Laser spectroscopic studies of gas diffusion in alumina ceramics

Hao Zhang; Sune Svanberg

doi:10.1364/OE.24.001986

1. Introduction

Gas diffusion is a very important mechanism for gas transport in various natural and man-made porous media. A lot of research has been pursued on gas diffusion through porous media, a field which has many scientific and engineering applications, including soil mechanics [1,2], plant respiration [3] and performance of polymer electrolyte fuel cells [4,5]. The most commonly used method to describe the phenomena is numerical simulation based on theoretical models, such as the Monte Carlo (MC) method [6] and the Lattice Boltzmann method (LBM) [7]. In recent years, some techniques have been developed to study the transport characteristics of gas diffusion in porous media, such as inverse gas chromatography [8], magnetic resonance imaging (MRI) [9], photothermal deflection [10], and nuclear magnetic resonance (NMR) [11,12].

In the present work, a high-resolution laser spectroscopy method, called gas in scattering media absorption spectroscopy (GASMAS), was used to experimentally study the process of gas diffusion in alumina ${(Al}_{2} O_{3})$ ceramics. Employing wavelength modulation and lock-in detection, GASMAS was used to detect the weak gas absorption imprint of free gas, dispersed in scattering materials, with a detection limit of 10⁻⁵ absorption fraction [13]. Actually, it has been shown that GASMAS can be used to study the characteristics of ${Al}_{2} O_{3}$ ceramic non-intrusively. The first laser spectroscopic studies of ceramics were demonstrations of line broadening due to wall interactions [14,15]. It was also demonstrated that porous strongly scattering ceramics could be used as a miniature multi-pass gas cells [16]. Mei et al. have used the GASMAS technique to study the optical porosity and optical properties (such as refractive index, reduced scattering coefficient) of ceramics combined with frequency domain photon migration [17].

In the present paper, gas diffusion was studied by first keeping a ceramic sample in a gas-flushed transparent plastic bag for a prolonged time, and then opening it for re-invasion of ambient air. GASMAS was used to estimate the steady-state optical path length through gas, and the time constant for gas diffusion through the ceramic by measuring the oxygen gas signal. The time constant τ is referred to as the time that the oxygen signal takes to reach a factor of 1/e of its value at the onset of the process. Moreover, the effect of wet pore spaces on gas diffusion was investigated by measuring the oxygen signals during gas diffusion through wet and dry ceramics, respectively. Porosity is one of the key parameters for the studied media, and an investigation of the influence of porosity on gas diffusion was performed by measuring the oxygen signal during gas diffusion through ceramics with 45% and 70% porosity.

2. Theory

The pore size of ceramics used in this study is about 200 µm, which is out of the predominant region for Knudsen diffusion, i.e., less than 100 nm at standard pressure and temperature conditions [18,19]. Therefore, Knudsen diffusion is assumed negligible. For one-dimensional gas diffusion, Fick’s second law of diffusion can be used to obtain the gas concentration in the transient process. Here we assume that gas diffusion is the only dynamic process and that the diffusion is one-dimensional. Then the diffusion equation along the z-axis can be expressed by

\frac{\partial C (z, t)}{\partial t} = D_{e f f} \frac{\partial^{2} C (z, t)}{\partial z^{2}},

where

C (z, t)

is the concentration and

D_{e f f}

is the diffusion coefficient (m²/s). For one-dimensional diffusion in a slab, assuming that the initial concentration is

C_{0}

and that the sample is surrounded by a constant concentration

C_{1}

, Eq. (1) can be solved, subjected to the initial and boundary conditions:

{\begin{array}{l} C (z, t) |_{t = 0} = C_{0} - d / 2 < z < d / 2 \\ C (z, t) |_{z = - d / 2} = C (z, t) |_{z = d / 2} = C_{1} t \geq 0 \end{array},

where d is the thickness of the ceramic sample. As described in [20], the method of separation of variables can be used to obtain one solution as [21]

C (z, t) = C_{1} - (C_{1} - C_{0}) \frac{4}{π} \sum_{n = 1}^{\infty} \frac{{(- 1)}^{(n - 1)}}{2 n - 1} \exp {- D_{e f f} t {(\frac{(2 n - 1) π}{d})}^{2}} \cos \frac{(2 n - 1) π z}{d},

where

C (z, t)

is the time dependent spatial concentration distribution.

Due to the multiple scattering of light in the scattering medium, a probability function $P (z; - \frac{d}{2}, \frac{d}{2})$ can be used to describe the probability that light passes the position z. M. Sjöholm et al. [21,22] have used a Monte Carlo method to quantify the sampling probability, and then spatially integrated the time- dependent gas concentration at different locations by multiplying with the corresponding sampling probability $P (z; - \frac{d}{2}, \frac{d}{2})$ , a time-dependent gas absorption imprint can be obtained according to

S (t) = A - \sum_{n = 1}^{\infty} B_{n} \exp (- {(2 n - 1)}^{2} \frac{t}{τ_{0}}),

where

τ_{0} = \frac{d^{2}}{D_{e f f} π^{2}} .

The time constant

τ

in the series expansion satisfies

τ = \frac{τ_{0}}{{(2 n - 1)}^{2}},

so the longest time constant in the expansion is

τ_{0}

. We note, that the next term in the expansion corresponds to a 9 times faster time constant, and with a weight which should be 9 times lower.

3. Materials and methods

3.1 Materials

Two types of porous alumina ${(Al}_{2} O_{3})$ ceramics with the porosity of 45% and 70% were manufactured by Cen-Lon Ceramics, Luoyang, China. The ceramic with the porosity of 45% was sintered with 0.5 µm size alumina powder, while the ceramic with the porosity of 70% was made out of 2 µm size alumina powder. The samples were produced by sintering the powder for 48 hours under a high temperature of 1700 degrees, and the average pore size is about 200 µm. The ceramics were shaped into discs of 10 cm diameter, and with the thickness of 5 mm, 10 mm, 15 mm, 20 mm and 25 mm. We used two samples of each thickness.

3.2 Experimental setup

The experimental setup is illustrated in Fig. 1. A diode laser (LD-0760-0100, Toptica, Munich, Germany) is operated at the oxygen absorption line at 760. 445 nm by employing a laser driver with a current and a temperature controller (LCD 201C and TED 200C, respectively, Thorlabs, Newton, New Jersey). The current is modulated with a superposition of a 5 Hz ramp wave and a 10295 Hz sinusoidal wave which is produced by a data acquisition (DAQ) card (NI6120, National Instruments, Texas, USA). The ramp wave is used to scan through the absorption line, and the sinusoidal wave is used to modulate the laser to acquire the harmonic signals. The output of the diode laser is coupled to an optical fiber with a core diameter of 600 µm, and transmitted light through the ceramic is collected by a photomultiplier tube (PMT, H10722-01, Hamamatsu Photonics, Hamamatsu, Japan). The output of the PMT is amplified by a variable-gain low-noise current amplifier (DLPCA-200, Femto, Berlin, Germany) and then recorded by an input channel of the DAQ card. The acquired data are further analyzed with the methods of digital lock-in amplification and intensity correction; the detailed signal processing is described in the next section.

Fig. 1 System schematics of the experimental setup for GASMAS. The system includes the measurement subsystem, the data acquisition subsystem and a data analysis system. System control and data analysis are achieved by a LabVIEW program. Abbreviations: PD (photodiode), LD (laser diode), PMT (photomultiplier tube), AO (analog output), AI (analog output), and DAQ (data acquisition).

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3.3 Signal processing

As shown in Fig. 1, the methods utilized for signal processing include digital lock-in amplification and intensity correction [23, 24], where the digital lock-in amplifier process is used to extract the desired harmonic signals, and intensity correction is used to obtain a signal which is not depending on the amount of light received by the detector.

The acquired raw data from the DAQ card can be recorded as $s (t) .$ Firstly, Fourier transformation is used to obtain the signal in the Fourier domain, i.e., $S (ω) = F (s (t)) .$ As displayed in Fig. 2(b), individual harmonics of the modulation frequencies appear in the Fourier spectrum. By performing band-pass filtering, the digital lock-in amplification action can be achieved to obtain a specific harmonic. Here, a super-Gaussian window centered at $ω_{m}$ with a width of $δ ω$ is used, and the corresponding signal $S_{n f} (ω)$ in the frequency domain is expressed by

S_{n f} (ω) = 2 \times S (ω) \times e x p (- {(\frac{ω - n \times ω_{m}}{δ ω})}^{8}),

where

ω_{m}

is the modulation frequency, and the factor 2 is used to compensate for the loss of the signal amplitude of the negative frequencies.

Fig. 2 (a) Raw data for molecular oxygen measured around 760 nm on a 1000 mm path through ambient air, and (b) its corresponding Fourier spectrum.

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Subsequently, the signal in the time domain is obtained by performing an inverse Fourier transformation, i.e. $s_{n f} (t) = F^{- 1} (S_{n f} (ω))$ . Since $s_{n f} (t)$ is a complex quantity, the real signal is achieved after offset removal and phase adjustment, as shown by

{\bar{s}}_{n f} (t) = Re {(s_{n f} (t) - m e a n (s_{n f} (t)) \times \exp (- i β_{n}))},

where

mean (s_{n f} (t))

is the mean value of the

n f

signal, and

β_{n}

is the phase of the

n f

signal. Furthermore, an intensity-corrected signal is obtained by performing intensity correction (i.e., normalizing the gas absorption signal with the direct signal

s_{D} (t)

), as given by

{\tilde{s}}_{n f} (t) = \frac{{\bar{s}}_{n f} (t)}{s_{D} (t)} .

3.4 Data analysis

As described in previous papers on GASMAS measurements [13,25–27], because of the strong scattering of the light travelling in porous media, the gas concentration in the medium is estimated by introducing an equivalent mean path length $L_{e q} .$ Frequently, molecular oxygen is studied and then $L_{e q}$ is the distance that the light travels in ambient air to get the same absorption imprint as that from the absorption of light in the porous media. According to the Beer–Lambert law, $L_{e q}$ can be described by the equation

C_{s} \times L_{s} = C_{a i r} \times L_{e q},

where

C_{a i r}

is the gas concentration in ambient air, L_s is the mean path length that light travels through gas in the sample, and

C_{s}

is the gas concentration in the sample. Therefore, the gas concentration can be determined by quantifying the

L_{e q} .

Here, a nonlinear Levenberg-Marquadt fitting model [23] is utilized to quantitatively analyze the intensity-corrected 2f signals, and the fitting model is given by

y_{s} (t) = p_{0} + p_{1} \times t + p_{2} \times t^{2} + k \times y_{r e f} (t - t_{0}),

where

y_{s} (t)

and

y_{r e f} (t)

is the intensity-corrected 2f signal in ceramics and 1000 mm ambient air, and the second-order polynomial is used for baseline correction. The factor k gives the fraction of the reference signal needed to describe the sample signal, and the shift parameter

t_{0}

is introduced to account the differences of the absorption location due to drifts in the operating temperature and current between calibration and actual measurements. Thus, the

L_{e q}

is calculated from

L_{e q} = k \times L_{a i r},

where

L_{a i r}

is the distance in ambient air, i.e., 1000 mm in this study.

4. Results and discussion

4.1 Studies of gas diffusion through pure nitrogen and oxygen filled ceramics

To demonstrate the possibility of using the GASMAS technique for gas diffusion in ceramics and the feasibility of the theoretical model, two 25 mm thick ceramic samples with 45% porosity were selected for measurements. Each sample was placed in a sealed nitrogen-flushed bag for 4 hours to obtain the pure nitrogen-filled pore space, and then the bag was opened to study the oxygen diffusion from the ambient air into the pure nitrogen-filled ceramic by measuring the invading oxygen absorption signals. After that, the same sample was placed in a sealed oxygen-flushed bag for 4 hours to obtain the pure oxygen-filled pore space, to study the oxygen diffusion from 100% oxygen-filled ceramic spread to the ambient air. The measured equivalent mean path length $L_{e q}$ of oxygen signals as a function of time is shown in Fig. 3.

Fig. 3 Measurements of the oxygen diffusion through two 25 mm ceramic samples with initially pure nitrogen and oxygen filled pores, respectively. Exponential curves were used to fit the experimental data, the blue and red curves are the fitting results for two ceramics originally filled with oxygen, while the black and green curves are the fitting results for two ceramics originally filled with nitrogen. The corresponding average time constant is about 253 s and 152 s.

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As can be seen in Fig. 3, the experimental data show an exponential change for both nitrogen- and oxygen-filled porous ceramics. When fitting the experimental data to a single exponential, the fitted R-square value is close to 1, which demonstrates that a good agreement between the theoretical model with a single decay constant and experimental results can be achieved. In the figure, the average time constant for the ceramics with pure oxygen-filled pore space is 253 s, which is longer than the average time constant - 152 s - for the ceramics with pure nitrogen-filled pore spaces. The results illustrate that it takes a longer time for the ceramics filled with oxygen to reach the steady-state than the same ceramic sample filled with nitrogen. The possible reason may come from the different molecular weight of oxygen and nitrogen. As is well-known, diffusion is the movement of molecules from a region of high concentration to a region of low region. The rate of diffusion is directly proportional to the pressure and temperature, and is faster for light molecules. While nitrogen is lighter than oxygen it thus travels faster for the same pressure and temperature, yielding faster diffusion. It is interesting that the same tendency has already been seen for fruit gas diffusion, as described in [28].

4.2 Validation of 100% humidity in pore spaces of water-immersed ceramics

To study the effect of wet pore space of ceramics for gas diffusion, the ceramic samples were immersed with liquid water for 20 hours to obtain a 100% humidity in the pore spaces. Firstly, water vapor measurements for one 25 mm thick and previously water-immersed ceramic sample with the porosity of 45% were performed to investigate if 100% humidity could be obtained. With the same experimental setup, a 937 nm diode laser was used to detect the water vapor signal in the ceramics, and further to demonstrate the 100% humidity of water vapor in the wet ceramics during measurements. In the laboratory, the temperature was about 25°C and the humidity was about 50%, as recorded by a digital temperature hygrometer. In addition, another sample of the same type of ceramic, now with dry pore spaces, was studied. The measured water vapor signals in the 25 mm thick ceramic samples with the porosity of 45% in the condition of dry pore spaces and wet pore spaces, respectively, are shown in Fig. 4(b).

Fig. 4 (a) Transmitted light intensity detected in the 25 mm wet ceramic sample. (b) The measured water vapor signals in the 25 mm dry gas-filled ceramic and wet ceramic samples. The time constant for ceramic sample with dry pore space is about 331 s.

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As discussed in [25] for wood drying measurements, the reduction of liquid water is accompanied with the increased scattering due to the lowered index-matching effect, and the transmitted light has a significant change. However, in this study, the transmitted light did not change much in the short measurement time (less than 1 hour), as shown in Fig. 4(a). It can be said that the scattering (i.e., index mis-matching effect) is not changing much during the measurements.

From Fig. 4(b), it can be seen that the measured $L_{e q}$ of the water vapor shows an exponential change during the gas diffusion through the dry ceramic material, which is similar to the measurements of oxygen signals. Instead, the $L_{e q}$ shows stability during the gas diffusion through the wet ceramic sample. The water vapor concentration is dependent on the relative humidity and the temperature, where the effect of temperature is given by the Arden-Buck equation [29]. In this study, the temperature is constant, and the changes in the water vapor concentration only depend on the relative humidity. Therefore, the stability of $L_{e q}$ means that the 100% relative humidity of saturated water vapor was achieved during the measurements. On the other hand, it can be clearly seen that the estimated time constant of water vapor signal is much larger than that of oxygen signals when comparing Fig. 4 with Fig. 3. As discussed before, it could be due to that water vapor has larger molecules than oxygen and nitrogen, thus yielding slower diffusion. Therefore, it is clear, that the GASMAS technique can be used to differentiate the diffusion processes for different gases.

4.3 Studies of the effect of moist pore space on gas diffusion

In this part of the study, only ceramic samples with pure nitrogen filled pores were selected for investigations of gas diffusion. At first, the dry air-filled porous ceramic samples with the porosity of 45% and the thickness of 10 mm, 15 mm, 20 mm and 25 mm were placed separately in sealed nitrogen-flushed bags for 4 hours, and then the sealed bags were opened for measurements of gas diffusion. The oxygen absorption signals were recorded continuously until equilibrium was achieved. After the measurements on ceramic samples with dry gas-filled pores, the samples were immersed with liquid water for 20 hours. As described above, 100% humidity of saturated water vapor in the pores of ceramics was achieved after immersing. Subsequently, the same procedure was performed for gas diffusion through ceramics with moist pore spaces. The recorded time evolution of the equivalent mean path length $L_{e q}$ from the 25 mm samples together with the fitted exponential curve is shown in Fig. 5, including the residuals in the bottom of the figure.

Fig. 5 Measured oxygen transient $L_{e q}$ from two 25 mm thick ceramic samples with dry gas-filled pores and 100% humidity of gas-filled pores. The blue and red curves are the exponential fitting results for two ceramics with dry gas-filled pores, and the black and green curves are the exponential fitting results for the same ceramics but with 100% humidity. The fitting residuals are presented at the bottom of the figure.

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As can be seen in Fig. 5, the 25 mm thick ceramic samples with dry gas-filled pores have slightly larger steady-state $L_{e q}$ values than those with wet gas-filled pores, while it needs slightly shorter time to reach the equilibrium than those with wet gas-filled pores. According to the fitted results, the $L_{e q}$ value and the time constant of ceramics with thickness of 10 mm, 15 mm, 20 mm and 25 mm were estimated, as shown in Fig. 6.

Fig. 6 Measured steady-state $L_{e q}$ and time constant for gas diffusion through dry and water-immersed ceramics with the thickness of 25 mm (red line), 20 mm (green line), 15 mm (blue line) and 10 mm (black line). The error bars for measurement on two ceramics of the same type are shown in the figure.

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It can be obviously seen that the steady-state $L_{e q}$ values are smaller for all ceramics immersed in water with different thicknesses when comparing with ceramics with dry pores. It can be explained by the fact that part of the pore volume of the ceramic is occupied by liquid water, while liquid water is an index-matching fluid resulting in a decrease of optical scattering. Actually, studies of wood drying as described in [25] have already shown that the equivalent mean path length $L_{e q}$ has a dramatic change during the drying process when the water-filled pores of wood are changing into air-filled. However, the time constants exhibit a large difference for ceramics in the condition of 50% (dry pore) and 100% (moist pore) humidity, which means that the presence of water in the pores is a critical factor for oxygen diffusion through porous ceramics. It could be explained by the fact that water-filling of the ceramic pore spaces would greatly reduce the effective area for gas diffusion, resulting in an increase of the time constant. Furthermore, it can be inferred that the effective diffusion coefficient of oxygen should be increased with the decreasing saturation of water vapor from 100%. For a given porous medium, although porosity and particle size distribution are usually constant, the water content often varies over time. Therefore, the effect of the saturation of water vapor on gas transport rate should be considered when studying the gas diffusion through porous media.

As expected, the time constant for different thickness of ceramics is different. Generally speaking, the time that it takes to achieve equilibrium becomes longer the thicker the ceramic is. As stated in Eq. (5), the time constant should show a characteristic quadratic dependence on the thickness. Actually, this behavior was not obtained in this study. The theoretical model has already been used to measure the gas diffusion coefficient by studying the gas diffusion through porous media, such as soils and photonic crystal fibers [30,31], the results have demonstrated that the measured gas diffusion coefficient is very similar with the reference diffusion coefficient. Many possible parameters could be involved in explaining the discrepancy, such as the inhomogeneity of materials, the inter-diffusion between multi-component gases in the air, the influence of confining walls. Further, the theoretical model may not be fully adequate. Especially for fractal porous media, the Fick’s second law is not applicable to unsteady gas diffusion. A modified Fick’s second law was established by introducing a correction factor $δ (t),$ $δ (t) = δ_{1} - \frac{δ_{2} t}{t + δ_{3}},$ where $δ_{1},$ $δ_{2}$ and $δ_{3}$ are constants related only to porosity, specific surface area, and fractal dimension, respectively [32,33]. Thus, the diffusion equation along the z-axis can be expressed by

\frac{\partial C (z, t)}{\partial t} = δ (t) D_{e f f} \frac{\partial^{2} C (z, t)}{\partial z^{2}} .

4.4 Studies of the effect of porosity on gas diffusion

The measurements were performed in a similar way as in Section 4.1 - the ceramic samples with the porosity 45% and 70% and the thickness of 10 mm, 15 mm, 20 mm and 25 mm were measured during the transition from full nitrogen filling to ambient air conditions. The time evolution of the equivalent mean path length $L_{e q}$ from the 25 mm ceramic samples is shown in Fig. 7, including the residuals in the bottom of the figure. We note that the fit is quite good, but in this particular case, at early times, there is a deviation, which could describe a weak component with a 9 times shorter time constant, as suggested by theory.

Fig. 7 The measured oxygen transient $L_{e q}$ values from two 25 mm thick ceramic samples with the porosity of 45% and 70%, respectively. The blue and red curves are the exponential fitting results for two ceramics with the porosity 45%, and the black and green curves are the exponential fitting results for two ceramics with the porosity 70%. The fitting residuals in the bottom of the figure show some evident deviation in the time range of 0-200 s.

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As expected from Eq. (3), the time evolution of the oxygen equivalent mean path length is found to follow an exponential curve. However, the steady-state $L_{e q}$ value between the 25 mm thick ceramics with the porosity of 45% and 70% has a large difference - about a factor of 2. Obviously, the time constant between the two kinds of ceramics is different as well. The quantified results of the steady-state $L_{e q}$ and time constant for the ceramics with the thickness of 10 mm, 15 mm, 20 mm and 25 mm are obtained by fitting the experimental data to exponentials with results shown in Fig. 8.

Fig. 8 The measured steady-state $L_{e q}$ and time constant for gas diffusion through the 45% and 70% porosity ceramics with the thickness 25 mm (red line), 20 mm (green line), 15 mm (blue line) and10 mm (black line). The error bars for measurement on two ceramics of the same type are shown in the figure.

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It can be seen from Fig. 8 that the $L_{e q}$ value is significantly larger for the ceramics with 70% porosity. The high porosity causes the increase of gas-filled pore volume, resulting in the increase of oxygen absorption signal. Actually, this is not the only reason for the major increase of $L_{e q}$ ; the increase of the reduced scattering coefficients related to the size of the alumina powder used will result in an increase of pore path length. Our ceramics with 45% porosity were sintered with 0.5 µm size alumina powder, while the ceramics with 70% porosity were sintered with 2 µm alumina powder. In [17], Mei et al. described that a large difference of pore path length was found for ceramics sintered with different size of alumina powder. As can be seen in Table 1, the reduced scattering coefficients increase dramatically for 70% porosity ceramics sintered with 2 µm size of alumina powder.

Table 1. Effective refractive index n_eff, reduced scattering coefficients µ^'_s, and absorption coefficients µ_a for the ceramics. (Data come from [17])

View Table

A lot of theoretical research has proved that the diffusion coefficient for gas diffusion in porous media is dependent on the air-filled porosity, while a uniform relational expression is not existing, because of the complication for gas diffusion in practice [34–36]. Figure 8 shows that the time constant is significantly smaller for the ceramics with 70% porosity, which means that the oxygen diffusion coefficient in 70% porosity ceramics is less than that in 45% porosity ceramics. In this study, the effect of porosity on gas diffusion through porous media is experimentally shown with laser spectroscopy. However, more research is required to study the relationship between diffusion coefficient and air-filled porosity.

As discussed above, the diffusion coefficient depends on air-filled porosity and water content in pore spaces. While it was not considered in the theoretical model used in this study as expressed by Eq. (1), here an optimized Fick’s second law [1,37] could be used to study the time- and space-dependent gas concentration by introducing a correction factor $θ_{e q}$ , as expressed by

θ_{e q} \frac{\partial C (z, t)}{\partial t} = D_{e f f} \frac{\partial^{2} C (z, t)}{\partial z^{2}},

where

θ_{e q}

is the equivalent diffusion porosity. It is defined here as

θ_{e q} = θ_{a} + H θ_{w},

where

θ_{a}

is the volumetric air contents,

θ_{w}

is the volumetric water contents, and H is a dimensionless constant [38]. As is well-know, the structure of ceramics is very complicated and inhomogeneous, it is possible that the theoretical model for fractal porous media is applicable to ceramics. If Eq. (10) could be used to exactly explain the gas diffusion through ceramics, when combined with Eq. (11), the time-dependent correction factor can be expressed as

δ^{'} (t) = δ^{'}_{1} - \frac{δ_{2} t}{t + δ_{3}},

where

δ_{1}^{'} = θ_{a} + H θ_{w} .

5. Conclusions

The present work demonstrates that the GASMAS technique could be used for investigations of gas diffusion in porous media, illustrated for the case of porous ${Al}_{2} O_{3}$ ceramics. The studies of gas diffusion were implemented by measuring the time evolution of the oxygen concentration, which is monitored by the changes in the equivalent mean path length $L_{e q} .$ The measured oxygen signal during gas diffusion through pure nitrogen and oxygen filled ceramics is a time-dependent exponential curve, and it is demonstrated that a good agreement is obtained between the theoretical model and experimental results. In particular, it was shown that a single exponential well describes the diffusion process. Moreover, the effect of water content in the pore space on the gas diffusion was investigated by comparing the measured results for different thicknesses of ceramics, with dry gas-filled pores and 100% saturated water-exposed pores, respectively. The effect of porosity on the gas diffusion was investigated by measuring the oxygen signal for different thicknesses of ceramics with 45% porosity and 70% porosity, respectively. It is found that liquid water accumulation in the pore spaces leads to a time extension of gas diffusion, while high porosity makes the time shorter. Based on the experimental results that the gas diffusion coefficient is dependent on the gas-filled porosity and water content in the pore space, a modified Fick’s second law could be used to further optimize the theoretical model.

In addition, the equivalent mean path length $L_{e q}$ obtained at steady-state is different for ceramics with different humidity and different porosity, respectively. The changes of $L_{e q}$ should come from the changed scattering characteristics. In particular, the dramatic change of reduced scattering coefficient induced by the particle size used for ceramic sintering is reflected in the measured $L_{e q},$ just as discussed by Mei et al. [17]. However, it should be noted that the time constant is not quadratically dependent on the thickness as expected from Eq. (5). That means that the theoretical model based on Fick’s second law is not perfect for gas diffusion in ceramics. Here we suggest to use a modified Fick’s second law by introducing a time-dependent correction factor, which is related to the porosity and pore structure.

The GASMAS technique was shown was shown to have the feasibility to study various aspects of gas diffusion through ceramics. In particular, it should be noted that GASMAS provides the potential for applications to a large variety of natural porous materials, provided that the bulk material absorption is not excessive. Another interesting thing is that the effective diffusion coefficient $D_{e f f}$ could be calculated by fitting the theoretical model with the experimental data, as mentioned in previous references [1,30,31].

Acknowledgments

The authors are grateful to Prof. Sailing He and Prof. Katarina Svanberg for support. This work was financially supported by a Guangdong Province Innovation Research Team Program (No. 201001D0104799318).

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Laser spectroscopic studies of gas diffusion in alumina ceramics

Abstract

1. Introduction

2. Theory

3. Materials and methods

3.1 Materials

3.2 Experimental setup

3.3 Signal processing

3.4 Data analysis

4. Results and discussion

4.1 Studies of gas diffusion through pure nitrogen and oxygen filled ceramics

4.2 Validation of 100% humidity in pore spaces of water-immersed ceramics

4.3 Studies of the effect of moist pore space on gas diffusion

4.4 Studies of the effect of porosity on gas diffusion

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Tables (1)

Equations (11)

Optics Express

Porosity	n_eff	µ^'_s/mm⁻¹	µ_a/mm⁻¹
45%	1.5	4.1	0
70%	1.44	44.7	0