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Variation of the Surface Stress-Charge Coefficient of Platinum with Electrolyte Concentration R. N. Viswanath,*,† D. Kramer,† and J. Weissmu¨ller†,‡ Forschungszentrum Karlsruhe, Institut fu¨ r Nanotechnologie, Karlsruhe, Germany, and Technische Physik, Universita¨ t des Saarlandes, Saarbru¨ cken, Germany Received October 25, 2004. In Final Form: March 11, 2005 We report an experimental study of the variation of surface stress with surface charge density for nanoporous Pt immersed in aqueous solutions of NaF of various concentration. As the concentration is reduced, we find initially an increase in the magnitude of the surface stress-charge coefficient, followed by saturation at a value of -1.9 V. Since specific adsorption is expected to be reduced as the solution becomes more dilute, the results support the notion that changes in the bonding at the metal surface play a decisive role in determining the change in the surface stress during double-layer charging.
Introduction Changes of the superficial charge density q at the surface of a conductive solid immersed in an electrolyte give rise to forces at the surface whichsat equilibriumsmust be balanced by stresses in the underlying bulk. Contemporary research, as reviewed in refs 1-3, aims at measuring the phenomenological parameter describing these forces, the surface stress tensor s, and at establishing an understanding of the underlying microscopic processes. An important phenomenological parameter is the surface stress-charge coefficient, the derivative of s with respect to q at constant values of the temperature T, of the tangential strain e, and of the chemical potentials µi of the components i in the electrolyte. Most experiments measure a scalar surface stress, f ) 1/2 trace s, and first experimental values have become available for the scalar surface stress-charge coefficient σ ) df/dq|T,e,µi. Numerical values of σ for Pt and Au have always been found negative, and they are roughly of the order of -1 V.3-5 In principle, several microscopic interaction terms can contribute to the charge-induced variation of s: electrostatic repulsion between the charge carriers in the spacecharge layers, changes in the bonding between the metal atoms, and forces either between neighboring adsorbate atoms or between adsorbates and the metal. While the contribution of electrostatic forces vanishes at the potential of zero charge (PZC), σ is found to be finite there, and the relative importance of the two remaining contributions in determining σ is the subject of debate. The issue is of relevance for the understanding of the electrochemistry of the metal surface, and the recent suggestion that changes in the surface properties of metals due to polarization may be exploited to create novel functional nanomaterials with tunable properties6,7 has added to its importance. † ‡
Forschungszentrum Karlsruhe, Institut fu¨r Nanotechnologie. Technische Physik, Universita¨t des Saarlandes.
(1) Cammarata, R. C.; Sieradzki, K. Annu. Rev. Mater. Sci. 1994, 24, 215. (2) Ibach, H. Surf. Sci. Rep. 1997, 29, 193. (3) Haiss, W. Rep. Prog. Phys. 2001, 64, 591. (4) Haiss, W.; Nichlos, R. J.; Sass, J. K.; Charle, K. J. Electroanal. Chem. 1998, 452, 199. (5) Weissmu¨ller, J.; Viswanath, R. N.; Kramer, D.; Zimmer., P.; Wu¨rschum, R.; Gleiter, H. Science 2003, 300, 312. (6) Gleiter, H.; Weissmu¨ller, J.; Wollersheim, O.; Wu¨rschum, R. Acta Mater. 2001, 49, 737. (7) Kramer, D.; Viswanath, R. N.; Weissmu¨ller, J. Nano Lett. 2004, 4, 793.
Spectroscopic experiments, such as electroreflectance8 and X-ray adsorption near-edge structure spectroscopy,9 give clear evidence for changes in the band occupancy at the metal surface during double-layer charging, the prerequisite for a variation of the lateral bond strength. Furthermore, Schmickler and Leiva10 have argued that, in weakly adsorbing electrolytes, the adsorbate-adsorbate and adsorbate-metal bond energies are typically too small to explain the empirical dependency of s on q in terms of forces acting on the adsorbates. This suggests that the dominant effect is the change in the bonding between the superficial metal atoms. This notion is also supported by a suggestion of Haiss,3 who argued that the variation σ when a given surface is in contact with different electrolytes is simply an effect of the more or less pronounced charge transfer to the adsorbate ions: conceptually, in the limit of very weak adsorption, the entire electronic charge remains localized in the spacecharge region of the metal surface, thereby affecting the surface bonds in the metal. In more strongly adsorbing electrolytes, the charge which is transferred from the metal surface to the adsorbate ions becomes unavailable for changing the bonding in the metal. This suggests that the magnitude of σ decreases with increasing strength of adsorption, in agreement with experimental observation. The importance of charge transfer in reducing σ has also been pointed out in the discussion of comparative data obtained in the capacitive double layer region of the cyclic voltammogram and in the OH-adsorption region, respectively:5 considerably larger values of σ are obtained in the former case. In contrast to the above concepts, there are arguments in favor of a decisive role of the surface chemistry in determining the surface stress. In aqueous electrolytes, dominantly capacitive behavior can only be observed within narrow windows of electrochemical potential and ionic concentration. Thus, adsorption will generally be present to some degree in most measurements of surface stress of the metal-electrolyte interface. In fact, Sieradzki and co-workers have suggested that for the adsorption of ClO4-sa weakly adsorbing ionson Au the double-layer contribution was negligible, so that changes in the surface stress are dominantly due to adsorption.11 This is com(8) Ko¨tz, R.; Kolb, D. M. Z. Phys. Chem. Neue Folge 1978, 112, 69. (9) Mukerjee, S.; Srinivasan, S.; Soriaga, M. P.; McBreen, J. J. Electrochem. Soc. 1995, 142, 1409. (10) Schmickler, W.; Leiva, E. J. Electroanal. Chem. 1998, 453, 61.
10.1021/la0473759 CCC: $30.25 © 2005 American Chemical Society Published on Web 04/14/2005
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patible with the well-established finding that submonolayer deposition of metals on metal surfaces in ultrahigh vacuumswhich changes the surface chemistry without surface chargingscan induce large changes in f.2 Recently, Vasiljevic et al.12 have estimated the value of the derivative ∂f/∂E (where E denotes the electric potential) for Au(111) at the PZC. These authors estimate the quantity ∂E/∂e|T,q,µi from the change in PZC accompanying the surface reconstruction and from an effective strain which may be associated with the reconstruction. In combination with the appropriate Maxwell relation, this yields an estimate for ∂f/∂E. Vasiljevic et al. report that the experimental ∂f/∂E in 0.1 M solutions of Na2SO4 and, to a lesser degree, in NaF are significantly above the estimated value, concluding that variations in surface chemistry control the surface forces. However, the conclusive experiment proposed in ref 12, based on a measurement of the shift in the PZC during purely elastic straining of the surface and avoiding the structural change associated with the reconstruction, is still outstanding. An alternative way of probing the importance of specific adsorption for the potential-induced variation in surface stress is to measure σ as a function of the electrolyte concentration X, thereby varying the relative contribution of capacitive processes versus specific adsorption to the double layer charging. When both the inner Helmholtz layer and the bulk electrolyte can be approximated as dilute solutions of the adsorbing species, then the superficial density of adsorbed ions at a given value of E varies as X whereas, according to the Gouy-Chapman theory, the double-layer capacitance varies as xX.13 Thus, capacitive charging becomes relatively more important at higher dilution. In the limit of purely capacitive charging, the double layer can be approximated as two space charge regions, electronic charge in the metal and a diffuse ionic layer in the solution, separated by the inner Helmholtz layer, a monolayer of water molecules which acts as a dielectric and which passivates the metal surface effectively against chemisorption.14 In this state the surface stress-charge coefficient should be entirely determined by the bonding in the metal and, via the dependency of the electronic structure of the metal surface on the dielectric constant of the surrounding medium, on the dielectric constant of the inner Helmholtz layer. Thus, in the limit of high dilution σ should become independent of the nature and concentration of the ions in solution. Here, we report measurements of the dependency of the surface stress of Pt on the surface charge density and on the electrolyte concentration in the weakly adsorbing electrolyte NaF. A crucial issue in such experiments using nanoporous materials is the slowing down of the charging kinetics at high dilution, the result of the decelerated transport of ions toward the surface and through the narrow pores due to the diminishing ionic conductivity. By using dilatometry to follow the surface-stress-induced expansion or contraction of porous bodies with a large specific surface area,5,7,15 we can detect the changes in the surface stress even when, at high dilution, only little charge is transferred to the pore surfaces. We show that the magnitude of σ increases substantially as the solution becomes more dilute. This finding indicates that the (11) Friesen, C.; Dmitrov, N.; Cammarata, R. C.; Sieradzki, K. Langmuir 2001, 17, 807. (12) Vasiljevic, N.; Trimble, T.; Dmitrov, N.; Sieradzki, K. Langmuir 2004, 20, 6639. (13) Bagotzky, V. S. In Fundamentals of Electrochemistry; Plenum Press: New York, 1993; Chapter 12. (14) Parsons, R. Solid State Ionics 1997, 94, 91. (15) Weissmu¨ller, J.; Cahn, J. W. Acta Mater. 1997, 45, 1899.
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capacitive processes contribute significantly to the value of σ, and it supports the importance of the charge-induced change in the bonding in the metal surface for determining f of metal-electrolyte interfaces. Experimental Methods A high surface area porous platinum sample was prepared by consolidation of commercial Pt black (Alfa Aesar) as described previously.5 A sample of dimensions 1.5 mm length and 2 mm diameter machined from the porous platinum body was placed as the working electrode in an in situ electrochemical cell operated in the sample space of a commercial dilatometer (Netzsch 402C).5,7 The cell, which has the comparatively small volume of about 2.5 mL, comprises a high surface area carbon fiber counter electrode and a Ag/AgCl reference electrode, separated from the sample compartment by a glass frit. Changes in sample length are transmitted to an inductive displacement sensor via a silica pushrod loaded by a contact pressure of 20 cN. The sample space of the dilatometer was temperature controlled to better than 0.1 K. Results of microstructure characterization by scanning electron microscopy, BET gas adsorption, and X-ray powder diffraction have been reported previously:5 The BET specific surface area and the mean crystallite size evaluated from the diffraction data are 25 m2/g and 6.0 ( 2.0 nm, respectively. These values are compatible when spherical particles with only little agglomeration are assumed. The mass specific surface area corresponds to the volume-specific area (per volume of the metal) R ) 0.573 nm-1. The cell and all glassware were cleaned with oxidizing acid (H2SO4/H2O2) before the experiments. Prior to immersion into NaF the sample underwent a cyclic scan of the potential in 0.5 M H2SO4 in the potential range -0.1 V < E < 0.6 V. This procedure was found necessary for obtaining reproducible results. It is known that oxidation/ reduction in H2SO4 produces a more ordered structure of polycrystalline Pt surfaces.16 The sample was then washed repeatedly with spectroscopy grade ultrapure water. The experiments were conducted with solutions of NaF (Merck, 99.9%) in ultrapure water as the electrolyte. We investigated electrolytes with concentrations X ) 1 M down to X ) 0.02 M, starting with the higher concentration, and washing the sample between changes of the electrolyte. For electrolytes of concentration smaller than 0.02 M the signal-to-noise ratio for dilatometry became too small for meaningful analysis of the data. All cyclic voltammograms were taken at the temperature 283 K, under control by a potentiostate (PGSTAT 100, EcoChemie), and at the scan rate 1 mV/s. The sample was held at the potential in the center of the scan interval for at least 2 h, and it was then subjected to 10 successive scans. The variation ∆q of the superficial charge density was computed by dividing the experimental value for the variation ∆Q in the total charge on the sample by its total surface area A. To obtain A, we multiplied the volumespecific area R by the volume of the metal as determined from the sample mass, m, and the mass density F of Pt. Thus, ∆q was obtained by
∆q ) ∆Q F/(mR)
(1)
Experimental Results Figure 1a shows the cyclic voltammograms obtained in various concentrations of NaF electrolyte. The dominant (16) Cervin˜o, R. M.; Triaca, W. E.; Arvia, A. J. J. Electroanal. Chem. 1985, 182, 51.
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Figure 1. Results of cyclic voltammetry for porous platinum in NaF electrolyte of various concentrations. (a) Cyclic voltammogram with wide potential window, involving specific adsorption. (b) Narrow potential window, probing the nominally capacitive region of the voltammogram. Concentrations are, in the order of decreasing current amplitude, 1, 0.5, 0.1, 0.06, and 0.02 M; subsequent curves alternate between full and dash-dotted style for better distinction. (c) Dependence of maximum charge transfer ∆Qmax on the electrolyte concentration X for the narrow window as in (b).
electrochemical features in the potential window between -0.25 and 0.5 V against the Ag/AgCl reference electrode are (i) formation and reduction of OH- ions at positive potentials above 0.25 V, (ii) adsorption and desorption of H+ ions below -0.1 V, and (iii) a weak ion adsorbing double layer region from -0.05 to 0.2 V. The features observed in NaF (1 M) are in agreement with the literature results for platinum in aqueous electrolyte.13 For increasing dilution the adsorption peaks shift toward larger potential magnitudes, and they eventually move outside the experimental potential window. By means of a series of scans at decreasing scan rate we found that this behavior does not represent a change in the equilibrium values of the adsorption potentials but that it results instead from the increasingly slow charging kinetics in the more dilute electrolytes. This is in agreement with the fact that the potentials for the onset of the adsorption of OH- and H+ are determined by the pH value, which depends only weakly on the NaF concentration. Since the present work was aimed at studying the change in properties of a porous platinum network induced by the double layer charge, in situ dilatometer experiments were carried out in the capacitive region of the voltammogram, -0.05 V < E < 0.12 V. Figure 1b shows exemplary cyclic voltammograms measured in the double layer region at a scan rate of 1 mV/s. Whereas the result at the highest concentration approaches the square shape expected for equilibrium charging of a capacitive surface, the voltammograms at the lower concentrations are again limited by kinetics. The net charge ∆Q transferred to the sample was determined by integration of the current. Figure 1c shows the plot of the amplitude ∆Qmax of the charge transfer (the difference between the maximum and the minimum values of ∆Q during the scan) on the electrolyte concentration X for scans in the narrow potential window. The graph illustrates the reduction in charge transfer due to kinetic slow down, in combination with a decrease in double-layer capacitance. The macroscopic dimension changes ∆l and the variation of the total surface charge ∆Q were studied simultaneously during the scans in the capacitive regime as shown in Figure 1b. As illustrated in the topmost two rows of Figure 2, the dependence of ∆l and ∆Q exhibits hysteresis when plotted against E; the width of the hysteresis loops increases on diluting the electrolyte. This indicates that we measured the change ∆l and ∆Q before a capacitive equilibrium is established between the porous platinum and the ions in the electrolyte. It is important to note that the loops are closed, indicating that the charging is reversible. Note also that each of the graphs in Figure 2 a-c shows a superposition of 10 successive cyclic volta-
mmogram scans. These appear as a single line in the graphs because the results are practically identical. The excellent reproducibility argues against significant impurity pickup at the surface during the scans. As a verification of reproducibility, Figure 3 shows a comparison of the results of in situ dilatometry for X ) 1 M before and after the experiments in the dilution series. It is seen that the ∆q versus electrode potential E as well as the graph of f(q) are in good agreement. The reproducibility indicates that the results are representative of the uncontaminated Pt surface. The apparent absence of noticeable contamination is remarkable in view of the high reactivity of the Pt surface. The finding is consistent with the fact that the electrolyte volume is quite small while at the same time the surface area of the samples is about 8000 cm2, 3 to 4 orders of magnitude more than in conventional experiments. Thus, the ratio of impurities in the electrolyte per surface atom is comparatively small. Data Analysis The generalized capillary equation for solids15
3V〈P - P0〉V ) 2A〈f〉A
(2)
relates the volumetric average 〈P〉V of the pressure in the bulk of the solid to the areal average 〈f〉A of the interface stress on all the surfaces. A denotes the total surface area, V the total volume of the solid, and P0 the pressure in the fluid in which the solid is immersed. The volumetric mean strain is given by ∆V/V ) 〈P - P0〉V/K with K the bulk modulus; for Pt, K ) 283 GPa.17 We have previously shown that the mean relative change in the lattice parameter due to change in surface stress in porous Pt samples prepared in the same way as those of the present study is identical to the relative change in the sample length as measured by dilatometry.5 As the mean lattice parameter measures the mean volumetric strain, we can infer ∆V/V from the dilatometer data. For small strain, ∆V/V ) 3∆l/l. This allows computing of the pressure in eq 2 except for an unknown constant, the pressure in the (arbitrary) reference state. On the basis of the experimental pressure values, and using the BET data for R, the surface area per volume, we determined the mean change in interface stress 〈∆f〉A from
〈∆f〉A ) -9K(∆l/l)/(2R)
(3)
(17) Macfarlane, R. E.; Rayne, J. A.; Jones, C. K. Phys. Lett. 1965, 18, 91.
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Figure 2. Exemplary results of in situ dilatometry on porous platinum immersed in NaF electrolyte for three selected concentrations: 1.0, 0.2, and 0.05 M electrolyte. Each of the figures (a-i) shows the results of 10 subsequent cycles superimposed. (a-c) variation ∆Q in total surface charge versus electrode potential E. (d-f) strain ∆l/l0 versus E. (g-i) variation ∆f of the surface stress versus variation ∆q of the surface charge density.
Figure 3. Results of in situ dilatometry on porous Pt immersed in NaF (1 M) electrolyte as in Figure 2 but measured before (solid line and closed symbols) and after (broken line and open symbols) the experiments in the dilution series: (a) variation ∆q in superficial charge density versus electrode potential E; (b) variation ∆f of the surface stress versus variation in superficial charge density ∆q. Results in a and b are averages over 10 successive cycles.
The bottom row of graphs in Figure 2 shows the results for ∆f, plotted versus the superficial charge density q. It is seen that the hysteresis is lost, in support of f being a function of q. We have obtained σ in the following way based on the experimental results for f(q). Each of the f(q) curve was fitted by a linear law f ) f0 + σq. The slope determined from the fitted straight line gives the surface stresscharge coefficient for each individual scan. This procedure
was applied individually to each scan, and σ was obtained from the average of the 10 scans. By comparison of eqs 1 and 3, it is readily seen that the experimental values for f and for q both depend in the same way on the specific surface area R. Therefore, when σ is determined from the slope of f(q), the result is independent of the value of R. In other words, the experimental error in R does not affect the result for σ; in fact, σ can be determined even if R is unknown. Furthermore, this means that σ can be determined even in the presence of contaminants that block the surface completely by adsorption if they are not electrochemically active in the potential range studied. As mentioned in the Introduction, investigations of the double-layer behavior in the dilution series face the fundamental problem that, as the solution becomes more dilute, the charging kinetics are slowing down due to the diminishing ionic conductivity. The hysteresis in our voltammograms testifies to this effect. Due to the slow ionic diffusion kinetics at high dilution, the surface in the interior of the porous samples does not reach equilibrium during the time scale of the experiment. Despite this limitation, it can be argued that our data supply meaningful values for the surface stress-charge coefficient: Since the voltammograms as well as the cyclic strain data are reversible and reproducible, the experiment probes reversible changes of state of the surface around the stable reference state which is established during the potential
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Figure 4. The negative of the surface stress-charge coefficient σ plotted versus the electrolyte concentration X.
hold time before the start of the cyclic scans. The effect of the kinetic limitations is to constrain the amplitude of the variation in the superficial charge density in the interior of the sample to values below the equilibrium value corresponding to the experimental voltage amplitude. As we are interested in small variations within the regime of weak adsorption, this effect is not undesirable. Furthermore, even for nonuniform changes in f along the pore surface, the external strain provides a valid measure for the mean pressure and, hence, for the mean value of f. In as much as σ can be approximated as a constant for small variations in q, a plot of mean f versus mean q provides the correct mean value of σ, even when kinetic limitations result in a nonuniform charge distribution. Figure 4 shows the graph of σ versus the concentration X. It is seen that σ is negative and that between X ) 1 M and X ) 0.02 M its value varies by a factor of more than 2. As the electrolyte is becoming more dilute, the magnitude of σ initially increases, up to X ) 0.2 M. Below 0.2 M, σ remains independent of the concentration within error bars, at the value -1.9 ( 0.2 V. To study the influence of specific adsorption on σ, we have carried out in situ dilatometry in the more concentrated solution, X ) 1 M, during scans with the larger potential interval, -0.25 to 0.5 V as in Figure 1a. This yields σ ) -0.6 ( 0.2 V. The result is comparable to the value reported for platinum in KOH electrolyte, -0.7 V,5 which is not surprising since in both cases OH adsorption is the dominant process. However, the σ value in the adsorption region at X ) 1 M is considerably smaller, roughly three times, than that obtained at high dilution when the adsorption is weak, indicating that charge transfer to the adsorbed OH ions reduces the σ value. Discussion and Conclusions The two essential findings for the surface stress-charge coefficient are (i) the value of σ measured in the nominally capacitive part of the cyclic voltammogram varies substantially as a function of the electrolyte concentration and (ii) at a given electrolyte concentration, the value of σ measured in the OH-adsorption regime is less than that in the capacitive regime. The finding that, in the dilute solution, a given amount of charge is more efficient in changing the surface stress than it is in the more concentrated one implies that the microscopic processes by which the electronic charge is accommodated in the metal surface change as the solution becomes more dilute. It appears obvious to relate this variation to the decreasing contribution of specific adsorption at the higher dilution. The increase in the magnitude of σ at higher dilution is in agreement with the notionssuggested in ref 3sthat, in less dilute electrolytes, part of the electronic charge is transferred into bonds
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with chemisorbed ions, thereby reducing the effectiveness of the electronic charge in changing the surface band structure of the metal. This argument is independently supported by the finding that the magnitude of σ during capacitive charging is larger than that measured during OH adsorption. Thus, the results of our study indicate that the charge accumulated in the space-charge region at the metal surface has a pronounced effect on the surface stress of the metal-electrolyte interface. In other words, the correlation between the extra charge in the metal and the bonding between the metal surface atoms plays a crucial role in determining the variation of the surface forces when the interface is polarized. A similar correlation between the charge transferred to the metal surface and the change in surface stress has also been invoked in discussing adsorption from the vapor,2 where the polarization of the interface results not from double layer charging in response to an applied potential but from the relocation of charge during the formation of bonds between the adsorbate and the original surface. This notion is well compatible with our observation, if it is borne in mind that in weak electrolytes, such as the ones investigated here, the ions in solution have a highly stable solvation shell and that the interaction of the solvated ion with the surface is considerably weaker than the interaction of the neutral atom with the metal-vapor interface. Remarkably, σ is found to become independent of the electrolyte concentration at the highest dilution. As argued in the Introduction, this would be expected if the idealized case of dominantly double-layer charging was reached. However, on the basis of Pajkossy and Kolb’s18 investigation of the Pt(111) surface in dilute solutions of KClO4 and NaF, one must conclude that some degree of adsorption remains present in all our experiments. Nevertheless, the plateau in σ(X) at high dilution is compatible with the assumption that the contribution of metal-adsorbate or adsorbate-adsorbate bonds on the surface stress is small, so that capacitive processes control the net value of σ even when some adsorption remains present. It is found that the strain of our samples and, hence, f exhibit hysteresis when plotted versus the potential but that there is no hysteresis in plots of f versus the charge. This implies that f is a function of q but not of E and that, consequently, σ ) ∂f/∂q may be both the more fundamental and the empirically more appropriate materials parameter of the surface as opposed to ∂f/∂E. It is also significant that our determination of σ rests on data for the macroscopic strain and for the total charge on the sample, which are readily measured, whereas it is not required to know the surface area A. Thus, the danger of artifacts due to errors in A arising from surface roughness or cracks in thin films, or from inaccurate modeling of BET data or coarsening of the pore structure in porous bodies, is greatly reduced when σ as compared to ∂f/∂E is measured. Furthermore, due to the independence of σ on the surface area A, the experimental value of σ is comparatively robust when the surface picks up contaminants that are inactive in the potential range used, thereby blocking the surface locally and reducing the overall active area A. We have discussed above that the reproducibility of the cyclic voltammograms argues against significant contamination in the present experiment. If a surface is covered with a thin, sticking, inactive adsorbate layer, it still has a certain double layer capacity and a potential dependence of f. For such a surface with an inert layer that prevents specific interactions of the (18) Pajkossy, T.; Kolb, D. M. Electrochem. Comm. 2003, 5, 283.
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double layer ions with the metal, f and σ depend on the charge density only, but not on the concentration of the electrolyte. We observed a concentration dependence, which also means that we can exclude that a large fraction of the surface is covered by such an inert layer. If we assume that a part of our platinum surface was covered by such a contamination adsorbate, an even larger concentration dependence would be expected for pure platinum. The surface areas of nanoporous samples are relatively large, and the potential window used for the determination of σ was rather narrow, which reduces the risk that contaminants contribute to the current. Due to all the facts discussed heresin addition to our precautions to keep the sample as clean as possiblesit is not likely that the result is affected by contamination which has always to be considered in the case of the highly reactive platinum. The surface stress-charge coefficient is defined as the derivative of f with respect to q at constant strain. However, the two experimental methods used to determine f, measurement of cantilever bending and of expansion of porous solids, determine changes in f by analysis of the resulting deformation. In other words, the experimental variation of f with q is not a variation at constant strain, an effect which is more severe in high surface area materials compared to cantilevers, since the strain is considerably higher in the former case. Our identification of the experimental ratio δf/δq with σ must therefore be subject of critical examination. In a hypothetical process, let us divide the change of state of the surface during charging into two steps, charging at constant strain followed by straining at constant charge. For isotropic surfaces the total variation in f is
δf ) σ δq + ∂f/∂e|q δe
(4)
where δe ) trace(δe) denotes the relative change in surface area. In the porous material, the surface strain can be approximated by δe ) 4Rδf/(9K) ) 4Rσδq/(9K). Thus, in our experiments
δf/δq ) σ[1 + 4RcS/(9K)]
(5)
where the parameter cS ) ∂f/∂e|q represents an excess elastic constant of the surface.19 No experimental values of cS have been reported so far, but the results of atomistic
simulations of the elasticity of {100} surfaces of Al and Si may provide guidance in estimating its magnitude: Miller and Shenoy20 considered flat plates of thickness h in tension; they found that the effective Youngs modulus Y of the plate scales with h and with the Youngs modulus Y0 of the bulk material according to Y ) Y0(h + 2h0)/h with h0 negative, of the order of -1 Å. In other words, the surface of the simulated materials is softer than the matter in the bulk, and the Youngs modulus of the plate equals that of a homogeneous material which is 1 Å less thick, at each surface, than the actual plate. By equating the force required to strain the plate to the appropriate combination of bulk stress and surface stress, one finds that cS ) 1/2h0Y/(1 - ν) with ν the Poisson number. Using the values for Pt, Y ) 170 GPa and ν ) 0.39, along with h0 ) 1 Å, one finds the estimate cS ) -14 N/m. By means of eq 5, this is found to imply δf/δq ) 0.987σ, in other words, the experimental δf/δq underestimates the magnitude of σ by of the order of 1%. Since this is well within the experimental uncertainty, it is concluded that the experimental values provide a valid approximation of the true value of σ. In principle, eq 5 could be used to measure cS by analysis of the variation of δf/δq with R when materials of different pore size are examined. However, materials with much larger specific surface area than presently available appear to be required in order to obtain a significant variation of δf/δq. We emphasize that our data represent averages over all surface orientations in the porous material. Thereby, they are less fundamental than data obtained by investigation of single crystal surfaces. Still, such experiments are of relevance, since the average values are exactly what is required for assessing the materials performance when one considers applications of porous materials as actuators, which have been proposed based on their attractive values of stroke and strain energy density.5,7,21 Acknowledgment. Support by Deutsche Forschungsgemeinschaft (Center for Functional Nanostructures) is gratefully acknowledged. LA0473759 (19) Gurtin, M. E.; Weissmu¨ller, J.; Larche´, F. Philos. Mag. A 1998, 78, 1093. (20) Miller, R. E.; Shenoy, V. Nanotechnology 2000, 11, 139. (21) Baughman, R. H. Synth. Met. 1996, 78, 339.
