doi: 10.1149/2.024211jes2012, Volume 159, Issue 11, Pages F683-F692.J. Electrochem. Soc. 

 G. S. Hwang and A. Z. Weber Fuel-Cell Gas-Diffusion LayersEffective-Diffusivity Measurement of Partially-Saturated

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© 2012 The Electrochemical Society

Journal of The Electrochemical Society, 159 (11) F683-F692 (2012) F6830013-4651/2012/159(11)/F683/10/$28.00 © The Electrochemical Society

Effective-Diffusivity Measurement of Partially-Saturated Fuel-CellGas-Diffusion LayersG. S. Hwang and A. Z. Weber∗,z

Environmental Energy Technologies Division, Lawrence Berkeley National Laboratory, Berkeley,California 94720, USA

In proton-exchange-membrane fuel cells, flooding of the cathode gas-diffusion layer (GDL) hinders the gaseous reactant transportand thereby limits cell performance. The understanding of the effective diffusivity of the reactants through the GDL is essential forperformance optimization and material design. In this paper, the effective diffusivities of unsaturated and partially-saturated GDLsare experimentally examined using an ex-situ electrochemical limiting-current method for various, uncompressed GDLs includingdifferent PTFE loadings. For unsaturated (including PTFE loadings) and partially-saturated (no PTFE) GDLs, the experimental resultsfollow a power law with respect to porosity and saturation, respectively. PTFE treatment favorably changes the liquid distribution forimproved gas-transport pathways, and a new correlation is proposed using a cumulative log-normal distribution function; however,the impact of PTFE on the overall effective diffusivity depends on the specific GDL structure. This work provides insights forfuel-cell models and transport phenomena, which can lead to the optimal GDL material design and cell operation.© 2012 The Electrochemical Society. [DOI: 10.1149/2.024211jes] All rights reserved.

Manuscript submitted July 17, 2012; revised manuscript received August 2, 2012. Published August 31, 2012.

Optimal water management in proton-exchange-membrane fuelcells (PEMFCs) is critical to enable high power and efficient energyconversion.1–3 For high-power generation, especially at lower temper-atures, one challenge is the flooding of the cathode gas-diffusion layer(GDL) by the produced and transported water. This flooding limits thegas diffusion to the electrochemical reaction sites, resulting in a de-crease in fuel-cell performance. Thus, the examination of the effectivediffusivity in a partially-saturated GDL promotes the understandingof the role of the liquid water on the gaseous reactant transport, whichis critical to improve the optimal GDL material design and enhancefuel-cell water management.

The GDL in PEMFCs is generally a porous material composed ofstacked or woven carbon fibers as shown in Figure 1a. Its porosityand pore-size distribution are designed for the desired gas diffusionand water management, while the carbon fibers provide the requiredelectrical/thermal conduction. The presence of the carbon fibers hin-ders the gas diffusion by increasing the tortuosity or gas-transportpath length (the degree of the meandering diffusion pathways). Poly-tetrafluoroethylene (PTFE) is often coated on the fiber surfaces toimprove water management, i.e., it wet-proofs the fibers to avoid wa-ter flooding, however, this coating reduces the porosity (see Figure 1b),resulting in more tortuous diffusion pathways. In a partially-saturatedGDL, water-filled pores further increase the diffusion pathways (Fig-ure 1c), and the water-filling behaviors are closely related to wet-proofing coatings, which could change the diffusion pathways (Figure1d). Thus, an optimal PTFE loading is required for improved watermanagement.4–6

In partially-saturated GDLs, the effective diffusivity can be ex-plained with respect to the porosity and liquid saturation as two sepa-rate functions,7,8

〈D〉D

= ε

τ= f (ε)g(S) [1]

where 〈D〉 is the effective diffusivity, D is the bulk diffusivity, ε isthe porosity (defined as the ratio of the void pore volume to the totalsample volume), S is the liquid saturation (ratio of water-filled volumeto the total pore volume), τ is the tortuosity, and f(ε) and g(S) are thenormalized functions of porosity and liquid saturation, respectively.The above formulation allows one to deconvolute the contributionsto the effective diffusivity of just the intrinsic pore structure and thatof the water-wetting behavior, which both affect the effective diffu-sivity in a different manner. Thus, both functions are scaled from 0 to1, where f(ε) describes the effective diffusivity for unsaturated (dry)GDLs (i.e., g(S) = 1), and g(S) describes the effective-diffusivity re-duction in the presence of liquid by normalizing the total effective

∗Electrochemical Society Active Member.zE-mail azweber@lbl.gov

diffusivity to that of the dry GDL. Note that the tortuosity is closelyrelated to the porosity, and therefore the effective diffusivity is of-ten modeled as solely a function of porosity; however, as mentioned,since liquid water seems to impact the effective diffusivity in a differ-ent manner than just the solid structure as discussed below, the twoeffects are explicitly accounted for in Eq. 1.

A widely used approach for f(ε) employs effective-medium theory,9

postulating that the effective diffusivity has a similarity to an electricalconductivity. This theory was originally developed for the electricalconductivity in the presence of spherical dielectric particles, obtainingm = 1.5 (Eq. I-1 in Table I),10 the so-called Bruggeman relation,which has been widely used for the effective diffusivity of GDLsin fuel-cell models.9,11 However, this value is only expected to bevalid when the size distribution of the spherical particles is large, and

(a) (b)

(c) (d)

EffectiveGas Diffusion

Fibers

PTFECoatings

Figure 1. Schematic of the typical GDL geometrical structures of (a) drywithout PTFE (b) dry with PTFE (c) partially-saturated without PTFE, and(d) partially-saturated with PTFE.

F684 Journal of The Electrochemical Society, 159 (11) F683-F692 (2012)

Table I. Literature expressions of the effective diffusivity for fuel-cell GDLs.

f(ε) g(S) Comments Label

εm 1 m = 1.5 (effective medium theory)9 I-1m = 3.8 (empirical)13 I-2

ε( ε−εpεo−εp

)α 1 εo = 1, α = 0.785, εp = 0.11 (network model with percolation threshold)19 I-3

1 − (1 − ε)0.46 1 network model20 I-4

1 − q(ε)[ 3(1−ε)3−ε

] 1 q(ε) = 1 (effective medium theory)21 I-5q(ε) = 2.76 ε cosh(3ε−1.92) (empirical)14 I-6

n = 2 (in-plane) n = 3 or 4 (through-plane) (network model)8 I-7f(ε) (1 − S)n n = 5 (network model)22 I-8

n = 1.5 (effective medium theory by analogy)11 I-9

f(ε) (1 − S)n( Sp−SSp

)k n = −1.7, k = 2.2, and Sp = 0.27 to 0.3 (network model)24 I-10

generally, it is not applicable for shapes other than spheres, especiallygreatly elongated or cavity-containing structures which is similar tothe typical GDL structures.12 To examine the true tortuosity, extensiveexperimental and modeling work has been performed for unsaturatedGDLs and, to a lesser extent, partially-saturated GDLs.

For unsaturated GDLs, various experimental and modeling re-sults have shown that the effective diffusivity is a strong function ofGDL porosity as summarized in Table I.4,6,13–15 Using electrochemicaldiffusimetry,6,16 the effective diffusivity was measured by correlatingthe effective ionic conductivity of electrolyte-soaked GDLs (conduc-tivity through the pores) with the gaseous transport (diffusivity throughthe pores), both for in-plane and through-plane diffusivities includingvarious compressions and PTFE loadings. Similarly, the effective dif-fusivity was also measured both for the cloth and paper type GDLs,showing that the fiber structures resulted in significant diffusion path-ways at the same porosity, with the cloth type being less tortuous.13

The suggested empirical relation for the paper type GDL is Eq. I-2with m = 3.8, while m = 1.5 predicts the effective diffusivity for thecloth-type GDL. Using humidity sensors, the effective diffusivity ofwater was measured in dry GDLs and similar results were obtained,and no apparent thickness effect was found.4 Another measurementapproach utilized a standard Loschmidt cell to examine the effectivediffusivity of GDLs and microporous layers, including various GDLPTFE loadings.15 Employing a limiting-current method with dilutedoxygen in the cathode, a total transport resistance was measured, andthe individual resistances were analyzed from different sources, suchas catalyst layer, MPL, GDL and gas channels.17 Another approachused a galvanic cell to measure dry GDLs, showing that the tortuositycannot be only function of porosity due to anisotropic diffusivity oftwo different samples,18 which is consistent with previous work.13

Extensive modeling work has also been performed for determiningthe unsaturated GDL tortuosity. Using a pore-network model with apercolation threshold, εp, Tomadakis and Sotirchos19 determined theexpression given as Eq. I-3, and for two-dimensional fiber-porous-structure simulation, they obtained α = 0.785 and εp = 0.11 forthrough-plane transport. In a random, 2-D pore-network model mim-icking a catalyst layer, the effective diffusivity was determined as Eq.I-4.20 Recently, the effective diffusivity was derived treating GDLsas an effective medium consisting of spherical particles to show themaximum effective diffusivity.21 The resulting relation was originallygiven as Eq. I-5, and this was modified using a realistic fiber-structurelater as Eq. I-6.14

For partially-saturated GDLs, the effective diffusivity has been alsoexamined through pore-network models. For a 2-D fiber structure,Eq. I-7 with n = 2 was calculated by Nam and Kaviany.8 In fact,the modeled 2-D fibrous structure was highly anisotropic, and theyobtained n = 2 for in-plane, and n = 3 or 4 for the through-plane di-rection depending on the liquid-water distribution. They addressedthat their configuration might have too much hindered the diffu-sion pathways, and suggested n = 2 for through-plane transport.

However, this relation overpredicts the effective diffusivity as de-termined in a further elaborated pore-network model, which foundn = 5 (Eq. I-8).22 Other studies also employed effective mediumtheory and used a Bruggeman expression for the effective diffusiv-ity, n = 1.5, for fuel-cell models.11,23 Another modeling approachemployed a Lattice-Boltzmann method in a randomly-placed, over-lapping spherical-particle domain, and they fitted the results usingEq. I-10 including a percolation threshold.24

The above relations for partially-saturated GDLs were all computa-tionally determined, and there is a need to provide experimental valida-tion for the expressions as they vary considerably. To date, there havebeen minimal experimental studies exploring these properties.25–27

In one study that used neutron imaging, the authors discuss thatEq. I-7 with n = 2 overpredicts the effective diffusivity,25 while theother neutron-imaging study concluded that n should be 5 to 6.26

However, these two studies were not comprehensive and plagued withuncertainties related to neutron imaging as well as accomplished withrelatively complicated setups where the water saturation was not wellcontrolled.28 The only other studies for measuring effective diffusiv-ities utilized an ex-situ setup with a galvanic oxygen sensor (as anoxygen absorber), and both commercial and customized heteroge-neous wetting GDLs were investigated.29,30 Using stacked GDLs (6 to130 samples), it was reported that the water-filling methodology wasimportant for determining the effective diffusivities for some samples,and also that GDL wettability alters the liquid-water distribution, lead-ing to significant changes in the effective diffusivity; however, neitherthe validation of the previous models nor any proposed relation forthe effective diffusivity were addressed.

In this paper, a novel ex-situ experiment for the effective diffu-sivity of partially-saturated GDLs is devised that utilizes a limiting-current measurement. Using this technique, the effective diffusivityis experimentally determined as a function of saturation for variouscharacteristic structures of GDLs including PTFE loadings. To thebest of the authors’ knowledge, this is the first attempt to measuresystematically the effective diffusivity of partially-saturated GDLs.Below, the technique is described and then the experimental resultsare given for both unsaturated and partially-saturated GDLs.

Experimental

Experimental setup.— The effective diffusivity is measured usingthe limiting current in an electrochemical hydrogen-pump cell. Thehydrogen-pump setup is chosen due to the fast and highly reversiblereaction kinetics of hydrogen on Pt and no production of water. Aschematic of the experimental setup is shown in Figure 2. The cellconsisted of a membrane-electrode assembly (MEA), GDLs, and flowchannels. The MEA (Ion Power, Inc.) was a NR212 membrane with aPt/C catalyst loading of 0.4 mg Pt/cm2 on both sides. The MEA wassandwiched with the dry GDL (SGL10AA in the reference/counter

Journal of The Electrochemical Society, 159 (11) F683-F692 (2012) F685

MEA GDLDry/Partially-

Saturated GDL

Dry/Partially-Saturated GDL

Counter/ReferenceElectrode

Working Electrode

Electrically-Insulating Gasket

Electrically-Conducting Gasket

CH

2H+

V

A

2H+ + 2e- → H2

CH2,o

LGDLx

CH2 = 0

H2 → 2H+ + 2e-

CH

H2H2

H2/Ar H2/Ar

2e-

2e- 2e-

+ -

2e-

Figure 2. Schematic of the electrochemical cell including the partially-saturated GDL in the working electrode. The hydrogen concentration profileof the GDL in the working electrode at the limiting current is also shown.

electrode) and dry or partially-saturated tested GDL in the workingelectrode. The flow fields had 1 × 1 cm2 parallel flow channels thatwere 0.4 mm wide by 0.2 mm deep and spaced 0.4 mm apart for bothsides. For the cell assembly, a 320 μm thick, electrically-insulatinggasket with a 1 cm2 square hole is used in the reference/counter elec-trode, whereas two types of gaskets are used in the working electrode;one is a 320 μm thick, electrically-insulating gasket with 1.905 cm-diameter circular hole (same size of GDL), and the other is a794 μm thick, electrically-conducting (silicon-based) gasket with a4.763 mm-diameter circular hole (area of 3.74 mm2) to control theeffective gas transport area. The small active area was used to en-sure that the ratio of the reactant supply to its consumption maxi-mizes and down-channel variations in the flow channels are minimal,so that the reactant supply is not a source of the limiting current.Such a small hole size could possibly introduce edge effect (in-plane hydrogen diffusion through the peripheral boundary), but theexpected contribution is limited since the aspect ratio of the active area(∼4.8 mm in diameter) to sample thickness (∼370 μm) is large. Al-though the validation on the minimal edge effect is not explicitlyshown by testing with the larger active areas since operating windowsare very limited, the measured diffusivities for dry GDLs are con-sistent with the available models and measurements as shown below.Thus, the edge effect is believed not to be a significant source ofmeasurement uncertainty.

To increase the supply/consumption ratio, one needs to increasethe supply flow rate; however, this may create turbulent flow next toand possibly within the GDL which makes data interpretation difficult.Thus, an optimal operation condition is required, which is determinedas 100 sccm for the electrodes (convective flow was determined tobe dominant with a flow rate of 200 sccm or greater in the currentsystem). It is noted that the limiting current by the hydrogen supplywas about 5 mA, and the measured current was carefully checkedto ensure that it was a diffusion-controlled limiting current and notlimited by hydrogen supply. For example, a single sheet of a thinsample with high porosity, i.e., Freudenberg H2315, is expected tohave a larger diffusivity, and in this case the current was limited by theH2 supply rate. To counteract this, for this test, a stack of two sampleswas used to increase the reactant diffusion resistance to have a limitingcurrent. Also, for the partially-saturated Freudenberg H2315, the testrange was confined to S > 0.5 to ensure that a well-defined limitingcurrent was obtained (the full S range was used in the other samplesfor the limiting current).

The cell was operated at room temperature with no back pressure.The above constraints led to a small operating and design window, butthe experimental results are consistent with previous works, as will bediscussed below, thus showing no major issues in the methodology.

In the catalyst layers, the reactions are given as

H2 → 2H+ + 2e− (working electrode) [2]

and

2H+ + 2e− → H2 (counter/reference electrode). [3]

In the working electrode, the reaction is limited by the diffusion ofhydrogen through the dry/partially-saturated GDL at high potentials.Thus, the hydrogen diffusion is related to the current by

NH2 = i2F

[4]

where NH2 is the molar flux of hydrogen, i is the current density, and Fis the Faraday’s constant. The hydrogen molar flux can also be relatedto Fick’s law (the mixture is essentially binary since both sides operatewith humidified gas to inhibit a water-vapor flux and evaporation),

NH2 = − ⟨DH2−Ar

⟩ ∇cH2 [5]

where 〈DH2−Ar〉 is the effective diffusivity of hydrogen through argonin a binary mixture in the GDL, and cH2 is the hydrogen concentration.At the limiting current, io, the hydrogen concentration at the catalystlayer is essentially zero; the catalyst layer is not limiting (diluted hy-drogen is completely depleted within 0.18 μm in the catalyst layerat the given cell potential of 0.35 V as calculated in the Appendix),and the concentration difference simply equals the hydrogen concen-tration in the flow channel, cH2,o. This assumption and treatment ofnegligible catalyst-layer influence on limiting currents has also beenseen previously in the literature.31 Combining Eq. 4 with Eq. 5, resultsin17

io

2F= 〈DH2−Ar〉 cH2,o

LGDLor 〈DH2−Ar〉 = io LGDL

2F cH2,o[6]

The above relation is used to calculate the effective diffusion co-efficient using the measured limiting current and known GDL samplethickness. It should be noted that the limiting current should be in-dependent of the other losses in the system (e.g., ionic conductionin the membrane), and Knudsen diffusion is neglected since the poresizes are greater than one would expect for this phenomenon; both ofthese assumptions are consistent with other subscale limiting currentexperiments.31 Here, only the through-plane diffusivity is measured.

Gas-diffusion-layer samples.— Three different types of GDLs(Toray-TGP-H-120, SGL Sigracet 10 series, and Freudenberg H2315)with various PTFE loadings, 0 to 20 wt-% were tested. The mea-sured porosity and thicknesses are summarized in Table II. The poros-ity was measured using mercury intrusion porosimetry by PorousMaterials Inc., or the available data in the literature were used.32,33

The characteristic structures of the GDLs are shown in Figure 3 us-ing scanning electron micrographs (SEMs). Toray-TGP-120 serieshave a mostly 2D structure with fibers arranged in multiple layers,Figure 3a to 3c, whereas SGL 10 series have a 3D structure with in-tertwined and curved fibers, Figure 3d to 3f. Freudenberg H2315 hasa similar structure to SGL 10 series, but not intertwined among thefibers, Figure 3g. The presence of the fiber structures increases the de-gree of the tortuosity through the interconnected pores. PTFE is coatedat fiber surfaces and/or vicinity among fibers, and this decreases thepore sizes, resulting in the reduced porosity as shown in Table II. Thisreduction becomes apparent as the PTFE loading increases, resultingin an expected reduction in the effective diffusivity.

Sample preparation.— Control of the liquid saturation of the testedGDL (working electrode) is essential to this study. The sample prepa-ration procedure was similar to a previous study for measuring freeze

F686 Journal of The Electrochemical Society, 159 (11) F683-F692 (2012)

Toray-120

0%

5%

20%

SGL 10 Freudenberg H2315

(a)

(b)

(c) (f)

(e)

(d) (g)

300 µm300 µm

300 µm

300 µm300 µm

300 µm

300 µm

Figure 3. SEM observations of the various GDL samples tested. (a) Toray-120A (b) 120B (c) 120D (d) SGL 10 AA, (e) 10 BA, (f) 10 DA, and (g) FreudenbergH2315; PTFE loadings (wt-%) are also shown.

rates in partially-saturated GDLs.34 A GDL sample was bored into a1.905 cm diameter disk, which was sufficiently large to minimize theweight measurement uncertainty on the saturation. The bored samplewas submerged in deionized water in a small container by placing itunder a porous PTFE membrane with small magnetic stir bars on top.Then the entire container was vacuumed in order to degas the samplein a home-built vacuum chamber for 1 hour at 9.4 kPa (absolute) tohave a saturated sample. With the highest tested PTFE loaded sample(20 wt-%), 2 hours was required to achieve a high saturation, S ∼ 0.8.Excess surface water was blotted with dust-free paper. Saturation wasdetermined gravimetrically before and after running each experimentusing

S = Vw

Vp= Vw

εV= mw/ρw

εV[7]

where Vw, Vp, and V are the water, pore, and total volumes, respec-tively, and mw and ρw are the mass and density of water, respectively.The desired saturation was controlled by starting with the maximum

saturation and then drying in ambient conditions before cell assem-bly. It is noted that some water evaporates during the cell assem-bly/disassembly and test, which was measured; the typical loss isS ∼ 0.05 to 0.1 with the higher losses occurring at higher saturations.No spontaneous water drainage was observed in agreement with previ-ous capillary-pressure-saturation measurements.35 For minimal GDLcompression in the cell assembly, a silicon-based soft gasket (men-tioned above) was used, and no apparent sample thickness change andwater loss by compression were observed after cell disassembly. Thecell was assembled at a through-plane pressure of about 0.1 MPa usinga torque wrench with 1.13 N-m force. After the assembly, a bubbleleak test was also done by pressurizing one side of the electrode up to5 bars with pure N2 to ensure no leakage through the gaskets.

Cell activation and test protocols.— For the diffusion-controlledlimiting current, one must determine the operating cell potential andalso try to minimize other potential losses (e.g., proton conductivitythrough the MEA). To undergo this analysis as well as a limited cell

Table II. Tested GDLs, including the measured porosity, thickness, PTFE loading, and normalized porosity function f(ε).

PTFE LoadingSample Porosity, ε Thickness (μm) (wt-%) f(ε)c

SGL Sigracet 10AA 0.814 390 0 0.584 (0.624/0.548)SGL Sigracet 10BA 0.794a 470 5 0.463 (0.472/0.451)SGL Sigracet 10DA 0.719 380 20 0.264 (0.320/0.234)Freudenberg H2315 0.802 33 200 0 0.366 (0.391/0.348)Toray-TGP-H-120A 0.763 32 370 0 0.320 (0.358/0.290)Toray-TGP-H-120B 0.731 370 5 0.325 (0.352/0.303)Toray-TGP-H-120C 0.728 370 10 0.316 (0.331/0.286)Toray-TGP-H-120D 0.648 370 20 0.283 (0.305/0.267)Toray-TGP-H-60A 0.763b 200 0 0.247 (0.286/0.210)

a this is interpolated from 0% and 20% values.b the same value as Toray-TGP-H-120A.c values in the parenthesis are the maximum/minimum.

Journal of The Electrochemical Society, 159 (11) F683-F692 (2012) F687

(a)

(b)

Hydrogen Crossover+ Oxygen Reduction

io

0.0 0.1 0.2 0.3 0.4 0.5

-60

-45

-30

-15

0

15

30

591 ppm H2 in Ar

Reference Electrode, 5% H2 in Ar 100 sccm

i, m

A

E (vs DHE), V

SGL 10 AA, T = 21oC, v = 50 mV/s

Ar

0 50 100 150 200 250 300-2

0

2

4

6

8

10

591 ppm H2 in Ar

Ar

SGL 10 AA, T = 21 oC

i, m

A

t, s

Figure 4. (a) Variations of the current from 591 ppm hydrogen oxidation inargon at the working electrode as a function of electrical potential at T = 21◦C.The sweep rate is 50 mV/s. (b) Variations of the current at E = 0.35 V from591 ppm hydrogen in Ar and pure Ar supplied at the working electrode as afunction of time at T = 21◦C. The limiting current io is shown.

conditioning procedure, a fresh MEA was activated by carrying outcyclic voltammetry (CV) of 50 cycles (using 50 mV/s between 0to 0.5 V), under humidified, 591 ppm H2 in Ar (Air Products) in theworking electrode and humidified 5% H2 in Ar in the reference/counterelectrode (5% was chosen for minimal hydrogen crossover throughthe membrane, yet sufficiently high for the reference electrode). Thegases flowed through humidity bottles at room temperature to maintaina fully vapor-equilibrated MEA for high proton conductivity and noevaporative water losses in the GDLs. The CV results with dilutedhydrogen in Ar and pure Ar in the working electrode are shownin Figure 4a, which are consistent with a similar previous work36

that used 100% humidified, 5% H2 in Ar for the counter/referenceelectrode and N2 in the working electrode, thus validating the cellassembly and test protocol. At E > 0.3 V (plateau region), the currentis controlled by hydrogen diffusion; hence, for the limiting current,the voltage needs to set above 0.3 V. However, the potential also needsto be maintained low to avoid any parasitic reactions such as platinumoxidation and carbon corrosion. Thus, a value of E = 0.35 V waschosen.

A steady-state limiting current was measured with chronoamper-ometry at 0.35 V for 5 to 6 minutes until it changes only +/−10μA for 1 min as shown in Figure 4b. With the diluted hydrogen, thetypical limiting currents were 0 to 4 mA depending on the type of

GDLs and its saturation level, whereas without hydrogen, the currentswere ±0.5 mA depending on hydrogen crossover through the MEAand some minor parasitic losses. The difference between the abovetwo limiting currents was used for the diffusion-controlled limitingcurrent through the GDL (Eq. 6). In the data analysis and cell design, afew assumptions were made such as a) hydrogen dissolution in wateris negligibly small for the steady-state limiting current (no significantnet change of hydrogen dissolution at steady state)37 and b) the elec-troosmotic flux of water across the membrane is likewise minimal(which would appear as an evaporative loss). Back-of-the-envelopecalculations show that the above effects would only result in about a0.06% reduction in the effective diffusivity.38

The overall measurement procedure can be generalized into thefollowing seven steps: 1. Measure the GDL sample weight; 2. As-semble the cell with dry or partially-saturated GDLs in the workingelectrode; 3. Feed the diluted hydrogen in the working electrode; 4.Measure the steady-state limiting current for 5 to 6 minutes; 5. Feedpure Ar in the working electrode; 6. Repeat step 4 to obtain the back-ground limiting current; and 7. Disassemble the cell and remeasurethe GDL sample weight.

Calibration.— The above procedure and setup were validated andcalibrated by removing the GDL and measuring the binary diffusivityof H2 in Ar (diluted H2), i.e., DH2−Ar, which is 0.823 cm2/s at 300 K.39

Using a copper electrode with a 6 mm diameter and 0.635 cm longhole, the limiting current was measured, and the concentration ofthe hydrogen was adjusted for the calibration. This diffusion coeffi-cient is used as a reference to obtain f(ε). The temperature-dependentdiffusivity is corrected using the relation of (T1/To)1.5, where To

= 300 K.

Results and Discussion

Effective diffusivity of dry GDL, f(ε).— The measured effectivediffusivity of various GDLs with respect to porosity is shown as anormalized function, f(ε), in Figure 5. The average f(ε) values with theminimum and maximum values are also summarized in Table II. f(ε)expotentially increases with increasing porosity as the more porousstructures lead to improved diffusion pathways. The obtained experi-mental results are consistent with the previous empirical correlations(Eqs. I-2 and I-6) as well as previous experimental results.4 The best

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0 TGP-H-120, [4] SGL10AA SGL10BA SGL10DA TGP-H-120A TGP-H-120B TGP-H-120C TGP-H-120D Freudenberg, H2315 Eq. (I-1) (I-2) (I-3) (I-6) (8)

f()

ε

ε

Figure 5. Normalized porosity function, f(ε), with respect to porosity of thedry, uncompressed, SGL 10, Toray-120, and Freudenberg H2315 GDLs (seeTable II for GDL properties). The predictions using relations from Table I andthe available experimental data4 are also shown.

F688 Journal of The Electrochemical Society, 159 (11) F683-F692 (2012)

fit to the data is

f (ε) = εm [8]

using m = 3.6, which is close to that measured by Martinez et al.(Eq. I-2 as shown in Figure 5); this agreement helps to validate themeasurement protocol. It is also observed that the SGL 10 seriesexhibit a more senstive effective diffusivity with respect to porositycompared to the Toray-120 series, meaning that the diffusion path-ways are related to both overall porosity as well as fiber structure13,18

(see Figure 3), where the Toray GDLs inherently have larger tortuos-ity (multiple layered structures do not lead to well-connected poresthroughout) compared to SGL 10 GDLs. The addition of PTFE toToray GDLs does not significantly change the tortuosity as it does forthe SGL 10 series. Nonetheless, at the porosity range of 0.7 to 0.8, theexperimental results among the different types of GDLs are consistentwithin the sample to sample variation and measurement uncertainties,although there are minor discrepancies in SGL 10AA and TGP-H-120D within f(ε) ∼ 0.05. This is not too surprising since all of theGDLs being studied are all paper-type with fibers, which are geomet-rically similar to a large extent as seen in Figure 3. Thus, the diffusivitycan be predicted solely using the porosity within the porosity rangesand samples tested in this study. Other often-used relationships (e.g.,Eqs. I-1 and I-3) overpredict the diffusivity significantly. This de-screpancy is related to the fact that they were derived for oversimpliedmodel structures such as spherical pack beds and cylindrical tubes,which do not reasonably represent the realistic GDL structures.22,40

To study the impact of thickness and multiple layers, two Toray-TGP-H-60A GDLs were stacked and tested, and the results comparedto the single-layer 120A GDL which should be nominally the samethickness and geometric structure as the stacked ones. The diffusivityof the two-layered 60A GDLs reduces by 23% compared to that ofthe single-layer 120A. In fact, the porosity distribution is non-uniformthroughout the through-plane direction where the maximum transportresistances (minimum porosity) are expected near the GDL surfaces.33

The additional interface from the two layers could be the source ofthe extra transport resistance. However, this deviation though is notvery significant as it is only slightly larger than the average differencebetween multiple samples (see Table II), which is in agreement withonly minor thickness effect observed in the literature both for thesame and different types of GDLs.4,15 Thus, the thickness effect onthe effective diffusivity is relatively small.

Effective diffusivity of partially-saturated GDLs, g(S).— The mea-sured effective diffusivities of the partially-saturated GDLs (no PTFE)are shown in Figure 6 as the normalized saturation function, g(S),with respect to saturation S. To obtain g(S), the measured diffusiv-ities are normalized by those of the dry GDLs (S = 0). Figure 6ashows the diffusivity for Toray-120A GDL, which decreases withincreasing saturation. This is caused by the fact that the water-filled pores hinder the diffusion pathways, resulting in decreasingthe effective diffusivity. This value sharply decreases at low satu-ration, and it becomes nearly zero at high saturation, S > 0.7 to0.8, where the unfilled pores are completely isolated without well-connected diffusion pathways. These findings are consistent amongthe different GDL types as shown in Figure 6b and 6c for SGL10 AA and Freudenberg H2315, respectively. Thus, the different geo-metrical structures do not significantly influence liquid-water channelformation associated with the diffusion pathways, in other words, thepartial- and complete-filling of the transport channels occurs in a sim-ilar fashion, which is not too surprising due to the similar nature ofthe materials. All the results are fitted using the relation,

g(S) = (1 − S)n [9]

where n = 3 with R2 = 0.98. In fact, the measured data of SGL10 AA at S < 0.3 follows n = 2, while n = 3 is reasonable S > 0.4 asshown in Figure 6b. This change is attributed to the fact that the struc-tural difference changes the liquid-water distribution and diffusion

pathways, but the resulting impact on g(S) is minor. As mentioned,previous modeling studies calculated various ranges of n = 1.5 to6 (Eqs. I-6 to I-9) as shown in Figure 6a, and the various resultingexpressions are associated with the different pore-size distributions,pore morphologies, and water-filling behaviors.8,22 The current ex-perimental results agree with one relation which was predicted usinga continuously-connected, water-filled-pore configuration in an ideallattice pore network, n = 3 (Eq. I-7), although the idealized pore struc-ture might not have been a physical representation of the actual GDL.It is noted that the randomly distributed, water-filled-pore configura-tion showed the lower effective diffusivity, i.e., n = 4, implying thatthe effective diffusivity of the random liquid distribution effectivelyreduces transport pathways. Another study showed that further real-istic pore-size distributions and pore morphologies for typical GDLscould further retard the effective diffusivity, i.e., n = 5. (Eq. I-8),and they addressed that this reduction was primarily caused by theoverestimated penalties from the complete blockages of the diffusionpathways in the water-filled pores. However, as recently observedusing computed X-ray tomography on partially-saturated GDLs,41–43

the partially water-filled pores still allow for gas transport, which isexpected to result in n < 5. Other studies seem to be much furtheroff than the experimental data due to their geometric and morphologyassumptions. In terms of comparison to in-situ experimental resultsusing neutron imaging (for saturation measurement) and limiting cur-rent (for effective diffusivity measurement) showed that the impactsof the liquid saturation were between n = 5 and 6,26 however, asmentioned, there might have been issues with resolution and quanti-tative accuracy, and the in-situ experiment might have included someparasitic transport resistances such as unavoidable interfacial wateraccumulation. Using a galvanic oxygen sensor in the previous ex-situexperiments,29 the results show significantly lower effective diffusiv-ities for the paper type GDL compared to the current data, especiallyat lower saturations (i.e., g(S) ∼ 0.5 at S ∼ 0) and their experimentaldata fit into n ∼ 11 using Eq. 9. In their measurements, this reductionmay be attributed to extra transport resistances from possible wateraccumulation near the interfaces among the stacked samples as wellas the different samples.

Effect of PTFE.— Wet-proofing of the GDL by addition of PTFEthrough dispersion and subsequent annealing is designed to increasethe portion of hydrophobic pores, which in turn increases the numberof unfilled pores for improved diffusion passages under wet con-ditions. However, none of the previous studies have explicitly ad-dressed the effects of PTFE on the effective diffusivity of the partially-saturated GDLs. In Figure 7a, the normalized function, g(S), for Toray-120B (PTFE 5 wt-%) is shown as a function of saturation, and g(S)= (1 − S)3 for the unteflonated GDLs is also plotted for comparison.The increased saturation results in monotonically decreasing the ef-fective diffusivity, but with a different trend, especially at S < 0.6.A possible explanation is that the PTFE coating hinders water fillingin the hydrophobic pores, thereby still allowing favorable diffusionpathways, i.e., a good phase separation, as shown in Figure 1d. Itis also consistent with the fact that good phase separation, in gen-eral, enhances gas transport in two-phase flow, which is demonstratedin hybrid-wetting GDLs30 and liquid-artery systems.44,45 The exper-imental results scatter much among the different samples, especiallyat 0.2 < S < 0.5, and this is probably related to the fact that PTFEheterogeneously coats the GDL fibers (see Figure 3),5 which resultsin a variable liquid distribution and thus different effective diffusiv-ities among the different samples. Nonetheless, a small amount ofPTFE (5 wt-%) significantly changes the wetting behavior and re-sulting effective diffusivity, as observed previously in the apparentwetting behavior and droplet adhesion-force changes on the GDLsurface.46 Similarly, Figure 7b shows the apparent increases for SGL10BA (PTFE 5 wt-%) at S < 0.2, although this is not as apparent asthat of Toray-120B. For both cases, the relation for no PTFE GDLsdoes not reasonably predict the experimental results. Here, an empir-ical relation based on a cumulative or integrated log-normal distribu-tion function is proposed to calculate g(S) for 5 and 20 wt-% PTFE

Journal of The Electrochemical Society, 159 (11) F683-F692 (2012) F689

(a) (b)

(c)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

2

Sample #1 #2 #3

n = 3

SGL 10 AA

g(S)

S

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

2

6

5

4

n = 1.5 Sample #1 #2 #3

TGP-120, 0%

g(S)

S

3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Sample #1 #2

g(S)

S

Freudenberg H2315

n = 3

Figure 6. Normalized saturation function, g(S), with respect to the liquid saturation for the partially-saturated GDLs (a) Toray-120A, (b) SGL 10 AA, and(c) Freudenberg H2315. The model predictions with different power laws (see Eq. 9), m = 1.5 to 6 and Eqs. I-7 and I-9 from Table I are also shown in (a). Thedifferent samples are marked with the different colors. The small bars (gray) represent the liquid-saturation changes before and after each test, and the saturationsare plotted at their average values.

loadings. Such a formulation was used to predict saturation as a func-tion of capillary pressure in GDLs, since the GDL pore-size distribu-tion is log normal.47 The resulting relation is given as

g(S) = 1

2

{1 + erf

[− ln(S) + a

b

]}[10]

where a = 1.27 and b = 0.82, with R2 = 0.90 for both Toray-120Band SGL 10BA, and a = 1.06 and b = 0.57, with R2 = 0.94 forToray-120D and SGL 10DA. These curve fits are shown inFigure 7 to predict the diffusivity of 5 and 20 wt-% loading GDLs, re-spectively. Figure 7c and 7d show that increased PTFE loadings (i.e.,10 and 20 wt-%) increase the normalized function for both the Torayand SGL samples, although the results for Toray 10 wt-% signifi-cantly scatter, yet remain bounded between the 5 and 20 wt-% data.Research is ongoing to understand the exact origins of the impactof PTFE.

The increased PTFE loadings improve g(S), but they also decreasef(ε) (see Table II and Figure 5). The overall effective diffusivity needs

to include both results as shown in Figure 8 for Toray-120 and SGL10 series with PTFE loading of 0 to 20 wt-%. In Figure 8a, Toray-120 series show that the PTFE loading increases the overall effectivediffusivity over the entire saturation range, and the optimal loading isbetween 0 and 5 wt-% for dry condition, whereas it is 20 wt-% at amoderate saturation level, 0.2 < S < 0.6. This improvement is alsoseen in the predicted results for PTFE loadings of 0, 5, and 20 wt-%,which are calculated using Eq. 9 and f(ε) = 0.320 (see Table II) for0 wt-%, and Eq. 10 and f(ε) = 0.325 and 0.283 (see Table II) for 5and 20 wt-%, respectively. However, this observation did not clearlyappear in the in-situ experimental results using similar samples.48,49

In that study, the optimal GDL choice was 0 wt-% PTFE, whichmeans that an optimal fuel-cell performance requires various con-siderations for a GDL selection such as mechanical, electrical, andwater-management aspects as well as the cell assembly/design andtest conditions. Thus, the reactant diffusion may not be only the factorto control the performance. In Figure 8b, the SGL 10 series showthat increasing PTFE loading monotonically decreases the overall ef-fective diffusivity over a wide range of saturation, implying that the

F690 Journal of The Electrochemical Society, 159 (11) F683-F692 (2012)

(a) (c)

(b) (d)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0g(

S)

S

Toray-120B, PTFE 5 wt. % Sample #1 #2 #3

Eq. (10) for 5 wt. %

No PTFE

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Eq. (10) for 5 wt. %

Sample #1 #2

No PTFE

SGL 10 BA, PTFE 5 wt. %

g(S)

S

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

5 wt.%

Eq. (10) for 20 wt.%

Sample #1 #2

SGL 10 DA, PTFE 20 wt. %g(

S)

S

No PTFE

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

5 wt. %

Sample #1 #2

Toray-120D, PTFE 20 wt. %

g(S)

S

Toray-120C, PTFE 10 wt. % Sample #1 #2

Eq. (10) for 20 wt.%

No PTFE

Figure 7. Normalized saturation function, g(S), with respect to the saturation for (a) Toray-120B, (b) SGL 10 BA, (c) TGP-120C and D, and (d) SGL 10 DA. Thepredicted results using Eqs. 9 and 10 for 0, 5, 20 wt-% PTFE loadings, respectively, are also shown. The small bars (gray) represent the liquid saturation changesbefore/after the experiments and the data are plotted at their average values.

(a) (b)

ε ε

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

20 wt. %

5 wt. %

PTFE 0 wt% 5 wt% 20 wt%

SGL 10

f()g

(S)

S

PTFE 0 wt. %

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

20 wt. %

5 wt. %

PTFE 0 wt. %

PTFE 0 wt% 5 wt% 10 wt% 20 wt%

Toray-120

f()g

(S)

S

Figure 8. Measured effective diffusivity, f(ε)g(S) (or <D>/D), as function of porosity and saturation. (a) Toray-120 series and (b) SGL 10 series for PTFE 0 to20 wt-%. The predicted results for 0, 5, 20 wt-% PTFE loadings using Eqs. 9 and 10 and f(ε) (see Table II) are also shown.

Journal of The Electrochemical Society, 159 (11) F683-F692 (2012) F691

(a) (b)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0/ SGL10BA (Dry/P-S)/ SGL10DA / TGP-120B/ TGP-120C/ TGP-120D G

3

<D>/

D

G

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0/ SGL10AA (Dry/P-S)/ TGP-120A/ Freudenberg, H2315 G

3.3

<D>/

D

Gε ε

ε

ε

Figure 9. Measured effective diffusivity (<D>/D) as function of gas-phase volume fraction for both dry and partially saturated (a) unteflonated and (b) teflonatedGDLs (5, 10, and 20 wt-% PTFE loadings). The empty symbols are for dry GDLs and the filled ones for partially-saturated (P-S) ones.

increased g(S) is outbalanced or outweighed by the decrease of f(ε),which is also readily seen at S = 0. This trend is also shown in thepredicted results for PTFE at 0, 5, and 20 wt-% loadings, which iscalculated using Eq. 9 and f(ε) = 0.584 (see Table II) for 0 wt-%,and Eq. 10 and f(ε) = 0.463 and 0.264 for the latter 5 and 20 wt-%.The different effect of PTFE loading on the different structures ofGDLs is mainly caused by the characteristic fiber structure differ-ence, i.e., multi-layered in-plane fibrous structure (highly anisotropic)versus random fibrous wool structure (isotropic), as shown inFigure 3.

Gas-phase volume fraction.— Similar to Figure 5, one can wonderhow the effective diffusion coefficient depends on the gas-phase vol-ume fraction, εG which accounts for both the saturation and porosity,

εG = ε (1 − S) [11]

Although we believe that it is more appropriate to look at thetwo functions separately (as in Eq. 1) since they are treating differentphenomena, it is still instructive to look at them as a function of thegas-phase volume fraction, which is often how they are modeled. Inthis fashion, the water-filling is treated as additional solids, hinder-ing gas diffusion. Figure 9 displays the various normalized effectivediffusivities, <D>/D, as a function of gas-phase volume fraction,where at zero saturation, Figure 5 is obtained. From Figure 9a, it isapparent that the water-filling similarly changes the tortuosity in asimilar manner as the fiber structures do, and the best fit to the data forthe unteflonated samples (Toray-120A, SGL10AA, and FreudenbergH2315) is m = 3.3 using Eq. 8, which is similar to that fit to only theunsaturated samples (m = 3.6). It is also clear that the SGL GDLsprovide less tortuous diffusion pathways than the other GDLs for agiven gas-phase volume fraction. However, in Figure 9b, the samecannot be concluded for the teflonated samples since there are deflec-tion points at the lower saturations (owing to the more complicatedway in which water fills the teflonated GDLs (see Figure 7)). Whilea power-law relationship can be fit to the data (m = 3), it is not agood fit. This is a reason why a two-term expression that separates thephenomena is preferred. From Figure 9b, it is obvious that due to theinitial minor effect at low saturations, a given effective diffusivity canbe obtained with several different gas-phase volume fractions, and, fora given gas-phase volume fraction, several effective diffusivities can

be obtained due to different PTFE treatments (i.e., there is no singlecorrelation).

Summary

An ex-situ technique for measuring the effective gas-phase dif-fusion coefficients of partially saturated gas-diffusion layers (GDLs)was discussed. The technique uses a limiting-current method with di-luted hydrogen gas to measure the effective through-plane diffusioncoefficients of commercially available GDLs such as Toray-TGP-H-120, SGL Sigracet 10 series, and Freudenberg H2315 with variousPTFE loadings, 0 to 20 wt-%. For dry GDLs, the porosity is thecritical factor that controls the effective diffusivity, with a secondaryimpact by the fiber structure. Increasing the PTFE loading reduces theporosity and the available diffusion pathways, which in turn resultsin a reduced effective diffusivity. It was shown that the popular effec-tive medium theory (Bruggeman relation) overpredicts the effectivediffusivity, and the relation f(ε) = ε3.6 is obtained for dry GDLs. Forpartially-saturated GDLs (no PTFE), liquid water hinders the diffusionpathways, resulting in a change of effective diffusivity from the dryGDL value of g(S) = (1 − S)3, with no significant variations amongthe different fiber structures. With PTFE, the liquid-percolation path-ways favorably change to enable better gas diffusion, resulting in anincreasing g(S) for a given saturation. An empirical cumulative log-normal distribution function is proposed for 5 and 20 wt-% PTFEloaded samples. Although further increasing PTFE loading enhancesg(S), the characteristic trend depends on fiber structure. This work isone of the first to measure effective diffusivities in partially saturatedGDLs, and the relations developed not only provide insights of theunderlying physics and material design, but also can improve fuel-cellmodels to consider correctly and optimize fuel-cell water and thermalmanagement under various operating conditions.

Finally, it should be noted that many fuel cells are assembledwith microporous layers in addition to the macroporous GDL. Theselayers are thought to enhance water management by minimizing waterflooding in the cathode catalyst layer, improving electrical contact,providing a better thermal response and larger thermal gradient, andrestricting the entrance points of water into the GDL (thus reducingits saturation). The measurements described in this manuscript allowone to understand better the possible roles or effects of a microporouslayer in that one can know how a GDL might behave in terms ofeffective diffusivity. In addition, the developed experimental setupand procedure can also be used to investigate the effective diffusivity

F692 Journal of The Electrochemical Society, 159 (11) F683-F692 (2012)

of microporous layers, however it is mostly confined to the dry onessince it is extremely challenging to partially saturate them (they arevery hydrophobic and a different saturation methodology is requiredto fill them). These and related studies including possibly liquid-wateraccumulation at interfaces are currently being investigated.

Acknowledgments

This work was supported by the Assistant Secretary for En-ergy Efficiency and Renewable Energy, Fuel Cell Technologies Pro-gram of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 and by CRADA agreement LB08003874 betweenLawrence Berkeley National Laboratory and Toyota Motor Company.

Appendix

To determine the hydrogen concentration at the catalyst layer / GDL interface whetherthe catalyst layer is at all limiting, a simplified model is accomplished. The calculationis performed for the maximum ratio, i.e., the largest hydrogen flux (or highest limitingcurrent density, ∼14.1 mA/cm2 from the setup without GDL). The transport resistanceratio is defined as

〈D〉C L /δp,C L

D/δp,bulk[A.1]

where <D>CL is the effective hydrogen diffusivity in the catalyst layer, D is bulk hydrogendiffusivity, and δp,CL is the hydrogen penetration depth (depletion depth), δp,CL is thehydrogen traveling distance (0.635 cm, the setup without GDL). The volumetric reactionrate is calculated using the Butler-Volmer relation given as

∇ · i = as io,re f

(cH2

cH2 ,re f

)γ [exp

(αF

RTη

)− exp

(− αF

RTη

)][A.2]

where i is the current density, a is the specific surface area (1.067 × 105 1/cm),50 io,ref is thereference exchange current density (3.0 × 105 A/cm2),51 cH2 is the hydrogen concentration(4.14 × 10−11 mol/cm3, representing 1 ppm which is negligibly small compared to thesupplied concentration at the channel), cH2,re f is the reference hydrogen concentration(2.66 × 10−5 mol/cm3),51 α is the transport coefficient (0.5),51 F is Faraday’s constant,R is the ideal-gas constant, T is the temperature (21◦C), γ is the reaction order (0.5),52

and η is the overpotential (0.35 V). The above reaction rate is integrated over the catalystlayer to obtain the current density,

io =∫ δp,C L

0∇ · i dx [A.3]

Assuming constant hydrogen concentration, the hydrogen penetration depth can beapproximated as

δp,C L = io

∇ · i[A.4]

The calculations result in δp,CL = 0.36 μm, and, approximating a linear hydrogenconcentration change, the effective penetration depth is 0.18 μm. Here, the maximum lim-iting current obtained from the experiment was io = 14.1 mA/cm2. In Eq. A.1, <D>CL/D= 0.05 at porosity of 0.4 for the catalyst layer is conservatively used (the calculationis done for oxygen and water, but we use it for an approximate).53 Thus, the transportresistance ratio is only 5.67 × 10−4, indicating the assumption is valid.

References

1. J. Larminie and A. Dicks, Fuel Cell Systems Explained, 2nd ed. (Wiley, Chichester,2003).

2. C.-Y. Wang, Chem. Rev., 104, 4727 (2004).3. A. Z. Weber and J. Newman, Chem. Rev., 104, 4679 (2004).4. J. M. LaManna and S. G. Kandlikar, Int. J. Hydrogen Energ., 36, 5021 (2011).5. Z. Fishman and A. Bazylak, J. Electrochem. Soc., 158, B841 (2011).6. R. Fluckiger, S. A. Freunberger, D. Kramer, A. Wokaun, G. G. Scherer, and

F. N. Buchi, Electrochim. Acta, 54, 551 (2008).7. M. Kaviany, Principles of Heat Transfer in Porous Media, 2nd ed. (Springer Verlag,

New York, 1995).

8. J. H. Nam and M. Kaviany, Int. J. Heat Mass Trans., 46, 4595 (2003).9. D. M. Bernardi and M. W. Verbrugge, J. Electrochem. Soc., 139, 2477 (1992).

10. D. A. G. Bruggeman, Annalen Der Physik, 416, 636 (1935).11. U. Pasaogullari and C. Y. Wang, J. Electrochem. Soc., 151, A399 (2004).12. R. E. D. L. Rue and C. W. Tobias, J. Electrochem. Soc., 106, 827 (1958).13. M. J. Martinez, S. Shimpalee, and J. W. Van Zee, J. Electrochem. Soc., 156, B80

(2009).14. N. Zamel, X. Li, and J. Shen, Energ. Fuel., 23, 6070 (2009).15. C. Chan, N. Zamel, X. Li, and J. Shen, Electrochim. Acta, 65, 13 (2012).16. D. Kramer, S. A. Freunberger, R. Fluckiger, I. A. Schneider, A. Wokaun, F. N. Buchi,

and G. G. Scherer, J. Electroanal. Chem., 612, 63 (2008).17. D. R. Baker, D. A. Caulk, K. C. Neyerlin, and M. W. Murphy, J. Electrochem. Soc.,

156, B991 (2009).18. Y. Utaka, Y. Tasaki, S. Wang, T. Ishiji, and S. Uchikoshi, Int. J. Heat Mass Trans.,

52, 3685 (2009).19. M. M. Tomadakis and S. V. Sotirchos, AIChE J., 39, 397 (1993).20. M. M. Mezedur, M. Kaviany, and W. Moore, AIChE J., 48, 15 (2002).21. P. K. Das, X. Li, and Z.-S. Liu, Appl. Energ., 87, 2785 (2010).22. J. T. Gostick, M. A. Ioannidis, M. W. Fowler, and M. D. Pritzker, J. Power Sources,

173, 277 (2007).23. H. Wu, X. Li, and P. Berg, Electrochim. Acta, 54, 6913 (2009).24. N. S. Martys, Mater. Struct., 32, 555 (1999).25. M. C. Hatzell, A. Turhan, S. Kim, D. S. Hussey, D. L. Jacobson, and M. M. Mench,

J. Electrochem. Soc., 158, B717 (2011).26. T. Shiomi, R. S. Fu, U. Pasaogullari, Y. Tabuchi, S. Miyazaki, N. Kubo, K. Shinohara,

D. S. Hussey, and D. L. Jacobson, in ASME Conf. Proc. (2010), pp. 667–676.27. D. Iwasaki, Y. Utaka, Y. Tasaki, and S. Wang, in ASME Conf. Proc. (2008),

pp. 1279–1284.28. D. S. Hussey, D. Spernjak, A. Z. Weber, R. Mukundan, J. Fairweather, E. L. Brosha,

J. Davey, J. S. Spendelow, D. L. Jacobson, and R. L. Borup, J. Appl. Phys. submitted(2012).

29. D. Iwasaki, Y. Utaka, Y. Tasaki, and S. Wang, in ASME Conf. Proc. (2008),pp. 1279–1284.

30. Y. Utaka, I. Hirose, and Y. Tasaki, Int. J. Hydrogen Energ., 36, 9128 (2011).31. N. Nonoyama, S. Okazaki, A. Z. Weber, Y. Ikogi, and T. Yoshida, J. Electrochem.

Soc., 158, B416 (2011).32. J. Lobato, P. Canizares, M. A. Rodrigo, C. Ruiz-Lopez, and J. J. Linares, J. Appl.

Electrochem., 38, 793 (2008).33. Z. Fishman, J. Hinebaugh, and A. Bazylak, J. Electrochem. Soc., 157, B1643

(2010).34. T. J. Dursch, M. A. Ciontea, C. J. Radke, and A. Z. Weber, Langmuir, 28, 1222

(2012).35. J. T. Gostick, M. W. Fowler, M. A. Ioannidis, M. D. Pritzker, Y. M. Volfkovich, and

A. Sakars, J. Power Sources, 156, 375 (2006).36. M. Prasanna, E. A. Cho, T.-H. Lim, and I.-H. Oh, Electrochim. Acta, 53, 5434

(2008).37. D. W. Green and R. H. Perry, Perry’s Chemical Engineers’ Handbook, 8th ed.

(McGraw-Hill, 2008).38. A. Z. Weber and J. Newman, J. Electrochem. Soc., 151, A311 (2004).39. K. R. Harris, T. N. Bell, and P. J. Dunlop, Can. J. Phys., 50, 1644 (1972).40. Z. Yu and R. N. Carter, J. Power Sources, 195, 1079 (2010).41. J. Eller, T. RoseFn, F. Marone, M. Stampanoni, A. Wokaun, and F. N. Buchi,

J. Electrochem. Soc., 158, B963 (2011).42. J. T. Gostick, H. P. Gunterman, B. W. Kienitz, J. S. Newman, A. A. MacDowell, and

A. Z. Weber, ECS Transactions, 33, 1407 (2010).43. P. Kruger, H. Markotter, J. Haußmann, M. Klages, T. Arlt, J. Banhart, C. Hartnig,

I. Manke, and J. Scholta, J. Power Sources, 196, 5250 (2011).44. G. S. Hwang, Y. Nam, E. Fleming, P. Dussinger, Y. S. Ju, and M. Kaviany,

Int. J. Heat Mass Trans., 53, 2662 (2010).45. G. S. Hwang, E. Fleming, B. Carne, S. Sharratt, Y. Nam, P. Dussinger, Y. S. Ju, and

M. Kaviany, Int. J. Heat Mass Trans., 54, 2334 (2011).46. P. K. Das, A. Grippin, A. Kwong, and A. Z. Weber, J. Electrochem. Soc., 159, B489

(2012).47. A. Z. Weber, J. Power Sources, 195, 5292 (2010).48. M. Prasanna, H. Y. Ha, E. A. Cho, S.-A. Hong, and I.-H. Oh, J. Power Sources, 131,

147 (2004).49. G.-G. Park, Y.-J. Sohn, T.-H. Yang, Y.-G. Yoon, W.-Y. Lee, and C.-S. Kim, J. Power

Sources, 131, 182 (2004).50. W. Yoon and A. Z. Weber, J. Electrochem. Soc., 158, B1007 (2011).51. N. P. Siegel, M. W. Ellis, D. J. Nelson, and M. R. von Spakovsky, J. Power Sources,

128, 173 (2004).52. Y. Wang and C.-Y. Wang, J. Electrochem. Soc., 152, A445 (2005).53. K. J. Lange, P.-C. Sui, and N. Djilali, J. Power Sources, 208, 354 (2012).