Review of Developed Methods for Measuring Gas Uptake and Diffusivity in Polymers Enriched by Pure Gas under High Pressure

Abstract

Gas emission and diffusion through polymeric materials play crucial roles in ensuring safety and monitoring gas concentrations in technology and industry. Especially, the gas permeation characteristics for O-ring material should be investigated for sealing application in a hydrogen infrastructure. To accommodate the requirements of different environments, we first developed four complementary effective methods for measuring the gas absorption uptake from polymers enriched by pure gas under high pressure and determining the gas diffusivity. The methods included the gravimetric method, the volumetric method, the manometric method, and gas chromatography, which are based on mass, volume, pressure, and volume measurements, respectively. The representative investigated results of the developed methods, such as gas uptake, solubility, and diffusivity are demonstrated. The measuring principles, measuring procedures, measured results, and the characteristics of the methods are compared. Finally, the developed methods can be utilized for testing transport properties, such as the leakage and sealing ability, of rubber and O-ring material under high pressure for hydrogen fueling stations and gas industry.

1. Introduction

Gas permeation in materials plays a crucial role in industrial applications [1,2,3,4,5,6,7] in various fields, such as polymer electrolytes, batteries, catalysts, protective coatings, fuel cells, gas membrane separation, gas monitoring sensors, food packing, gas storage vessels, and O-ring seals. Permeation is the penetration of a permeate through a polymer membrane. Gas permeability is a physical and chemical process [8,9,10] that includes the absorption, diffusion, and desorption of gas molecules. Permeation works through diffusion, which is the key process for determining the permeability and solubility of a gas. The diffusion process is modeled by Fick’s law of diffusion [11,12,13]. Diffusion is the process in which permeation is driven by a gas concentration gradient. Thus, the gas molecules move from high concentration regions to low concentration regions.

There are various methods for characterizing gas transport properties, such as solubility, diffusivity, and permeability [14,15,16,17,18,19]. Generally, researchers utilize the differential pressure, gas chromatography, and magnetic suspension balance methods.

A standard method for determining gas permeability is the differential pressure method, using a gas permeation test cell according to ISO 15105-1 [18,20,21,22]. The permeation cell consists of two cells separated by a testing sheet. One cell is the high feed concentration side, which receives the testing gas from the feed and vents it. The second cell is the lower permeate concentration side, which receives the corresponding gas and transports it to a gas pressure detector. This commercial apparatus has been utilized in various applications. However, the equipment, including the vacuum pumps, is rather complex and specific to certain gases. Under steady state conditions, the method generates permeability, diffusion coefficient, and solubility data.

Gas chromatography is also an appropriate method to qualitatively and quantitatively evaluate gases [23,24]. Detectors such as pulsed discharge detectors (PDDs) and thermal conductivity detectors (TCDs) can be used depending on their sensitivity to the single gas or mixed gas to be analyzed. Another detection method for gas transport properties is the volumetric method, typically with differential pressure applied in the cell [25]. However, these methods require large instruments, elaborate and fine control strategies, vacuum control/monitoring, and periodic maintenance.

In previous works, four simple and effective methods for measuring the time-varying gas concentration released from polymer specimens enriched by various gases in the desorption process were developed, and the diffusivity was determined by employing a dedicated diffusion analysis program. The methods are based on the gravimetric measurement [26,27], volumetric measurement [27,28], manometric measurement, and thermal desorption analysis–gas chromatography (TDA–GC) method [24,27] for quantitative evaluation of the released gas. The gravimetric method (GM) uses commercial electronic balances with minute resolution to quantize the amount of gas emitted in environments with well-maintained temperature and humidity. The volumetric method (VM) monitors decreased water levels in graduated cylinders to measure the uptake and diffusivity of the emitted gas. We employed a manometric method (MM) based on pressure measurements in a gas-enriched sample container with a simple pressure logger. The increase in gas pressure caused by the released gas was used to measure the gas concentration and diffusivity with the data pressure logger. Finally, we established an elaborate GC procedure to determine the H₂ transport parameters for H₂ emitted from polymeric materials enriched under high-pressure conditions.

The previous research was related the diffusivity and solubility obtained from the experiments for five gases (H₂, He, N₂, O_2,, and Ar) to the kinetic diameter and critical temperature, respectively. These gases could be appropriate candidates for the investigation of gas transport properties, because they were cheap and easily available. In this review article, the representative three types of gases (H₂, N₂, and O₂) in the experiment and analysis were chosen. The measuring principles, measuring procedures, and representative results for the three gases obtained by four methods are described. The performance and characteristics of the GM, VM, MM, and GC method are reviewed. They included the investigations of various gases uptake and diffusivity obtained by methods. The comparisons among the four methods, together with features of these methods, are also contained.

2. Sample Preparation and Gas Exposure under High Pressure Conditions

To measure gas uptake and diffusivity in polymeric rubber specimens filled with carbon black, such as ethylene propylene diene monomer (EPDM), nitrile butadiene rubber (NBR), and fluoroelastomer (FKM), used as sealing materials in the O-ring, are employed in this work. Low-density polyethylene (LDPE) was also included as a specimen for experimental investigations. The composition and density of the specimens are listed in previous literatures [24,29]. NBR, EPDM, FKM, and LDPE specimens with cylindrical and spherical shapes are used.

The heat treatment was conducted at 343 K for more than 48 h under the atmosphere according to the CHMC 2 standard [30] to remove the outgassing from the rubber specimen. Then the relative change in the mass of the specimen was measured using an electronic balance, and the measured change was less than 5 wt∙ppm over 24 h. This confirmed that outgassing from the specimen was completely removed.

A 316 stainless steel (SUS) chamber with an outer rectangular shape (length 150 mm × width 100 mm × height 150 mm) and inner cylindrical shape (diameter 50 mm × height 60 mm), as shown in Figure 1, was used for gas exposure under high pressure conditions at 298 K. The pressure chamber was purged at least three times with the corresponding gas below 3 MPa before testing the gas exposure. Then we exposed the specimen to the gas for 24–36 h at the corresponding pressure. Gas exposure for 36 h was regarded as sufficient for reaching equilibrium for gas absorption. The purities of the pure gases in the review work are as follows: H₂: 99.99%, He: 99.99%, N₂: 99.99%, O₂: 99.99%, and Ar: 99.99%.

3. Measuring Principle and Procedure for the Four Methods

We describe briefly the measuring principle and procedure of the four methods, such as GM, VM, MM, and GC. In the last section, the diffusion analysis program for obtaining diffusion parameters and algorithm, which is applied commonly to all methods, is demonstrated.

3.1. Gravimetric Measurement of the Gas Emitted by Enriched Specimens

After exposure to H₂ gas for the set time, the gas in the chamber (Figure 1 and Figure 2) was released by opening the needle valve. After decompression, the elapsed time is recorded from the moment (t = 0) at which the high pressure of the gas chamber decreased to atmospheric pressure.

As depicted in Figure 2, the concentration of H₂ gas released from a specimen is measured in real time by electronic balances with a resolution of 10 μg with a GPIB-interfaced PC over appropriate time intervals until mass equilibrium was reached. An electronic balance (Figure 2) is placed in a stable temperature and humidity chamber, with the temperature and humidity controlled and maintained within 298 ± 1.0 K and 10 ± 3%, respectively. The residual mass (C_R) of the specimen versus the elapsed time is calculated as follows [27]:

$$C_R\left(t\right)\left[wt·ppm\right]=\frac{M\left(t\right)-M_0}{M_0}\times {10}^6$$

(1)

where $M\left(t\right)$ is the mass of the specimen after decompression for an elapsed time $t$. and $M_0$ is the mass of the specimen before H₂ exposure in the high-pressure chamber. The inevitable time delay (lag) between decompression and the start of the mass measurement is approximately 5 min. Thus, the missing H₂ content released from t = 0 min to t = 5 min after decompression is determined by extrapolation in the diffusion analysis program, which is described later.

3.2. Volumetric Measurement of the Gas Emitted by Enriched Specimens

A VM using a graduated cylinder is utilized, as shown in Figure 3. After gas exposure in the high-pressure chamber and decompression, a specimen is loaded into the upper air space of the graduated cylinder, as shown in Figure 3. The main measurement system consists of a high-pressure chamber for gas exposure and a graduated cylinder immersed partially in a water container. Figure 3 illustrates the two volumetric methods employed to observe the water level: one method uses a digital camera, as shown in Figure 3b, and the other uses a capacitance meter, as shown in Figure 3c.

In the method shown in Figure 3b using a digital camera, the gas emitted from the rubber specimen leads to a reduction in the water volume in the graduated cylinder over time. The position of the water level is measured by the digital camera. The volume (V) and pressure (P) of the gas in the cylinder vary by emitted gas over time. The gas in the cylinder follows the ideal gas equation, PV = nRT, where $R$ is the gas constant, 8.20544 × 10⁻⁵ m³·atm/(mol·K), T is the absolute temperature of the gas in the upper part of the cylinder, and n is the mole number of the emitted gas in the cylinder. The time-dependent $P\left(t\right)$ and $V\left(t\right)$ of the gas in the cylinder are formulated as [27,29,31]:

$$P\left(t\right)=P_o-\rho gh\left(t\right), V\left(t\right)=V_o-V_s-V_h\left(t\right)$$

(2)

where $P_o$ is the atmospheric pressure outside the cylinder, $\rho$ is the density of the distilled water, g is gravitational acceleration, $h\left(t\right)$ is the level (height) of the water volume in the cylinder based on the water level in the water container, $V_o$ is the total volume of both water and gas in the cylinder based on the water level in the water container, the time-varying $V_h\left(t\right)$ is the water volume in the cylinder measured based on the water level in the water container, and $V_s$ is the specimen volume.

The concentration of the gas emitted from the specimen is obtained by measuring the water volume $\left[V_h\left(t\right)\right]$ over time. Thus, the total mole number $\left[n\right(t\left)\right]$ of the emitted gas is obtained by measuring the total gas volume $\left[V\right(t\left)\right]$ in the cylinder, i.e., the reduction in the water volume.

$$\begin{gathered} n(t)=\frac{P\left(t\right)V\left(t\right)}{RT\left(t\right)}=\frac{P\left(t\right)\left[V_a+V_g(t)\right]}{RT\left(t\right)}=\frac{P_0\left[1+\beta \left(t\right)\right]\left[V_a+V_g(t)\right]}{R{T}_0\left[1+\alpha \left(t\right)\right]} \\ \cong \frac{P_0}{R{T}_0}\left[V_a+V_g\left(t\right)+V\left(t\right)\left(\beta \left(t\right)-\alpha \left(t\right)\right)\right]=n_a\left(t\right)+n_g\left(t\right), \end{gathered}$$

(3)

$${\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} n}_a\left(t\right)=\frac{P_0}{R{T}_0}{V}_a, n_g\left(t\right)=\frac{P_0}{R{T}_0}\left[V_g\left(t\right)+V\left(t\right)\left(\beta \left(t\right)-\alpha \left(t\right)\right)\right]$$

$$\alpha (t)=\frac{T\left(t\right)-T_0}{T_0}, \beta (t)=\frac{P\left(t\right)-P_0}{P_0}$$

where $T_{0 }$ and $P_0$ are the initial temperature and pressure of the gas inside the cylinder, respectively, $V\left(t\right)$ is the sum of the initial remaining air volume ($V_a)$ and the emitted gas volume ${[V}_g\left(t\right)]$, i.e., $V\left(t\right)=V_a+V_g\left(t\right)$, $n_a$ is the initial air mole number, and $n_g(t)$ is the time-varying gas mole number corresponding to the gas volume increase by the emitted gas. Thus, $n_g(t)$ can be used to determine the emitted gas concentration $\left[C\left(t\right)\right]$ per mass from the rubber as follows [32]:

$$\begin{gathered} C\left(t\right)\left[\mathrm{wt}·\mathrm{ppm}\right]=n_g(t)\left[\mathrm{m}\mathrm{o}\mathrm{l}\right]\times \frac{m_g\left[\frac{\mathrm{g}}{\mathrm{m}\mathrm{o}\mathrm{l}}\right]}{m_{sample}\left[\mathrm{g}\right]}\times {10}^6 \\ =\frac{P_0}{R{T}_0}\left[V_g(t)+V\left(t\right)\left(\beta \left(t\right)-\alpha \left(t\right)\right)\right]\left[\mathrm{m}\mathrm{o}\mathrm{l}\right]\times \frac{m_g\left[\frac{\mathrm{g}}{\mathrm{m}\mathrm{o}\mathrm{l}}\right]}{m_{sample}\left[\mathrm{g}\right]}\times {10}^6 \end{gathered}$$

(4)

where $m_g$ [g/mol] is the molar mass of the gas, for instance, for H₂, $m_{H2}$ [g/mol] = 2.016 g/mol, and $m_{sample}$ is the specimen mass. According to Equations (3) and (4), the time-dependent gas mole number, $n_g(t)$, can be used to obtain the gas mass concentration, $\left[C\left(t\right)\right],$ by multiplying, $k=\left[\frac{m_g }{m_{sample}}\right].$ $n_g(t)$ and $C\left(t\right)$ are dependent on the variations in pressure and temperature. Thus, the variations in temperature and pressure are compensated for, leading to precise measurements.

We can also obtain $C\left(t\right)$ in emitted gas through the capacitance measurement to determine the water level (Figure 3c). A capacitor made with two semicylindrical electrodes mounted to the outer face of an acrylic tube is employed, as shown in Figure 4 [32]. The inner part of the acrylic tube is filled with a mixture of water and gas. The electrodes attached to the outer part of the acrylic tube are fabricated with a thin copper material. The capacitance in the acrylic tube depends on the dielectric permittivity of the media between the two capacitive electrodes. The dielectric permittivity of distilled water is approximately 78 times larger than that of the gas in the cylinder. Thus, the change in the water level in the two capacitor electrodes causes an appreciable change in the capacitance. Consequently, we measure the change in the actual capacitance caused by the changed water level. Thus, the water level corresponding to the measured capacitance is determined by the precalibration equation between the measured capacitance and water level [32]. The obtained water level can be used to determine the emitted gas mole number and gas concentration per mass according to Equations (3) and (4).

3.3. Manometric Measurement of the Gas Emitted by Enriched Specimens

Figure 5 illustrates the MM to measure the content of the released gas at 298 K; this apparatus consists of a high-pressure chamber for gas exposure and a cylindrical specimen container with a USB-type pressure/temperature logger and rubber seal.

After exposure under high pressure and decompression in the chamber, the specimen is moved into the cylindrical container, as depicted in Figure 5. The time lag between decompression and the start of the measurement is determined. The gas emitted from the specimen increases the pressure in the specimen container over time. Thus, the pressure $\left[P\right(t\left)\right]$ and temperature $\left[T\left(t\right)\right]$ of the gas inside the sample container varies with time. The gas in the container follows the ideal gas equation, PV = nRT, where n is the mole number of the released gas in the specimen container.

The gas released from the specimen is obtained by measuring the increase in pressure $\left[P\left(t\right)\right]$ over time with the manometric measurement method at constant volume in the container. Thus, the total number of moles $\left[n\right(t\left)\right]$ can be obtained by measuring the increase in the gas pressure $\left[P\right(t\left)\right]$ due to the released gas in the cylindrical container as follows:

$$\begin{gathered} n(t)=\frac{P\left(t\right)V_0}{RT\left(t\right)}=\frac{P\left(t\right)V_0}{RT\left(t\right)}=\frac{{[P}_0{+\mathsf{\Delta }P\left(t\right)]V}_0}{R{T}_0\left[1+\alpha \left(t\right)\right]} \\ \cong \frac{P_0{V}_0+\mathsf{\Delta }P\left(t\right)V_0}{R{T}_0}\left[1-\alpha \left(t\right)\right]=n_0+\mathsf{\Delta }n\left(t\right), \end{gathered}$$

(5)

$${\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} n}_0=\frac{P_0{V}_0}{R{T}_0}, \mathsf{\Delta }n(t)=\frac{V_0}{R{T}_0}[\mathsf{\Delta }P(t)-\alpha \left(t\right)P_0-\alpha \left(t\right)\mathsf{\Delta }P(t)]$$

$$\alpha (t)=\frac{T\left(t\right)-T_0}{T_0}$$

where $T_0$, $V_0$ and $P_0$ are the initial temperature, initial air volume and initial pressure of the gas inside the cylindrical sample container, respectively, $P\left(t\right)$ is the sum of the initial air pressure $\left(P_o\right)$ and time-varying released gas pressure $\left[\mathsf{\Delta }P\left(t\right)\right]$ from the specimen, i.e., $P\left(t\right)=P_o+\mathsf{\Delta }P\left(t\right)$, $n_0$ is the initial air mole number, and $\mathsf{\Delta }n\left(t\right)$ is the time-varying gas mole number corresponding to the increase in gas pressure due to the released gas. $\alpha \left(t\right)$ is the change rate of the temperature with regard to the initial temperature. Thus, $\mathsf{\Delta }n\left(t\right)$ can be transformed into the released gas concentration $\left[\mathsf{\Delta }C\left(t\right)\right]$ performance mass for the polymeric specimen as:

$$\begin{gathered} \mathsf{\Delta }C\left(t\right)\left[\mathrm{wt}·\mathrm{ppm}\right]=\mathsf{\Delta }n(t)\left[\mathrm{m}\mathrm{o}\mathrm{l}\right]\times \frac{m_g\left[\frac{\mathrm{g}}{\mathrm{m}\mathrm{o}\mathrm{l}}\right]}{m_{specimen}\left[\mathrm{g}\right]}\times {10}^6 \\ =\frac{V_0}{R{T}_0}[\mathsf{\Delta }P(t)-\alpha \left(t\right)P_0-\alpha \left(t\right)\mathsf{\Delta }P(t)]\left[\mathrm{m}\mathrm{o}\mathrm{l}\right]\times \frac{m_g\left[\frac{\mathrm{g}}{\mathrm{m}\mathrm{o}\mathrm{l}}\right]}{m_{specimen}\left[\mathrm{g}\right]}\times {10}^6 \end{gathered}$$

(6)

where $m_g$ [g/mol] is the molar mass of the gas used, for instance, for H₂, $m_{H2}$ [g/mol] = 2.016 g/mol; for N₂, $m_{N2}$ [g/mol] = 28.001 g/mol; and $m_{specimen}$ is the mass of the specimen.

The first term, $\mathsf{\Delta }P\left(t\right)$, in Equation (6) is the pressure increase due to the gas emitted by the specimen. Two terms, $[-\alpha \left(t\right)P_0-\alpha \left(t\right)\mathsf{\Delta }P(t\left)\right]$, in Equation (6) indicate the pressure change caused by temperature variation $\alpha \left(t\right)$, regardless of the emitted gas. According to Equations (5) and (6), the time-dependent gas mole number, $\mathsf{\Delta }n\left(t\right)$, can be transformed into the gas mass concentration, $\left[\mathsf{\Delta }C\left(t\right)\right],$ by multiplying by $k=\left[\frac{m_g }{m_{specimen}}\right].$ $\mathsf{\Delta }n\left(t\right)$ and $\mathsf{\Delta }C\left(t\right)$ are influenced by temperature variations. Thus, we must compensate for variations caused by temperature changes to obtain precise measurements. The changes in the gas volume and pressure due to the released gas are monitored based on the pressure, which is measured with the pressure data logger in the container vessel, as shown in Figure 5.

3.4. Gas Chromatography of the Gas Emitted by Enriched Specimens

To measure the concentration of the emitted gas, a GC method is applied, as shown in Figure 6. The pressure in the high-pressure chamber was reduced to atmospheric pressure by opening a needle valve, and the specimen was loaded into a quartz tube connected to a GC injector. The time lag between decompression and the start of the GC measurement was approximately 7 min.

The TDA–GC method quantitatively and qualitatively analyzes the testing gas by measuring the position and area of the identified separated GC signal [24], as shown in Figure 6b. The flow rate of the carrier gas (helium) is controlled using a mass flow controller. The gas emitted from the specimen is mixed with the helium carrier gas and sent to the capillary GC column through the injector. Then a PDD produces electrical signals corresponding to the separated gas components. Oxygen and nitrogen signals are not released by the specimen but are temporarily observed initially because of contact with air as the specimen is moved from the high-pressure chamber to the quartz tube, as shown in Figure 6b. TDA–GC has been set up for selectively H₂ gas measurement. Thus, we have only provided the H₂ results later. Diffusion analysis is applied based on the H₂ peak released from the specimen.

Under the experimental conditions for the GC method at 1 atm, 298 K, and a helium carrier gas flow rate of 1.67 × 10⁻⁷ m³/s, the mass concentration of H₂ per second, $C_{\mathrm{mass}}\left(\frac{\mathrm{wt}·\mathrm{ppm}}{\mathrm{s}}\right)$, can be determined according to Equation (7) [27],

$$C_{mass}\left(\frac{\mathrm{wt}·\mathrm{ppm}}{\mathrm{s}}\right)\cong 1.392\times {10}^{-5}\frac{C_{mol}(\mathrm{m}\mathrm{o}\mathrm{l}·\mathrm{ppm})}{m_{sample}\left(g\right)}$$

(7)

where C_mol (mol·ppm) is the molar concentration of H₂ in ppm obtained from GC measurements, $m_{\mathrm{sample}}$ is the mass of the specimen, and the molar mass of H₂ is 2.016 g/mol.

3.5. Diffusion Analysis Program for Obtaining Diffusion Parameters and its Algorithm

Assuming that the gas emitted from the gas-enriched specimens follows Fickian diffusion, the concentration $C_{\mathrm{E}}(t)$ of the released gas in emission mode is formulated as [33,34]:

$$\begin{array}{ll}{C}_{\mathrm{E}}(t)/C_{\mathrm{\infty }}& =1-\frac{32}{{\pi }^2}\times \left[{\displaystyle \sum _{\mathrm{n}=0}^{\mathrm{\infty }}}\frac{\mathrm{e}\mathrm{x}\mathrm{p}\left\{\frac{{-\left(2\mathrm{n}+1\right)}^2{\pi }^{2}Dt}{l^2}\right\}}{{\left(2\mathrm{n}+1\right)}^2}\right]\times \left[{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{\infty }}}\frac{\mathrm{e}\mathrm{x}\mathrm{p}\left\{-\frac{D{\beta }_{\mathrm{n}}^{2}t}{{\rho }^2}\right\}}{{\beta }_{\mathrm{n}}^2}\right]\\ & \begin{array}{ll}=1-\frac{32}{{\pi }^2}& \times \left[\frac{\exp \left(-\frac{{\pi }^{2}Dt}{l^2}\right)}{1^2}+\frac{\exp \left(-\frac{3^2{\pi }^{2}Dt}{l^2}\right)}{3^2}+\dots ,+\frac{\exp \left(-\frac{{\left(2\mathrm{n}+1\right)}^2{\pi }^{2}Dt}{l^2}\right)}{(2{\mathrm{n}+1)}^2}+\dots ,\right]\\ & \times \left[\frac{\exp \left(-\frac{D{\beta }_1^{2}t}{{\rho }^2}\right)}{{\beta }_1^2}+\frac{\exp \left(-\frac{D{\beta }_2^{2}t}{{\rho }^2}\right)}{{\beta }_2^2}+\dots ,+\frac{\exp \left(-\frac{D{\beta }_{\mathrm{n}}^{2}t}{{\rho }^2}\right)}{{\beta }_{\mathrm{n}}^2}+\dots ,\right]\end{array}\end{array}$$

(8)

where ${\beta }_{\mathrm{n}}$ is the root of the zeroth-order Bessel function J₀ (β_n). Equation (8) is the solution to Fick’s second diffusion law in the case of a cylindrical specimen. Equation (8) is an infinite series expansion consisting of two summations. The product of the two parentheses becomes $\frac{{\pi }^2}{32}$ at t = 0. Thus, the value $\frac{32}{{\pi }^2}$ is inserted in front of the equation so that the gas concentration at t = 0 becomes C_E (t = 0) = 0. Moreover, C_E (t = ∞) = $C_{\mathrm{\infty }}$ is the total gas uptake obtained at infinite time. D is the gas diffusion coefficient, and $l$ and $\rho$ are the thickness and radius of the cylindrical specimen, respectively. In the remaining mode, Equation (8) can also be applied for the cylindrical specimen. The total gas uptake is defined as C₀ at t = 0.

The residual concentration c(t) of the gas in the remaining mode for the spherical specimen is expressed as follows [29,33].

$$C_R\left(t\right)=\frac{6}{{\pi }^2}{c}_0{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}\frac{1}{n^2}\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{D{n}^2{\pi }^{2}t}{a^2})$$

(9)

Equation (9) is the solution of Fick’s second diffusion equation for a spherical specimen. $a$ is the radius of the spherical specimen and $D$ is the diffusion coefficient. C₀ is the total content of emitted H₂ at t = 0. In Equations (8) and (9), more summation terms are needed to determine precise values for $D$ and C₀. Thus, we developed a dedicated diffusion analysis program using Visual Studio, which allows us to precisely calculate D and C₀, with up to 50 terms included in the summation in Equations (8) and (9).

We developed a diffusion analysis program using a nonlinear optimization algorithm [35]. Figure 7 shows the flowchart of the diffusion analysis program developed to precisely analyze the $C_E\left(t\right)$ and $C_R\left(t\right)$ data using Equations (8) and (9), respectively, with the Nelder–Mead simplex nonlinear optimization algorithm [18,24]. The data type (remaining, emission, and transmission mode) and specimen shape (cylinder, sphere, and sheet) are used to determine the appropriate diffusion equation. The algorithm analyzes the gas uptake and diffusivity according to the solutions of the diffusion equation of Equations (8) or (9), and we choose the appropriate diffusion model corresponding to the specimen shape and the number of superposition models.

The diffusion analysis program is applied to a cylindrical-shaped specimen in the remaining mode, and the GM is used to obtain the C₀ and D values of H₂, as shown in Figure 8. Figure 8 shows the representative analytical results using the analysis program for a cylindrical NBR specimen (radius of 5.0 mm and height of 2.0 mm) exposed to a pressure of 35 MPa. The D and C₀ values are obtained by substituting the remaining relative H₂ mass (C_R) at each time from Equation (1) into Equation (8) and optimizing each parameter with the least squares method. Finally, we determine that D = 1.86 × 10⁻¹¹ m²/s and C₀ = 986 wt∙ppm. The red line and × symbol indicate the fit obtained using Equation (8) and the experimental data, respectively. The arrow in the unknown parameter list indicates the diffusion coefficient, D, obtained from the diffusion analysis program. The figure of merit (FOM) of 3.3% is the standard deviation between the measured data and Equation (8). The arrow on the y-axis in Figure 8 corresponds to the value of C₀ obtained at t = 0 by extrapolating the fitted line.

4. Results and Discussion

We demonstrate the representative results in the four methods based on uptake and diffusivity obtained by applying diffusion analysis program.

4.1. Gravimetric Method

For the GM using an electronic balance (Figure 2), the residual mass (C_R) of the specimen based on the H₂ emitted over time is obtained by measuring the change in mass of the specimen according to Equation (1). The measured results are shown in Figure 9. The diffusion analysis program is used to obtain the C₀ and D values of H₂ for spherical-shaped NBR specimens exposed to hydrogen at a high pressure of 10.2 MPa [27]. The D and C₀ values are obtained by substituting the residual H₂ concentration after the elapsed time determined from Equation (1) into Equation (9) and optimizing each parameter by the least squares method. Thus, the values D = 1.27 × 10⁻¹⁰ m²/s and C₀ ≈ 696 wt∙ppm are determined by extrapolation. The dashed line and filled squares indicate the fitted line of Equation (9) and the experimental data, respectively. The arrow in Figure 9 corresponds to the value of C₀ obtained at t = 0 by extrapolating the fitted line.

In addition, Figure 10 shows the representative result of H₂ uptake content and diffusivity versus pressure for the spherical NBR, EPDM, and FKM specimens determined by the GM [27]. The H₂ uptake followed Henry’s law [36,37], as shown by the black dashed lines in Figure 10a. The obtained slopes for the NBR, EPDM, and FKM specimens are shown in Figure 10a. The determined diffusivity in Figure 10b does not show pressure-dependent behavior for the specimens. Thus, the average diffusivity is taken as a representative value for the investigated pressure. The average diffusivity of the NBR, EPDM and FKM specimens is determined, as shown by the black dashed horizontal line in Figure 10b. The error bars in (a) and (b) indicate the standard deviation of the measured value.

4.2. Volumetric Method

The emitted N₂ gas content and its diffusivity are obtained through the VM using a graduated cylinder (Figure 3b). Figure 11 displays the two procedures for obtaining the N₂ gas diffusion parameters in the EPDM sample with a digital camera [32]. Figure 11a shows the time-dependent water level (emitted gas volume) measured by a digital camera. Figure 11b depicts the time-dependent mass concentration determined according to the measured water level and Equation (4). The nitrogen emission is saturated 60,000 s after decompression.

By applying the diffusion analysis program results shown in Figure 8, D and C_∞ are obtained by fitting the emitted H₂ content to Equation (8) and optimizing the parameters with the least squares method. The black line in Figure 11b indicates the line fitted with Equation (8) based on the experimental data. Thus, the value D = 5.35 × 10⁻¹¹ m²/s is found. The blue solid line represents the total compensated emission curve used to restore the missing content caused by the time lag, and C_∞ = 3382 wt∙ppm is the value obtained at t = ∞ by extrapolating the fitted line.

On the other hand, Figure 12 shows the procedure for acquiring D and $C_{\infty }$ for the same EPDM cylindrical specimen by using two semicylindrical electrodes (Figure 3c and Figure 4) [32]. Figure 12a represents the precalibration result according to the second polynomial equation between the water level and the measured capacitance by quadratic regression. Figure 12b shows the water level determined according to the measured capacitance, where the black and blue squares indicate the measured capacitance and water level, respectively. Figure 12c shows the diffusion parameters, D and $C_{\infty }$, obtained using the diffusion analysis program and Equation (8). The blue line in Figure 12c represents the total compensated emission curve, including the missing content due to the lag time. The investigated results in Figure 11 and Figure 12 are consistent within the experimental uncertainty.

With the volumetric method using Figure 3b, we measure the emitted hydrogen content at pressures up to 90 MPa for one neat EPDM specimen and nine EPDM composites blended with fillers [38]. Figure 13 represents the H₂ uptake versus pressure for four representative EPDM specimens. Figure 13a–d show the relationship between the pressure and H₂ uptake for the neat EPDM, EPDM composites compounded with silica fillers (40 phr and 60 phr), EPDM HAF40, and EPDM SRF40 specimens, respectively.

The H₂ uptake (C_∞) for the neat EPDM and EPDM S20 (Figure 13a,b) linearly increases with increasing pressure up to 90 MPa, satisfying Henry’s law. This is attributed to H₂ absorption into the polymer matrix. However, in Figure 13c,d, the H₂ uptake in the EPDM HAF40 and SRF40 deviates from Henry’s law at pressures above 15 MPa; this behavior is responsible for the H₂ adsorbed at the CB filter. Thus, dual sorption is observed for all CB-blended EPDM composites. The dual-mode sorption phenomenon is observed over the entire pressure range up to 90 MPa can be expressed as [36,37,39,40,41,42,43]:

$$C_{\mathrm{\infty }}=kP+\frac{abP}{1+bP}$$

(10)

$C_{\infty }$ indicates total H₂ gas uptake. The first term is related to Henry’s law and has a coefficient k. The second term indicates the Langmuir model with $a$ representing the maximum adsorption quantity and $b$ representing the adsorption equilibrium constant.

4.3. Manometric Method

The emitted O₂ content is measured by the MM, as shown in Figure 5. The representative result for the O₂ emitted over time is shown in Figure 14a for a cylindrical LDPE specimen at a pressure of 6.0 MPa. The O₂ uptake and diffusivity are determined by applying the diffusion analysis program and Equation (8). In Figure 14a, single-mode emission/diffusion behaviors for LDPE plastic are observed based on the time-varying gas uptake data. The single-mode oxygen emission for the LDPE specimen is caused by O₂ diffusion, which is attributed to the absorption of O₂ in the amorphous phase. The emitted oxygen content is obtained from t = 6 min after decompression due to the time lag. Thus, the missing amount of oxygen emitted from t = 0 min to t = 6 min after decompression is determined with the diffusion analysis program. The missing content is included in the oxygen emission data, as shown by the blue line of Figure 14a.

The oxygen emitted by the LDPE specimen in Figure 14b linearly increases with increasing pressure, following Henry’s law. However, the O₂ diffusion coefficient for the LDPE specimen in Figure 14c remains constant at 4.17 × 10⁻¹¹ m²/s within the experimental uncertainty (± 0.5 × 10⁻¹¹ m²/s), regardless of the pressure.

Single-mode emission and diffusion behaviors of N₂ for the cylindrical LDPE plastic specimen are also observed in Figure 15a. The single-mode N₂ emission for LDPE is caused by N₂ diffusion. The nitrogen emitted by the LDPE specimen in Figure 15b is proportional to the pressure, following Henry’s law. Moreover, the nitrogen diffusivity for the LDPE specimen in Figure 15c remains constant at 2.56 × 10⁻¹¹ m²/s, regardless of pressure. In summary, two pressure-dependent emission and diffusivity behaviors for O₂ and N₂ gases are very similar.

4.4. Gas Chromatography

The emitted hydrogen from specimen is measured by a GC system within the corresponding time interval, as shown in Figure 6. Figure 16a shows representative time-varying H₂ concentration data determined according to the mass concentration per second based on the GC data and Equation (7) for EPDM specimens at a pressure of 4.85 MPa. The data in Figure 16a over time was used to obtain the saturation value, i.e., the amount of H₂ absorbed by the EPDM specimen, as shown in Figure 16b. The extrapolated H₂ concentration is 103 wt∙ppm, which is the saturated value at infinite time. The content is obtained from the time delay of 6 min. Thus, the missing amount of hydrogen released up to time delay after decompression was determined by applying the diffusion analysis program [24].

By subtracting the emission values (Figure 16b) from the exposed emission value (103 wt∙ppm), we obtain the remaining H₂ concentration. Then, we obtain D and C₀ through the diffusion analysis program [24]. The application window of the diffusion analysis program is shown in Figure 17a. The analysis results show two diffusion characteristics, i.e., slow and fast, and the experimental data fit the sum of these two components, as shown in Figure 17b. D and C₀ are as follows: D_fast = 3.51 × 10⁻¹⁰ m²/s, C_0-fast = 161 wt∙ppm, D_slow = 2.64 × 10⁻¹¹ m²/s, and C_0-slow = 43.8 wt∙ppm. After the cylindrical EPDM specimen was exposed to H₂ at 4.85 MPa, we determine (C₀ = 204.7 wt∙ppm) the total H₂ uptake. The total uptake value, C₀, is the value on the y-axis at t = 0, which is obtained by extrapolating the fitted line with the diffusion analysis program.

5. Comparisons of the Measured Results and Characteristics of the Four Methods

The developed methods for measuring H₂ uptake and diffusivity are validated by comparing the measured values for the same specimen. The H₂ uptake, solubility and diffusivity results obtained with the GM, VM, MM and GC method are consistent within the experimental uncertainty, as shown in Figure 18 [27]. The solubility is obtained from the slope of the uptake data with pressure shown in Figure 10a. The uncertainty estimations for solubility and diffusivity are well described in prior investigations [24,26]. The relatively large uncertainty in the GM is caused by variations due to temperature, humidity and instability of the electronic balances.

Based on the measured results and analysis, we review the performance of the four developed methods. The characteristics of the measuring methods are presented in terms of measurand, resolution, stability, uncertainty, figure of merit (FOM), sensitivity, and features in

Table 1. Comparison of the characteristics and performance of the four developed methods.

Measuring Methods	GM	VM	MM	GC
Measurand	mass	volume	pressure	volume
Resolution	0.3 wt∙ppm	0.1 wt∙ppm	0.1 wt∙ppm	0.01 wt∙ppm
Stability	<0.22%	<0.16%	<0.15%	<0.13%
Expanded uncertainty	<13%	<10%	<10%	<10%
Figure of merit (FOM)	1.5%	0.8%	0.7%	0.5%
Sensitivity to temperature and pressure	sensitive	sensitive	very sensitive	less sensitive
Features	very simple and inexpensive	simple, inexpensive and effective	very simple, inexpensive and effective	complicated and delicate

. The resolutions, indicating the lowest readable digits for the four methods, were obtained from the specifications of the equipment and measurand for each method. Compared with the other methods, the lowest resolution of 0.01 wt∙ppm was estimated for the GC method, corresponding to the area of the smallest hydrogen GC peak obtained from the GC signal array versus time after decompression. The stability can be represented as the standard deviation obtained by repeated measurements for 24 h after the H₂ emission is finished, which were approximately 0.13~0.22%. The expanded uncertainty indicates the uncertainty in the diffusivity measurement, represented as the product of the combined standard uncertainty and coverage factor. The expanded uncertainties for the VM, MM, and GC method are less than 10%, but the expanded uncertainty for the GM is higher. The relatively large uncertainty for the GM is due to the large uncertainty in the temperature/humidity, uncertainty in the electronic balance measurements and standard deviation of the fitting result. The FOM is the standard deviation between the measured data and the fits from Equations (8) and (9). A FOM value less than 1%, as shown by three methods, implies good agreement between the measured and theoretical values. The sensitivity to temperature and pressure of the methods is also presented in Table 1.

An electronic balance detects minute-level changes in electric resistance, proportional to the deforming force acting by the principle of the Wheatstone bridge. Thus, the GM is a sensitive technique that is influenced by the offset of the electronic balances and the stability of the temperature and humidity. Although the GM is simple and inexpensive, the electronic balance for GM must be operated in a controlled environment that maintains both temperature and humidity.

The VM using a graduated cylinder is an effective approach for determining gas transport parameters. This simple method could be used for on-site evaluations of gas diffusion because of the simple equipment employed. The VM for detecting the water level can be applied in real time by using a digital camera or capacitance meter with two capacitor electrodes, which requires precalibration between capacitance and water level prior to the measurements.

On the other hand, the MM is also simple, inexpensive, and effective. This method is more appropriate for on-site evaluations in the field than the VM. However, this method is very sensitive to temperature and thus requires pressure compensation to address temperature variations.

GC is a sophisticated method with complicated procedures for obtaining H₂ uptake and diffusivity data according to individual GC peaks. This technique could be used to quantitatively evaluate the H₂ content in small samples because of its good resolution. This sensitive method is the best candidate for precise analysis of H₂ gas properties with multiple gas components. In summary, the main features of the four methods are described in the last line of Table 1.

6. Conclusions

Four methods for measuring gas uptake and diffusivity in polymers enriched with gas under high pressure up to 90 MPa are developed. These methods are based on measurements of the mass, volume, and pressure of the emitted gas and include the gravimetric, volumetric, manometric, and gas chromatography methods. The representative results of gas uptake, solubility, and diffusivity obtained from four methods are demonstrated in the several polymer specimens. In addition, the pressure-dependent gas uptake is associated with Henry’s law or Langmuir model depending on the polymer characteristics. The gas uptake and diffusivity obtained by these methods are consistent, and the characteristics and performance are reviewed. The features of these methods are presented as follows:

• The three developed methods are inexpensive and simple techniques, except for gas chromatography, for evaluating the gas uptake and diffusion of gas-enriched polymer materials under high-pressure conditions.
• The four developed methods are insensitive to variations in temperature and pressure, regardless of the specimen size, specimen shape, and testing gas species.
• All the methods are exactly calculable by applying the diffusion analysis program, which includes more than 50 terms in the summation of provided equation.
• The volumetric and manometric methods are flexible techniques in which the sensitivity, resolution, and range can be flexibly changed.
• All the methods are independent techniques, without any interaction between the testing gas molecule and gas sensor.
• The volumetric method is visible because the entire process of gas release and leakage can be observed by monitoring the change in water level.

The proposed techniques for effectively measuring the transport property of gas in sealing elastomers have different characteristics. Thus, the complementary four methods could be utilized in the testing of permeation properties of rubber material and O-ring under high pressure for hydrogen fueling stations.