Patent Publication Number: US-2010109651-A1

Title: Device for conductivity measurement in a controlled environment and method thereof

Description:
This application claims priority based on the U.S. Provisional Application Ser. No. 61/080,018 filed on Jul. 11, 2008, which is incorporated herein by reference in its entirety. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention is related to a device for measuring the conductivity of materials in a controlled environment and a method thereof. It finds particular application in conjunction with the measurement of a sample such as powder in an environmental medium such as an air/water vapor mixture, and will be described with particular reference thereto. However, it is to be appreciated that the present invention is also amenable to other like applications. 
     In response to requirements of the fuel cell industry, numerous materials have been studied in recent years as low-humidity proton conductors in different temperature regions. These include acid surface functionalized nanostructures, heteropolyacids, phosphate and/or silicate glasses, hydrogen sulfates and phosphates, and ceramic doped double oxides. In order to make fuel cell membranes, these materials are often converted into composites with better mechanical properties. 
     Nevertheless, the conductivity of single phase non-castable ionic materials often needs to be measured in the powder form before the best candidates for the composite development can be selected. Therefore, the intrinsic conductivity of pure powdered materials is of a significant interest. 
     The conductivity of powders is usually measured using ac impedance after compressing the powder into a pellet and depositing electrodes on or pressing electrodes against the two flat sides of the pellet. This design does not allow for a quick equilibration between the water/oxygen content in the sample and that in the surrounding, and complicates the analysis of the sample&#39;s impedance spectra due to the presence of two interfacial capacitances. The latter problem is exacerbated by the small thickness of a typical pellet, and, thus, smaller resistance and higher ac frequencies needed for accurate resistance measurements. Also, this approach is applicable only to materials that form free-standing pellets which do not disintegrate due to the Rehbinder effect in the studied range of humidity. Reported in Holmberg, B. A.; Yan, Y. S.  Journal of the Electrochemical Society  2006, 153, A146-A149 (hereinafter “Holmberg and Van”) was a new cell design that allows for conductivity measurements on a loose powder placed into a cylindrical hole and squeezed between two fixed rod electrodes. This design, however, does not resolve three problems including the interfacial impedance, the humidity equilibration and the loss of contact between the sample and the electrode upon sample shrinking. 
     Advantageously, the present invention provides an improvement over Holmberg and Yan&#39;s design by using, for example, four electrodes, a porous cell material and spring-loads on the current collector electrodes to overcome the three aforementioned problems. The invention may be used in for example measuring conductivity of powders under variable humidity and temperature. 
     BRIEF DESCRIPTION OF THE INVENTION 
     One aspect of the invention is to provide a device for measuring the conductivity of a sample, which comprises:
         (i) a sample comprising a first material,   (ii) an environmental medium comprising a controlled level of a second material, and   (iii) a separator;   wherein the separator isolates the sample from the environmental medium;   the separator substantially prevents the first material from migrating into the environmental medium; and   the separator allows the second material to migrate between the environmental medium and the sample.       

     Another aspect of the invention provides a method of measuring the conductivity of a sample, for example, the conductivity of a deformable sample as a function of humidity and/or temperature, which comprises a step of using the device comprising:
         (i) a sample comprising a first material,   (ii) an environmental medium comprising a controlled level of a second material, and   (iii) a separator;   wherein the separator isolates the sample from the environmental medium;   the separator substantially prevents the first material from migrating into the environmental medium; and   the separator allows the second material to migrate between the environmental medium and the sample.       

    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows the “4-point probe circuit” as embodied in a device according to an embodiment of the invention; 
         FIGS. 2 and 3  show a device comprising a sample holder made from porous zirconium phosphate according to an embodiment of the invention; 
         FIG. 4  shows the dependence of the conductivity of a powder sample on pressure between two outer electrodes; 
         FIG. 5  shows the conductivity of a powder sample measured with two and four electrodes at different ac frequencies; 
         FIG. 6  shows the dependence of the phase shift between current and voltage on ac frequency in impedance measurements with two and four electrodes; and 
         FIG. 7  shows conductivities of a film sample and a powder sample both of which were prepared from the same material as functions of relative humidity. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     As the separator allows the second material to migrate between the environmental medium and the sample, there is an equilibrium between the second material in the environmental medium and that in the sample, such as water equilibrium. In exemplary embodiments, the environmental medium is in a phase selected from gas phase, liquid phase, solid phase, a phase there-in-between, and any combination thereof. When the environmental medium is a liquid phase, it can be selected from aqueous phase, non-aqueous phase, and any combination thereof. 
     In exemplary embodiments, the second material is selected from ions such as H +  and OH − ; neutral molecules such as water, acids, bases, and solvents; complexing agents; and any combination thereof. 
     In exemplary embodiments, the environmental medium is a gas phase and the second material is selected from inert or reactive gases and vapors such as water vapor, hydrogen, oxygen, and any combination thereof. 
     In exemplary embodiments, the environmental medium is in gas phase; the second material is water vapor; and the level of the water vapor in the gas phase is controlled by a humidity controller, with which the invention provides a method of measuring the conductivity of a sample as a function of humidity. A suitable humidity controller may be selected from a saturated salt solution, a bubble humidifier, a membrane humidifier, and any combination thereof. 
     In exemplary embodiments, the environmental medium is in gas phase; and the separator is a sample holder made of a material permeable to the second material such as water vapor. 
     Examples of material permeable to the second material include, but are not limited to, ceramic materials, plastic materials, metallic materials, porous materials such as zirconium phosphate and borosilicate glass, and any combination thereof. For example, the utilization of a porous sample holder allows for easy control of the water activity in the sample. 
     In exemplary embodiments, the invention provides a device for measuring the conductivity of a sample, which comprises:
         (i) a sample comprising a first material,   (ii) an environmental medium comprising a controlled level of a second material,   (iii) a separator, and   (iv) at least two electrodes;   wherein the separator isolates the sample from the environmental medium;   the separator substantially prevents the first material from migrating into the environmental medium;   the separator allows the second material to migrate between the environmental medium and the sample; and   the pressure controller controls the contact pressure at the interface between the sample and the at least two electrodes.       

     In exemplary embodiments, the device of the invention comprises at least four electrodes, wherein at least two of the at least four electrodes are inner electrodes located within the sample, and at least two of the at least four electrodes are outer electrodes located on the sides of the sample. In a specific embodiment, the device of the invention embodies a circuit, a “4-point probe circuit” as shown in  FIG. 1 . 
     With reference to  FIG. 1 , two inner electrodes  301  and  302  and two outer electrodes  201  and  202  can be viewed to separate the sample into three segments  110 ,  120 , and  130 . As a part of the measurement of the sample segment  120 , resistance R 120  of the sample segment  120  can be calculated by measuring the voltage V between the two inner electrodes  301  and  302 , and the current I between the two outer electrodes  201  and  202 , according to the equation R 120 =V/I. 
     In the “4-point probe circuit” as shown in  FIG. 1 , every electrode/sample such as electrolyte interface can be approximated for the purposes of its electric properties as a parallel combination of a double layer capacitance, Cd, and a faradaic resistance, Rf. The Cd is due to the fact that neither electrons nor ions can cross the interface and they accumulate on the surface of the respective materials. Rf is due to electrochemical reaction, which allows for the flow of electrons across the interface. Rf is large for small voltages (no electrode reactions) and Rf is small for high voltages (electrolysis). By measuring the current between  201  and  202  and the voltage between  301  and  302 , the impedance of  120  can be obtained. For all practical purposes, the current flowing between two inner electrodes  301  and  302  is zero. The currents through  110 ,  120 , and  130  are the same. 
     The invention thus can provide a simple method to measure, for example, the conductivity of powders using four-electrode ac impedance spectroscopy, which is not affected by the dimensional changes of the sample. The problems of interfacial impedance and of maintaining constant inter-electrode distance are resolved by the 4 point nature of the cell. One reason for using AC is to reduce the “battery” effect, i.e. the voltage difference between the two inner electrodes which is due to different chemical environment, temperature and pressure differences. 
     In exemplary embodiments, the pressure controller controls both the contact pressure at the interface between the sample and the outer electrodes; and the stress of the sample applied on the inner electrodes. For example, the pressure controller may be selected from spring, elastomeric member, hydraulic pump, pneumatic pump, and any combination thereof. 
     In an example, the invention takes advantage of a spring-loaded contact electrode, which prevents the loss of contact upon sample shrinking. The implementation of four-electrode AC impedance measurements effectively reduces the inaccuracy due to interfacial capacitance of the probes. 
     In exemplary embodiments, the device of the invention comprises a distance detector that measures the distance between the two inner electrodes for the final acquisition of conductivity value. Suitable distance detectors may be selected from X-ray Computed Axial Tomography (CAT), X-ray Projection Photography, Magnetic Resonance Imaging, and any combination thereof. 
     In exemplary embodiments, the invention provides a device for measuring the conductivity of a sample, which comprises:
         (i) a sample comprising a first material,   (ii) an environmental medium comprising a controlled level of a second material,   (iii) a separator, and   (iv) a temperature controller;   wherein the separator isolates the sample from the environmental medium;   the separator substantially prevents the first material from migrating into the environmental medium;   the separator allows the second material to migrate between the environmental medium and the sample; and   the temperature controller controls the temperature of the sample and of the environmental medium.       

     In exemplary embodiments, the invention provides a method of measuring the conductivity of a sample as a function of temperature, which comprises a step of using such a device. As such, the invention is particularly useful for low-humidity proton conductors for different temperature regions. 
     Examples of suitable temperature controllers include, but are not limited to, electrical heating tape, sand bath, oil bath, oven, and any combination thereof. 
     The conductivity to be measured may be selected from electronic conductivity and ionic conductivity such as cationic conductivity, anionic conductivity, and any combination thereof. In exemplary embodiments, the cationic conductivity is selected from conductivities of H + , Ag + , Li + , Cd 2+ , Hg 2+ , and any combination thereof; and the anionic conductivity is selected from conductivities of O 2− , F − , S 2− , and any combination thereof. In preferred embodiments, the conductivity is protonic (H + ) conductivity. 
     In exemplary embodiments, the conductivity is AC conductivity or DC conductivity. 
     The present invention may be used on various samples such as powder, monolith, ceramic, metal, elastomeric material, gel, plastic, liquid, and any combination thereof. In preferred embodiments, the sample comprises a proton conductor. Examples of proton conductors include, but are not limited to, ionic polymers, sulfonated polymers, acid surface functionalized nanostructures, ionic liquids, acid-doped polymers, heteropolyacids, phosphate and/or silicate glasses, hydrogen sulfates and phosphates, ceramic doped double oxides, and any combination thereof. In preferred embodiments, these proton conductors can be in powder form. 
     The invention can be used to measure the conductivities of powdered samples. Because the powder is exposed to the environment, the invention can be used to measure the conductivity as a function of temperature and relative humidity of the environment. 
     With the present invention, protonic conductivity of powders can be measured under variable and controlled humidity conditions using a porous holder for the sample. 
     EXAMPLE 1 
     A Test Device 
     This example comprises a rectangular box with cylindrical holes drilled in porous zirconium phosphate. One large hole is drilled lengthwise through the box. A sample such as a powder sample was put into this cylindrical channel. Two outer electrodes with a pressure controller such as two spring loaded metal electrodes are inserted into the openings at the ends and pushed against the powder with constant pressure. Two smaller holes are drilled perpendicular in the middle of the box, intersecting the large channel. Inner electrodes such as Pt wires are inserted into these smaller channels and penetrate through the entire width. 
     The device consists of three pieces as shown in  FIGS. 2 and 3 . The top  401  and bottom  402  pieces are identical in shape and made of e.g. polyetherimide (PEI, ULTEM 1000, McMaster #7612K15). A separator such as a sample holder e.g. center piece  403  is made from porous zirconium phosphate (Aremco #502-1550, with μm range pore size and 30% porosity). Porous borosilicate glass (Ace Glass #D129888, porosity E: 4-8 μm) as the material for the central piece can also be used and test proved it was successful. Reference electrode wire holders  404  and  405  are shown disassembled for side view only. 
     The powder sample was loaded into one of the two vertical straight-through holes  501  in the central piece. The central piece is placed between the cavities made in the plastic pieces, and the whole assembly is held together by two threaded stainless steel rods  531  that go through the outer holes  530  in the plastic pieces. 
     The powder in the hole  501  in the central piece  403  is compressed between two Pt-rods  600  (for example, 1.50 mm diameter, 15.0 mm long, Tanaka Kikinzoku Kogyo) that enter each hole  501  through inner holes  604  in the plastic pieces. The pressure on each rod is produced by a compression stainless steel spring  605  (McMaster #9662K14, 0.125″ OD, 8.0 mm free length), one of each which is placed into the inner holes  604  of the plastic pieces as shown in  FIGS. 2 and 3 . The pressure was applied to the spring  605  using a screw  606  (e.g. UTS 8-32, 1″ long) threaded into hole  604 . To assure a better mechanical and electrical contact between the spring  605  and the rod  531 , a stainless steel ball bearing  607  (e.g. ⅛″ inch diameter, McMaster #9291K14) is placed in between. Since for the torques used in this work the spring is under full compression, a spring with a larger Hooke&#39;s coefficient may be used. This however will require larger diameter inner holes in the plastic pieces. 
     The two Pt rods  600  inserted in holes  501  through holes  604  are used as the outer electrodes in the four-electrode impedance cell. The inner electrodes are two Pt wires  503  (e.g. 0.15 mm thick) that go through horizontal holes  502  in the center piece for example 5.0 mm apart. Each wire  503  is held in place by two polyetherimide holders ( 404  and  405 ) attached with screws (e.g. UTS 6-32 0.7″ long) through holes ( 705  and  706 ) to the two opposite sides on the corresponding plastic piece ( 401  for the upper wire and  402  for the lower wire). Each back holder  404  has a fixed screw  707  to which a Pt wire  503  is attached. From the point of attachment the wire  503  goes vertically into a wire hole  708  where it bends horizontally and directed through a centerpiece hole  502 . On the other side of the center piece the wire  503  is attached through hole  710  to a machine head  709  on the front holder  405 . Two enantiomorphs of both the front and back holders are used at the opposite corners of the center piece  403 . 
     There are no specific limitations on the dimensions of the device. For example, vertical straight-through holes  501  in the central piece can have a diameter of from about 0.1 mm to about 10 mm such as 1.50 mm diameter. The threaded stainless steel rods  531  can have a length of from about 1 cm to about 100 cm, such as 10.0 cm. The distance from an edge of pieces  401  and  402  to the center of the outer holes  530  can be from 0.1 mm to 1000 mm, such as 7.8 mm. Pt-rods  600  can have a diameter of from about 0.1 mm to about 10 mm, such as 1.50 mm diameter and a length of from about 1 mm to about 100 mm, such as 15.0 mm. Similarly, top and bottom pieces  401  and  402  can have a length of from about 5 mm to about 1000 mm, such as 40 mm, a thickness of from about 1 mm to 1000 mm, such as 25 mm and a width of from about 1 mm to 1000 mm, such as 70 mm. The distance from the edge of pieces  401  and  402  to the center of inner holes  604  can be from about 1 mm to about 1000 mm, such as 25 mm. The hole  604  is composed of two openings, on a top of each other, and can be from 1 mm to 1000 mm, such as 27.5 mm. The larger opening accommodates a screw, a spring and a stainless steel ball. This larger opening can have a length of from 1 to 1000 mm, such as 20 mm, and can have a diameter of from 0.1 mm to 1000 mm, such as 3.5 mm. The length of the threaded part of the larger opening can be from 1 mm to 1000 mm such as 10 mm. The smaller opening of  604  can have a length of from 1 to 1000 mm such as 7.5 mm and a diameter from 0.1 mm to 1000 mm, such as 1.5 mm. Similarly, central piece  403  can have a length from 1 mm to 1000 mm, such as 40 mm, a thickness of from 0.1 mm to 1000 mm, such as 10 mm and a height of from 1 mm to 1000 mm, such as 25 mm. The vertical straight-through holes  501  can have a diameter of from 0.01 mm to 1000 mm, such as 1.5 mm. Reference electrodes or Pt-wires can be 0.1 to 1000 mm apart, such as 5 mm, and can have a diameter of from 0.01 mm to 100 mm, such as 0.25 mm. Polyetherimide holders  404  and  405  can have dimensions of from 0.1×0.1×0.1 mm to 1000×1000×1000 mm, such as 19×19×12.7. Similarly, holes  706  and  705  can have diameters of from 0.1 mm to 1000 mm, such as 4 mm. The distance between the edges of  404  and  405  holders to the centers of the  706  and  705  holes can be from 0.1 mm to 1000 mm, such as 4 mm. The diameter of the  708  hole can be from 0.01 mm to 100 mm, such as 1 mm. Likewise, the hole which accommodates a screw  707  can have a diameter from 0.1 mm to 1000 mm, such as 3 mm. The distance between the centers of  706  and  705  holes can be from 0.1 mm to 1000 mm, such as 9 mm. 
     EXAMPLE 2 
     Conductivity Measurement 
     Using the device of Example 1, a comparison between the conductivities of an extruded Nafion film and a Nafion powder at different humidities has been made. The data shows that the conductivity of a powder is several times lower than the conductivity of a film, which is likely due to the porous nature and intergrain resistances in the powder. 
     The conductivity of the Nafion film was measured using a four-electrode conductivity clamp (BekkTech BT-110). Both the powder cell and the film clamp were placed into a 1 L custom-made high-pressure environmental chamber (Parr, Moline, Ill.) equipped with a humidity-temperature sensor (Vaisala HMT330), electrical feedthroughs and gas inlet/outlet valves. The chamber temperature can be maintained using a temperature controller (Omega CN 8200) connected to an electrical heating tape (Barnstead international #BIH101-060) and several thermistors positioned inside the chamber. All data reported was obtained at room temperature (23±0.5° C.). The relative humidity in the chamber was controlled using saturated salt solutions placed in a glass beaker on the bottom of the chamber and it was stable within 0.2%. Impedance data was acquired using a Solartron 1287 Electrochemical Interface and 1255B Frequency Response Analyzer under ZPlot software (Scribner Associates, Inc.). The frequency of the ac perturbation was step-scanned in the region from 1 MHz to 1 Hz with the amplitude of 100 mV. 
     The powder Nafion was obtained by grinding a wet film of Nafion 950 (equivalent weight EW=950 g/mol H + ) (IonPower Inc.) in liquid nitrogen using a stainless steel mortar and a glass pestle. Both powder and film Nafion samples were boiled in 3% H 2 O 2  for 4 hours, in deionized water (18.2 MΩ·cm conductivity) for 2 hours, in 1.0 M HClO 4  for 2 hours and then again in deionized water 3 times. 
     Prior to a powder sample loading, the two Pt wires  503  and the first Pt rod  600  were inserted into one of the center piece holes  501  with the rod touching the lower wire from below. The powder was poured into the vertically held hole via the upper opening until it covered the upper wire (5-10 mg). The second rod  600  was inserted into the upper opening and the central piece was turned upside down and the first rod  600  as taken out. Another portion of the powder (2-5 mg) was poured in and the powder in the hole was finger-pressed between the rods. The process of pouring and compressing was repeated until the distance between the two Pt rods reached 10-15 mm (4-8 times). If needed, the second hole  501  in the center piece was filled in the same way. 
     After the filling process, the center piece  403  was sandwiched between the top  401  and bottom  402  fixture pieces. The two treated stainless steel rods  531  were inserted into side opening  530  and tightened with two nuts to hold the three fixture pieces together. A ball bearing  607 , a spring  605 , a nut, a washer and a screw  606  were inserted in the hole  604  in the top piece and finger-tightened; then the procedure was repeated for the bottom part. After this, the screws  606  were tightened to the desired torque using a torque screwdriver. 
     After assembly, the two Pt wires  503  were extended through holes  708  and  710  of the back  404  and front  405  holders, respectively. The holders were screwed though smooth holes  706  to threaded holes  705  of the top  401  and bottom  402  pieces. The two Pt wires  503  were wrapped around screws  707  of the back holders  404 . Then, the other end of each wire  503  was wrapped around screw  709  of the front holder  405 . The distance between Pt rods in the assembled cell was measured, if needed, using X-ray Computed Axial Tomography (CAT). 
     The cell with powder Nafion and BekkTech Conductivity Clamp with film Nafion and saturated salt solution was loaded into the pressure vessel. The vessel was sealed and vacuum was applied to speed up the diffusion of water vapor from the salt solution to the samples. The humidity sensor typically showed stable relative humidity values within 5 hours after the vacuum valve was closed. In some experiments a non-flammable mixture of 4% v/v H 2  in He was used inside the chamber. The H 2  stabilized the dc potential on the reference Pt electrodes  503  and the He increased the heat conductivity inside the chamber, thus, yielding more uniform temperature distribution. 
     EXAMPLE 3 
     Effect of the Applied Stress 
     Holmberg and Yan reported a strong effect of the torque applied to rods on the conductivity of their ceramic powder squeezed between the rods. This may be understood as the resulting pressure increasing the contact area and decreasing the spacing between the powder particles. Using the device of Example 1, this effect was investigated for Nafion powder. 
     The sample was loaded into a test fixture hole  501 , the screws  606  were tightened to the lowest desired torque using a torque screwdriver, and a full impedance spectrum was acquired. After this, the torque was increased and a new impedance spectrum was recorded. The cycle was repeated until conductivity became independent of torque. In order to determine the compressive stress on the sample, P, the minimal torque required to loosen the screw, T backward , was measured in separate experiments with the washer of screw  606  removed. 
     The stress of the sample powder was estimated using the following relationships between T forward  , T backward , and the torques due to static (i=s) or dynamic (i=d) friction of the screw, T i   friction , and due to compressive stress of the sample, T compr : 
         T   forward   =T   d   friction   T   compr   (1) 
         T   backward   =T   s   friction   −T   compr   (2) 
     Since it was observed that the value of T forward  is much higher than the value of T backward  in all cases, the following equation applies: 
       T compr ≈T s   friction   (3) 
     Assuming T s   friction ≈T d   friction , one obtains: 
         T   compr   ≈T   forward /2  (4) 
     For an infinitely small increment of the rotation angle eq. (1) yields: 
         T   forward   dφ=T   d   friction   dφ+T   compr   dφ   (5) 
     which based on the definition of stress, P, and eq. (4) reduces to 
         T   forward   dφ ≈2  PA δ( dφ/ 2π)  (6) 
     where δ is the screw thread pitch ( 1/32″) and A is the cross-sectional area of the powder column with radius r. Therefore, 
         P≈T   forward /(δ* r   2 )  (7) 
     Then, using the relationship [N·m]=141.6 [inch·oz·force], it can be concluded that 
         P [MPa]≈ 15.8  T   forward  [inch·oz·force]  (8) 
     Holmberg and Yan used a different approach to calculating the stress from the torque, based on the estimation of the friction force via a tabulated friction coefficient and making no distinction between dynamic and static friction. They obtained values of stress of about half of the values calculated based on eq. (8). 
     A typical dependence of powder conductivity on applied stress is shown in  FIG. 4 . Specifically,  FIG. 4  shows the dependence of the conductivity of Nafion 950 powder on pressure between Pt rod electrodes. Experimental conditions were 23° C., 17.3% RH, and four-electrode measurements. As expected, the conductivity of the Nafion powder increased with applied pressure until or up to about ca. 200 MPa and remains constant at higher pressures up to at least 350 MPa. This behavior was reproducible with fresh powder samples at all humidities studied. All subsequent data was acquired in the plateau region, i.e. at 300 MPa. 
     EXAMPLE 3 
     Comparison Between Two- and Four-Electrode Conductivity Measurements 
     The double-layer capacitance at the interface between an electron conductor and an ion conductor imposes restrictions on the ranges of conductivities and frequencies that can be accurately measured using ac impedance spectroscopy. Increasing the ac frequency reduces the effect of the capacitance on the impedance, but in practice the effect of parasitic impedances and of the finite rise time of operational amplifiers makes such measurements inaccurate as can be seen from  FIGS. 5 and 6 .  FIG. 5  shows the conductivity of Nafion 950 powder measured with two (d=20. mm thickness, open circles) and four (d=5.0 mm thickness, solid squares) electrodes at different ac frequencies under experimental conditions of 23° C., 32.1% RH, 300 MPa. and A=7.065 mm 2 .  FIG. 6  shows the dependence of the phase shift between current and voltage on ac frequency in impedance measurements with two (open circles) and four (solid squares) electrodes under experimental conditions of 23° C., 32.1% RH, and 300 MPa. 
     The resistance of the sample can be increased (thus shifting the RC transition frequency to lower values) by using thicker samples. This approach works reasonably well for Nafion sample also as can be seen in  FIGS. 5 and 6  for the two-electrode data, but it may be of little use for samples with higher conductivity. i.e. 0.1 S/cm such as for Nafion under full hydration. 
     Even at 32.1% RH for Nafion, the phase shift and the modulus of the two-electrode impedance deviate from a pure resistive behavior below 300 Hz as shown in  FIGS. 5 and 6  (open circles), apparently due to the presence of a serial double-layer capacitance combined with the phase-error of the lock-in amplifier (the latter is noticeable below 10 Hz in  FIG. 5 ). The use of a four-electrode arrangement allows for extension of distortion-free measurements to lower frequencies ( FIGS. 3 and 4 , solid squares). All conductivity, σ, data were calculated from the real part of the measured four-electrode impedance at a frequency that shows the smallest modulus of the phase shift (typically in the 80-2000 Hz range). 
     EXAMPLE 4 
     Comparison Between Powder and Film Conductivities at Different Humidities 
       FIG. 7  shows conductivities of film and powder samples prepared from the same piece of a commercial Nafion 950 membrane as a function of relative humidity. More specifically,  FIG. 7  shows the dependence of the conductivity of Nafion with EW=950 powder (solid squares) and film (open circles) on relative humidity. Experimental conditions: 23° C., 300 MPa, four-electrode measurement. Also shown are literature data for films of Nafion with EW=1100: in-plane at 20-80° C. (open up triangles), through-plane at 20° C. (filled down triangles), and through-plane at 80° C. (filled up triangles) (see Kreuer, K. D.; Schuster, M.; Obliers, B.; Diat, O.; Traub, U.; Fuchs, A.; Klock, U.; Paddison, S. J.; Maier, J.  Journal of Power Sources  2008, 178, 499-509). 
     The values for the film are close to the previously reported values for through-plane conductivity of Nafion with EW of 1100 in the range 20-80° C. (See Lin, J.-C.; Kunz, H. R.; Fenton, J. A. In  Handbook of Fuel Cells: Fundamentals, Technology and Applications;  Vielstich, W., Lamm, A., Gasteiger, H., Eds.; Wiley, 2003; Vol. 3. Ch. 36; pp 456-463; and Lepiller, C.; Gauthier, V.; Gaudet, J.; Pereira, A.; Lefevre, M.; Guay, D.; Hitchcock, A.  Journal of the Electrochemical Society  2008, 155, B70-B78). On the other hand, the conductivity of the powder is 3.5-5.5 times lower than the conductivity of the film under the same conditions and closer to the values reported recently for through-plane conductivity of Nafion 1100 at 20° C., as reported in Marechal, M.; Souquet, J. L.; Guindet, J.; Sanchez, J. Y.  Electrochemistry Communications  2007, 9, 1023-1028. Three effects can contribute to the lower conductivity of the powder: anisotropy of Nafion conductivity, intergrain resistance, and voids/tortuosity in the powder structure. The latter two are also responsible for the pressure dependence of the powder conductivity. 
     The exemplary embodiments have been described with reference to the preferred embodiments. Obviously, modifications and alterations will occur to others upon reading and understanding the preceding detailed description. It is intended that the exemplary embodiment be construed as including all such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.