JOURNAL OF APPLIED PHYSICS

VOLUME 96, NUMBER 6

15 SEPTEMBER 2004

Quantitative impedance measurement using atomic force microscopy Ryan O’Hayre Department of Materials Science and Engineering, Rapid Prototyping Laboratory, Stanford University, Building 530, Room 226, Stanford, California 94305-3030

Gang Feng Department of Materials Science and Engineering, Stanford University, Building 550, Stanford, California 94305

William D. Nix Department of Materials Science and Engineering, Stanford University, Building 550, Stanford, California 94305

Fritz B. Prinz Department of Mechanical Engineering, and Department of Materials Science and Engineering, Rapid Prototyping Laboratory, Stanford University, Building 530, Room 226, Stanford, California 94305-3030

(Received 16 March 2004; accepted 9 June 2004) Obtaining quantitative electrical information with scanning probe microscopy techniques poses a significant challenge since the nature of the probe/sample contact is frequently unkown. For example, obtaining quantitative kinetic data from the recently developed atomic force microscopy (AFM) impedance technique requires normalization by the probe/sample contact area. In this paper, a methodology is proposed that enables the extraction of quantitative information from the AFM impedance technique. This methodology applies results from nanoindentation experiments and contact mechanics theory to characterize AFM probe contacts. Using these results, probe/sample contact forces (which can be accurately measured in the AFM) may be converted into probe/sample contact area estimates. These contact area estimates, when included in model of the probe/sample contact, enable the extraction of quantitative data. This methodology is applied to the recently developed AFM impedance measurement technique, enabling a quantitative study of the oxygen reduction reaction (ORR) at nanometer length scales. Using the AFM impedance system, kinetic data for the (ORR) at nanoscale Platinum/Nafion contacts is extracted. The kinetic data obtained from the AFM impedance technique match previous bulk measurements—affirming the technique’s quantitative potential. © 2004 American Institute of Physics. [DOI: 10.1063/1.1778217] I. INTRODUCTION

In this paper, a methodology for the extraction of quantitative data from the AFM impedance technique is developed. While this work focuses specifically on application to the AFM impedance technique, it may be easily extended to a variety of other local-probe measurements. The methodology depends critically on the ability to quantitaively characterize the AFM-tip/sample contact. Concepts and results from the fields of nanoindentation and contact mechanics provides this critical link. Using results from nanoindentation experiments, tip/sample contact forces measured in the AFM can be converted into tip/sample contact area estimates. A model for the tip/sample contact is then proposed which allows quantitative assignment of the impedance results based on the contact area estimates. The methodology is validated by AFM impedance studies of the oxygen reduction reaction (ORR) at nanoscale platinum/Nafion contacts. The quantitative kinetic data obtained from the AFM impedance technique is shown to match well with previous bulk measurements. Furthermore, measurements acquired over a wide range of tip/sample forces show good linearity with little scatter; after correcting for the estimated tip/sample contact areas, they all produce

Recently, several research groups have developed impedance microscopy systems that allow highly localized ac measurements to be acquired at the submicron length scale.1–5 So far these systems have been used mainly in qualitative imaging modes to produce two-dimensional (2D) maps or images of impedance variations across sample surfaces. The technique has been applied to visualize electronic conductors, electroceramics, and ionic conductors with submicron resolution. Nanometer scale visualization and measurement of impedance promises to be a valuable tool for a wide variety of fields, including solid state ionics, semiconductors, coatings and corrosion research, and battery and fuel cell systems. Unfortunately, the technique is currently limited by an inability to acquire quantitative impedance information. More generally, many other local-probe techniques currently suffer from the same limitation. Extracting quantitative information from scanning spreading resistance microscopy,6–9 conductive or current sensing atomic force microscopy (AFM), 10–12 tunneling-AFM13,14 (TUNA), and scanning capacitance microscopy15–17 has also proved difficult. 0021-8979/2004/96(6)/3540/10/$22.00

3540

© 2004 American Institute of Physics

Downloaded 14 Sep 2004 to 203.197.98.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

O’Hayre et al.

J. Appl. Phys., Vol. 96, No. 6, 15 September 2004

FIG. 1. AFM impedance measurement. (a) General concept. Impedance is measured between a local probe (the AFM tip) and a bulk electrode. A significant spreading resistance contribution at the AFM tip/sample contact point ensures local characterization. (Shown schematically by the hemispherical lines.) (b) Detail of the Pt/Nafion impedance experiment. ORR kinetics at the interface between a Pt-coated AFM tip and a Nafion electrolyte membrane are measured using AFM impedance. Electrochemical bias is applied relative to the bulk bottom electrode (a reversible hydrogen electrode) which is hermetically sealed and supplied with hydrogen gas.

approximately the same kinetic result with a small standard deviation—illustrating the technique’s capability to achieve reliable quantitative results.

3541

FIG. 2. Simple equivalent circuit model of the tip/sample contact. Rtip represents the resistance contribution from the tip, Rcont accounts for the tip/ sample interfacial resistance, which may include Schottky or Faradaic effects if the contact is nonohmic. C represents the tip/sample capacitance. RSR accounts for the spreading resistance associated with a small volume of material just beyond the tip/sample contact.

the following section, a model for the tip/sample contact is introduced which facilitates this understanding. B. Tip/sample contact model

II. METHODOLOGY A. AFM impedance measurement

A general schematic of the AFM-impedance concept is shown in Fig. 1(a). The system is constructed from a commercially available AFM coupled to an electrochemical measurement instrument. The AFM unit is equipped with a conductive tip, which serves as the probe electrode for the desired electrical/electrochemical measurements. (In addition to impedance, the system can be used to perform other measurements). The signal measured by the system depends critically on the nature of the tip/sample contact. For a homogeneous sample, the spreading resistance associated with a small volume of material near the tip/sample contact point often dominates the total impedance response of the system. Assuming a circular contact of radius r between the tip and sample, this spreading resistance 共RSR兲 varies with 1 / r:18 RSR =

␳ , 4r

共1兲

where ␳ is the resistivity of the sample. In electrochemically active systems, the impedance response of the tip/sample contact may scale with the tip/sample contact area. For example, Faradaic resistance and double layer capacitance scale with the electrode/electrolyte contact area: Rf =

R0f , Arxn

共2兲 共3兲

Cd = C0dAcontact , R0f ⫽specific

Faradaic rewhere R f ⫽Faradaic resistance 共⍀兲; sistance 共⍀ cm2兲; Arxn⫽the available reaction surface area 共cm2兲; Cd⫽double layer capacitance (F); C0d⫽specific interfacial capacitance 共F / cm2兲; and Acontract⫽the true electrode/ electolyte contact area 共cm2兲. Thus, the interpetation of AFM impedance data depends critically on our understanding of the tip/sample interface. In

Figure 2 proposes a simple equivalent circuit model of the tip/sample contact. The contact is modeled as a parallel RC element with additional series resistors included to account for the tip and sample spreading resistances. The resistance of the tip (Rtip) is often significant given the small dimensions of AFM probes. A conductive diamond coated tip, for example, may have Rtip ⬇ 3000 ⍀. The tip/sample contact resistance (Rcont) models the interfacial resistance between the tip and the sample. Obviously this contact resistance is a strong function of the tip/sample contact area. For a purely ohmic contact, Rcont will be a simple resistor, although it may be a function of the tip/sample pressure. For a nonohmic contact, Rcont might also be a function of any dc bias between the tip and the sample (i.e., the tip/sample contact may exhibit Schottky behavior). In the case of an electrochemically active interface, Rcont represents the resistance to Faradaic charge transfer at the tip/sample interface. As shown previously, this Faradaic resistance scales with interfacial area. For electrochemically active interfaces, the tip/ sample capacitance (C) contains a contribution from the double layer capacitance Cd. As shown earlier, Cd scales with contact area. However, C also contains a nonlinear contribution from stray capacitive coupling to the extended tip area. The final resistor in the model (RSR) represents the spreading resistance associated with a small volume of sample just beyond the tip/sample contact. As discussed earlier, this spreading resistance is a function of the tip/sample contact radius. Quantitative interpretation of AFM electrical measurements made through this tip/sample contact therefore requires quantitative information about the tip/sample contact size. In a standard AFM, tip/sample contact size is not directly quantifiable but tip/sample contact force can be precisely measured. By using nanoindentation data and contact mechanics theory, however, tip/sample force measurements can be converted into tip/sample contact area estimates. Perhaps surprisingly, this conversion process is reasonably insensitive to the exact tip geometry—making it applicable for most AFM probes. The experimental issues related to tip/

Downloaded 14 Sep 2004 to 203.197.98.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

3542

O’Hayre et al.

J. Appl. Phys., Vol. 96, No. 6, 15 September 2004

sample contact force control and the force to contact area conversion process will now be addressed. C. Contact force control

Fortunately, AFM systems excel at providing force control—the most common AFM operation mode is the constant-force mode, where feedback control loops are used to maintain a constant force setpoint between the tip and the sample. By operating the AFM in constant force mode during measurement, feedback loops ensure that a constant tip/ sample force (and thus a constant tip/sample contact area) is maintained at all times. By straightforward adjustment of the force setpoint from test to test, an entire set of scaling measurements as a function of contact force (and therefore as a function of contact area) can be quickly produced—allowing an examination of the linearity and reproducibility of measurements over a range of contacts.

four-sided pryamid Vickers indenter).19,20 The hardness defined in Berkovich-based nanoindentation experiments is simply equal to Pm [following Eq. (5) above]. In general, hardness is tip-geometry dependent. Because the tips used in an AFM differ from the standard Berkovich or Vickers geometry, the hardness relations defined in Eqs. (4)–(6) above are not immediately applicable to AFM experiments. However, by applying a conical approximation to the standard AFM tip geometry, further progress can be made. Conical tips possess the property of geometric similarity; therefore a simplified version of Johnson’s expanding cavity model can be used to relate hardness to the materials properties of the sample and the sharpness of the tip:22

冤

Er + 4共1 − 2␯兲 Y tan ␪ 2 4 H共␪兲 = Y + Y ln 3 3 6共1 − ␯兲 =

D. Contact area estimation from contact force data

Assuming that the hardness 共H兲 of a material is constant, contact area estimates can be obtained from contact force data using the following simple relation:19,20 H=

P , A

共4兲

where H⫽hardness 共N / m2兲; P⫽applied load (applied normal force) 共N兲; and A⫽projected or surface area of contact 共m2兲. H is a constant for noncrystalline materials and geometrically similar probes (such as Vickers and Berkovich indenters). H is also constant for crystalline materials with small indentation size effect ISE.21 H may be based either on the projected area of contact (e.g., Knoop and Berkovich Hardness) or the surface area of contact (e.g., Brinell, Vickers Harndess). More fundamentally, H is related to the mean presure, Pm that develops around an indentation Pm =

P , Aproj

共5兲

where Pm⫽mean pressure 共N / m2兲; P⫽applied load (applied normal force) 共N兲; and A⫽projected area of contact 共m2兲. Hence, Knoop and Berkovich hardness values are directly related to the mean pressure developed under an indenter. However most micro hardness experiments provide the Vickers hardness number, H␯, which is based on contact surface area rather than projected surface area. A simple geometric correction factor relates H␯ to Pm:19,20 Pm = 0.010 787H␯ ,

共6兲

where Pm has units of Gpa. The Vickers hardness is available for a wide range of materials or can be easily obtained from standard microhardness experiments. More recently, nanoindentation (a variant of depth-sensing indentation) has been increasingly used to test the mechanical properties of materials at length scales as small as 20 nm. Nanoindentation experiments generally employ a three-sided pyramid Berkovich indenter, which can be considered as a conical indenter with an effective half apex angle of 70.3° (the same as a

冥

2Y 关2 + ln共␩兲兴, 3

共7兲

where H共␪兲⫽material hardness measured by a conical tip of sharpness ␪共N / m2兲; Y⫽yield strength of the sample 共N / m2兲; Er⫽reduced modulus or indentation modulus 共N / m2兲; ␪⫽tip semiangle from the vertical 共°兲; and ␯⫽Poisson’s ratio of the sample. Er, the reduced modulus, is defined as 1 1 − ␯2t 1 − ␯2 , = + Er Et E

共8兲

where ␯t⫽Poisson’s ratio of the tip material; Et⫽Young’s modulus of the tip material 共N / m2兲; ␯⫽Poisson’s ratio of the sample; and E⫽Young’s modulus of the sample 共N / m2兲. In general, Et Ⰷ E, so Er ⬇ E / 共1 − ␯2兲, and

␩⬇

冋

册

1 E + 4共1 − 2␯兲 . 2 6共1 − ␯兲 共1 − ␯ 兲Y tan ␪

共9兲

For most metallic materials, ␯ is about 1 / 3; for most polymers, ␯ is about 0.5, thus we have

␩metal ⬇

9E 1 + 32Y tan ␪ 3

共10a兲

and

␩polymer ⬇

4E 9Y tan ␪

共10b兲

Figure 3 shows H / Y versus tip semiangle for different E / Y, based on Eq. (7). As the figure indicates, H decreases with increasing tip semi angle. The figure also demonstrates that the results are insensitive to ␯. For a specific tip semi angle 共70.3°兲, Figure 4 shows curves of (H for v)/(H for ␯ = 0.5) versus ␯ for various values of E / Y, indicating again that H is relatively insensitive to ␯. Since quantification of AFM measurements often requires an estimate of the true contact surface area between the tip and the sample rather than the projected contact area, an additional modification is required. We define Hsurf共␪兲 as a

Downloaded 14 Sep 2004 to 203.197.98.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

O’Hayre et al.

J. Appl. Phys., Vol. 96, No. 6, 15 September 2004

3543

(7)–(10) to generate an approximate hardness function. More accurately, nanoindentation experiments on the sample of interest can be used to obtain Er and H共␪兲 values at known tip angles ␪. For example, Berkovich nanoindentation experiments give H共70.3°兲. Estimating the Poisson’s ratio of the material, Eq. (7) may be solved to produce the yielding stress Y. Combining Eqs. (7) and (12), Hsurf共␪兲 estimates can then be obtained at any tip angle. Several significant assumptions have been made to derive these relations. Among the most important are the following.

FIG. 3. H / Y and Hsurf / Y Vs the tip semiangle for different E / Y; E / Y = 10 (typical of polymers, ceramics), E / Y = 100, E / Y = 1000 (typical of ductile metals). Based on Eqs. (7) and (12).

surface-area related hardness value, which provides an estimate of the tip/sample contact surface area (rather than the projected area) for a given load Hsurf共␪兲 =

P , Asurf

共11兲

where Asurf is the contact surface area 共m2兲. For the case of perfect conical tip geometry, we have Hsurf共␪兲 = H共␪兲sin ␪ .

共12兲

The second set of curves in Fig. 3, then, are curves of Hsurf共␪兲 versus tip semiangle for different E / Y. As the figure indicates, a broad maximum occurs in the Hsurf共␪兲 curves. Around this maximum, Hsurf共␪兲 is insensitive to the tip geometry. At E / Y = 10 and 30° ⬍ ␪ ⬍ 70°, for example, Hsurf共␪兲 is nearly constant. This fortuitous tip-geometry insensitivity is due to two opposing geometric correction factors that partially cancel one another out. This result implies that the contact area conversion methodology should be applicable for most AFM probe geometries if the AFM tip angle is not far from the curve maximum. Eqs. (7)–(12) provide the necessary adjustments to produce contact surface area estimates for conical AFM tips of arbitrary sharpness based on nanoindentation hardness measurements or materials properties. If the materials properties of the sample are well known, they can be inserted into Eqs.

(1) AFM tip geometry is perfectly conical. (2) Hardness is constant (No ISE). (3) Material is elastic-perfectly plastic. For large contact depths (large contact areas), the first two assumptions are quite reasonable. However, due to sample surface effects and the finite sharpness of real AFM tips, these assumptions become problematic for extremely small contacts. Most AFM tips have tip radii on the order of 10 nm. Therefore, for contacts below 10 nm (radius) in size, spherical rather than conical contact mechanics theory should likely be applied. Eq. (7) may be replaced by22

冤 冤

E rr + 4共1 − 2␯兲 2Y YR H共r兲 ⬇ 2 + ln 3 6共1 − ␯兲

冥冥

,

共13兲

where r, contact radius 共m兲; R= radius of the spherical tip 共m兲. However, at the 10 nm length scale, surface effects and sample roughness likely become significant. Thus, the methodology developed in this paper is recommended only for contact sizes larger than about 10 nm (contact areas ⬎100 nm2). For the third assumption, Johnson22 shows that the results above can be applied to strain-hardening materials by replacing the yielding stress Y with the compression flow stress at a representative strain ⑀r

⑀r ⬇

0.2 . tan ␪

共14兲

For materials with small work hardening rates, however, the elastic-perfectly plastic assumption is acceptable and Eqs. (7)–(12) produce reasonable estimates. III. EXPERIMENT A. Validation study: Experimental setup

FIG. 4. (H for ␯)/(H for ␯ = 0.5) vs ␯ for different material tested with a Berkovich indenter.

The ability to capture quantitative impedance information from AFM measurements is validated in this paper by an experimental study of the ORR kinetics at nanoscale Pt/Nafion contacts. The experimental configuration for the Pt/Nafion kinetic study is shown in Fig. 1(b). The AFM unit is equipped with a platinum coated AFM tip, which serves as the probe electrode for the electrochemical measurements. This AFM tip is brought into contact with the surface of a Nafion 115 polymer electrolyte membrane sample in the presence of air. On the backside of the Nafion membrane

Downloaded 14 Sep 2004 to 203.197.98.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

3544

O’Hayre et al.

J. Appl. Phys., Vol. 96, No. 6, 15 September 2004 TABLE I. Summary of nanoindentation results for Nafion 115 using a Berkovich indenter. Number Strain rate 共s−1兲 of indents 0.01 0.05

5 8

v

H共70.3°兲a (GPa)

Eb (GPa)

0.5 0.5

0.032± 0.001 0.039± 0.001

0.320± 0.01 0.320± 0.001

Y c (GPa) H共40°兲共GPa兲 Hsurf共40°兲共GPa兲 0.0149 0.0199

0.044 0.055

0.028 0.035

a

H is the average value over indentation depth from 1000 to 2000 nm. E is the value obtained from the unload at 2000 nm. c Y is obtained by solving Eq. (6). Here we assume there is no work-hardening effect, i.e., the flow stress Y at the representative strain 0.2/ tan共40°兲 is the same as that at the represntative strain 0.2/ tan共70.3°兲. b

sample, hydrogen gas is provisioned to a sealed anode compartment. (The anode is a standard polymer electrolyte fuel cell catalyst electrode, details given else where23,24) Applying a dc bias to the platinum coated AFM tip (the bias is applied relative to the anode electrode, which is pseudoreversible because of its much larger size) causes a Faradaic charge trnasfer reaction to occur, with protons from the membrane combining with oxygen from the air to produce water. Essentially, the Pt-coated AFM tip, in contact with the electrolyte membrane, becomes a tiny fuel cell cathode. The kinetics of the cathode reaction will be proportional to the size of the Pt-tip/Nafion contact. The size of the Pt-tip/Nafion contact is in turn dependent on the magnitude of the force applied between the tip and the sample. The contact force applied between the Pt-tip and the Nafion membrane can be very accurately controlled by the AFM system. By changing this force, the area of the Pt/Nafion contact can be varied, allowing systematic measurement of the kinetics of the Pt/Nafion interface as a function of size. If the force-to-area conversion methodology developed in this paper is correct, the calculated kinetics of the Pt/Nafion contacts should be independent of their size—providing a check on the reliability and accuracy of the force-to-area quantitative methodology. B. Validation study: Measurement details

All experiments were conducted with a modified Molecular Imaging Picoplus™ AFM unit. A Gamry PC4 / 750 potentiostat-impedance system was used for all electrochemical measurements. A complete description of the combination AFM/Impedance system is detailed elsewhere.5 Tests were conducted at room temperature and atmospheric pressure. Dry hydrogen gas was delivered to the sealed anode compartment at low flow rate (H2 flow rate Ⰶ1 SCCM) (SCCM denotes cubic centimeter per minute at STP) so as to minimize disturbance. The bare electrolyte top surface probed by the Pt-coated AFM tip was exposed to ambient air. No attempt was made to control the ambient air humidity, which typically ranged from 30% to 60%. The AFM hardware was enclosed in a vibration isolation chamber to limit acoustic noise, and placed inside a Faraday cage to limit electronic noise. Measurement cables were shielded and cable lengths were balanced and minimized to limit stray capacitance. Impedance measurements were conducted over frequency ranges from 100 kHz to 10 mHz, with ac excitation signals ranging from 10 to 100 mV, and cathodic dc bias ranges from 0 to 0.5 V. The impedance measurements used in the determination of j0 data were all ob-

tained at short circuit 共Ecell = 0 V兲. Measurement under short circuit provided an easily obtainable and reproducible point of comparison for all samples. Furthermore, it provided the best possible signal to noise ratio, since the electrochemical reaction proceeds at its maximum rate (highest current) at short circuit. The measured impedance spectra were fit to the tip/ sample contact model proposed in Figure 2 using a complex nonlinear least squares fitting algorithm. In all cases, the sum of 共Rtip + RSR兲 turned out to be negligible compared to Rcont. Since the investigated system was electrochemically active, it was assumed that Rcont could be wholly attributed to the Faradaic resistance 共R f 兲 of the Pt/Nafion interface. Following procedures detailed in a previous study,23 these fits were then used to extract the Faradaic resistances of the Pt/Nafion contacts. Because significant distortion was observed in some of the impedance responses (i.e., the Faradaic impedance loops were not perfectly circular; refer to Fig. 7), impedance fitting was accomplished with a constant phase element 共␾CPE兲 in place of a pure capacitor. (This changes the capacitor C in Fig. 2 to a constant phase element, CPE). The use of a CPE in place of a capacitor is common in the electrochemical literature; it describes time constant dispersion that may arise due to the nonuniform nature of an electrode/electrolyte interface. Measurements were conducted over a range of force setpoints spanning from 4.4 nN to 5.38 ␮N. To access such a wide range of forces, it was necessary to employ two different types of cantilever probes: a stiff cantilever geometry (spring constant, k = 45 N / m) and a soft cantilever geometry 共k = 0.19 N / m兲. Using the force to area conversion procedure detailed earlier, the contact force setpoints were converted into contact area estimates. The Faradaic resistance 共R f = Rcont兲 values obtained from the impedance measurements were then used to compare the kinetic scaling of the Pt-tip/Nafion contacts as a function of estimated contact size. At each contact force setpoint, five impedance measurements were taken. The R f versus contact size data presented in this study represent the mean values from this measurement procedure. Because hardness data on Nafion is lacking in the literature, a series of nanoindentation experiments were performed to extract the H共␪兲 values for Nafion samples under identical experimental conditions (ambient humidity, temperature) as the AFM impedance tests. The nanoindentation experiments were performed with a Nanoindenter™ XP using a constant strain rate loading schedule. Table I summarizes the results from two sets of nanoindentation experiments conducted at

Downloaded 14 Sep 2004 to 203.197.98.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

O’Hayre et al.

J. Appl. Phys., Vol. 96, No. 6, 15 September 2004

3545

FIG. 5. H共␪兲 and Hsurf共␪兲 vs tip semiangle ␪ for two sets of indentations on Nafion with different strain rates. Curves calculated from nanoindentation experiments using a Berkovich indenter, following Eqs. (7) and (12).

different locations across a Nafion sample over a time scale of several hours. The data shows low variability, suggesting that the Nafion properties are homogeneous and time stable (i.e., not varying across the surface or due to thermal or humidity fluctuations in the time scale of the experiments). Based on Table I, we obtain H共␪兲 and Hsurf共␪兲 curves for Nafion as shown in Fig. 5. As the data in Table I and Fig. 5 indicate, Nafion’s modulus is strain-rate insensitive but Nafion’s hardness is modestly strain-rate sensitive. Better accuracy should be obtained by matching the strain rates experienced during impedance measurement and nanoindentation. The low strainrate data more accurately reflects the conditions imposed by the impedance measurement system, and is therefore applied to these experiments. Both the constant-hardness and elastic-perfectly plastic assumptions used to develop the force-to-area conversion methodology can be reasonably applied to Nafion. A depth sensitive analysis of the Nafion hardness data (using the technique described by Nix and Saha25) indicates that the hardness decreases by approximately 35% at the surface. This surface-localized hardness decrease is largely due to sample roughness effects. Nafion, like most polymers, also exhibits reasonable elastic-perfectly plastic behavior. Even assuming a worst-cast strain hardening rate of E / 100, the calculated Nafion hardness values change by less than 3%, indicating that the elastic-perfectly plastic assumption is valid. Scanning electron microscopy micrographs of the AFM probes used in the impedance experiments indicated conical tip geometries, with an average tip semi angle ␪ of around 40°. According to Table I and Fig. 5, Hsurf共40°兲 is about 2.8⫻ 107 Pa for the low strain-rate case. This value is used as the force to area conversion coefficient for all AFM impedance experiments in this study according to Eq. (11). IV. VALIDATION RESULTS A. Evidence for the Faradaic reaction

As stated previously, it is assumed that the measured contact impedance Rcont can be wholly attributed to the Faradaic resistance 共R f 兲 of oxygen reduction at the Pt/Nafion interface. Thus it is important to establish that the signal ac-

FIG. 6. Polarization behavior of the Pt/Nafion tip interface at two force setpoints. (a) I - V plot of Pt/Nafion tip interface at 1.9 ␮N force 共䊏兲 and 0.9 ␮N force 共䉱兲. (b) Tafel plot of Pt/Nafion tip interface at 1.9 ␮N force 共䊏兲 and 0.9 ␮N force 共䉱兲. Polarization measured according to the experimental configuration in Fig. 1(b). Tafel plots yield transfer coefficients ␣ of 0.29 and 0.28, respectively. Force setpoints of 1.9 and 0.9 ␮N correspond to contact area estimates of 0.063 and 0.029 ␮m2, respectively.

quired by the AFM impedance technique truly comes from oxygen reduction at the Pt-tip/Nafion contact interface. In these measurements, the tip/sample contact radii are on the order of tens of nanometers, resulting in gigaohm impedances and picoamp currents. At such levels, spurious electronic signals can easily overwhelm the desired electrochemical response. Experimental results from a previous contribution5 strongly argue that the electrochemical response measured at the AFM-tip/Nafion interface can indeed be attributed to the oxygen reduction reaction. Among the arguments, a Faradaiclike charge transfer loop is seen in the impedance response only if all of the following conditions are satisfied. (1) (2) (3) (4)

A Pt-coated tip is used. H2 gas is provisioned to the anode compartment. A cathodic dc bias is applied. The tip is in hard physical contact with the electrolyte membrane.

Observation of this response only when all four conditions are met indicates that the loop most likely corresponds to a Faradaic reaction at the tip-electrolyte interface. This conclusion is also supported by the bias-dependent behavior of the impedance loop. The measured impedance loop shrinks with increasing dc bias, highly characteristic of a Faradaic charge transfer reaction. Furthermore, as shown in Figs. 6(a) and 6(b), the polar-

Downloaded 14 Sep 2004 to 203.197.98.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

3546

O’Hayre et al.

J. Appl. Phys., Vol. 96, No. 6, 15 September 2004 TABLE II. Comparison of the AFM impedance data obtained in the present work to previous literature data for ORR kinetics in similar systems. Electrode Pt AFM-nanoelectrode Pt microelectrode Pt microelectrode Pt microelectrode Pt microelectrode Pt disk Pt/ C

Electrolyte

T(K)

␣a

j0共A / cm2兲

Reference

Nafion Nafion Nafion Nafion Nafion 9.5 M TFMSA Nafion

298 298 323 303 303 308 368

0.29 0.5 0.29 0.237 0.345 0.5 0.47

2.73⫻ 10−7b 7.8⫻ 10−7b 10−9 − 10−8b Not reported 1.7⫻ 10−8c 1.48⫻ 10−7b 0.56⫻ 10−8d

This work 26 27 28 29 30 31

a

Transfer coefficients calculated for the reported high current density (low potential) region Tafel slopes. j0 is calculated according to geometric area. c j0 based on high current densitly region, normalized to geometric area using reported roughness value. d j0 based on electrochemically active surface area of Pt/ C catalyst powder. b

ization behavior of the Pt-tip/Nafion interfaces is Tafel like over nearly the entire potential range. The two sets of curves in these figures were acquired at force set points of 1.9 and 0.9 ␮N respectively. Following the procedure detailed in the appendix, the curves yield calculated transfer coefficients 共␣兲 of 0.28 and 0.29, respectively—nearly independent of force, as should be expected. Also as described in the Appendix, exchange current densities 共j0兲 may be calculated from the Tafel plots based on contact area estimates from the two force setpoints. The results are j0 = 2.5⫻ 10−7 A / cm2 for Fcontract = 1.9 ␮N, and j0 = 2.6⫻ 10−7 A / cm2 for Fcontact = 0.9 ␮N. Despite the fact that the measurements were taken at significantly different force setpoints, they produce nearly identical kinetic values. Furthermore, on an absolute scale, these results are comparable to previously reported measurements for ORR kinetics at Pt/Nafion interfaces. This is shown in Table II, where the ␣ and j0 values derived from Fig. 6 are compared to previous literature values.26–31 The existence of Tafel-like behavior over the entire potential range suggests a lack of iR and/or mass transport effects—this is a typical scaling feature of extremely small electrochemical microsystems.28,32–34 It is also consistent with experimental conclusions that (Rtip + RSR) is negligible compared to R f . The current acuracy limits of the Gamry potentiostat system are on the order of 5 pA, therefore data below this current threshold are probably inaccurate. Additionally, because the potentiostat has a non zero current offset (i.e., a finite input impedance), the apparent OCV of the system is decreased compared to the ⬇1.0 V expected from the oxygen reduction potential. Due to the tiny electrochemical area of the measured system, even a subpicoamp offset current can affect the measured OCV by several hundred millivolts. The minuscule currents measured in these polarization curves illustrate the challenges of working with nanoscale electrochemical phenomena.

element.23 The loop distortion might be associated with inhomogeneties in the tip/sample contact, resulting in a distribution of tip/sample capacitance. As expected, increasing the contact force decreases the impedance loop diameter due to the increase in tip/Nafion contact area. The impedance plots in Fig. 7 qualitatively show the decrease in impedance due to the increase in contact area. The force to contact area conversion methodology developed in Sec. II D allows us to put the measurements on a quantitative basis, as shown in Fig. 8. This figure gives a complete summary from impedance measurements spanning more than three orders of magnitude in tip/sample contact area. (Corresponding to three orders of magnitude in tip/sample contact force.) In this figure, the R f values calculated from complex nonlinear least squares fits of the impedance data are plotted versus the estimated tip/Nafion contact areas for each experimental force setpoint. The data are broken into two clusters: one set is obtained from a stiff Pt-coated cantilever 共k = 45 N / m兲 while the second set is obtained from a soft Pt-coated cantilever 共k = 0.19 N / m兲. Both sets of data support a consistent trend despite the fact that they were acquired with different cantilever geometries. As can be seen from a power-law fit of the experimental data, the Faradaic impedance of the Pt-tip/Nafion contact scales almost precisely with the inverse of the estimated contact area. This

B. Area-based kinetics

Figure 7 shows a set of impedance spectra for Pt-tip/Nafion contacts acquired at four different force setpoints ranging from about 1 to 5 ␮N. (Measurements acquired from 100 kHz to 100 mHz with a 30 mV excitation.) Although these impedance loops are not perfect semicircles, they can be fit reasonably well with a constant phase

FIG. 7. Nyquist spectra of the Pt-tip/Nafion impedance response as a function of the Pt-tip/Nafion contact force. Impedance spectra are measured according to the experimental configuration in Fig. 1(b) over the frequency range 100 kHz to 100 mHz with a 30 mV excitation signal. The impedance loop shrinks with increasing tip/sample contact force.

Downloaded 14 Sep 2004 to 203.197.98.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

O’Hayre et al.

J. Appl. Phys., Vol. 96, No. 6, 15 September 2004

3547

TABLE III. Statistical summary of the AFM impedance data. Data acquired at various force setpoints ranging from 4.4 nN to 5.38 ␮N. The j0 values for the oxygen reduction reaction were calculated from the measured impedance values by using the estimated contact area at each force setpoint (see the Appendix for details). While both stiff and soft cantilever geometries delivered roughly the same results, the standard deviation in the stiff cantilever measurements is significantly lower. Measured Force 共␮N兲 R f 共⍀兲

FIG. 8. Summary of Faradaic impedance vs estimated tip/sample contact area for the Pt-tip/Nafion system. As indicated by the power law fit, the Faradaic impedance scales with the inverse of the contact area. 共䊏兲 Data acquired using stiff cantilever geometry 共k = 45 N / m兲. 共⽧兲 Data acquired using soft cantilever geometry 共k = 0.19 N / m兲.

follows exactly the prediction of Eq. (2) if the entire area of the Pt-tip/Nafion contact is assumed to be active for the ORR. For the extremely small contacts examined by this study, it is reasonable to believe that the entire contact area is accessible to oxygen gas for reaction. In other words, diffusion of oxygen to the interface is not a bottleneck. It should be noted that straightforward areas based scaling is not always observed for platinum/Nafion contacts. In a recent Pt microelectrode study,23 R f was observed to scale with perimeter length for micron scale 共400 nm⬍ rmicroelectrode ⬍ 40 ␮m兲 Pt microelectrodes on Nafion membranes. The change from perimeter to area based scaling may be due to a change in the relative balance between the rates of oxygen reaction and oxygen transport. For small Pt/Nafion contacts, the entire interfacial area participates in the reaction because oxygen diffusion through the membranes is fast compared to reaction, but for large contacts the center becomes a “dead-zone” because the rate of oxygen diffusion is too slow. This issue will be explored in a forthcoming paper by modeling coupled reaction and diffusion processes to predict the area-to-perimeter transition. Returning to Fig. 8, while the fidelity of the stiff cantilever data is extremely good, the soft-cantilever data shows significantly greater scatter. This is confirmed by Table III, which gives a complete statistical summary of the impedance measurements. The table shows j0 values determined from the R f measurements and contact area estimates according to procedures detailed in the Appendix. The data acquired with the stiff cantilever yield an average calculated j0 value for the ORR of 2.61⫻ 10−7 A / cm2 with a standard deviation of ±4.1%. The data acquired with the soft cantilever yield an average j0 value for the ORR of 2.94⫻ 10−7 A / cm2, but with a standard deviation of ±33.4%. There are several possible explanations for the increased scatter in the soft cantilever measurements. First, the contact areas are extremely small—therefore the measured impedance values are extremely high. For the smallest contacts, the measured impedances approach the quoted range limit of the analyzer (around 1011 ⍀). Therefore, some of this scatter

Calculated Area 共␮m2兲 J0共A / cm2兲

Stiff cantilever 2.19⫻ 108 0.179 333 3 2.31⫻ 108 0.162 666 7 2.86⫻ 108 0.146 3.10⫻ 108 0.129 333 3 3.59⫻ 108 0.112 666 7 4.25⫻ 108 0.096 5.26⫻ 108 0.079 333 3 6.26⫻ 108 0.062 666 7 8.08⫻ 108 0.046 1.41⫻ 109 0.029 333 3

2.64⫻ 10−7 2.77⫻ 10−7 2.49⫻ 10−7 2.59⫻ 10−7 2.57⫻ 10−7 2.55⫻ 10−7 2.49⫻ 10−7 2.65⫻ 10−7 2.80⫻ 10−7 2.52⫻ 10−7

Soft cantilever 0.0264 5.41⫻ 1010 0.000 88 0.022 6.50⫻ 1010 0.000 733 3 0.0176 9.06⫻ 1010 0.000 586 7 0.0132 7.51⫻ 1010 0.000 44 0.0088 9.15⫻ 1010 0.000 293 3 0.0044 1.66⫻ 1011 0.000 146 7 Average j 0 (stiff cantilever): Standard deviation (stiff cantilever): Average j0 (solt cantilever): Standard deviation (soflt cantilever):

2.18⫻ 10−7 2.18⫻ 10−7 1.96⫻ 10−7 3.15⫻ 10−7 3.87⫻ 10−7 4.28⫻ 10−7 2.61⫻ 10−7 4.10% 2.94⫻ 10−7 33.40%

5.38 4.88 4.38 3.88 3.38 2.88 2.38 1.88 1.38 0.88

may be associated with errors in the impedance measurement. Second, uncertainties associated with the contact area estimates will grow as the contact size is reduced due to the increasing relative importance of sample roughness and surface effects. Finally, there may be a fundamental change in the kinetics at extremely small contact sizes. The smallest contacts measured had contact radii less than 10 nm. At the 10 nm and below, particle size effects have frequently been reported in the fuel cell literature for platinum catalyst systems. V. CONCLUSIONS

Nanoindentation experiments and contact mechanics theory have been successfully applied to quantify the nature of the tip/sample contact in AFM experiments as a function of contact force. Accurate conversion of contact force data to contact area estimates allows quantitative interpretation of electrical or electrochemical measurements from AFM experiments. As a validation, this methodology has been applied to the study of ORR kinetics at nanoscale Pt/Nafion contacts. The kinetic data obtained from nanoscale AFM impedance experiments matches previous data obtained from bulk and microelectrode-based experiments. These results also illustrate the potential of a combined AFM/nanoindenter or “scanning nanoindentaiton” instrument. In addition to mapping mechanical properties, such an

Downloaded 14 Sep 2004 to 203.197.98.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

3548

O’Hayre et al.

J. Appl. Phys., Vol. 96, No. 6, 15 September 2004

instrument could prove ideal for acquiring electrical, electrochemical, or other spectroscopic information across sample surfaces as well. Crossing discipline boundaries and dramatically expanding characterization capabilities, such multifunctional instruments would likely find applications in numerous fields.

i共t兲 = i0 exp

Z=

共15兲

where i⫽Current 共A兲; i0⫽Exchange current 共A兲; ␣⫽Transfer coefficient; n⫽Electrons transferred; F⫽Faraday’s constant 共96 500 C / mol兲; ␩⫽Overpotential Ecell − E0共V兲; R⫽Gas constant 关8.314 J / 共mol K兲兴; and T⫽Temperature (K). For the Pt/Nafion contacts, we assume that iR and mass transport effects are negligible. In other words, the I - V behavior of the Pt/Nafion contacts is dominated by Tafel kinetics over the entire potential range. We also neglect the anode processes, since the anode is highly reversible and much larger in size than the cathode. Under these assumptions, ␣ and i0 may be extracted from a logarithmic fit of the I - V curve: ln i = ln i0 −

␣nF共Ecell − E0兲 . 2.303 RT

共16兲

Finally, the exchange current density j0 may be calculated from the exchange current i0 by normalizing by the estimated surface area of the Pt/Nafion contact j0 =

i0 , Asurf

共17兲

where Asurf is the contact surface area 共m2兲. Assuming a particular value for transfer coefficient, ␣, j0 values may also be obtained from impedance data. During an impedance measurement, a sinusoidal voltage excitation Eexc of angular frequency ␻ is superimposed on the fixed dc cell voltage E共t兲 = Ecell + Eexce j␻t . The resulting current response is

共18兲

1 E共t兲 = , 1 i共t兲 + j␻Cdl Rf

Rf =

共20兲

RT . ␣nFi0e−␣nF␩/RT

共21兲

Since the impedance measurements in this study were conducted at short circuit, Ecell = 0 and therefore ␩ = −E0. Equation (21) above may then be solved for i0

APPENDIX

i = i0e共− ␣nF␩/RT兲,

共19兲

where R f is identified as the Faradaic resistance and is equal to

i0 =

Tafel analysis is used to extract kinetic data (␣ and j0 parameters) from the current-voltage response of the Pt/Nafion contacts. For overpotential excursions greater than 50 mV, the Tafel equation (a simplified form of the ButlerVolmer equation), may be employed to describe an activated electrochemical reaction process

+ j␻CdlEexce j␻t ,

where Cdl is the double layer capacitance (F). The impedance Z may be calculated as

ACKNOWLEDGMENTS

The authors would like to warmly thank the members of the Rapid Prototyping Laboratory and especially the fuel cell team for their support. In particular, the authors would like to recognize Minhawn Lee, Dr. Turgut Gur, and Dr. Seong Ihl Woo for their helpful insights and assistance with the study. This work was supported under a Stanford Graduate Fellowship. The Stanford Global Climate Energy Project (GCEP) provided funding for this research.

−␣nF关共Ecell+Eexce j␻t兲−E0兴 RT

RT ␣nFR f e␣nFE0/RT

共22兲

where R f is obtained from complex least squares fits of the impedance spectra and we assume ␣ = 0.285, a representative average value for the experiments in this study. Finally, the exchange current density j0 is calculated from the exchange current i0 by normalizing by the estimated surface area of the Pt/Nafion contact as before [Eq. (17) above]. 1

R. Shao, S. V. Kalinin, and D. A. Bonnel, Appl. Phys. Lett. 82, 1869 (2003). 2 S. V. Kalinin and D. A. Bonnell, Appl. Phys. Lett. 78, 1306 (2001). 3 S. V. Kalinin and D. A. Bonnell, J. Appl. Phys. 91, 832 (2002). 4 A. Layson, S. Gadad, and D. Teeters, Electrochim. Acta 48, 2207 (2003). 5 R. O’Hayre, M. Lee, and F. B. Prinz, J. Appl. Phys. 95, 8382 (2004). 6 P. De Wolf, J. Snauwaert, L. Hellemans, T. Clarysse, W. Vandervorst, M. D’Olieslaeger, and D. Quaeyhaegens, J. Vac. Sci. Technol. A 13, 1699 (1995). 7 P. De Wolf, M. Geva, T. Hantschel, W. Vandervorst, and R. B. Bylsma, Appl. Phys. Lett. 73, 2155 (1998). 8 P. De Wolf, J. Snauwaert, T. Clarysse, W. Vandervorst, and L. Hellemans, Appl. Phys. Lett. 66, 1530 (1995). 9 A. C. Diebold, M. R. Kump, J. J. Kopanski, and D. G. Seiler, Proc.Electrochem. Soc. 94, 78 (1994). 10 T. W. Kelley, E. L. Granstrom, and C. D. Frisbie, Adv. Mater. (Weinheim, Ger.) 3, 261 (1999). 11 A. Bietsch, M. A. Schneider, M. E. Welland, and B. Michel, J. Vac. Sci. Technol. B 18, 1160 (2000). 12 M. Freitag, M. Radosavljevic, W. Clauss, and A. T. Johnson, Phys. Rev. B 62, R2307 (2000). 13 S. J. Oshea, R. M. Atta, M. P. Murrell, and M. E. Welland, J. Vac. Sci. Technol. B 13, 1945 (1995). 14 A. Olbrich, B. Ebersbergerm, and C. Boit, Appl. Phys. Lett. 73, 3114 (1998). 15 C. C. Williams, W. P. Hough, and S. A. Rishton, Appl. Phys. Lett. 55, 203 (1989). 16 C. C. Williams, J. Slinkman, W. P. Hough, and H. K. Wickramasinghe, Appl. Phys. Lett. 55, 1662 (1989). 17 A. C. Diebold, M. R. Kump, J. J. Kopanski, and D. G. Seiler, J. Vac. Sci. Technol. B 14, 196 (1996). 18 J. W. Orton and P. Blood, The Electrical Characterization of Semiconductors (Academic, London, 1990), p. 53. 19 D. Tabor, The Hardness of Metal (Clarendon, Oxford, 1951). 20 B. Bhushan, Handbook of Micro/Nano Triblology, 2nd ed. (CRC Boca Raton, FL, 1999). 21 W. D. Nix and H. J. Gao, J. Mech. Phys. Solids 46, 4577 (1998). 22 K. L. Johnson, Contact Mechanics (Cambridge University Press, Cambridge, 1987).

Downloaded 14 Sep 2004 to 203.197.98.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

O’Hayre et al.

J. Appl. Phys., Vol. 96, No. 6, 15 September 2004 23

R. O’Hayre and F. B. Prinz, J. Electrochem. Soc. 151, A756 (2004). M. S. Wilson and S. Gottesfeld, J. Appl. Electrochem. 22, 1 (1992). 25 W. D. Nix and R. Saha, Acta Mater. 50, 23 (2002). 26 A. Parthasarathy, C. R. Martin, and S. Srinivasan, J. Electrochem. Soc. 138, 916 (1991). 27 S. Mitsushima, N. Araki, N. Kamiya, and K. Ota, J. Electrochem. Soc. 149, A1370 (2002). 28 F. A. Uribe, T. E. Springer, and S. Gottesfeld, J. Electrochem. Soc. 139, 765 (1992). 29 P. D. Beattie, V. I. Basura, and S. Holdcroft, J. Electroanal. Chem. 468, 180 (1999). 24

3549

30

M. A. Enayetullah, T. D. Devilbiss, and J. O. Bockris, J. Electrochem. Soc. 136, 3369 (1989). 31 S. Murkerjee, S. Srinivasan, M. P. Soriaga, and J. McBreen, J. Phys. Chem. 99, 4577 (1995). 32 A. J. Bard and L. R. Faulkner, Electrochemical Methods, 2nd ed. (Wiley, New York, 2001). 33 R. M. Wightman and D. O. Wipf, p. 267 (Marcel Dekker, New York, NY, USA, 1989). 34 M. Fleischmann, S. Pons, D. R. Rolison, and P. P. Schmidt, Ultramicroelectrodes (Datatech Systems, Morgantown, NC, 1987).

Downloaded 14 Sep 2004 to 203.197.98.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp