Efficient X-ray CT-based numerical computations of structural and mass transport properties of nickel foam-based GDLs for PEFCs

Nickel foams are excellent candidate materials for gas diffusion layers (GDLs) for polymer electrolyte fuel cells (PEFCs) and this is due to their superior structural and transport properties. A highly computationally-efficient framework has been developed to not only estimate the key structural and mass transport properties but also to examine the multi-dimensional uniformity and/or the isotropy of these properties. Specifically, multiple two-dimensional X-ray CT images and/or numerical models have been used to computationally determine the porosity, the tortuosity, the pore size distribution, the ligament thickness, the specific surface area, the gas permeability and the effective diffusivity of a typical nickel foam sample. The results show that, compared to the conventionally used carbon substrate, the nickel foam sample demonstrate a high degree of uniformity and isotropy and that it has superior structural and mass transport properties, thus underpinning its candidacy as a GDL material for PEFCs. All the computationally-estimated properties, which were found to be consistent with the corresponding literature data, have been presented and thoroughly discussed.


Introduction
The polymer electrolyte fuel cell (PEFC), which directly converts the chemical energy of hydrogen and oxygen into electrical energy, is a favourable power source for a multitude of portable, stationary and automotive applications [1][2][3][4][5][6].A wet-proofing, porous layer located between the catalyst layer and flow-field plate is termed as the gas diffusion layer (GDL) [7,8].The main functions of the gas diffusion layers are to: (i) supply reactant gases of oxygen and hydrogen to the active areas in the catalyst layer, (ii) mitigate the detrimental impact of excess water and (iii) electrically connect between the catalyst layers and the flow field plates to power the external load [9][10][11].
Water (mostly in the form of liquid) is produced as a by-product of the oxygen reduction reaction (ORR) taking place at the cathode catalyst layer.Furthermore, the reactant gasses fed to the fuel cell are mostly humidified to ensure the initial humidification of the membrane electrolyte.Insufficient or excess amount of water are both detrimental for the operation of the fuel cell.Namely, an insufficient amount of water causes the membrane to dry out, and subsequently reduce the membrane ionic conductivity.On the other hand, an excess amount of water, which is likely to occur at high current densities, induces flooding which may partially/completely block the voids within the GDL and hinder the transport of the reactant gases to the catalyst layers, thus resulting in mass transport losses [12][13][14][15][16].These losses are more detrimental while the PEFC is operated at the saturated conditions.To this end, water management in PEFC (particularly at its cathode) is highly crucial for its sustainable operation.
The commercially available GDLs are customarily made of electrically conductive carbon fibre-based materials and they are often made in the form of carbon papers or carbon cloths [17][18][19].However, such materials are likely to be, due to their low mechanical strength, subject to different types of degradation: mechanical degradation due to compression; thermal degradation arising from freeze/thaw cycles; dissolution; and erosion caused by the flow of gases [20][21][22].Hence, many studies have been conducted to explore and investigate alternative materials for the GDLs.
Metal foam-based materials have attracted a great deal of attention as promising components in a variety of energy and engineering related applications such as: filtration and separation, heat exchangers, flow distributors, thermal energy storage, heat pipes, electrolysers, catalyst supports, lithium-ion batteries, etc. [23][24][25][26][27][28][29][30][31][32].Recent studies have reported that nickel foams are promising materials that could be used as PEFC cathode flow field plates (FFPs) and/or GDLs and this is, compared to the conventionally-used materials, due to their higher: volume-to-weight ratio, cost effectiveness, electrical and thermal conductivity, porosity and specific surface area [33][34][35][36].Some recent investigations experimentally explored the potential use of the nickel foam as a cathode FFP and showed that such a FFP demonstrates better fuel cell performance and durability than those of the conventional graphite FFPs [35,[37][38][39][40][41][42].For instance, Shin et al. [41] experimentally tested a PEFC equipped with various nickel foam-based cathode FFPs and compared it with that running with a conventional graphitic serpentine FFP.They showed that the fuel cells operating with nickel foam cathode FFP are more stable and demonstrates a 50.6% better performance than that of the graphitic serpentine FFP.Likewise, similar findings were reported by Liu et al. [42] who indicated that the catalytic activity and gas diffusion are likely to increase when using cathode nickel foam FFPs.
The PEFC performance is highly influenced by the structural and effective transport characteristics of the GDLs and as such it is important to estimate these characteristics and subsequently understand how various quantities are transported within the GDLs [43].A variety of methods are used to estimate the structural GDL characteristics and the following are some examples [3].Mercury intrusion porosimetry (MIP) is widely used to determine the GDL porosity and the pore size distribution; see for example [44][45][46].Under vacuum conditions, gradually-increased pressure is applied to force the non-wetting mercury to penetrate the pores and subsequently determine the pore size distribution and the porosity of the porous media.The superficial two-dimensional images generated by the scanning electron microscopy (SEM) are often used to measure the thickness of the fibres; see for example [47].However, these superficial images do not necessarily provide accurate measurement of the fibre thickness.The Brunauer-Emmett-Teller (BET) absorption method is normally used to estimate the specific area of the GDL materials, e.g.Refs.[48,49].Most of the characterisation techniques, including the above mentioned ones, provide some "global" properties of the investigated material [50]; however, they do not resolve the heterogeneity of some materials that often lead to significant anisotropic transport properties as is the case for typical GDL materials.
X-ray computed tomography (CT) has been a handy tool to resolve the nano-or micro-scale local structural variation within the investigated PEFC materials, including the GDLs.Compared to other similar techniques, the X-ray CT technique is (i) non-destructive, (ii) quick and accurate, (iii) cost-effective and (iv) enable the examination under the realistic working conditions [51][52][53].The micro-structure of the carbon fibre based GDL has been extensively investigated using X-ray CT.For example, Zenyuk et al. [54], Fishman et al. [45,55,56], James et al. [57], Meyer et al. [58,59], Fazeli et al. [60], and Garcia-Salaberri et al. [61] used X-ray CT to determine the structure-related properties such as porosity, tortuosity and pore size distribution.On the other hand, the microstructure of nickel foams as alternative materials for PEFCs was investigated using X-ray CT in several studies [62][63][64][65].For instance, Fly et al. [63] performed imaging using the μ-X-ray CT and conducted electrochemical and mechanical tests to investigate the potential use of nickel foam as a FFP.The X-ray CT images show that the contact area between the nickel foam and the carbon paper GDL is ten times higher than that between the nickel foam and the stainless-steel FFP.Furthermore, the X-ray CT-based 3D models have also been employed to determine the structural and transport characteristics of the carbon-based and metal foam-based materials [66][67][68].However, these 3D models require high computational resources and are very computationally expensive.To reduce the computational time, some researchers [69,70] have investigated the structural properties of some conventional GDL materials using a two-dimensional Lattice-Boltzmann model that was created by employing SEM images.However, the SEM images do not provide information regarding the interior structure of the investigated material.
To the best of the authors' knowledge, there have been no in depth studies on the structural and the transport characteristics of the nickel foams as potentially-used GDLs in PEFCs.Therefore, the characteristics of a typical nickel foam have been estimated using a workflow that involves imaging using an X-ray CT facility, image processing and numerical modelling.In this paper, the non-uniformity of the material has been innovatively resolved through solving multiple computationallyeconomic two-dimensional models representing some local slices of nickel foam within the imaged structure.The estimated properties using the above workflow (i.e.imaging, image processing and numerical modelling) have been presented and discussed.

X-ray CT and image processing
X-ray CT images were captured using the SKYSCAN 1172 X-ray microscopy platform (Bruker, US).The platform comprises: an X-ray source, a rotational stage with a specimen holder and an X-ray detector to measure the X-ray density (Fig. 1).A commercially-available nickel foam sheet with 99.5% purity (GoodFellow Cambridge Ltd., the UK) was punched, creating 12.7 mm-diameter samples.The specimen was fixed on the rotational stage in the through-plane direction by a plastic-made apparatus.The emitted X-ray from the source crosses throughout the sample and then the detector that is located next to the rotational stage collects it.The scan was performed every time that the stage was rotated by 0.7 • until the full 180 • rotation was reached.The exposure time was 0.885 s and 1025 projections (i.e.two-dimensional slices) were collected.The source voltage was 80 kV and the beam current was 124 μA.The image resolution was 7 μm per pixel.
The captured slices were 2D grey shadows and were processed by the NRecon Reconstruction (SKYSCAN, Belgium) software to reconstruct the cross-section images from the X-ray CT projection images and Bruker CTan Micro-CT software to threshold the slices.To discriminate between the gaseous and solid phases, Otsu's thresholding method was used.The processed images were imported to COMSOL Multiphysics® 5.5 using MATLAB® and LiveLink™ for MATLAB®.

Numerical modelling
The continuity equation and conservation of momentum equation were, using COMSOL Multiphysics® 5.5, solved for 20 two-dimensional computational domains representing 20 equally-spaced scanned slices taken within 350 μm distance of the scanned nickel foam sample.The length and height of each slice are 6.65 and 1.05 mm, respectively.The flow within the computational domains was assumed to be steady-state, laminar (Reynolds numbers ≪ 2300) and incompressible (Mach numbers ≪ 0.3) and as such the continuity and momentum equations are expressed as follows: where ∇ is the operator ( ∂ ∂x + ∂ ∂y ),ρ is the fluid density (kg/m 3 ), and u is the velocity vector.
where p is the pressure, I is the identity matrix, and μ is the dynamic viscosity (Pa⋅s).The conservation of chemical species (oxygen gas in our case) is given by: where D O2 is oxygen diffusion coefficient (cm 2 /s), C O2 is the oxygen concentration (mol/m 3 ), and R is the source term which is zero in our case (there is no reaction taking place within the GDL).The 2D meshing procedure is conducted for each of the 20 slices using the COMSOL Multiphysics® 5.5 software.To ensure meshindependent solutions, the number of elements has been systemically changed and the property of interest has been computed for all the various meshes and the computation time has been observed for the all the corresponding meshes.For example, Fig. 2 shows how the computed through-and in-plane gas permeability and the corresponding computation time changes with the number for elements for the 200 th CT slice.The figure shows that the computed permeability values become almost insensitive to the number of elements with around 600,000 elements (the corresponding times are reasonably short: 100-200 s).The mesh independency test has been performed for all the 20 equally-spaced CT nickel foam slices and the minimum number of elements that achieves the mesh-independent solution was found to be between 400,000 and 650,000.The fluid flow was assumed to be air (mimicking the real-life situation in which air is fed to the cathode side of the fuel cell) and some arbitrary values of air velocity and oxygen concentration were used for the boundary conditions.Namely, the velocity was prescribed at the inlet of the domain (0.1 m/s) and the pressure was set at the outlet of the domain (0 Pa).Note that the bulk oxygen diffusivity coefficient D O2 at 20 • C and 1 atm is around 0.219 cm 2 /s [71].In addition, it is assumed that the inlet air comprises 21% oxygen and 79% nitrogen.As for the conservation of the species, the inlet oxygen concentration, C in,O2 , is 8.73 mol/m 3 and the outlet oxygen concentration is 7.73 mol/m 3 .It has been shown in the supplementary material how the inlet concentration of oxygen has been calculated.The governing equations ( 2) and (3) were solved independently for each direction (i.e.through-plane direction (Fig. 3a) and in-plane direction (Fig. 3b)) to respectively solve for the distributions of velocity and oxygen concentration within the computational domain and subsequently estimate the gas permeability and effective diffusivity for each principal direction.It is noteworthy that the convective term is not considered in Equation ( 3) as the objective is to estimate the effective diffusivity of the porous media and for this to be achieved, one only needs to solve for the concentration, and not the velocities.It should be noted that symmetry boundary conditions were used for the left and the right sides (through-plane case) and wall boundary conditions were used for top and bottom sides (in-plane case).The solid-phase within the domains (Fig. 3) are the white areas and the no-slip boundary conditions were used for the walls of these areas.The simulations were performed assuming that the temperature of the inlet is 20 • C. The models were numerically solved using a small workstation (Inter® Xenon® CPU E3-1246 v3@ 3.50 GHz, 16 GB installed RAM) and the computational time for each modelled slice was found to be between 140 and 220s.The estimated properties using the above workflow (i.e.imaging, image processing and numerical modelling) have been presented and discussed.Figure S1 in the supplementary material is a flow chart that schematically shows the methodology and the order of the investigations.

Structural properties 2.3.1. Porosity
The porosity (ε) is the ratio between the void volume and the total volume of the porous medium.All the transport properties of the porous media are a strong function of the porosity.For example, the mass transport properties (e.g.gas permeability and diffusivity) increase with increasing porosity while the electrical and thermal conductivity decrease with increasing porosity [72].The porosity values are often determined using mercury intrusion porosimetry (MIP) or other standard porosimetry methods.The maximum reported value for the porosity of a commercially available carbon fibre-based GDL is 0.9 while the porosity of nickel foam could be as high as 0.98 [3,73,74], signalling higher transport properties of the latter structure (i.e. the nickel foam).The porosity of the nickel foam sample was estimated for each imaged slice by using the COMSOL Multiphysics® 5.5 to measure the void phases and solid phases.The porosity of the nickel foam sample was estimated for each imaged slice within the Geometry node of the COMSOL Multiphysics® 5.5 by separately measuring the areas for each of the void and the solid phases.Evidently, the porosity for each slice is estimated by dividing the area of the void by the total area (which is the sum of the void and solid phases [75]:

Tortuosity
The tortuosity (τ) is, as implied by its name, a measure of how tortuous is the pathway for the transported fluid; the higher is the tortuosity, the longer is the pathway for the fluid.It also provides insight into the interconnections between the pores of the porous media.The following Equations ( 5) and ( 6) (which were found to be suitable for two-dimensional domains) could be (and have been in this work) used to estimate the tortuosity values [70,76]: where u mag is the velocity magnitude averaged over the computational domain and u y and u x are, respectively, the velocity components in the through-and in-plane directions, averaged over the computational domain.

Pore size distribution
The "pore" term means a small void/space/perforation that connects the cells.The pore size distribution strongly impacts on the transport of reactant gases and liquid water within the GDLs, particularly those at the cathode side, and subsequently impacts the performance of the fuel cell [19,41].The pore size distribution was estimated for the selected imaged 20 slices using an open-source ImageJ/Fiji software package [77].All the pores are manually measured and counted through the slices using ImageJ/Fiji software and the pore size distributions have been given in Section 3.3.It should be noted that, with SEM images, one could only compute the pore size distribution on the surface of the scanned structure while the X-ray CT enables one to obtain the multiple local interior pore size distribution of the structure.

Ligament thickness distribution
The ligament or the strut is a solid metal rod that forms the cellular structure in the nickel foam.The ligament thickness directly affects the electrical and the thermal conductivity, and the mechanical strength of the nickel foam.As with the pore size distribution, the range of the ligament thickness of the nickel foam and its distribution was determined for the selected imaged 20 slices using an open-source ImageJ/ Fiji software [77].Likewise measuring the pore size, the ligaments have been manually measured and counted through the slices, and the ligament thickness distributions have been presented in Section 3.4.

Specific surface area
The specific surface area is the surface area of the material per unit of volume [78].As with the other above mentioned structural properties, the specific surface area of the nickel foam affects the mass, heat and charge transport distributions [79,80].The nickel foams inherently possess a very high surface area and this is due to their tetradecahedron structure [81].The Brunauer Emmett Teller (BET) volumetric and gravimetric methods are often used to measure the specific surface area of porous materials, including the metal foams [82,83].Further, the knowledge of the porosity and the pore size distribution allows for the estimation of the specific surface area as demonstrated in Refs.[84,85].The total length of the borders and the area of the solid phase were calculated within COMSOL Multiphysics® 5.5 and when these two quantities are divided, the specific surface area is obtained for the investigated slice.M. Ercelik et al.

Mass transport properties 2.4.1. Gas permeability
The gas permeability is, as implied by the name, a measure how permeable is the porous medium to gases.The low permeability of the GDL leads to higher gas pressure gradients which leads to higher water saturation in the cathode GDLs and potentially undesirable water flooding phenomena particularly at the cathode side [3,18,[86][87][88][89][90][91].Furthermore, the high gas permeability increases the convective flow, which translates into an increased amount of the reacting gas supplied to the catalyst layer and ultimately better fuel cell performance [8,12,[92][93][94][95][96][97][98].Therefore, it is always desirable for the GDL materials to demonstrate high permeability values.For sufficiently low velocities, the viscous resistance is dominant and the inertial resistance is negligible and this enables one to use the simple linear equation of Darcy's Law (rather than the Forchheimer equation) to estimate the permeability of the porous media, K, as follows [93,94,99]: where ΔP is the pressure difference across either the thickness (throughplane) or the length (in-plane) of the GDL material(L) and μ is the dynamic viscosity of the flowing fluid.Following the solution of the conservation of the momentum equation, as described in Section 2.2, the pressure drop was computed for each of the 20 modelled nickel foam slices in both the through-and in-plane directions; this enables one to calculate the respective gas permeability values.It should be noted that u has been set to be the velocity prescribed at inlet of the computational domain (i.e.0.1 m/s).It should be noted that the gas permeability is an intrinsic property of the material and it is therefore invariant with velocity.Figure S2 in the supplementary material show that the pressure gradient changes linearly with the inlet velocity and Figure S3 shows that the gas permeability values (in both directions) for arbitrarily selected CT slices are almost insensitive to the inlet velocity.The through-plane gas permeability of the nickel foam was also experimentally estimated using an in-house setup [8,95] to validate the corresponding computed permeability values.The setup consists of top and bottom fixtures, a flow controller (HFC-202, Teledyne Hastings, UK) and a differential pressure sensor (PX653, Omega, UK).Seven circular samples (25.4 mm in diameter) were punched out of a nickel foam sheet and each sample was placed between the fixtures.Nitrogen gas was passed through the sample and the pressure drop was recorded for each flow rate.Finally, using all the knowledge of all the parameters shown in Equation ( 7), the gas permeability of each sample was calculated and then the gas permeability values for all the samples were averaged.It should be noted that the dynamic viscosity of nitrogen is 1.751 × 10 − 5 Pa.s at 20 • C which was the temperature of the room in which the test was conducted [100].

Effective diffusivity
Diffusion is the main mode of transport for the gases in the conventional carbon fibre based GDLs and this is due to their relatively low permeability (>10 − 13 m 2 ) [96] and is, at least, expected to play a crucial role in transporting the chemical species within the nickel foam based GDLs.The effective diffusivity is a measure on how diffusive the chemical species into each other within the porous medium.The effective diffusivity of the species i into the species j within the GDL,D eff ij , could be estimated using the following form of Fick's law: where J is the molar flux (mol/(m 2 ⋅s)), and ΔC is the concentration difference between the inlet and outlet across either the thickness (through-plane) or the length (in-plane) of the GDL material (L).The molar flux was, following the solution of the conservation of the chemical species equation as described in Section 2.2, computed for each of the 20 modelled nickel foam slices in both the through-and inplane directions.With this and with the knowledge of ΔC (1 mol/m 3 ) and L (1.05 mm for the through-plane direction and 6.65 mm for the inplane direction), the respective effective diffusivities could be calculated.It should be noted that the species i and j were taken to be oxygen and nitrogen in this study, mimicking the real-life situation in PEFCs where air is typically fed into the cathode of the fuel cell.In addition, it is worth mentioning that the concentration difference does not have an impact on the estimated effective through-or in-plane diffusivity of oxygen as the molar flux proportionally changes with the concentration difference, thus maintaining the proportionality factor (the effective diffusivity of oxygen) constant; see Table S

Results and discussion
Figs. 4 and 5 show the velocity profiles, for the through-and in-plane flow directions, respectively, within one of the modelled CT slices.It is evident that the random position and the size of the ligaments of the foam significantly influence the velocity magnitude and direction within the modelled CT slice, thus resulting in highly non-uniform velocity profiles.Similar profiles were obtained for the other modelled CT slices (not shown).Notably, a slightly more uniform velocity profile is obtained for the in-plane flow compared to the through-plane direction.As will be shown in Section 3.2, this could be attributed to the smaller tortuosity values as demonstrated by the nickel form in the in-plane direction than in the through-plane direction.The following subsections present the computationally-estimated structural and mass transport properties of the nickel foam.

Porosity
Fig. 6 shows the porosity values for each of the equally-spaced selected 20 slices.The range of the porosity is from 0.844 to 0.873 and the mean values for the porosity values of all the slices is 0.856 with a 95% confidence interval of ±0.003.These porosity values are in agreement with those reported in the literature [101][102][103][104][105][106].For instance, the porosity of the nickel foam was experimentally determined by Oun and Kennedy [103] as 0.88.Also, the porosity of the nickel foam was reported by Khayargoli et al. [104] to be between 0.83 and 0.90, and by Slade [107] to be between 0.85 and 0.97.Moreover, Vicente et al. [106] computed the porosity of some nickel foam samples employing 3D CT images and found that it lies between 0.87 and 0.93.These relatively wide ranges for the porosity values and/or the slight discrepancies between the values/ranges reported by the various research groups could be attributed to the potentially different manufacturing settings used [50].The multiple 2D CT images using computational method, unlike the conventional experimental methods and the 3D numerical X-ray CT studies, reveal how uniform the nickel foam is in terms of porosity.This uniformity and the high values of porosity of nickel foams is expected to facilitate mass transport within the PEFCs.

Tortuosity
The through-and in-plane tortuosity values for the 20 equally-spaced two-dimensional CT slices are presented in Fig. 7.The average throughplane tortuosity is 1.175, which is between those values reported in the literature by Khayargoli et [108].The average in-plane tortuosity (i.e.1.124) was found to be slightly less than that in the through-plane direction and in line with those reported in the literature by Brun et al. (i.e.1.09-1.13)[109].It is noteworthy that the latter authors estimated the tortuosity making use of the X-ray CT images generated for a nickel foam material.The results shows that the tortuosity of the nickel foam are significantly less than those of the conventionally-used carbon substrates; the smallest reported tortuosity for an SGL and Toray carbon substrates were found to be 1.33 and 2.50, respectively [110,111].This suggests that, compared to the carbon substrates, the reacting gases and/or excess water would flow through more straight pathways when using nickel foams as the GDLs, thus reducing the mass transport resistance and potentially mitigating the water flooding, particularly beneath the ribs of the flow-field plates.

Pore size distribution
The pore size distribution histograms for 10 CT slices are shown in Fig. 8. Considering all the slices, the average pore size was calculated to be around 390 (±16) μm and the range of the pore size is between 100 and 750 μm.The above average pore sizes are in line with those reported in the literature.Namely, it lies between that reported by Khayargoli et al. [104], i.e. 360 μm, and that reported by Milazzo et al. [112], Oun and Kennedy [113], and Hellmann et al. [113], i.e. 450 μm.The average pore size of the nickel foam is two orders of magnitude higher than those of the conventional carbon fibre based GDLs [110], which suggests that, as with the tortuosity, that less mass transport resistance is demonstrated by nickel foam based GDLs.It is noteworthy that the nickel foam exhibits a slightly larger pore size compared to other metal foams and this due to the relatively thinner ligament thickness of the former foam [114].Large pore size in the porous material is beneficial for effective mass transport [115]; Crosnier et al. [116] stated that the larger is the pore diameter, the smaller is the pressure drop and the higher is the  permeability.As the pore size of nickel foam decreases, the specific surface area increases which creates an additional flow resistance.

Ligament thickness distribution
The ligament thickness distribution histograms for 10 CT slices are shown in Fig. 9. Considering all of these 10 slices, the average ligament thickness was calculated to be around 99 (±3) μm and the range of the ligament thickness is between 60 and 180 μm.This calculated average ligament thickness is consistent with those reported in the literature.Namely, Tsolas et al. [117] experimentally and numerically evaluated the properties of nickel foams and reported the ligament thickness as 95 μm.Further, Miwa and Revankar [101] experimentally evaluated the structural characteristics of nickel foam and reported an average ligament thickens that is almost identical to that reported in this study, i.e. ~ 99 μm.It should be noted that the average ligament thickness of nickel is significantly larger than the average carbon fibre thickness of the conventionally used GDLs; the carbon-based GDLs fibre diameter has been reported to be between 7 and 10 μm [118,119].

Specific surface area
Fig. 10 shows the specific surface area (SSA) for 20 equally-spaced two-dimensional CT slices.As seen from the figure, the range of the SSA is between around 40,500 and 46,100 m − 1 .The average SSA for all the slices is around 43,560 (±310) m − 1 which is in accordance with those reported in the literature.Namely, Yang et al. [120] experimentally investigated nickel foam as a cathode electrode for aluminium-hydrogen peroxide fuel cells and they reported the SSA of the nickel foam as 42,800 m − 1 .Similarly, Langlois and Coeuret [121] experimentally assessed the structural properties of nickel foams and they, using a BET analysis, reported that the SSA of the nickel foam sample to be around 41,000 m − 1 .Compared to the conventionally used carbon substrates, nickel foam possess significantly higher SSA; this implies that they have superior heat and electrical conduction due to the better contact to the flow-field plates [113].

Gas permeability
The through-and in-plane permeability was calculated as described in Section 2.4.1 for all the 20 equally-spaced two-dimensional CT slices; see Fig. 11.The average through-plane and in-plane permeability values are, considering the permeability of all the individual modelled slices, 1.26 × 10 − 9 and 1.39 × 10 − 9 m 2 , respectively.The range is between 9.81 × 10 − 10 and 1.51 × 10 − 9 m 2 for the through-plane permeability and is between 8.63 × 10 − 10 and 1.84 × 10 − 9 m 2 for the in-plane permeability.The above computationally-estimated through-plane permeability is in very good agreement with those experimentally estimated values using the in-house set-up which is 1.40 × 10 − 9 m 2 .Further, the above average permeabilities are in excellent agreement with those reported in the literature.Namely, Khayargoli et al. [104] and Medraj et al. [122] experimentally estimated the through-plane permeability of nickel foam as 1.30 × 10 − 9 m 2 .The average in-plane permeability computed in this study lies between that reported by Hugo et al. (1.38 × 10 − 9 m 2 ) [102] and that reported by Miwa and Revankar (1.45 × 10 − 9 m 2 ) [123].Overall, the gas permeability of nickel foam is at least two orders of magnitudes higher than those of the conventionally used carbon fibre based substrates whose values are normally between 10 − 11 and 10 − 12 m 2 [110].This signifies that nickel foam based GDL would demonstrate significantly higher convective flows than the conventionally used GDLs, thus improving the supply of the reactant gases to the catalyst layers and/or the removal of excess water from the catalyst layers.Another observation is that the gas permeability of nickel foam was found to be more isotropic than that of the commonly used carbon fibre based carbon substrates; the in-plane permeability of the latter material is larger than the through-plane permeability by almost one order of magnitude [12,124].

Effective diffusivity
Fig. 12 shows the through-and in-plane effective diffusivity of oxygen into nitrogen corrected for the structure of the nickel foam for the all the 20 equally-spaced two-dimensional CT slices.The average throughand in-plane effective diffusivity are, considering all the 20 CT slices, 0.154 and 0.166 cm 2 /s, respectively.The range is between 0.142 and 0.165 cm 2 /s for the through-plane effective diffusivity and is between 0.156 and 0.176 cm 2 /s for the in-plane effective diffusivity.As with the gas permeability of nickel foam, the effective diffusivity of nickel foam shows a good degree of isotropy as evidenced from the comparable values for the effective diffusivity in the through-and in-plane directions.This should be compared with the effective diffusivity of the commonly used carbon substrates where the in-plane diffusivity is higher than the through-plane value by a factor of 1.3-3 [61,125].Moreover, the nickel foam-based GDL yields a much higher effective diffusivity for both the through-and in-plane directions than those of the conventional carbon-based GDLs which are typically between 0.066 and 0.120 cm 2 /s (for through-plane direction) and 0.085 and 0.124 cm 2 /s (for in-plane direction), respectively [126].

Conclusions
Nickel foams have a great potential to be used as gas diffusion layers in polymer electrolyte fuel cells and this is due to their superior structural and transport properties.In this work, some key structural (porosity, tortuosity, pore size distribution, ligament thickness distribution, and specific surface area) and mass transport (gas permeability and effective diffusivity) of a typical nickel foam sample have been determined using equally-spaced multiple computationally-efficient two-dimensional X-ray CT images and/or corresponding numerical models.This is, in addition to saving computational time through not solving computationally expensive three-dimensional images and/or numerical models, performed to check for the multidimensional uniformity of the nickel foam samples.The main findings could be summarised as follows: • All the computationally-determined properties of the nickel foam, compared to those of the conventionally-used carbon substrates, demonstrate better uniformity in the in-plane and through-plane directions and better isotropy as evidenced from the comparable individual values computed for all the two-dimensional CT slices.Further, all of these computed properties were found to be in agreement with experimental and/or computationally-determined literature data.• The mean porosity and the mean pore size of the nickel foam sample were computed to be around 0.86 and 390 μm respectively; these   values are, compared to those of the conventionally-used carbon substrates, high; thus facilitating the mass transport of the gases and liquid water to/from the catalysts layers.Likewise, the relatively low through-plane (~1.18) and in-plane (~1.12) tortuosity lowers the mass transport resistance of the chemical species.
• The mean ligament thickness was computed to be around 99 μm which is one order of magnitude higher than the mean carbon fibre diameter of the conventionally-used carbon substrates; thus implying better mechanical support by nickel foam for the delicate membrane electrode assembly of the fuel cell.
• The computed mean through-plane (1.26 × 10 − 9 m 2 ) and in-plane (1.39 × 10 − 9 m 2 ) gas permeability was found to be at least one order of magnitude higher than those of the normally-used carbon substrates, signifying higher contribution of convective flow to the mass transport of gases within the nickel foam based GDL.Likewise, the computed mean effective through-plane (0.154 cm 2 /s) and inplane (0.166 cm 2 /s) gas diffusivity was found to be higher than those of the carbon substrates by an order of magnitude, meaning more effective diffusive transport to/from the catalyst layer and beneath the ribs of the flow field plates.

Fig. 1 .
Fig. 1.Photographs of (a) the SKYSCAN X-ray CT platform and (b) the obtained two-dimensional slices.

Fig. 2 .Fig. 3 .
Fig. 2. Change of through-plane (a) and in-plane (b) permeability (and the corresponding computation time) of the 200 th slice with number of elements.Yellow strip shows the selected number of elements which is around 600,000.

Fig. 4 .
Fig. 4. The numerical-estimated velocity profile for the through-plane flows within the modelled CT nickel foam slice 200 (a and b), 500 (c and d) and 800 (e and f).

Fig. 5 .
Fig. 5.The numerical-estimated velocity profile for the in-plane flows within the modelled CT nickel foam slice 200 (a and b), 500 (c and d) and 800 (e and f).

Fig. 12 .
Fig. 12. Computationally computed (a) through-plane and (b) in-plane effective diffusivity of oxygen into nitrogen for 20 equally-spaced CT nickel foam slices.
Abbreviations BET Brunauer-Emmett-Teller CT Computed tomography FFP Flow field plate GDL Gas diffusion layer MIP Mercury intrusion porosimetry ORR Oxygen reduction reaction PEFC Polymer electrolyte fuel cell SEM Scanning electron microscopy SSA Specific surface area (m − 1 ) .1, Table S.2 and Figure S.4 in the supplementary material.