Water balance model for polymer electrolyte fuel cells with ultrathin catalyst layers.

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Cite this: Phys. Chem. Chem. Phys., 2014, 16, 2106

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Water balance model for polymer electrolyte fuel cells with ultrathin catalyst layers Karen Chan and Michael Eikerling* We present a water balance model of membrane electrode assemblies (MEAs) with ultrathin catalyst layers (UTCLs). The model treats the catalyst layers in an interface approximation and the gas diffusion layers as linear transmission lines of water fluxes. It relates current density, pressure distribution, and

Received 18th November 2013, Accepted 2nd December 2013

water fluxes in the different functional layers of the assembly. The optimal mode of operation of UTCLs

DOI: 10.1039/c3cp54849j

to avoid flooding of the gas diffusion layers. The model provides strategies for increasing the critical

is in a fully flooded state. The main challenge for MEAs with UTCLs is efficient liquid water removal, current density for the onset of flooding, via liquid permeabilities, vaporization areas, and gas pressure

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differentials. Finally, we discuss methods to identify regimes of transport via water flux measurements.

1 Introduction In recent years, ultrathin catalyst layers (UTCLs) for polymer electrolyte fuel cells (PEFCs) have enabled enormous improvements in Pt-specific current density over conventional catalyst layers (CLs).1 In contrast to conventional CLs, UTCLs are ionomer-free and exhibit considerably reduced thicknesses (20–500 nm). Research in UTCL fabrication has explored a variety of alternative support materials and structural designs. Recent examples include UTCLs fabricated by deposition of Pt onto nanostructured supports (e.g. 3M’s organic perylene whiskers or carbon nanotubes2,3), or by direct deposition of nanoporous Pt onto the gas diffusion layer (GDL) or polymer electrolyte membrane (PEM).4–6 3M nanostructured thin films (NSTFs) have been the most widely studied UTCLs in the past decade. 3M NSTFs consist of an organic whisker material coated with a continuous film of Pt. Since the whisker material is non-conductive, support corrosion is eliminated; the bulk-like nature of the Pt leads to increased activity for the oxygen reduction reaction.7 Experimental studies suggest that Pt mass loss via Pt dissolution is strongly suppressed in 3M NSTFs.2 This enhanced catalyst stability could be due to a combination of the bulk-like nature of the Pt and the lower proton concentration within the UTCL.8,9 Overall, 3M NSTFs enable considerable improvements in electrochemical performance and durability of PEFCs, and enable a crucial cost reduction. However, a hitherto unresolved challenge is the increased propensity of membrane electrode assemblies (MEAs) that employ 3M NSTF to exhibit increased water management challenges. Department of Chemistry, Simon Fraser University, Burnaby, BC, Canada. E-mail: [email protected]; Fax: +1 778 782 3765; Tel: +1 778 782 4463

2106 | Phys. Chem. Chem. Phys., 2014, 16, 2106--2117

3M NSTF MEAs show poor performance at low RH and low temperatures, and an increased propensity for cell reversal under load transients, i.e. a decrease of the cathode voltage to o0 V when the current density is abruptly ramped from close to 0 to 1 A cm2.10–12 The causes of the water management issues in UTCL MEAs are not well established. Previous models assumed that water can only undergo vapour diffusion within the gas diffusion layers (GDLs), so that they do not flood with liquid water. Water management issues such as poor steady state performance and cell reversal during load transients were therefore attributed only to the flooding of the catalyst layers. In fact, previous models assumed that the oxygen diffusion coefficients within the UTCLs drop to zero once the UTCL saturates with water,13 even though oxygen can very well diffuse through water. Based on existing theoretical results, experimental work has focussed primarily on structural modifications to the UTCL.11 However, modeling studies of flooded UTCLs8 that include realistic diffusion coefficients of oxygen in water show that flooded CLs should not lead to limiting current behaviour or MEA shutdown, which have been observed in UTCL MEAs. In fact, the ionomer-free UTCLs rely entirely on water for proton conduction. We postulate that the poor performance in MEAs with UTCLs arises primarily from the flooding of the cathode gas diffusion layer (GDL). We study water transport in MEAs containing UTCLs using a simple, one-dimensional water balance model of the MEA. We extend the half-PEFC model by Baghalha and Eikerling14 to consider liquid water transport in the anode and vaporization in the GDL via a transmission line model of water fluxes.15 The model relates transport properties and operating conditions to capillary pressure distributions, current density, liquid and vapor water fractions out the anode and cathode, and the onset

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of catalyst layer and GDL flooding. We evaluate strategies for increasing the current density for the onset of GDL flooding and discuss identification of the regimes of water transport via water flux measurements.

2 Model formulation Published on 02 December 2013. Downloaded by McMaster University on 23/10/2014 16:33:30.

2.1

Model assumptions

We consider a 1D water balance model approach14 to water fluxes in the UTCL MEA. Fig. 1 shows a schematic of the MEA and the water fluxes. Red arrows show the sources of water in the cathode catalyst layer (cCL), i.e. the electro-osmotic drag of water across the PEM and oxygen reduction. Blue arrows show fluxes of water away from the cCL, via permeation, vaporization, and subsequent diffusion. We make the following simplifying assumptions: (1) Flow rates in flow field channels are high, such that relative humidities (RHs) at the MEA boundaries are fixed and liquid water does not accumulate in the channels. G (2) Total gas pressures at the anode and cathode, pG a and pc , are constant, due to high convective flux of gases in the diffusion media. (3) GDLs are treated with a transmission line model of coupled liquid and vapor fluxes.15 We make a 3-state approximation for the capillary pressure–saturation curves of the GDLs, illustrated in Fig. 2. Saturation and liquid permeability are zero at capillary pressure pc o 0, constant at 0 r pc o pcGDL,fl, and reaching flooded conditions with maximal saturation at pc = pcGDL,fl. (4) The cathode microporous layer (MPL) has constant vapor diffusivity and liquid permeability. Negligible vaporization occurs in the MPL.

Fig. 1 Schematic of water transport paths in the MEA. GDL = gas diffusion layer, MPL = microporous layer, CL = catalyst layer, PEM = polymer electrolyte membrane. a and c indicate anode and cathode, respectively. Red arrows show the sources of water in the cCL, i.e. the electro-osmotic drag of water across the PEM and oxygen reduction. Blue arrows show fluxes of water away from the cCL, via permeation, vaporization, and subsequent diffusion.


Fig. 2 3-state approximation for the capillary isotherm of the GDLs. Saturation and liquid permeability are zero at capillary pressure pc o 0, constant at pc Z 0, and reaching flooded conditions with maximal saturation at pc = pcGDL,fl. Experimental data adapted from ref. 17, where the capillary isotherm was measured for the entire diffusion medium, GDL + MPL. We assumed that the measured variation in saturation was mainly due to GDL saturation. Capillary pressures required to penetrate the nanosized, hydrophobic pores of the MPL were considered to be high. We further assumed that the entire GDL is saturated at the maximum capillary pressure applied, and normalized the data to the saturation at this maximum capillary pressure.

(5) Anode and cathode catalyst layers (aCL and cCLs) are approximated as infinitesimally thin interfaces. The CLs flood at a negative capillary pressure, since they are hydrophilic. (6) Water is transported through the PEM via electroosmotic drag and hydraulic permeation, with a constant drag coefficient, neo, and a permeability that depends linearly on water content, w.16 In (4), we neglect dynamic breakthrough and eruptive transport phenomena,18,19 as well as hysteresis arising from the mixed wettability of the GDLs.17 In (5), we assume that water flows through hydrophilic cracks in the MPL.17,20 Given the nanometer-sized (10–100 nm) and hydrophobic MPL pores,21 we do not consider flooding in the MPL, and assume vaporization rates in it to be negligible. We do not include an MPL at the anode side, as anode MPLs were found to be detrimental to the performance of 3M NSTF MEAs.22 As we discuss below, this is consistent with our finding that liquid transport in the diffusion media of NSTF MEAs prevails; an anode MPL would create greater water transport resistance out the anode side, leading to increased propensity to flooding at the cathode side. In (6), the interface assumption is based on the 2–3 orders of magnitude lower thickness of UTCLs compared to other MEA components, which should result in essentially constant pressure distributions along the UTCL thickness. Therefore, unlike conventional CLs, the UTCL plays little role in steering liquid fluxes within the MEA.23,24 The assumption of a single flooding capillary pressure in the UTCL corresponds to assuming it to have monodisperse pores. In (7), we neglect water transport via diffusion in the PEM, though its effect could be incorporated into the hydraulic permeation term.25

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Several of the above assumptions could be relaxed in a more complex, all-encompassing model.26 However, such models require a detailed knowledge of transport parameters and properties of MEA components, which are not available for 3M NSTF MEAs. The focus of the present work is on the qualitative trends and results, not in the exact reproduction of fluxes and pressure distribution within any particular MEA.


2.2

Regimes of transport

With the assumed model of the GDL, there are three regimes of water transport at each electrode, denoted as regime ‘‘V’’, where all liquid water is vaporized at the CL; only vapor is transported out of the CL. regime ‘‘Vg’’, where liquid water is transported out of the CL into the MPL and GDL, and vaporized completely in the GDL; there is no liquid flux out from the GDL to the flow fields. This corresponds to a case where the liquid water is not transported along the entire extent of the GDL. regime ‘‘M’’, where a mixture of liquid water and vapor flows out of the GDL and into the flow fields. There are 9 possible regimes of transport, with V, Vg, or M on either the anode or cathode side. In labeling the transport regime, we state first the transport regime at the anode, then the transport regime at the cathode, e.g. VMT means that only vapor (V) flows out of the aCL and mixed liquid–vapor (M) flow out of the cGDL; T stands for transport. Fig. 3 shows these regimes with possible paths for the transitions amongst them as current density increases. As current density and liquid production in the cathode increase, each electrode goes from V to Vg and eventually to the M regime; current densities at these transitions are determined by the transport properties of MEA components and operating conditions. In practice, we can neglect the case of liquid transport in the anode prior to the cathode (e.g. the VgVT regime), unless a large differential in gas pressure across the MEA is applied (DpG = G pG c pa > 70 kPa with the base case parameters assumed in this work), or with unusually impermeable diffusion media at the cathode side. In Fig. 3, less likely paths are marked in grey while more likely paths are marked in red. We denote the current density at the onset of liquid flux out V

of the cCL and out of the cGDL at the cathode by jc g and j M c , respectively. For the current densities referring to the transitions at the anode, the subscript is a. For example, for the path shown in red

Fig. 4 Schematic of a polarization curve with a typical sequence of transport regimes, transition and cGDL flooding current densities marked out. The model does not consider current densities above jflcGDL, the onset of flooding at the cGDL|MPL interface.

V

V

in Fig. 3, jc g o jcM o ja g o jaM . The onset of cGDL flooding, where pc = pcGDL,fl at the MPL|cGDL interface, is denoted by jflcGDL. Fig. 4 shows a schematic of a polarization curve with these current densities marked out in black dashed lines. The present model does not include analysis at jo > jflcGDL, viz. above the onset of flooding of the cGDL. Above this j flcGDL, given the 4–5 orders of magnitude decrease of the O2 diffusion coefficient in water relative to air, the increasing liquid water saturation in the GDL is expected to cause severe oxygen transport losses. With the focus on flooding mitigation, jflcGDL is the key optimization parameter of the present model. 2.3

Governing equations

In what follows, we show the equations of water balance in the aCL and cCL, and flux in the MEA components. Symbols, parameters, and their definitions are given in Table 1. Pressure gradients are the driving force for water flux. Vaporization rates, diffusivities, permeabilities, MEA component thicknesses, and water viscosity are lumped into effective resistances R to water flux. We impose flux and pressure continuity at component interfaces. For the diffusion media, we only show the flux equations for cathode components, indicated by a subscript c; the anode equations are identical (subscript a), with the additional condition that RLMPL = RVMPL = 0, since we do not include an MPL on the anode side. 2.3.1 Fluxes of water within the PEM. Mass fluxes of water produced in the cCL via ORR and arriving via electro-osmotic drag in the PEM are, respectively, Jo ¼

jo M w ; 2F

Jeo ¼

jo neo Mw ; F

(1)

where jo is the Faradaic ORR current density. Liquid flux in the PEM is given by L ¼ JPEM

Fig. 3 The 9 possible regimes of transport within the current model. Possible paths for transitions amongst the regimes, as current density increases, are shown in arrows. Likely paths are marked in red; less likely paths, corresponding to liquid transport in the anode relative to the cathode, are marked in grey.


w dpL RLPEM dyp

;

RLPEM ¼

mw LPEM ksat PEM rw

(2)

where w is the water content of the PEM with range 0 r w r 1, normalized by the saturated water content, and yp the variable of displacement along the PEM axis, normalized by the thickness of


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PCCP Table of physical constants, variables, and base case parameters and resistances

Symbol DMPL, DGDL F jo j fli M jM a , jc


V

V

ja g ; jc g Ja, Jc JLi JVi Jo Jeo ka, kc kMPL ksat PEM LaTL, LcTL la , lc LGDL LMPL LPEM Mw neo pc pcfl,CL pcfl,GDL pG pL pV,eq pV,eq N r R RLa, RLc RLMPL RLPEM RLV CL RLV GDL RVa, RVc RVMPL RHa, RHc T VM ya, yp, yc DpG g kLV mw rw y xLV CL xLV GDL

Definition

Value

Vapor diffusion coefficient of MPL, aGDL, cGDL Faraday constant Faradaic current density Current density at the onset of flooding in medium i Current density at onset of M regime in aGDL, cGDL Current density at onset of Vg regime in aGDL, cGDL Total mass water flux out aGDL, cGDL Mass liquid flux in medium i Mass vapor flux in medium i Mass flux of water produced in ORR Mass flux of electro-osmotic drag Permeability of aGDL, cGDL Permeability of MPL Permeability of saturated PEM Extent of anode and cathode transmission lines LaTL/LGDL, LcTL/LGDL respectively Thickness of gas diffusion layers Thickness of microporous layer Thickness of PEM Molar mass of water Electro-osmotic drag coefficient Capillary pressure pL–pG Flooding capillary pressure of CL Flooding capillary pressure of GDL Gas pressure Liquid pressure Equilibrium vapor pressure Equilibrium vapor pressure at infinite pore radius CL pore radius Gas constant aGDL, cGDL liquid permeation resistance (m s1) Cathode MPL liquid permeation resistance (m s1) PEM liquid permeation resistance (m s1) aCL, cCL vaporization resistance (m s1) aGDL, cGDL vaporization resistance (m s1) aGDL, cGDL vapor diffusion resistance (m s1) MPL vapor diffusion resistance (m s1) Relative humidity at anode, cathode Temperature Molar volume of water Normalised axes along aGDL, PEM, cGDL G Gas pressure difference pG c pa Surface tension of water Vaporization rate constant Viscosity of water Density of water Contact angle of water|Pt Liquid|vapor interfacial area in CL Liquid|vapor interfacial area in GDL

the PEM, LPEM. We have assumed a linear dependence of permeability with w.16 The relationship of w to equilibrium vapor pressure pV,eq and capillary pressure pc are given by empirical fits to capillary and sorption isotherms of Nafion 112,27,28 !0:2 !4 3 pV;eq 11 pV;eq w¼ þ (3) 14 pV;eq 14 pV;eq 1 1 w = 0.5(1 + tanh(log| pc| + 2.4))

(4)

c L G where pV,eq N is the saturated vapor pressure, and p = p p , given in bars. In eqn (3), the coefficients denote surface and bulk water fractions of a saturated membrane. These isotherms are shown in Fig. 5.


Ref. 6

2

6.3 10 m s 96 485 C mol1

1

38

1 1017 m2 1.2 1019 m2 2.4 1019 m2

39 14 39

225 mm 60 mm 20 mm 0.018 kg mol1 2 2.5 bar 6 kPa

17

See eqn (5) See eqn (5) 50 nm 8.314 J K1 mol1 6.6 106 2.0 108 3.3 107 2.5 108 2.5 108 5.7 106 1.4 106 50% 323 K 1.8 106 m3 mol1 0 kPa 0.0626 Nm 4.13 109 kg Pa1 s1 m2 See eqn (6) 1 103 kg m3 0 1 m2 mgeo2 1 m2 mgeo2

Eqn Eqn Eqn Eqn Eqn Eqn Eqn

(8) (15) (2) (13) (9) (7) (14)

40 23 41

The saturated vapor pressure is given by the empirical Antoine equation log pV;eq 1 ¼a

b T þc

(5)

29 where a = 4.6543, b = 1435.624, c = 64.848 for pV,eq N in bars. 30 The variation in viscosity with temperature is given by

B

mw ¼ A 10TC ; B ¼ 247:8 K;

A ¼ 2:41 105 Pa s;

(6)

C ¼ 140 K:

2.3.2 Fluxes of water within the GDLs. We treat liquid and vapor fluxes within the anode and cathode GDL with a transmission


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2.4

Boundary conditions


Mass balance equations at cCL and aCL are

Fig. 5 Empirical fits to capillary (top) and vapor sorption (bottom) isotherms of Nafion 112.27,28

at cCL: Jo + Jeo = Jc + J LPEM,

(10)

at aCL: Jeo + Ja = J LPEM

(11)

where Jo is the flux produced with the oxygen reduction reaction (ORR), Jeo the electroosmotic flux across the PEM from anode to cathode, JLPEM the back permeation in the PEM, and Jc and Ja the total water fluxes out the cGDL and aGDL, respectively. The boundary conditions for the transmission line GDL model are given by pressure and flux continuity, which depend on the transport regime. Given the large GDL pores (1–10 mm),21 we can assume pcc(1) = 0, i.e. pLc(1) = pG c . Where the transmission line does not extend to the GDL|flow field interface (the Vg regime), 0 o lc o 1, the vapor flux out of the GDL and into the flowfield is V ¼ JcFF

V;eq pV c ð1Þ p1 RHc ; V Rc ð1 lc Þ

(12)

and liquid flux out is zero. The vaporization rate at the cCL is23 LV JcCL ¼

Fig. 6 Transmission line model of the cGDL, showing the effective resistances to liquid and vapor fluxes and liquid|vapor conversion. The axis yc is normalised to LcTL, the extent of the transmission line and penetration of liquid water in the GDL. Lc is the width of the cGDL. Dyc denotes an infinitesimal element along yc.

line model,15 illustrated in Fig. 6. To condense notation, we denote properties/fluxes within the cGDL by the subscript c, and those within the aGDL by the subscript a. We introduce here the flux equations within the cGDL; the anode ones are identical with the subscript c replaced by a. In the current three-state assumption of the GDL, liquid water penetrates the GDL progressively, and the length of the transmission line, LcTL, is variable. The axis along the transmission line, yc, is normalized by LcTL. We introduce a dimensionless parameter lc = LcTL/Lc, where 0 r lc r 1. The vapor and liquid fluxes in the cGDL are, JcV ¼

1 dpV c ; RV c lc dyc

RV c ¼

RTLc ; Dc Mw

(7)

1 dpL ; RLc lc dyc

RLc ¼

mw Lc : kc rw

(8)

JcL ¼

Vaporization converts liquid to vapor flux, and is given by dJcV dJ L pV;eq pV ¼ c ¼ 1 LV c ; dyc dyc Rc =lc

RLV c ¼

1 : kLV xLV c

RLV CL ¼

1 ; kLV xLV cCL

(13)

where pV,eq(w) is given implicitly by eqn (3). In the anode, without an MPL, pVaCL = pVa(0) and pLaCL = pLa(0). Vapor diffusion and liquid permeation flux in the cathode MPL are V JMPL ¼

V pV pV;eq ðwÞ pV c ð0Þ cCL pc ð0Þ ¼ ; V LV þ R RMPL RV MPL CL

RV MPL ¼

RTLMPL DMPL Mw (14)

L ¼ JMPL

pLcCL pLc ð0Þ ; RLMPL

RLMPL ¼

mw LMPL kMPL rw

(15)

the 2nd equality of eqn (14) is given by pressure and flux continuity at the cCL|MPL interface, J VMPL = J LV cCL from eqn (13).

3 Model solution In the following, we first discuss the solutions to the PEM and GDL models in terms of liquid and/or vapor pressures at their respective boundaries. When these solutions are combined with mass balance (eqn (10) and (11)), and pressure and flux continuity at the MEA component boundaries, we obtain the relationships between jo and capillary pressure (e.g. at the cCL and cGDL boundaries, pccCL and pc(0)), anode and cathode fluxes, and critical current densities at CL and cGDL flooding. 3.1

(9)

pV;eq ðwÞ pV cCL ; RLV CL

Solution to PEM model

The total water flux in the cGDL is Jc = J Vc + J Lc.

We make the approximation of constant water content w within the PEM, since pressure differences of \10 bars are required for Dw B 0.2 (cf. Fig. 5); we evaluate this assumption a posteriori



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with the computed pressure drop across the PEM. In this case, eqn (2) reduces to w L w c L JPEM ¼ L p pLaCL ¼ L p pcaCL þ DpG (16) RPEM cCL RPEM cCL G where DpG = pG c pa . Where there is liquid in either electrode, c c i.e. paCL and/or pcCL Z pcfl,CL, w = 1 throughout the PEM. Where water is transported out both the anode and cathode in vapor form (VVT regime), w o 1, and this variable must be solved for via mass balance and eqn (3) and (4).

3.2

d2 JcV RV LVc lc2 JcV ¼ 0: 2 dyc RGDL

(17)

V LV As discussed below, estimates for RVc, RLV c indicate that Rc { Rc . V LV Where Rc t Rc , we can linearize the solutions to eqn (17). At yc = 0 and 1,

JcV ð0Þ ¼

V pV lc V V c ð0Þ pc ð1Þ LV 3pV;eq 1 pc ð1Þ 2pc ð0Þ V lc R c 6RGDL

JcL ð1Þ ¼

pLc ð0Þ pG lc V V c LV 3pV;eq 1 pc ð1Þ 2pc ð0Þ : (21) lc RLc 6RGDL

We now show Jc and relevant capillary pressures in the 3 regimes, V, Vg, and M. 3.2.1 Solution in the V regime. The V regime corresponds to the trivial case of lc = 0, with vapor transport only in the GDL, and the transmission line model is not required. Continuity of pressures and fluxes at the cCL|MPL and MPL|cGDL boundaries gives the total flux Jc ¼

pV;eq ðwÞ pV;eq 1 RHc : V V RLV þ R MPL þ RcGDL CL

(22)

In the anode case, RVMPL = 0. Where there is liquid in either the anode or cathode GDL, w = 1 and pV,eq(w) = pV,eq N . In the VVT regime, w is determined from the mass balance equations and eqn (3) and (4). 3.2.2. Solution in the Vg regime. In the Vg regime, where 0 o lc o 1, continuity of fluxes and pressures gives: JVc(0) = JVMPL

(eqn (14) and (18))


(eqn (21)).

(23)

c V,eq pccCL = (A2 + A3)pV,eq N (1 RHc) = pc(0) + A3pN (1 RHc), (24)

where the factors A1, A2, A3 depend on lc (determined implicitly via mass balance equations) and transport resistances. These factors are given in Appendix A. In the anode case, where no MPL is considered, pcaCL = pca(0) and the corresponding A3 = 0. 3.2.3 Solution in the M regime. In the M regime, lc = 1 and pVc(1) = pV,eq N RHc. Applying continuity of fluxes and pressures, i.e. the 1st and 3rd conditions given above for the Vg regime above, we have 2 equations for 2 unknowns pVc(0), and pcc(0). We can then write the total, vapor, and liquid fluxes out the GDL, Jc ¼ JcV þ JcL ¼

RLc

pccCL þ B1 pV;eq 1 ð1 RHc Þ; þ RLMPL

JVc(1) = B2 pV,eq N (1 RHc), JcL ð1Þ ¼

(19)

pLc ð0Þ pG lc V V c þ LV 3pV;eq (20) 1 pc ð1Þ 2pc ð0Þ lc RLc 6RGDL

(eqn (15) and (20))

Jc = JVc(1) = A1 pV,eq N (1 RHc),

pV ð0Þ pV lc V V c ð1Þ JcV ð1Þ ¼ c þ LV 3pV;eq 1 pc ð1Þ 2pc ð0Þ : lc RV 6R c GDL

JcL ð0Þ ¼

J Lc(0) = J LMPL

This provides 4 equations for the 4 unknowns, pLcCL, pLc(0), pVc(0), pVc(1), which give

(18)

From mass conservation, the total water flux in the cGDL, Jc, is fixed. Substituting JLc = Jc JVc into eqn (8) and integrating from yc = 0 to 1, we obtain

(eqn (12) and (19))

J Lc(1) = 0

Solution to the transmission line model of the GDL

We first give the solution to eqn (7)–(9) in terms of pVc and pLc at the transmission line boundaries, yc = 0 and 1. Eqn (7) and (9) give a 2nd order ODE for JVc;

J Vc(1) = J VcFF

pccCL RLc þ RLMPL

þ ðB1 B2 ÞpV;eq 1 ð1 RHc Þ

(25) (26) (27)

where pccCL is related to pcc(0) via pccCL ¼

RLc þ RLMPL c pc ð0Þ þ B3 pV;eq 1 ð1 RHc Þ: RLc

(28)

The constants B1, B2 and B3, given in Appendix A, depend on the transport resistances. Again, in the anode case, RLMPL = RVMPL = 0 and the corresponding B3 = 0. 3.3

Solution of the mass balance equations

At any of the 9 water transport regimes (cf. Fig. 3), relations among jo, relevant pc, and water fractions are obtained by combining the solutions to the PEM and GDL transmission line models with mass balance equations (eqn (10) and (11)). Given either jo or the capillary pressure at a point within the MEA, e.g. pccCL, pcc(0), or pcaCL, the corresponding transport regime is not known a priori. It can, in principle, be determined via trial and error with the transmission line solutions in various regimes until a consistent scenario is found, i.e. the determined capillary pressures are consistent with the transport regime assumed. An additional challenge is that mass balances involving Vg on either side give implicit functions of lc, pccCL, and/or jo. These cases rely on Newton solution methods for the mass balance equations that require good initial guesses to obtain the physical solution. We therefore consider Vg regimes last, and only when M and V cases do not give consistent solutions. This solution procedure is detailed in Appendix B.


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4 Results and discussion In what follows, we perform a parameter study of the effective transport resistances, RH, and gas pressures. We evaluate the impact of these parameters on the pressure distribution in the MEA, transition and flooding current densities, and anode and cathode water and liquid fractions.


4.1

Base case parameters

Base case transport coefficients, operating conditions and effective transport resistances are listed in Table 1. We obtained transport coefficients from various literature sources for the base case, as there is no comprehensive published data on the transport properties of the components of MEAs with ultrathin catalyst layer. The focus of this work is qualitative results and trends, not the reproduction of exact results within any particular UTCL MEA. A particular source of uncertainty is the dependence of the permeability of diffusion media on saturation. Recent work from Hussaini and Wang suggest the relative permeability of carbon paper to show an empirical s5.5 dependence, where s is the saturation, and carbon cloth to follow 0.01s3, while the saturated permeability of both materials was B2 1011 m2. Since we assume the GDL to have continuous liquid water paths within the extent of the transmission line, they must then have a saturation corresponding to at least that at the breakthrough pressure. Ref. 17 and 31 found s B 1–10% at breakthrough; assuming s = 5%, either of Hussaini’s relations give a total permeability of kGDL B 1 1017 m2. For the MPL permeability kMPL, we assume the fitted value from the water balance model of ref. 14. Assuming a monodisperse pore size distribution in the CL, it should either be completely unsaturated or fully flooded, so the CL liquid|vapor interfacial area, xLV CL, should always be about 1. We therefore assume a base case value of xLV CL = 1. Dispersion in pore sizes would result in partial saturation at certain current densities and an increase of xLV CL. To obtain a rough, base case estimate of xLV , we assume that the water is GDL transported through MPL cracks and that the GDL saturates first in proximity of these cracked regions.17 From the estimated area density of MPL cracks, ncrack = 810 cm2, and crack perimeters, pcrack = 0.42 cm, extracted from ref. 32, we estimate xLV GDL B ncrack pcrackLGDL B 1. We consider the impact of varying LV xLV CL and xGDL on the onset of CL and GDL flooding. The capillary pressure at CL flooding, pcfl,CL, is estimated from the CL pore radius r via the Young–Laplace equation pc ¼

2g cos y ; r

4.2

Flooding of ultrathin catalyst layers

The degree of the liquid water saturation in the UTCL is a source of controversy.10 For the base case, the assumed r = 50 nm corresponds to a pcCL,fl = 25 bar via eqn (29). Fig. 7 shows the variation of the cCL flooding current density, jflcCL, with temperature for the base case, as well as for various r, RH, and RLV CL (we assumed RHa = RHc = RH). A comparison of the base and r = 5 nm ( pcCL,fl = 250 bar) cases shows only a slight sensitivity of jflcCL to r; this arises from the large capillary pressure variations required to vary the saturation, cf. Fig. 5. A comparison of jflcCL corresponding to low (0%) and high (95%) RH shows the expected trend of higher jflcCL for lower RH, which corresponds to higher vaporization rates at the CLs. We have considered variation of RLV CL by factors of 0.2 and 5, since we expect about an order of magnitude uncertainty in RLV CL; ref. 23 and 33 provide vaporization rates, kLV, that differ by about an order of magnitude, and xLV CL could increase with dispersion in pore sizes. Variation in RLV by factors of 0.2 and 5 give rise to CL fl significantly different j flcCL at high T > 320 K. At 0.2 RLV CL, j cCL 2 approaches 1 A cm at T > 350 K; in this case, the catalyst layer would not be fully flooded at jo o 1 A cm2. Since protons require water for conduction, an unsaturated CL would lead to increased proton transport losses, as suggested by the polarization data of ref. 10. Operation at sufficiently high RH is required to keep CL pores flooded and to minimize proton transport losses. 4.3

Water and liquid fractions and transition current densities

The fraction of water produced in the ORR and transported across the PEM from cathode to anode is the anode water fraction,

(29)

where g is the surface tension of water and the contact angle y = 0 due to hydrophilicity of Pt. For the base case, we assume r = 50 nm, which is around the size of the NSTF whiskers. The capillary pressure at the onset of GDL flooding, pcfl,GDL is estimated from the capillary pressure isotherms of ref. 17.


Fig. 7 The variation of j flcCL with temperature for the base case with r = 50 nm, RH = 50% (at anode and cathode), as well as for various pore radii r, RH, and vaporization resistances RLV CL.

AWF ¼

Ja : Jo

(30)

Where the cathode is in the ‘‘M’’ regime, the fraction of water transported out the cathode flow fields in liquid form is the cathode liquid fraction, CLF ¼

JcL ð1Þ : Jc

(31)


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Where the anode is in the ‘‘M’’ regime, the anode liquid fraction is


ALF ¼

JaL ð1Þ : Ja

(32)

Fig. 8 shows the AWF, CLF, and ALF vs. jo for base case transport parameters and T = 323 K, RH = 50%. The various regimes of transport are also marked. The AWF curve shows that a substantial fraction of water produced in the cathode is evacuated out the anode; CLF and ALF curves show that the water is evacuated out both electrodes in liquid form, except at jo o 0.25 A cm2. Given that we have assumed RLV GDL to be similar in magnitude as RLV CL, the intermediate Vg regimes occur at a small range of jo, i.e. liquid flux out the flow fields occurs at current densities slightly above the onset of liquid flux into the GDL. The cusps in AWF arise from transitions between regimes at the anode side. In the VVT case, with the assumption of constant w, the AWF is V V V;eq Jo RLV CL þ RMPL þ RcGDL þ p1 ðRHc RHa Þ LV AWF ¼ : (33) V V Jo 2RCL þ RaGDL þ RMPL þ RV cGDL At the base case parameters assumed, AWF E 1/2 in the VVT regime, as shown. In the VVgT and VMT regimes, the AWF is AWF ¼

LV JaCL ; Jo

(34)

and JLV aCL is given by eqn (13) with w = 1, and is fixed as Jo varies. In the VgMT and MMT regimes, where liquid is also present in the anode GDL, the AWF is determined by the liquid pressure distributions that vary with jo. Fig. 9 shows the base case transition current densities, V

V

jc g ; jcM ; ja g ; jaM , and the cGDL flooding current density j flcGDL as

Fig. 8 Water and liquid fractions CWF, CLF, and ALF vs. the jo for base case transport parameters (Table 1) and T = 323 K, RHa = RHc = 50%.


V

V

Fig. 9 Transition current densities, jc g ; jcM ; ja g ; jaM , and cathode GDL flooding current density jflc as a function of T; the various regimes of transport are also marked. Assumed base case transport parameters of Table 1.

a function of T. At T o 330 K, we would expect liquid transport out both the anode and cathode (MMT regime) for all jo > 0.2 A cm2. The reduced flooding current density at lower T arises from primarily the decrease in pV,eq N , which decreases vaporization rates, and also the decrease in mw, which increases the permeation resistances. This dramatic reduction in j flcGDL with temperature is in line with increased issues with flooding observed in UTCL MEAs at low and moderate temperatures.11,12 4.4 Effect of transport resistances, operating conditions on onset of flooding at the GDL LV Fig. 10 shows j flcGDL vs. R/R(base) for R = RLV CL and RGDL (on both the cathode and anode sides). In both cases, there is a dramatic increase in j flcGDL as the R/R(base) decreases (i.e. vaporization rates increase). This effect has been explored in 3M NSTF MEAs by Kongkanand et al.11 These authors employed catalyst layers consisting of NSTF as the main active layer and an additional conventional CL with low Pt loading. They found a dramatic improvement of steady state and transient performance at low temperatures. This improvement may arise from the increased vaporization area provided by the conventional CL (increased xLV CL). Modeling studies suggest that conventional CLs, comprised of 3-phase composites of Pt nanoparticles, carbon black, and ionomer, have a large liquid|vapor interfacial area due to their partially saturated bimodal porous structure.23,24 Increasing xLV GDL via, e.g., engineering of GDLs with an optimal hydrophilic–hydrophobic bimodal porous structure, may also lead to further increases in performance. Given that liquid water transport predominates over relevant ranges of jo, liquid transport resistances have a dramatic impact on j flcGDL. Fig. 11 shows the variation in j flcGDL as RLPEM, RLcMPL, and RLcGDL are varied. We note that in certain cases the anode can also begin to flood. However, given the 6 orders of magnitude difference between HOR and ORR exchange current densities,34,35 we expect H2 diffusion limitations in


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LV Fig. 10 j flcGDL vs. R/R(base) for RLV CL and RGDL (on both cathode and anode).

Paper

Fig. 12 j fliGDL as a function of DpG at RHa = RHc = 50%, T = 323 K, and base case transport parameters.

possible via introducing ordered, hydrophilic laser-perforations in GDLs.36 The distribution of liquid fluxes is affected by the gas pressure G fl differential DpG = pG c pa . Fig. 12 shows j cGDL as a function of G Dp . As the cathode pressure increases relative to the anode one, anode water fractions increase and j flcGDL increases. The change in slope at DpG = 75 kPa corresponds to a VMT to VgMT transition; in the VMT regime, anode water flux is solely in vapor phase and the gas pressure no longer has an effect on the water flux distribution. The increase in j flcGDL with a positive DpG is consistent with preliminary experimental results from 3M,22 where improvements in steady state performance were observed with sub-atmospheric anode pressures. Fig. 11 j flcGDL (solid lines), j flcGDL (dotted lines) vs. R/R(base) for RLPEM, RLMPL, RLcGDL, and RLaGDL are varied.

the aGDL to pose negligible overpotential losses compared to those from O2 diffusion limitations. Therefore, the impact of anode flooding on performance should be more or less insignificant. Since RLPEM and RLMPL, are order(s) of magnitude higher than RLcGDL and RLaGDL, they determine the liquid flux distribution. By either decreasing RLPEM or increasing RLMPL, the water fraction to the anode is increased and j flcGDL increases. RLcGDL has little effect on the overall liquid flux distribution, but decreasing RLcGDL does decrease j flcGDL, since lower pc differences across the GDL are required for a given amount of liquid flux. These results suggest two methods for mitigating cathode GDL flooding via modification of liquid permeabilities of MEA components: in maximizing the RLMPL/RLPEM ratio to steer more liquid to the anode side, or decreasing RLcGDL to facilitate water removal via the cathode. In recent years, PEM thicknesses have already been reduced by an order of magnitude, so an increase in RLMPL/RLPEM should likely arise from an increase in RLMPL, i.e. through reducing the hydrophilic cracks that form in MPLs during MEA fabrication.17,20,32 Decreasing RLcGDL without conceding substantial decreases in O2 diffusivity and pcfl,GDL may be


4.5

Anode water fractions

Water flux measurements at varying DpG can help identify the water transport regimes. Fig. 13 shows the AWF vs. DpG for jo = 0.2, 0.5, and 0.8 A cm2 at RH = 50%, T = 323 K, and base case transport parameters. As DpG increases, the transport regimes shift in the sequence VMT - VgMT - MMT MVgT - MVT. The four transition points for each jo are marked with dotted lines in the same color as the AWF curve. Where there is liquid phase in both the anode and cathode GDLs (Vg or M on either side), DpG = pLc(1) pLa(1), i.e. the gas pressure difference determines the liquid pressure difference at the ends of cathode and anode transmission lines, and thus has a direct influence on the water flux distributions. In either the VMT or MVT regimes, the water flux distribution is determined by the vaporization rate on one side of the PEM, which is unaffected by DpG. In the MMT regime, where liquid flux is present on both sides, the slope of AWF vs. DpG is slope ¼

1 ; Jo RLPEM þ RLaGDL þ RLcGDL þ RLMPL

(35)

i.e. it is inversely proportional to the total liquid transport resistance of all MEA components. With Vg on either side, there is no explicit expression for the slope of AWF vs. DpG.


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G 2 Fig. 13 AWF vs. DpG = pG at RHi = c pa for jo = 0.2, 0.5, and 0.8 A cm 50%, T = 323 K, and base case transport parameters. As DpG increases, the transport regimes shift in the sequence VMT - VgMT - MMT - MVgT MVT. The four transition points for each jo are marked with dotted lines in the same color as the AWF curve.

However, as Fig. 13 shows, DpG does exert an influence on AWF as long as there is liquid water within the GDL. Preliminary 3M water flux measurements at jo = 1 A cm2, T = 20–50 C, and DpG = 0 to 150 kPa suggest a linear dependence of AWF on DpG,22 and hence the presence of liquid water in the GDLs on both electrodes, consistent with the base case parameters assumed in this work.

5 Conclusions In this work, we presented a water balance model of a UTCL MEA, where the PEM was considered via a hydraulic permeation model37 and the GDL via a transmission line model of water flux.15 The model provides relationships between current density and pressure distributions, current densities at transitions between liquid and vapor transport regimes, and current densities at the onset of CL and GDL flooding. We have analysed the model in the relevant case of high vaporization resistances, where liquid water transport dominates in the MEA at moderate RH and T. Model results suggest that UTCL MEAs require sufficiently high RH at high T B 80 C to retain flooded CLs for proton conduction, but also efficient liquid transport paths out of the MEA at low to moderate T to avoid flooded conditions in diffusion media. This is in contrast to previous models of UTCL MEAs, where poor steady state and transient performance was attributed to the flooding of the UTCL, and GDLs were considered to be free of liquid water.10,13 The current density at the onset of cathode GDL flooding can be increased by: Increasing the ratio of MPL to PEM permeation resistances to steer liquid flux to the anode side, or by a cGDL of increased liquid permeability but fixed flooding capillary pressure (e.g. introduction of hydrophilic channels via laser perforation).


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A gas pressure differential, which steers liquid flux to the anode side. Increasing vaporization areas via hybrid NSTF/conventional catalyst layers, or by increasing the vaporization rates within diffusion media via a specifically designed pore space morphology and wettability. Preliminary experiments on 3M NSTF MEAs are consistent with several of these recommendations; increased steady state performance has been observed in hybrid NSTF/conventional catalyst layer electrodes,11 in MEAs of reduced PEM thickness, and where gas pressure differentials between cathode and anode have been applied.22 Finally, water flux measurements can help evaluate the model and identify water transport regimes. The model predicts transitions among transport regimes as the gas pressure differential is varied, which are reflected in slope changes of the anode water fraction. Where liquid is present on both GDLs, the anode water fraction is expected to show a linear dependence on the gas pressure differential, and no dependence on gas pressure when water transports out of either catalyst layer in vapor form. The former dependence has been suggested in preliminary 3M NSTF water flux measurements,22 which further suggests the dominance of liquid transport in the diffusion media of 3M NSTF MEAs.

Appendix A: constant factors in solution of transmission line water flux model The constant factors in the solution to the transmission line model in the Vg case, eqn (23) and (24), are V 3 1 LV LV V 2 RCL þ RV MPL Rc lc þ 4Rc Rc lc S LV LV 2 V ; þ 12 RLV CL þ RMPL Rc lc þ 12 Rc

(36)

LV 1 L 2 LV V LV R lc 6 RCL þ RV MPL Rc þ 4Rc Rc lc S c V 2 V ; þ RLV CL þ RMPL Rc lc

(37)

LV 1 L V V LV RMPL lc RLV CL þ RMPL Rc þ 6Rc Rc lc S V 2 V ; þ RLV CL þ RMPL Rc lc

(38)

A1 ¼

A2 ¼

A3 ¼

with the denominator S V V 2 LV V LV 3 4 S = (RVc)2(RLV CL + RMPL)lc + (Rc ) ((RCL + RMPL) 4Rc )lc LV V V 2 + 4RVcRLV c (2(RCL + RMPL) + Rc )lc V V LV LV 2 LV V V + 12(RLV CL + RMPL)Rc Rc lc + 12(Rc ) ((RCL + RMPL) + Rc ). (39)

In the anode case, where no MPL is considered, RLMPL = RVMPL = 0.


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The constant factors in the solution to the transmission line model in the M regime, eqn (25)–(28), are B1 ¼

6RLV c

RLc þ RLMPL

L L RV c RMPL þ 3Rc

V RLV CL þ RMPL

LV LV V ; V V LV LV 2 RLc þ RLMPL 3RV c Rc þ 3 RCL þ RMPL Rc þ RCL þ RMPL c

Table 2

Table relating transport regimes and transition current densities V

V

jo jc g

jc g o jo o jcM

jo > j M c

jo ja g

V

VVT

VVgT

VMT

V ja g

o jo o jaM j o > jM a

VgVT

VgVgT

VgMT

MVT

MVgT

MMT


(40) B2 ¼ 2 LV LV V V LV V V 12 RLV þ12 RLV c CL þ RMPL Rc þ 4Rc Rc þ Rc RCL þ RMPL LV ; V LV V LV LV V 4RLV 3RV c c Rc þ 3 RCL þ RMPL Rc þ RCL þ RMPL R (41) B3 ¼ " # V V Rc þ 3 RLV RLMPL CL þ RMPL LV LV V V LV LV V 2 3RV c Rc þ 3 RCL þ RMPL Rc þ RCL þ RMPL Rc (42) Again, in the anode case, RLMPL = RVMPL = 0.

Appendix B: determination of the water transport regime In this section, we detail the approach for determining the transport regime corresponding to a given jo. All calculations were done in MATLABr 2008 using standard functions; implicit equations were solved using the MATLAB fsolve function, which implements the Levenberg–Marquardt algorithm. We first require the pccCL at the transitions between transport regimes. At the transition from V to Vg, the capillary pressure at the cCL pccCL = 0. At the transition from Vg to M, the capillary pressure pccCL,M is found via eqn (21) with j Lc(1) = 0 (for the anode, the subscripts c are replaced by a). From these capillary pressures, we determine the transition current densities by trial and error of the mass balance equations, where regimes that include Vg are tested last, e.g., V Determination of jc g . pccCL = 0 (1) solve mass balance with VVT (2) if pcaCL determined is inconsistent with VVT, i.e. pcaCL > 0, solve mass balance with MVT (3) if pcaCL determined is inconsistent with MVT, i.e. pcaCL o pcaCL,M, solve mass balance with VgVT c c Determination of j M a . paCL = paCL,M (1) solve mass balance with MVT (2) if pccCL determined is inconsistent with MVT, i.e. pccCL > 0, solve mass balance with MMT (3) if pccCL determined is inconsistent with MMT, i.e. pccCL o pccCL,M, solve mass balance with MVgT V Likewise, ja g and j M c can also be determined. With the transition current densities, we determine the transport regime corresponding to a given jo via the following table (Table 2).


For the determination of current densities at the onset of GDL flooding, the procedure is similar. For j flcGDL, we consider the capillary pressure at the MPL|cGDL boundary, i.e. pcc(0). The capillary pressure at the Vg to M transition, pcc,M(0), can be determined from eqn (27) and (28), with jLc(1) = 0. The determination of j flcGDL is as follows: if pcfl,GDL > pcc,M(0), (1) solve mass balance with VMT (2) if pcaCL determined is inconsistent with VMT, i.e. pcaCL > 0, solve mass balance with MMT (3) if pcaCL determined is inconsistent with MMT, i.e. pcaCL o pcaCL,M, solve mass balance with VgMT c if pfl,GDL o pcc,M(0), (1) solve mass balance with VVgT (2) if pcaCL determined is inconsistent with VVgT, i.e. pcaCL > 0, solve mass balance with MVgT (3) if pcaCL determined is inconsistent with MVgT, i.e. pcaCL o pcaCL,M, solve mass balance with VgVgT Similarly we can determine the current densities at the onset of flooding in the anode GDL, and at both catalyst layers.

Acknowledgements Funding from NSERC via a Strategic Project Grant and a Vanier Scholarship is gratefully acknowledged. The authors thank Anatoly Golovnev and Pierre-Eric Alix Melchy for their feedback on the manuscript.

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