\frac{A}{V}=\frac{\pi {\stackrel{-}{w}}^{2}/2+\pi \stackrel{-}{w}h}{\pi {\stackrel{-}{w}}^{2}h/4}=4\left(\frac{1}{2h}+\frac{1}{\stackrel{-}{w}}\right).$$

For a cylinder with height much greater than its diameter, \(A/V\cong 4/\stackrel{-}{w}\) and hence the fitting parameter \(b\) should be \(\stackrel{-}{v}/\pi =141 \mathrm{G}\mathrm{H}\mathrm{z}\bullet \mathrm{n}\mathrm{m}\), which is very close to the experimentally obtained fitting parameter. Consequently, the pores in our samples are better modeled as cylinders with large heights compared to their diameters than spherical pores. However, the fitting parameter might vary if the samples have different morphology.

The model based on pores of long cylindrical shape, introduced in this work, is found to yield much better agreement, not only with the measurements presented in this work, but also with measurements reported in literature [21, 30, 32, 33]. Previously, a model based on pores of spherical shape is used in Ref [30]. and it obtains an average pore size, extracted from the wall-collision broadening, almost double that measured by MIP. Figure 10 shows wall-collision broadening as a function of average pore diameter measured using MIP for this work as well as those presented in Ref.[30]. The figure shows three curves of the form \(b/\stackrel{-}{w}\); one is the fit of Fig. 9, one is using parameter \(b\) for long cylindrical pores, and the last one is using parameter \(b\) for spherical pores. It is clear that the measurements are better modeled with pores of long cylindrical shape than pores of spherical shape. It should be noted that the MIP measurements usually give smaller average diameter than the real ones [8, 31]. If this factor is taken into consideration, the model may even yield better fitting. Certainly, further investigation is needed to have better understanding of the relationship between the average pore shape and the wall-collision broadening.

Fig. 10
figure 10

Comparison between measurements and models based on kinetic theory of gases. The dashed curve is for broadening due to wall collision using a model with spherical pores, the dotted curve is for broadening due to wall collision using a model with long cylindrical pores, and the solid curve is the fit of Fig. 9

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5 Conclusions

The GASMAS method is employed to investigate absorption line broadening due to wall collisions of entrapped oxygen gas in nanoporous solid discs. The molecular oxygen line P9P9 at 763.8 nm is selected to study commercial and homemade porous alumina discs with pore sizes over the range 70–150 nm. One sample with an average pore diameter of about \({10}^{3}\) nm, which is much larger than the mean free path of intermolecular collisions in open-air, is also investigated; however, no significant wall-collision broadening is observed. The overall absorption line shape is found to be approximately Lorentzian, which is similar to that in open-air. This indicates that the broadening due to wall collisions can be also approximated to be Lorentzian. As predicted by a model based on the kinetic theory of gasses, the wall-collision broadening is found to be inversely proportional to the average pore diameter measured by MIP. The fitting parameter is found to be more consistent with a model of the samples having pores of cylindrical shapes of large heights rather than pores of spherical shapes. Certainly, more work is needed to reach definitive conclusion on the relation between the fitting model and pore morphology.