Determination of the in-plane components of the electrical conductivity tensor in PEM fuel cell gas diffusion layers

Electrochimica Acta 85 (2012) 665–673

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Determination of the in-plane components of the electrical conductivity tensor in PEM fuel cell gas diffusion layers David R.P. Morris, Jeff T. Gostick ∗ Department of Chemical Engineering, McGill University, H3A 2B2 Montréal, Québec, Canada

a r t i c l e

i n f o

Article history: Received 11 June 2012 Received in revised form 21 August 2012 Accepted 23 August 2012 Available online 30 August 2012 Keywords: Gas diffusion layer Electrical conductivity Anisotropy Tensor Fiber direction

a b s t r a c t The in-plane components of the electrical conductivity tensor for gas diffusion layers (GDLs) used in polymer electrolyte membrane fuel cells were measured using an alternative method that has not previously been applied to GDLs. This method uses a square electrode configuration, which offers many practical and theoretical advantages over conventionally used linear four point probe methods. Results obtained using this method were in excellent agreement with reported values where applicable. Pronounced anisotropy was found in the in-plane electrical conductivity for all samples, in agreement with other findings. Conductivity measurements were also performed on samples rotated relative to a fixed axis to determine the full extent of anisotropy, assumed to be due to fiber alignment, allowing the intrinsic components of the conductivity tensor to be found. The maximum and minimum conductivities were found at a rotation angle different from the main directions of the GDL sheet from which the tested samples were cut. The average in-plane conductivity of GDL samples was independent of rotation angle. Because the direction of maximum conductivity was found to differ from the main sheet axis, the measured conductivity tensor was rotated to yield the corrected 2-dimensional tensor relative to the main GDL axis. The method for performing this correction is discussed and an experimental method for measuring the necessary data from only a single sample is proposed. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction Polymer electrolyte membrane fuel cells (PEMFCs) are receiving significant interest in the automotive sector, however increasing their power density to reduce stack size and cost remain necessary for commercialization. Improving power density requires a deeper understanding of the many coupled transport processes in the cell, and this in turn requires detailed measurement of the relevant transport parameters of the electrode materials. The gas diffusion layer (GDL) is a thin (200–300 ␮m) sheet of carbon fiber paper and is an integral component of PEMFCs electrode assemblies. One key feature of GDL materials is their anisotropic structure. The carbon fibers are highly oriented in the plane of the paper (termed the inplane or IP direction), resulting in significantly different properties in the in-plane vs. the through-plane (TP) direction. The impact of this fiber alignment on transport properties, such as gas permeability [1], tortuosity [2], heat [3] and electron conduction [4–6] have

Abbreviations: 4PP, four-point probe; CD, cross-machine direction; FD, fiber direction; FS, full scale; IP, in-plane direction; MD, machine direction; MEA, membrane electrode assembly; TP, through-plane direction; VDP, van der Pauw. ∗ Corresponding author at: Room 3060, Wong Building, 3610 University Street, Canada. Tel.: +1 514 398 4301; fax: +1 514 398 6678. E-mail address: [email protected] (J.T. Gostick). 0013-4686/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.electacta.2012.08.083

received recent attention. In addition to the prominent IP alignment of fibers, there can also be an appreciable amount of fiber alignment within the plane of the paper due to the manufacturing process. This results in different transport properties between the two orthogonal IP directions, termed the machine direction (MD) and the cross-machine direction (CD). Some studies of gas permeability [1] and electrical conductivity [4–6] have reported the transport properties measured in the two orthogonal IP directions. Because the manufacturing process is somewhat random, however, the predominant direction of fiber alignment or fiber direction (FD) does not necessarily correspond exactly with the MD [7]. This has notable consequences on the measurement of transport properties whose values depend on fiber direction and are described by a tensor. If the FD is not aligned with the MD, then measurement of properties relative to the MD and CD would not yield the maximum and minimum values of the measured properties. This in turn means that the measured values would not be the true or “intrinsic” transport properties, but are convoluted by the very tensorial nature of the material that was sought to be measured. In the present work, an experimental technique and analysis procedure for measuring the intrinsic components of the electrical conductivity tensor in the two IP directions is developed for GDL materials. One of the roles of the GDL is to conduct electrons from the bipolar plates to the catalyst layer, thus the electrical conductivity is a key transport property of the GDL. High electrical conductivity is

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desired in the IP direction of the GDL because it can compensate for low TP conductivity [6]. High IP conductivity facilitates the distribution of electrons to areas of the catalyst layer that lie under the flow channels, and away from areas that are masked by water blockages or have become delaminated from the membrane. Numerous measurements of conductivity are reported in the literature [4–6], but none of these looked closely at IP anisotropy and fiber alignment. Some recent studies have reported the conductivity in two in-plane directions (MD and CD), and appreciable IP anisotropy has been seen in many common GDL materials. Kleeman et al. demonstrated that the in-plane electrical conductivity in the MD and CD differed for four distinct types of GDL [4]. They used a linear four-point probe (4PP) method via a printed circuit board with embedded electrodes. This research was focused on the effect of compression on electrical conductivity, rather than the effect of fiber alignment on anisotropy [4]. Ismail et al. [6] also reported the in-plane electrical conductivity in the MD and the CD of some GDL sampled using a 4PP method developed by Smits [8]. They also observed that GDL samples exhibited significant in-plane anisotropy, with conductivities varying by a factor of 2 between in the MD and the CD. Han et al. [5] used a two-point probe device to assess the in-plane electrical conductivity in the MD and CD. A two-point probe device is generally thought to be unfavorable compared to a 4PP device, which eliminates internal resistances [9]. They found anisotropy in the IP electrical conductivity with ratios up to 1.95. In addition to electrical conductivity, Han et al. [5] also studied the impact of IP anisotropy on the bending stiffness of GDLs and found that it was highly sensitive to fiber alignment; much more so than conductivity. This finding clearly demonstrates the important influence that fiber alignment has on other GDL properties beyond the direct link with electrical conductivity. All the above mentioned studies measured the directional conductivities referenced to the MD, and no study to date has investigated the impact of the FD differing from the MD. The collinear four-point probe (4PP) method is the conventional method for measuring the IP electrical conductivity of GDL materials. The ASTM standard (F1529) for the method was discontinued in 1998. Other ASTM standard test methods that employ the collinear four-point probe method are used for specific applications such as thin metallic films (F390) and thin film conductors (F1711). Williams et al. [10] used ASTM C611 to test the conductivity of GDLs. The standard is specific to carbon and graphite articles and thus is possibly inappropriate for GDLs, as Ismail et al. have mentioned [6]. All these methods are related and consist of four probes placed in a linear array across the sample material, as illustrated in Fig. 1(a). The two outer probes are used to provide current to the test sample, while the inner probes are used to measure the electric potential across the sample. To appraise the anisotropy of the sample, the probes are rotated 90◦ and the measurements are taken again; this often means cutting a second sample from the master sheet at a rotation of 90◦ from the first. There are two main issues with the commonly used 4PP method. Firstly, porous materials such as GDLs possess a random structure with varying properties between locations. This spatial dependence can be verified by the fact that the position of the electrode array on the sample affects the resistance measurement. Koon and Chan found that local increases in the resistivity of the sample will actually lower the measured resistance, thus decreasing the apparent resistivity of the sample [11]. By applying bars instead of point probes heterogeneities can be averaged out [4]. Secondly, in materials with any degree of IP anisotropy, the 4PP method produces results that are somewhat ambiguous. For instance, when the current probes are aligned precisely with the FD, a 1D conduction scenario will result and placement of voltage probes at any location will yield consistent results (notwithstanding heterogeneity). If, however, the current probes and fibers are misaligned, for instance

if fibers run diagonally in the setup shown in Fig. 1(a), a 2D current distribution will result since the current will tend to follow the FD. In this case the voltage probes will read a different value depending on where in the 2D current field they are placed. The extent of error incurred in this situation may or may not be important, but one should be aware of the possible problems. An alternative approach to measure conductivity in thin materials is a square 4-point probe method, such as the approach developed by van der Pauw [12,13]. The square electrode array shown in Fig. 1(b) avoids the issues presented above and as such is well suited for inhomogeneous and anisotropic materials [14]. Firstly, the conductivity in two orthogonal directions can be measured by merely switching the leads connected to the electrodes, thus eliminating the need to cut samples in different directions or orient the electrode array differently. Secondly, with the van der Pauw method the sample size and shape is entirely flexible. In fact the method works equally well on the small coupons used in this study, full MEA-size samples or on entire sheets, making the method useful for quality control purposes. Indeed, the sample dimensions and probe spacing are not even parameters in the equation(s). Thirdly, the effects of heterogeneities are included in the resistivity result, by virtue of taking measurements across the whole sample, rather than of just a section of the sample. Finally, and most importantly, the van der Pauw method provides valid conductivity values regardless of fiber alignment affects, whereas the linear 4PP setup is only valid when the probes are aligned with (or perpendicular to) the fiber direction. The van der Pauw method is often used to test the in-plane electrical conductivity of various thin materials, including graphite [15,16]. It was recently used by Cunningham et al. to determine the conductivity of bipolar plates in PEMFCs [17]. They followed ASTM F76-86, which is designed for single-crystal semiconductors, but is applicable to many different materials. One important limitation of the van der Pauw method is that it can only be used to determine the average conductivity of a material (that possesses IP anisotropy); this is discussed in Section 2.3. Montgomery improved upon van der Pauw’s method and extended it such that the directional components of conductivity in anisotropic materials could be found. Both methods use the same electrode configuration and produce the same raw data; only the mathematical analysis of the data differs. The present study aims to demonstrate that the Montgomery method is a robust method for measuring and appraising the anisotropy of the in-plane electrical conductivities of GDLs.

2. Theoretical background The IP conductivity is typically an order of magnitude larger than the TP conductivity of GDLs [4,6,18]. In this work it is assumed that this behavior can be solely attributed to the effects of fiber alignment. Specifically, the large anisotropy between the IP and TP directions is understood by the fact that electrons can travel long distances in the IP direction along a single fiber, while in the TP direction no such direct paths exist. Transport in the TP direction occurs by transfer between stacked fibers, requiring movement in the IP direction along shorter fiber segments to reach relatively infrequent fiber intersection points. Such a highly extended transport path leads to significantly lowered conductivity. This same picture of extended transport lengths can also be applied to IP electron transport when IP fiber alignment is present, since conduction in any direction other than the FD will require fiber–fiber transfers and extended paths. With this picture in mind, anisotropic conductivity is herein interpreted strictly in terms of fiber alignment effects. GDLs are manufactured using papermaking techniques that can result in the fibers being approximately orientated in the MD,

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Fig. 1. (a) Schematic diagram showing the arrangement of probes in a typical 4PP method. (b) Schematic diagram showing the placement of electrodes in the van der Pauw/Montgomery method used in this work. Rotating the leads by 90◦ permits measurement of Vy and Iy . (c) Schematic diagram illustrating the dimensions of a GDL sample. Using these dimensions, the resistivity would be measured in the x-direction, as the cross-sectional area is normal to x.

Fig. 2. (Left) Schematic diagram of a GDL sheet defining the sheet directions (MD, CD), corresponding angles and fiber direction (FD) and (right) a diagram of a GDL sample defining the sample directions, angles and coordinates used in this work.

however, since the process is somewhat random it is possible that the FD differs from the MD [7]. Fig. 2 shows the relationship between MD, CD and FD, along with the definition of several pertinent angles. The angle formed between the MD and FD is defined as ˇ; angles referenced to FD are termed the orientation angle ˛, and thus ˛ = 0◦ at FD; angles referenced to the MD are termed the rotation angle . The GDL sheet axis is therefore defined by FD and ˛, as shown in Fig. 2. The coordinate system used for GDL samples is also presented in Fig. 2. The in-plane directions are x and y, which are relative to the edges of the sheet or cut sample. Therefore, x and y correspond to the MD and CD only if the sample is cut parallel to the edges of the GDL sheet. When the GDL sample is cut at a sample rotation angle  other than 0◦ (or a multiple of 180◦ ), the directions x and y do not correspond to the MD or CD. FD is bounded by the sector 0◦ ≤  ≤ 180◦ since conduction along fibers is symmetric (i.e. not direction dependent). Finally, the angle formed with the x-axis in the cut sample and the FD is ˇ .

The only tensor elements that are typically reported for GDLs are those in the main diagonal, namely  xx and  yy . This is because conductivity measurements are conveniently performed parallel to the edges of the GDL sheet (MD and CD) and it is assumed that MD and FD coincide. If this is the case then the reported values do indeed correspond to the intrinsic conductivities of the GDL and the off-diagonal components in Eq. (2) are zero. However, if the FD does not coincide with the MD, then the off-diagonal elements of the conductivity tensor,  xy and  yx , will be non-zero and the main diagonal elements will not be the intrinsic conductivity values. Hence, the off-diagonal elements of the tensor account for the offset, if any, between the FD and MD. Even if the conductivity in the MD and CD direction are only of interest, performing measurement in these two directions alone is insufficient since the off-diagonal components will influence their value. The salient point is that the tensorial nature of conductivity must be considered not only when modeling electron transport, but also when experimentally determining the electrical conductivity tensor itself. Although this is superficially done in most literature reports of GDL conductivity it is not safe to assume that the FD and MD coincide and therefore populating the conductivity tensor from measurements made in the MD and CD is not fully accurate. Since the FD of a material is not known a priori, one cannot assuredly claim that the off-diagonal components are zero [19]. This consideration not only impacts electrical conductivity, but also thermal conductivity in the fibrous backbone, as well as all pore space transport properties of the GDL, which are also impacted by fiber alignment and structure. The main consequence of misalignment between MD and FD is that the purely diagonal tensor measured relative to FD (termed FD-aligned tensor) must be manipulated to find the MD-aligned tensor, which is presumably more useful since fuel cells would be assembled from GDLs cut parallel to the MD. Assuming one knew the FD a priori and was able to contrive experimental conditions such that the true FD-aligned tensor was obtained, the following would result:

2.1. Conductivity tensor A material’s conductivity  [S m−1 ] is an operator in the general expression of Ohm’s law: J =  E

(1)

where E is the electric field [V] vector, and J is the current density vector [A m−2 ]. Eq. (1) can be expressed for a two-dimensional system with the conductivity described as a second-rank tensor [16] as follows:

  Jx

Jy



=

xx

xy

yx

yy



Ex Ey



(2)

 

FD

=

xx

0

0

yy

 (3)

/  yy It must be reiterated that this tensor is only valid where  xx = for a measurement with the electrode assembly oriented in the FD of the material being tested. Eq. (3) can be converted to the desired MD-aligned tensor by applying a rotation of −ˇ (the angle formed between FD and MD). In general terms, a rotation of the FD-aligned tensor by any angle ˛ is expressed mathematically by multiplying

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a rotation matrix R by the tensor , and then by the transpose RT :

 (˛) = RRT =

 ×

cos(˛) −sin(˛) sin(˛)

cos(˛)



cos(˛)

sin(˛)



xx

0

0

yy



(4)

−sin(˛) cos(˛)

After multiplication of the matrices, the following form is found:



(˛) =

 =

xx cos2 (˛) + yy

2

(␴xx − yy )cos(˛)sin(˛)  xx

 xy

 xy

 yy

(xx − yy )cos(˛)sin(˛)

sin (˛)



yy cos2 (˛) + xx



2

sin (˛) (5)

By using Eq. (5) it is possible to determine the conductivity tensor at any angle ˛, provided the intrinsic properties (i.e. the diagonal components of the FD-aligned tensor,  xx and  yy ) are known. A procedure for the determination of FD is presented in Section 4.2. Eq. (5) shows that the off-diagonal components of the tensor are equivalent, in accordance with Onsager’s theorem [20]. The offdiagonal components are equal to zero when the sample is aligned in the FD (˛ = 0◦ ) and Eq. (3) is recovered. The off-diagonal components also disappear for an isotropic material since  xx =  yy . In order to distinguish the tensors in Eqs. (2) and (3), which represent the intrinsic conductivity values, with that in Eq. (5), primes have been added to the tensor elements. Therefore, elements with primed components have been rotated away from FD by an angle ˛. In other words  xx and  yy are the intrinsic conductivities of the  and   would be the conductivities relative to material, while xx yy the MD and CD directions when MD and FD do not coincide. 2.2. Average in-plane conductivity It is common to report the average in-plane conductivity  IP of GDLs, rather than the main diagonal components of the FD-aligned tensor (since the latter is not trivial to determine). It is often claimed that the in-plane conductivity of a thin film is the geometric mean of the conductivities in the two perpendicular in-plane directions, i.e. [12,13,21,22]: IP (˛ = 0◦ ) =



xx yy

(6)

This is only true, however, if the measurement of  xx and  yy is taken in the FD. The true average in-plane conductivity is actually the square root of the determinant of the conductivity tensor [22]: IP =





det (˛) =

   − (  )2 xx yy xy

(7)

A rather useful property of the average in-plane conductivity is that it is independent of orientation angle ˛ and therefore it is constant for all samples of a given material. 2.3. van der Pauw method A simple method for measuring the in-plane electrical conductivity of thin isotropic materials was developed by van der Pauw in 1958 [13]. The van der Pauw (VDP) method uses four probes in a square configuration to measure the resistance in the two orthogonal in-plane directions, as shown in Fig. 1(b). Two adjacent probes are used to apply a constant current while the other two probes are used to measure the electric potential. The VDP method allows testing of samples of virtually any size or shape. The method requires that the probes be placed on the periphery of the sample and their contact areas on the sample must be as small as possible.

The errors associated with finite electrode contact area, electrode placement, finite sample thickness, among other factors, have been studied extensively [11,14,23] and it has been found that the VDP method is quite robust. To interpret the data obtained by the square probe arrangement, van der Pauw developed an equation relating the resistances measured in the x and y directions (Rx and Ry ) and the sheet resistance RS , which is found by iterative solution of Eq. (8): e



R

− Rx s



+e



Ry

− R s



=1

(8)

The sheet resistance Rs [] is a measure of the average in-plane resistance of a thin material. The sheet resistance can be derived from the volume or bulk resistivity relationship for a thin material, as demonstrated in Eq. (9): R = IP

IP L L L = = Rs A t W W

(9)

where IP is the average in-plane resistivity of the material, and L, W, t and A are the length, width, thickness and cross-sectional area of the sample, as illustrated in Fig. 1(c). Therefore, current flows along the length L of the thin material and the cross-sectional area A is defined as product of the two remaining dimensions, t and W. Hence, IP represents the average in-plane resistivity of the material and for a thin material, the sheet resistance Rs is simply the in-plane resistivity divided by the thickness of the sample. The sheet resistance for a square sample of a thin material is given by Eq. (10), where L = W: Rs =

IP 1 = t IP t

(10)

Using sheet resistance as a metric for conductivity is rather useful as it is calculated directly from any 4PP measurement (square or linear), whereas conductivity must be calculated using the thickness of the sample. The thickness of a sheet of GDL material is quite variable and thus the thickness measurement introduces uncertainty into resistivity or conductivity calculations. Therefore, the conductivity of samples of the same nominal thickness can be compared using sheet resistance. Moreover, many GDL manufacturers report sheet resistance rather than resistivity or conductivity; therefore it is often advantageous to determine the sheet resistance of samples. 2.4. Montgomery method Montgomery extended van der Pauw’s method by developing a more advanced mathematical treatment of the measured resistances to account for anisotropy [24]. The probe configuration and experimental methodology are identical to the VDP method; however the ensuing data analysis is not. Montgomery’s approach uses conformal mapping to model anisotropic materials as isotropic with different dimensions [14,16,19,24]. The original method employs a simple and somewhat imprecise graphical approach to calculate the resistivity of anisotropic materials. Recently, dos Santos et al. rendered the Montgomery method more tractable by modeling the graphs [16]. As mentioned, the Montgomery method uses conformal mapping of an anisotropic solid to model it as an isotropic solid. Therefore, two sets of dimensions are used to describe the solids: those that are primed represent the dimensions of actual anisotropic solid, whereas those that are not denote the equivalent isotropic solid. The resistivity 1 in direction 1 of the anisotropic solid is calculated using Eq. (11): 1 = H1 ER1

(11)

where H1 is a function of the ratio of the isotropic equivalent dimensions L1 /L2 and E is the effective thickness of the isotropic equivalent

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Fig. 3. (a) Expanded view of sample holder, (b) top view of base with GDL sample and (c) bottom view of plunger. The electrodes sit in the grooves and touch the corners of the sample. (d) Top and (e) side views of the assembled holder for electrode compression (contact resistance) tests. The middle ring sits on top of the electrodes and does not compress the sample directly. The position of the bolts is shown in the top view (d). The plunger is not shown for clarity. The bolts are not shown in the side view (e). (f) The top view of base with small GDL sample; the electrodes are extended further when the small sample is used. (g) The bottom view of the modified plunger shows that the corners have been removed to accommodate the electrodes.

sample. For thin samples, E is equal to the thickness t(= L3 ). Directions 1 and 2 are thus the IP directions. An approximate relationship between the ratio L1 /L2 and the measured resistance ratio R1 /R2 was devised by dos Santos et al. and is presented in Eq. (12). Direction 2 was arbitrarily chosen to denote the direction in which fiber alignment is greater. Thus, R2 is smaller than R1 , and thus R1 /R2 ≥ 1, which is a requirement of Eq. (12).

L1 ∼ 1 = L2 2



1 R1 + ln  R2



1 R1 +4 ln  R2

 1 = t 8

(12)

 t 8

(13)

L2



L1

L1 L2



L  2

sinh 

L1

R1

(14)

L  1

L2

L  2

sinh 

L1

R1

(15)

An equation similar to Eq. (15) was used to calculate the resistivity in direction 2: 2 =

 t 8

L  2

L1

L  2

sinh 

L1

R1

(16)

Therefore, 1 and 2 are related by: 2 = 1





Square samples were used in the present study such that L2  /L1  is equal to unity, and thus Eq. (14) is reduced to: 1 =

Montgomery developed a graphical method to determine H1 . dos Santos et al. [16] provided an approximation of this graphical method:  L2 H1 ≈ sinh  8 L1

Finally, using Eq. (12) to solve for H1 in Eq. (13), the resistivity can be expressed as:

 L 2 2

L1

(17)

Other relevant equations and relationships are available in [16].

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3. Methods 3.1. Experimental set-up A custom sample holder for measuring IP electrical conductivity using a square electrode array was designed and built as shown in Fig. 3. The sample holder consists of a base with a cavity for the GDL sample, a middle ring to secure and compress the electrode leads against the sample, and a plunger used to keep the electrodes on the periphery of the sample. The plunger is not used to compress the electrodes, however applying force to the plunger can be used to compress the GDL sample, although this aspect is not explored here. The components of the holder were made of glass fiber filled polyether ether ketone (PEEK), which has good compressive strength and is electrically inert. Four rectangular gold-plated copper electrodes (1 in. × 0.825 in. × 0.125 in.) were placed on the corners of the square GDL sample. Leads were inserted into holes bored into the electrodes, which were then tin-soldered. 3.2. Procedure A power supply (HP 6210B) was used to provide a constant dc current to the GDL samples through the electrodes and the current was measured using a digital multimeter (Keithley 197A) with 0.2% dc current accuracy (FS: 200 mA and 2 A, where applicable). The potential drop V across the other two electrodes was also measured using a digital multimeter (Tektronix DMM 4020) with 0.01 + 0.004FS% dc voltage accuracy (FS: 200 mV and 2 V, where applicable). The current was varied between 50 mA and 500 mA, with a step size of 50 mA, for each experiment. The results at each current gave the same resistance value within experimental accuracy of the equipment and were used only to provide an average value. The absence of current dependence confirms that no significant ohmic heating was occurring. All tests were performed at ambient conditions. After collecting the data for one probe configuration, the leads were shifted 90◦ and the process was repeated, yielding measurements in two orthogonal directions. In order to reduce experimental error, additional measurements were taken by rotating the probes 90◦ twice more to replicate the first set of orthogonal measurements and average values were obtained. The difference between the replicate tests was always negligible. 3.3. Interfacial contact resistance The effect of the interfacial contact resistance between the goldplated copper electrodes and the GDL sample was evaluated by varying the pressure on the electrodes using the middle ring of the assembly, as shown in Fig. 3(d) and (e). The torque of four equally spaced bolts on the middle ring was varied from 1 to 4 N m, which compressed the electrodes onto the sample. The contact area of each electrode and the GDL sample was very small (2.5 mm2 ) such that it can be assumed that the pressure distribution was uniform. The goal of compressing the electrodes was to minimize the effect of material asperities and surface roughness on the voltage measurements. The sheet resistances of the three types of GDL were measured as a function of the electrode compression at dc currents between 400 and 500 mA, with a 50 mA step size. The sheet resistance as a function of bolt torque is provided in Fig. 4. The sheet resistance did not vary significantly (<0.2%) in the range of torque applied per bolt. However, the sheet resistance decreased slightly at 2 N m for every type of GDL tested. Successive tests with increasing compression proved to have inconclusive effects. For instance, the sheet resistance of SGL-10AA decreased at 2 N m, but subsequently increased at 3 N m: this behavior was expected as electrode compression could damage fibers and reduce conductivity. The increase in sheet resistance was however very

Fig. 4. Sheet resistance as a function of bolt torque (electrode compression) for three types of GDL.

small (<0.02%) and could have been experimental error. The nonmonotonic decrease of sheet resistance of the other two GDL samples is also small (<0.2%) and is within experimental accuracy. Thus, the effect of interfacial resistance between the electrodes and the GDL samples was marginal for the electrodes used in the present study. Nevertheless, a bolt torque of 2 N m was used for all tests to ensure repeatability and to reduce contact resistance. Further, the risk of damage to a GDL sample at this torque level was low. 3.4. Materials The in-plane conductivities were measured for one type of GDL supplied by Toray Inc. (TGP-H-090), one from Freudenberg GmbH (H2315-T30A), and one from SGL (Sigracet 10AA). The values for the PTFE loading of the GDL samples were provided by the suppliers and are given in Table 1 along with thickness. The thicknesses of the samples were measured using a Mitutoyo digital micrometer (0.001 mm accuracy), using the average of five measurements. The sample sizes of the GDLs were the same size as the cavity (1.375 in. × 1.375 in.) as shown in Fig. 3(b). 4. Results and discussion 4.1. Method validation The square probe arrangement used here has not been previously applied to GDL materials. To confirm the validity of this method the present results are compared directly to a variety of literature values in the sections that follow. Table 1 Properties of GDL materials. GDL

PTFE content (wt.%)

Thickness (␮m)

TGP-H-090 H2315-T30A SGL-10AA

0 30 0

298 193 365

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Table 2 Experimental and reported values of the average in-plane electrical conductivity for various GDLs. Standard error is reported alongside experimental values. Replicates were performed on at least three separate samples, each taken from the same sheet of material. Reported values are provided by the manufacturer, except where noted. Values reported as sheet resistance were converted to conductivity using the reported thickness. Average in-plane conductivity (S m−1 )

GDL

TGP-H-090 H2315-T30A SGL-10AA

Van der Pauw

Reported

14,376 ± 156 6322 ± 32 3789 ± 80

17,860 6360 4000 [6]

4.1.1. Average in-plane conductivity The average in-plane conductivity was determined for each material listed in Table 1 using the van der Pauw method by Eq. (8). Experimental and reported values of the average in-plane conductivity are presented in Table 2. The results obtained using both methods for H2315-T30A and SGL-10AA are in excellent agreement with reported values. Results obtained for TGP-H-090 are about 22% lower than the reported value; this discrepancy is discussed in the section below. The difference amongst the conductivities of the different types of GDLs can be attributed to a number of factors including carbon fiber diameter, fibrous structure and topology, sample porosity, and the degree of graphitized vs. amorphous carbon, among others. Overall the data obtained using the new square probe arrangement, analyzed using the van der Pauw approach, produces excellent agreement with accepted values, lending credence to this approach. 4.1.2. Anisotropic in-plane electrical conductivity In addition to determining the average in-plane conductivity, the square probe arrangement can be used to measure the directional conductivity in the two orthogonal IP directions of the sample when the Montgomery method is used to analyze the data. For these tests the samples were cut parallel to the edges of the main GDL sheet so the orthogonal directions correspond to MD and CD. This comparison basis was chosen rather than results measured relative to the FD since was assumed that the reported values were also measured relative to the MD. Table 3 shows the measured results with reported literature values in parentheses. The MD and CD conductivity values are in good agreement with reported values. Results for SGL-10AA were similar to those found by Ismail et al. [6]. The measurements of Ismail et al. were performed using a linear 4PP method, which as mentioned above can be erroneous in anisotropic samples. The small differences between their reported values and the present results suggest that the error incurred may not be substantial. The different types of GDL exhibit varying anisotropy ratios, which is likely due to different fiber alignment. Note that the MD conductivity value obtained for TGP-H-090 using the Montgomery method closely matches the average in-plane conductivity reported in Table 2 (within 7%). One possible explanation is that the value reported by the manufacturer is actually the direction-dependent value measured in the MD. The close Table 3 Experimental and reported values of the two in-plane electrical conductivities for selected GDL samples. Replicates were performed on at least three separate samples, each taken from the same sheet of material. Standard error is reported alongside the values. Reported values were provided by the manufacturer unless specified. GDL

In-plane conductivity (S m−1 ) MD

TGP-H-090 H2315-T30A SGL-10AA

16,646 7793 (7700) 5073 (5100) [6]

CD 12,417 5134 (5250) 2877 (3200) [6]

 1.34 1.518 (1.47) 1.76 (1.6) [6]

Fig. 5. Anisotropy and sheet resistance as a function of rotation angle.

agreement between the results obtained with the present method and the values reported in the literature for average in-plane, MD and CD conductivities seems to confirm that the square probe method is suitable and accurate for GDL materials.

4.2. Determination of conductivity tensor One of the main goals of this study was to determine the impact of IP fiber alignment in the GDL on the electrical conductivity tensor. Due to the possibility that FD and MD are not coincident, and knowing the impact this has on the components of the tensor, an attempt was made to determine the angle ˇ between the MD and FD of the material. This was accomplished by cutting a series of GDLs from a master sheet at different rotation angles and measuring the x and y conductivity values for each. According to Eq. (5)  and   will vary as the this means that the measured values of xx yy sample is rotated. The off-diagonal components will also vary as the sample is rotated; however, it is not possible to measure these  and directly. The off-diagonal components will be zero and the xx  values will be their respective maximum and minimum ( yy xx and  yy ) values only in a sample cut parallel to FD. By finding the sample in which this occurs, it is possible to infer ˇ from the sample rotation angle. To facilitate this analysis, it is convenient to define the anisotropy ratio between the conductivity values measured in the two orthogonal directions as : =

  yy xx =   yy xx

(18)

The anisotropy ratio  defined in this way is dependent on orientation angle ˛, and it is equal to the maximum anisotropy ratio  max of the material at ˛ = 0◦ . Fig. 5 shows experimentally measured log() values as a function of rotation angle  determined for 12 samples cut from a master sheet at 7.5◦ intervals between 0◦ and 90◦ . The most striking feature of the data shown in Fig. 5 is that the maximum anisotropy occurs in samples cut at a rotation angle  of 16◦ rather than 0◦ . Since the maximum anisotropy likely corresponds to the maximum fiber alignment in the sample, this reveals that the papermaking process led to variability in the in-plane fiber alignment. This has notable implications on the measurement of all tensorial transport properties.

672

D.R.P. Morris, J.T. Gostick / Electrochimica Acta 85 (2012) 665–673

The anisotropy ratio data in Fig. 5 can be empirically fit as a function of rotation angle using a cosine curve with a phase shift 2 = 0.979) as follows: (Radj log() = log(max )cos(2( − ˇ))

(19)

This simple description captures the two most critical features of the data. Firstly, the amplitude describes the maximum extent of anisotropy in the GDL sheet and this is given by  max ; secondly, the angle  at which this maximum amplitude occurs corresponds to the FD and this is given by the phase shift of the fit, or ˇ. The period of the fit was halved since log() is symmetric about 90◦ rather than 180◦ . This is because  is a ratio of two perpendicular measurements, thus rotating samples beyond 90◦ replicates data between 0◦ and 90◦ (i.e. log(()) = log(1/( + 90◦ ))). In fact, measurements beyond 45◦ were redundant since the data are symmetric about the -axis in Fig. 5, but they help to reduce experimental uncertainty by providing more points for the fitting of Eq. (19). The scatter of the data around the fitted waveform seen in Fig. 5 can be attributed to the fact that each sample was cut from a different location on the master sheet. An alternative way to model the data in Fig. 5 is to use the rotation dependent conductivity tensor formulation presented in Eq. (5). The anisotropy ratio  can be expressed as the ratio of the main diagonal elements of Eq. (5): (˛) =

 xx cos2 (˛) + yy sin2 (˛) xx =  yy yy cos2 (˛) + xx sin2 (˛)

(20)

The data in Fig. 5 can be fit using this equation with ˛ =  − ˇ, 2 = 0.979). At ˛ = 0◦ and the same value of ˇ = 16◦ was found (Radj the anisotropy ratio is maximal: (˛ = 0◦ ) =

xx = max yy

(21)

4.2.1. Calculation of the MD-aligned conductivity tensor From a practical perspective, the GDLs used in a PEMFC will likely be cut parallel to the MD, which in this case does not correspond to ˛ = 0◦ and thus the off-diagonal components of the tensor are non-zero. It is nevertheless critical that PEMFC simulations incorporate the full MD-aligned electrical conductivity tensor for such scenarios. As described in Section 2.1 the MD-aligned tensor can be determined from



(−ˇ) =

2

xx cos2 (−ˇ) +  sin (−ˇ)

(xx − yy )cos(−ˇ)sin(−ˇ)

(xx − yy )cos(−ˇ)sin(−ˇ)

yy cos2 (−ˇ) + xx sin (−ˇ)

2



(22)

where ˛ =  − ˇ and  = 0◦ for the MD. Before using Eq. (22) however, the intrinsic conductivity values ( xx and  yy ) and fiber orientation angle ˇ must be found as described in the previous section. This is accomplished by fitting Eq. (19) or (20) to at least 2 anisotropy ratio measurements to determine the orientation angle ˇ and the maximum anisotropy  max . The intrinsic conductivity components  xx and  yy are simply found by relating Eq. (21) with the measured average in-plane conductivity of the sample using the VDP method ( IP from Eq. (6)), which is independent of orientation angle: IP =



xx yy =



(max yy )yy = yy



max

(23)

Therefore,  yy is determined from Eq. (23), and then  xx can be found from Eq. (21). With this information in hand it is now possible using Eq. (22). to calculate the MD-aligned tensor  MD The above procedure was performed for data given in Fig. 5. The average in-plane conductivity of the samples was 14,309 S m−1 , calculated via Eq. (8). From Eq. (19),  max = 1.37, and from Eq. (23)

 yy = 12,466 S m−1 and  xx = 17,129 S m−1 . With ˇ = 16◦ and  = 0◦ , the MD-aligned tensor for TGP-H-090 are

 (˛ =  − ˇ = −16◦ ) = 

MD

=

16, 741

−1300

−1300

12, 856

 (24)

The errors incurred by neglecting the off-diagonal components of the conductivity tensor are not large in this case (≈3%), but this increases as the MD and FD differ and as the anisotropy ratio increases. These measurements were conducted on TGP-H-090, which has the least amount of IP anisotropy of the three GDLs used in this study, so errors may be more significant in other material types. 4.2.2. Effect of rotation angle on average in-plane conductivity Average in-plane conductivity is not a function of the rotation angle. This is illustrated by the fact that the determinant of the conductivity tensor is independent of the orientation angle ˛ as in Eq. (7), and thus remains constant. The sheet resistance, which is inversely proportional to in-plane conductivity, is presented as a function of rotation angle  in Fig. 5 and it is clearly independent of the rotation angle. No replicates were possible for these experiments. This is because the fiber alignment in a sheet of GDL material varies spatially due to random fluctuations in the sheet, despite the sizable samples used (12 cm2 ). Thus samples cut at the same rotation angle may have significantly different properties. To illustrate this dependency, four samples cut at  = 0◦ were taken from the four corners of another sheet of TGP-H-090 material (over 30 cm apart). The average anisotropy of the four samples is 1.44 with a standard deviation of 0.132 (9%). This degree of variance exemplifies that indeed the fiber alignment, and thus anisotropy, is somewhat spatially dependent. 4.2.3. Determination of conductivity tensor from a single sample To obtain the dataset shown in Fig. 5, 12 samples were required. According to the Nyquist sampling theorem, however, this characteristic anisotropy waveform can be determined a sheet of GDL by measuring the anisotropy ratio of only two samples at different rotation angles. This can be demonstrated using the present data by taking the MD data ( = 0◦ ) and data from any other single rotation ( = / 0◦ ) and using them alone to determine the fitting parameters of Eq. (19). It was found that the difference between the amplitude ( max ) and phase shift (ˇ) calculated for any two rotations varied from the overall curve on average by 6% and 15% respectively, confirming the feasibility of using of only two samples. In fact, it is actually possible to collect both necessary measurements from a single sample with the holder presented here. After measuring the anisotropy ratio for a given sample (the sample cut at 37.5◦ was used in this case), a new smaller square sample was obtained by cutting the corners from the original sample thus effecting a 45◦ sample rotation. The smaller sample was tested using a modified plunger to allow the electrodes to contact the corners, shown in Fig. 3(f) and (g). The conductivities in the x and y directions were calculated using the Montgomery method, and the anisotropy of the new smaller sample was determined. Therefore, two data points for the characteristic anisotropy waveform were generated and they were 45◦ apart. From these data, the parameters  max and ˇ were determined using Eq. (19). The parameters were in good agreement with the full data: the  max was 8% lower and ˇ was about 4.5◦ lower. Of course, these differences can be attributed to the inherent heterogeneity of a GDL sheet: fiber alignment is spatially dependent, as discussed above, and as such a moderate degree of error was expected. Calculating the MDaligned tensor from this information simply requires repeating the

D.R.P. Morris, J.T. Gostick / Electrochimica Acta 85 (2012) 665–673

procedure used to generate Eq. (24) with  = 0◦ and the measured values of  max = 1.41 and ˇ = 12.5◦ . In this case the result is:



( = 0◦ ) = 

MD

=

16, 953

−1057

−1057

12, 426



(25)

Symbols ˛ ˇ ˇ 

Using a single sample to determine the tensor saves effort, but also increases experimental uncertainty; performing this procedure on 2 or more samples may be desired, but then the heterogeneity of the GDL sheet would influence the results.

  

5. Conclusion

 

The in-plane electrical conductivities of different types of GDLs have been determined using the van der Pauw and Montgomery methods. The average in-plane conductivity results obtained with the two methods are in excellent agreement with reported values. The Montgomery method was further used to extract values of the direction-specific in-plane conductivity in two perpendicular directions. Samples cut at different angles from the same sheet of GDL material exhibited different anisotropies. Indeed, the anisotropy of the sample was related to the rotation angle by a cosine waveform. The maximum and minimum conductivity values and the intrinsic conductivity values of a GDL sheet were determined. All four in-plane components of the electrical conductivity tensor were determined for the sheet oriented in the MD. Maximum anisotropy occurred in a direction other than the MD, which suggests that fibers were oriented in that direction, rather than the MD. This finding suggests that other transport property tensors previously investigated should be reassessed to determine the impact of fiber alignment. Future work will investigate the effect of compression on the conductivity, which can be easily implemented with the sample holder designed for this work. Acknowledgments The authors acknowledge the support of the Automotive Fuel Cell Cooperation and the McGill University Faculty of Engineering S.U.R.E. program. The authors would also like to thank Mr. Mark Belchuk at Freudenberg FCCT and Mr. Wayne Triebold at Toray Industries for useful discussion and provision of data. Appendix A. Nomenclature

A E H I J L L R R Rs t V W

cross-sectional area (m2 ) electric field (V m−1 ) montgomery equation function electric dc current (A) current density (A m−2 ) length of the mapped isotropic specimen (m) length of the anisotropic specimen (m) resistance () rotation matrix sheet resistance () specimen thickness (m) electric potential (V) specimen width (m)



673

orientation angle; angle relative to the FD (◦ ) angle between the MD and FD (◦ ) angle between the x-axis of a sample and the FD (◦ ) rotation angle; angle relative to the edges of a GDL sheet (◦ ) anisotropy ratio electrical resistivity ( m) electrical resistivity tensor ( m) electrical conductivity (S m−1 ) electrical conductivity as a function of orientation angle (˛) (S m−1 ) electrical conductivity tensor (S m−1 )

Superscripts T transpose Subscripts max maximum IP (average) in-plane x, y in-plane tensor components through-plane tensor component z References [1] J.T. Gostick, M.W. Fowler, M.D. Pritzker, M.A. Ioannidis, L.M. Behra, Journal of Power Sources 162 (2006) 228. [2] R. Flückiger, S.A. Freunberger, D. Kramer, A. Wokaun, G.G. Scherer, F.N. Büchi, Electrochimica Acta 54 (2008) 551. [3] J.G. Pharoah, K. Karan, W. Sun, Journal of Power Sources 161 (2006) 214. [4] J. Kleemann, F. Finsterwalder, W. Tillmetz, Journal of Power Sources 190 (2009) 92. [5] K. Han, B.K. Hong, S.H. Kim, B.K. Ahn, T.W. Lim, International Journal of Hydrogen Energy 35 (2010) 12317. [6] M.S. Ismail, T. Damjanovic, D.B. Ingham, M. Pourkashanian, A. Westwood, Journal of Power Sources 195 (2010) 2700. [7] T. Naito, in: R.E. Mark, C. Habeger, J. Borch, M.B. Lyne (Eds.), Handbook of Physical Testing of Paper, vol. 1, CRC Press, New York, NY, 2001. [8] K.-H. Kim, K.-Y. Lee, H.-J. Kim, E. Cho, S.-Y. Lee, T.-H. Lim, S.P. Yoon, I.C. Hwang, J.H. Jang, International Journal of Hydrogen Energy 35 (2010) 2119. [9] J.G. Webster, Electrical Measurement, Signal Processing, and Displays, CRC Press, Boca Raton, FL, 2003. [10] M.V. Williams, E. Begg, L. Bonville, H.R. Kunz, J.M. Fenton, Journal of the Electrochemical Society 151 (2004) A1173. [11] D.W. Koon, W.K. Chan, Review of Scientific Instruments 69 (1998) 4218. [12] B.J. Horkstra, L.J. Van Der Pauw, N.V. Philips, Journal of Electronics and Control 7 (1959) 169. [13] L.J. Van Der Pauw, Philips Research Reports 13 (1958) 1. [14] D.W. Koon, C.J. Knickerbocker, Review of Scientific Instruments 63 (1992) 207. [15] E.J. Zimney, G.H.B. Dommett, R.S. Ruoff, D.A. Dikin, Measurement Science & Technology 18 (2007) 2067. [16] C.A.M. dos Santos, A. de Campos, M.S. da Luz, B.D. White, J.J. Neumeier, B.S. de Lima, C.Y. Shigue, Journal of Applied Physics 110 (2011) 083703. [17] B.D. Cunningham, J.H. Huang, D.G. Baird, Journal of Power Sources 165 (2007) 764. [18] T.H. Zhou, H.T. Liu, Journal of Power Sources 161 (2006) 444. [19] N. Athanasopoulos, V. Kostopoulos, Composites Part B: Engineering 42 (2011) 1578. [20] W. Jones, N.H. March, Theoretical Solid State Physics: Non-equilibrium and Disorder, Dover Publications Inc., New York, NY, 1986. [21] W.L.V. Price, Journal of Physics D: Applied Physics 5 (1972) 1127. [22] L.J. Van Der Pauw, Philips Research Reports 16 (1961) 187. [23] D.W. Koon, Review of Scientific Instruments 60 (1989) 271. [24] H.C. Montgomery, Journal of Applied Physics 42 (1971) 2971.