Hydrogen diffusion through polymer membranes

This experimental study determines H2 transfer properties of High Density PolyEthylene (HDPE) or epoxy membranes. Two different techniques are compared. The pressure gradient technique is analyzed for permeability or diffusion. Results show that the main phenomenon involved is diffusion. The second technique involves a gas concentration gradient. Although implying significantly slower kinetics, this classical technique is consistent with results obtained with the faster pressure gradient technique.


Introduction
Polymer membranes are used in the industry as watertight sealing barriers for low-level radioactive waste storage. They are generally made of High Density PolyEthylene (HDPE) or epoxy. Waste may contain metals (e.g. aluminum) or organic materials. The presence of water in contact with this waste, whether accidental or residual, may trigger gas-producing reactions, particularly hydrogen gas [1], and lead to explosion risks. Although watertight, polymer membranes may allow gas transfer, first without significant gas pressure gradient ΔP, and, with time, under ΔP. In both instances, gas transfer properties of polymer membranes must be quantified to assess the H2 accumulation risk (or leakage rate) by coupling with numerical simulation codes. For HDPE or epoxy, very few data are available in the literature, relating to transport of pure hydrogen under pressure or concentration gradients. Van Krevelen et al. [2] provide diffusion coefficients for pure HDPE or epoxy, but solely for CO2, O2 or N2 gases. Moreover, diffusion coefficients of polymers are obtained through very lengthy experiments, and they are dispersive in nature, depending on the actual polymer structure and/or filler minerals [2,3]. In this research, we investigate both scenarii for H2 presence in contact with polymer membranes, by subjecting samples either to a concentration gradient, or to a pressure gradient of hydrogen. Two original experiments are designed, carefully calibrated and setup in the laboratory. First, a custom permeameter is operated at different concentrations and gas pressures. It allows to quantify gas flowrate and to analyze whether gas transfer is governed by permeation (convection) or by diffusion. Particular precautions are taken in relation to handling hydrogen, which limit experimental possibilities mainly in terms of maximum gas pressure. Tests are also conducted with nitrogen gas, to investigate the effect of gas nature. Moreover, very low gas flowrates are measured. It is a research domain where our laboratory has wide experience, since we are able to measure gas permeabilities down to 10 -22 m 2 [4].

Materials and methods
HDPE membranes are made of a patented mix of pure HDPE and filler minerals of 2 mm thickness (Fig. 1a) , and 0.97 g/cm 3 (+/-0.02) density, provided by AGRU-Franck GmbH (Wölfersheim, Germany). This small thickness has imposed a specific design of the permeameter test bench, which is detailed hereafter. No information is provided by the manufacturer on their chemical composition, and on the filler nature potentially modifying its transport properties compared to pure HDPE [2]. Three membranes 1, 2 and 3 (10 cm diameter) are tested with the permeameter, and one membrane n. 4 (17.5 x 4.1 cm 2 ) with the classical diffusion test.  Patented epoxy resin membranes (Fig. 1b), reinforced with glass fibers, are also tested, with 1.25 mm thickness and 1.30 g/cm 3 (+/-0.02) density.

Pressure gradient technique
This test set-up is designed to fit several constraints, namely 1) to use a wide exchange surface area between gas and polymer membrane to maximize gas flowrates, 2) to measure very low gas flowrates, and 3) at maximum gas pressure of 1.7 MPa. The flowrate measurement technique is deduced from small pressure variations in a chamber of small volume Vr, carefully calibrated and located after the membrane (downstream chamber) (Fig. 2). The lateral sealing of the system is ensured by a strong clamping of the upper and bottom set-up metal parts and by 4 O-ring joints. On the upstream membrane side, the injection pressure P1 is kept constant, while on the downstream side, inside Vr, pressure increases from atmospheric pressure P2 by ΔP, of only a few hundreds of Pa. This observation also eliminates a potential leakage scenario. The main assumption is to analyze gas transfer as unidirectional, i.e. only through the exchange surface of area A, as indicated in Fig. 2, while neglecting any lateral parasitic flow (owing to sample clamping). Any volume change of reservoir Vr is also neglected, thanks to a porous sintered disk placed after the membrane, which hinders polymer deformation. Volume gas flowrate Q is estimated by measuring a slight increase in pressure ΔP during time Δt in downstream reservoir Vr. The corresponding mass flowrate during Δt is Qmass = Q ρmean = Δm/Δt = (Δρ Vr / Δt). The perfect gas law is then applied inside Vr, assumed at an average pressure P2 (with ΔP <<P2, on the first order, P2+ ΔP ≈ P2) and at constant temperature T, so that the mean gas density ρmean is equal to (Δρ P2)/ΔP. Finally, Q is written as [4][5][6][7][8][9]: If this flowrate is due to permeation, Darcy's law applies and leads to the Hagen-Poiseuille law [4][5][6][7][8][9]: (2) Kapp is apparent gas permeability (in m 2 ), µ is gas dynamic viscosity (in Pa.s), and e is membrane thickness. The experiment is also analyzed by assuming that gas flow is solely due to diffusion, through an isotropic polymer medium. As downstream overpressure ΔP is very low (a few hundreds of Pa), on the first order, gas concentration gradient is assumed equal to c/e, where c is the upstream gas concentration in mol/m 3 : On the downstream side, if pressure variation ΔP is due to the diffusion of N gas moles, then: During time variation Δt, gas molar flowrate ΔΦ in mol/s is: Finally, for steady state unidirectional gas transfer, Fick's law writes so that diffusion coefficient D (in m 2 /s) is: Three P1 values are tested with N2, and six P1 values with pure H2. For comparison purposes, a single value P1=1.5 MPa is used with pure Ar or pure N2.

Concentration gradient technique
A classical diffusion test is carried out at atmospheric pressure P1 = P2 with a specific original cell (Fig. 3), regardless of gas nature. Although of rectangular shape, sample surface are A' is similar to that in the pressure gradient technique. In this experiment, a mass spectrometer is calibrated for each gas (here, N2 and H2) and used to detect and quantify gas particles passing through the polymer membrane. The cell is divided into two parts. The upstream volume is filled with a mixture of N2 / H2 (at 2.5 / 5 or 10 mol% H2). Diffusing particles are quantified in the downstream volume. It is flown through by a carrier gas (N2), which atomic number is sufficiently far from H2 to improve resolution. Because the gas mixture is of low H2 concentration, combined with a low diffusive membrane, a static method has been devised to limit test duration. It consists in isolating the downstream volume from ambient air for 2 to 3 days, and collecting a detectable gas concentration, on the order of 0.1% (reliable measure with the spectrometer). The measurement is repeated 3 times for reproducibility. For a gas with m mol% H2, upstream concentration cup (in mol/m 3 ) is: where P1 is equal to 10 5 Pascal (atmospheric pressure). After time Δt, in the carefully calibrated downstream chamber of volume Va, a small downstream concentration (compared with m) is measured, equal to p (mol%). This corresponds to M diffused moles: To apply Fick's law through the membrane ( ), a constant concentration gradient equal to (cup / e) is assumed (p<<m). During Δt, mean gas flowrate ΔΦ (in mol/s) is: ΔΦ= M/Δt (9) So that (10) Where A' is the membrane surface area.

Results and discussion
Whatever the polymer considered, each permeameter experiment lasts 4h, whereas the classical diffusion technique requires 2-3 days. Therefore, the permeameter experiment is an accelerated measurement of gas transfer properties through polymer membranes.

Gas transfer phenomenon during pressure gradient test
As shown in Appendix, when gas is applied as a pressure gradient and flows steadily through the polymer membrane, several definitions exist for permeability, either Kapp in the Darcy's sense, as described in Eq. (2) [4][5][6][7][8][9] or Pe according to Eq. (A10) [10][11][12][13][14][15][16]. In Darcy's sense, i.e. if gas flows by advection through the polymer membrane, gas flowrate Q scales with P1 2 (square of the imposed pressure on the upstream side) (Eq. (2)). This is described through apparent gas permeability Kapp. Comparatively, if gas diffuses according to Fick's first law, Q is proportional to pressure P1 (Eqs. (1) and (6)). This is described by permeability coefficient Pe in [10][11][12][13][14][15][16] or directly by diffusion coefficient D (see Sub-section 2.1). Pe and D are directly related (Appendix 1.2, Eq. (A12)). For HDPE membranes subjected to pure H2 flow, Fig. 4 shows that, at different gas injection levels P1, gas volumetric flowrate Q is a linear function of P1. This is characteristic of diffusion-driven transport. This observation is identical whatever the gas nature (H2, He, Ar or N2), and for both HDPE and epoxy membranes. It is concluded that during pressure gradient test, polymer membranes are mainly subjected to diffusion. In the case of argon flow through HDPE membranes at P1=1.5MPa, diffusion coefficient D is calculated at an average value of 6.5 x 10-13 m 2 /s +/-0.3. This corresponds to Pe values of 9.8 x 10 -16 m 2 .s.Pa -1 +/-3.9. When comparing with values for HDPE from the literature [14] (see also Appendix 1.2), Pe is significantly lower than for HDPE containing 21% amorphous matter. Our result is attributed to a greater crystallinity (or to a greater mineral charge content) of the patented HDPE tested in this research.

HDPE diffusion properties
With the permeameter, HDPE diffusion coefficients with N2 (or Ar) and H2 (or He) differ by approximately one order of magnitude, indicating a much slower N2 (or Ar) transfer than that of H2 (or He), see Table 1. A greater diffusion coefficient is obtained with Ar when compared to N2, owing to a smaller atom size than N2 molecule. Comparatively, He and H2 diffusion coefficients are close, as their respective atom or molecule size suggests. This is consistent with results from [2]. Results analyzed as apparent gas permeability Kapp are significantly more dispersed than those analyzed with diffusion. Moreover, Kapp values are very low, ranging between 4.2 10 -24 -1.32 10 -22 m 2 . This is also indicative that the main gas transfer phenomenon is diffusion rather than advection.  With the classical gas concentration technique (Table 2), good agreement is obtained between the average D(H2) value (3.3 10 -12 m 2 /s +/-1. 4 10 -12 ) and that measured with the permeameter (average D(H2) = 2.0 10 -12 m 2 /s +/-0.78 10 -12 ), with less scatter with the latter technique. This is another way to demonstrate that the permeameter test may be interpreted as an accelerated diffusion test, by neglecting convective transport.

Epoxy diffusion properties
With the accelerated diffusion experiment (using the pressure gradient technique in steady state), results on HDPE are compared to those for epoxy with either argon or helium (Fig.  5). Whereas helium diffusion is quite high and very close for both polymer membranes (with average values of 1.2 x 10 -12 m 2 /s for HDPE and 0.8 x 10 -12 m 2 /s for epoxy), argon diffuses with more difficulty through epoxy than through HDPE (with average values of 7.9 x 10-13 m 2 /s for HDPE and 0.3 x 10 -13 m 2 /s for epoxy). This means that gas diffusion through epoxy is more sensitive to gas nature, i.e. that epoxy is a more selective polymer to gas diffusion.

Conclusion
This study combines 1) the design of two original experimental devices and 2) the fast reliable measurement of HDPE and epoxy membrane transport properties to H2 gas (and Ar, N2, He). With both pressure and concentration gradient techniques, Fick's law provides similar diffusion coefficients D(H2) (uniform isotropic medium), at 2.0 10 -12 m 2 /s +/-0.78 10 -12 (pressure gradient) or 3.3 10 -12 m2/s +/-1.4 10 -12 (concentration gradient). The pressure gradient technique is faster (4h instead of 2-3 days), because higher gas pressure increases H2 concentration on the sample upstream side. Gas nature plays a major role, as e.g. N2 gas flowrate is ten times smaller than for H2, under equivalent pressure conditions (average D (N2) = 1.3 10 -13 +/-0.68 10 -13 ). Given the very small quantities of gas involved, it is also noted that diffusion through HDPE is almost independent of gas concentration or pressure. This is not the case with epoxy, which diffusion coefficient shows a greater dependency on gas nature. where is fluid density (in kg/m 3 ), is gas velocity (in m.s -1 ), ϕ is porosity, is fluid dynamic viscosity (in Pa.s), K is permeability in m 2 , and P is local pressure. Let add the state equation for gas, assumed perfect: where R is the perfect gas constant (equal to 8.31 J.mol -1 .K -1 ), T is temperature (in K) and Mf is fluid molar mass (in kg/mol). Combining Eqs. A1, A2 and A3 provides diffusivity equation: Under steady state flow, Eq. A4 leads to , which is integrated for a unidirectional flow along axis x as: (Eq. A5)