J . Phys. Chem. 1987, 91, 3559-3564 TABLE II: Electron Populations and Overlap Populations of Interacting Orbitals for the Adsorption of a Hydrogen Molecule to Ni Surfaces
electron population overlap orbital Ni surface H2 population sym antisym sym antisym
1.120 1.943
SYm
1.317
antisym sym
antisym
1.092 1.857 1.943 1.334 1.861
2.043'
-0.117
0.045 2.006 0.138
-0.074
2.044 0.044 2.018 0.132
0.034 0.082 -0.128 0.033 -0.098 0.078
aThis is due to the application of Mulliken population analysis. The partitioning of the total electron population into the fragment interacting orbitals is only tentative. thicker film may be appropriate to represent the repulsive interaction. Table I1 compares the (1 10) long bridge site and the (1 11) surface for two different H-H distances, 0.74 and 1.00 A. When the H-H bond is stretched to simulate the dissociation of the hydrogen molecule, the electron population and negative overlap population of the symmetric pair are reduced more efficiently on the (1 10) long bridge site than on the (1 11) surface. Electron delocalization is facilitated in both cases. Though the changes are yet very small at this stage, a similar calculation on clusters has shown that this trend is strengthened rapidly as the H-H separation is increased further. Conclusion
Though we should refrain from deriving a definitive conclusion from the present crude M O calculations, it is very likely that the experimentally observed preference of (1 10) surface to the (1 11) surface14dveis due primarily to the overlap repulsion.2o The (100) surface and the (1 10) short bridge site are not active for electron (20) The Fermi level was calculated to be -8.70 eV for the (1 11) surface and -8.91 eV for the (1 10) surface. This suggests that the (111) surface has a strong ability for electron donation.
3559
donation to the adsorbate. Unlike the other surfaces, electron delocalization takes place significantly from the second layer in the adsorption onto the (1 10) long bridge site. This is obviously due to the structure of the surface. The Ni (1 10) long bridge site is not so much closely packed by the Ni atoms as the (1 11) surface, and therefore, the Ni atoms in the second layer are exposed to the attack of the hydrogen molecule. This kind of delicate balance between the bonding and antibonding interactions for the variation of surface-adsorbate separation seems to be a clue to the origin of activation of molecules by irregular defects in surfaces. Clusters and periodic surfaces have been compared in calculating the adsorption energy and other physical q ~ a n t i t i e s . ' ~The ~,~ problems related to the use of cluster MO's in studying the surface-adsorbate interactions were reviewed by Messmer.lo We tried here to minimize the possible discrepancy between the cluster and crystal studies by carrying out the recombination of the fragment orbitals to evaluate properly the contributions of all the MO's to the interaction.2' The cluster and its extended system give rise to localized interacting orbitals which appear to be very similar to each other, indicating that chemical interactions are more or less localized in a certain number of atoms around the interaction site. Even if the orbitals look very similar, their potentials for electron donation and electron acceptance would be different, depending on the fields that span those orbitals. An attempt to get projected density of state on our localized interacting orbitals is underway to study this problem. We may note here that the low-lying bands participate more significantly in electron delocalization as the adsorbate comes closer to the surface. This signifies that the reactive region begins to be localized on the surface, as we have seen a similar trend in the reactions of smaller organic Acknowledgment. We are grateful to the Ministry of Education for financial assistance under a Grant-in-Aid for Scientific Research (No. 60550572). A part of the calculation was carried out at the Computing Center, Institute for Molecular Science. Registry No. H,, 1333-74-0; Ni, 7440-02-0; ethylene, 74-85-1; 1,3,S-heptatriene, 2196-23-8; 1,3,5,7,9-decapentaene,2423-91-8; polyacetylene, 25067-58-7. (21) The interacting orbitals are derived not only from the AO's but also directly from the canonical MO's.
Electrochemistry at Pt Band Electrodes of Width Approaching Molecular Dimensions. Breakdown of Transport Equations at Very Small Electrodes Rae1 B. Morris, David J. Franta, and Henry S. White*+ Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455 (Received: October 20, 1986; In Final Form: March 12, 1987) The electrochemical behavior of platinum and gold band electrodes, O S to 1 cm long and 20 to 500 8, wide, is reported. Quasi-steady-statevoltammograms are observed at all electrodes for the oxidation of ferrocene (in acetonitrile) and ferrocyanide (in water). Limiting currents obtained at band electrodes of width >200 8, are in quantitative agreement with values predicted by semiinfinite diffusional transport of a soluble redox species to an infinitely long hemicylindrical electrode. For electrodes of width 200 A. The results of only 200- and 500-&thick Au films are reported here. Electrical contact was made by attaching a fine copper wire to the metal film by using Ag conductive paint. The entire assembly was placed in an inverted rubber septum used as a cylindrical mold and the mold was filled with epoxy (Magen Scientific Inc., CM-161 resin and hardener). The epoxy was allowed to cure to hardness at room temperature for 1-2 days. The hardened cylinder was removed from the mold and ground flat (perpendicular to the Pt film) with silicon carbide paper to 600 grit. This procedure yielded band electrodes nominally 1 cm long and 20 to 500 A wide, Figure 1. Instrumentation and Chemicals. Electrochemical results were obtained in a conventional three-electrode cell arrangement using equipment previously described.1° The counter electrode was a large Pt wire and the reference electrode was a sodium saturated calomel electrode (SSCE). Tetrabutylammonium perchlorate (TBAP) was recrystallized twice from acetone/ether, dried under vacuum at 100 OC, and stored in a drying oven at 90 OC. Acetonitrile (Aldrich, HPLC grade) was stored over molecular sieves. Decamethylferrocene (DMFc), ferrocene (Fc), and potassium ferrocyanide were used as received.
Results and Discussion Kovach et a1.I4 quantitatively examined the current-time behavior for the oxidation of soluble redox species at 20-pm-wide Pt band electrodes. These authors found that the i-t response following a potential step to induce a diffusion-limited response was accurately described by the asymptotic long-time limit of the expression derived by for radial flux to a hemicylindrical solid boundary: i ( t ) = nFDIC%/ln [4Dt/ro2]
(1)
Here, D is the diffusion coefficient, F is the Faraday, 1 is the electrode length, is the concentration of redox species in bulk solution, t is time, and ro is the effective electrode radius. In calculating values of i(t), it was assumed that the band electrode width w was sufficiently small that the diffusion current was described by a cylindrical flux to the electrode surface, allowing substitution of ro by w/a,where w is the electrode width.14 This assumes equivalent behavior of a hemicylindrical electrode of radius ro with a band electrode of the same area and length. Equation 1 predicts that the magnitude of the current at microband electrodes is remarkably insensitive to the electrode width. cm2/s, t = 100 s, and all other values For instance, taking D = constant, we find that eq 1 predicts a factor of 2 decrease in the magnitude of the current upon reducing the electrode width from 10000 to 10 A. The validity of eq 1, however, for very small electrode dimensions ( w < 500 A) has not been quantitatively examined. In an investigation of heterogeneous charge-transfer rate measurements at small electrodes, Wehmeyer et al. noted, but did not explore, an anomalously low limiting current for a single 5-nm Pt band e 1 e ~ t r o d e . l ~ ~ Figure 2 shows the electrochemical response a t Pt band electrodes in H 2 0 containing 10 mM Fe(CN)6e as the redox active (23) Jeager, J. C. Proc. R . SOC.Edinburgh 1942, A b l , 223. (24) Delahay, P. New Instrumental Methods in Electrochemistry; Interscience: New York, 1954.
1
Ferrocyanide
4
0
A
I
50 nA
L
1 T 10 nA
I
20A
0.2
0.0 V
vs
0.4
SSCE
Figure 2. Voltammetric response of 20-, 40-, 80-, 100-, and 500-&wide Pt band electrodes in aqueous solution containing 9.71 mM K,Fe(CN), in 0.1 M KCI. The electrode lengths were 8.0, 9.1, 11.5, 10.5, and 11.0 mm, respectively. Scan rate: 20 mV/s. Limiting currents measured with respect to the zero current base line at the end of the positive sweep.
species and 0.1 M KCI as supporting electrolyte. Sigmoidal shaped voltammograms, corresponding to the oxidation of Fe(CN)64-, were obtained at electrodes of nominal width between 20 and 500 A. Each voltammogram represents a typical curve from a set of measurements obtained at a single electrode. Between measurements, the electrode was repolished on silicon carbide paper. The variance of limiting currents for a set of measurements (typically 3-5 runs) that differ only in polishing was typically on the order of 50%. The variance in the average value of a set of measurements using different electrodes of the same nominal width is of the same magnitude. The point here is that the uncertainty in absolute currents is due as much to the effects of polishing as to the film deposition procedure. Voltammograms obtained for the oxidation of ferrocene in acetonitrile (0.1 M TBAP) are qualitatively and quantitatively similar to those shown in Figure 2. Limiting currents from both sets of measurements are plotted in Figure 3 as i/nFID@a vs. 1/[In (4Dt/ro2)].The solid line in Figure 3 represents Jaeger's s o l ~ t i o n , eq 2 ~ 1; ~ ~the ~ slope of this line is 1. Equation 1 describes the i-t behavior following a large amplitude potential step, where the boundary condition24at the electrode surface is C(0,t) = 0 for t > 0. Its use in predicting the voltammetric limiting current, where the surface concentration is only zero at potentials 100 mV positive of the half-wave potential, is limited to slow scan rates. In plotting experimental points in Figure 3, we have used a value of t = 25 s, which corresponds to the time required to scan the potential over a 0.5-V range at 20 mV/s. The choice o f t = 25 s is somewhat arbitrary and other values (e.g. 20 s) could have easily been used. The particular value of t does not affect the trend of experimental points outlined by the dashed line in Figure 3. However, the choice of t does shift the position of the experimental points slightly along the abscissa, but for 15 < t C 30 s this error is small. The reason for this is that eq 1 predicts a weak dependence of the diffusion current on t for large values of Dt/ro2. The smallest value of Dt/ro2used in lotting experimental points in Figure 3, corresponding to a 500-1-wide electrode in Fe(CN)64- solution, is 10'. Finally, we have compared the limiting voltammetric current with the current at 25 s following a potential step from 0 to 0.4 V in a Fe(CN),4- solution. These values agree to within 5%.
3562 The Journal of Physical Chemistry, Vol. 91, No. 13, 1987 I
6
1
I
I
I
I
I
I
Morris et al.
I
T j
Ferrocene
i C 3.6
4.0
4.4
4.8
5.2
Figure 3. Plot of normalized limiting current ( i / n F l D @ ~ )vs. {In [4Dt/ro2])-' obtained at Pt and Au band electrodes. Solid line (slope = 1) corresponds to eq 1. Dashed lined drawn through experimental points obtained for the oxidation of 1-10 mM ferrocene (0)in acetonitrile (0.1 M TBAP) and 1-10 mM ferrocyanide ( 0 ) in H 2 0 (0.1 M KCI).
Electrode widths are indicated on the plot. Number of independent measurements for ferrocyanide: 10 (20 A), 7 (40 A), 3 (60 A), 6 (80 A), 5 (100 A), 4 (500 A); for ferrocene: 4 (40 A), 2 (100 A), 2 (200 A), 5 (500 A); Error bars indicates one standard deviation from mean. See text for discussion of D and t values used in plotting experimental points.
Figure 3 shows that the limiting currents obtained for the oxidation of ferrocyanide in H 2 0 and ferrocene in MeCN at 500-&wide Pt and Au electrodes are in excellent agreement with eq 1. In plotting experimental points we have used reported diffusion coefficients of 0.7 X and 2.4 X cmz/s for ~~ Beginning with electrodes Fe(CN)64-25 and F c , respectively. of 200 A width, the limiting currents deviate strongly from the theoretical values. At even smaller electrode dimensions ( 0, used in obtaining eq 1, leads to a physical inconsistency for electrodes of width approaching molecular dimensions. The problem can be qualitatively stated as follows. While eq 1 predicts a diffusion current depending on In ro, the electrode area depends linearly on r,. Therefore, as ro decreases, the current density and concentration gradient at the electrode surface become increasingly larger. When the electrode radius is reduced to very small dimensions, significant variations in concentration are predicted to occur over lengths that correspond to molecular dimensions. For instance, the concentration gradient is infinite at a line electrode (zero width). We show that for values of ro corresponding to band electrode widths less than 100 8,the surface concentration of the reacting redox species necessarily deviates from a zero value, yielding a current smaller than predicted by eq 1 . Consider the reversible one-electron oxidation of a reduced species R at a hemicylindrical electrode. The limiting current given by eq 1 is assumed to hold. The concentration of R in the solution layer immediately adjacent to the electrode (Figure 4 ) , and of thickness 22, is given by
where NRz is the number of moles of R in the surface layer and P is the volume of the layer. The superscript z is used here to denote a surface layer that defines the surface concentration of R. Because ions and molecules occupy a finite volume, the value of z should correspond roughly to a characteristic dimension of (25) von Stakelberg, M.; Pilgram, M.; Toome, V. 2.Elektrochem. 1953, 57, 342. (26) Kuwana, T.; Bublitz, D. E.; Hoh, G.J . Am. Chem. Soc. 1961, 82, 5811.
b '
r Figure 4. Top: Schematic drawing indicating the magnitude of reaction zone thickness z , relative to electrode radius, ro. Bottom: Qualitative sketch of concentration profiles at a hemicylindrical electrode as a function of z / r o at constant t .
R. Throughout this paper, we will assume that z is approximately equal to the radius of R. If the oxidation of R is diffusion limited, NR' is given by2' N
RZ
- 1''idt - nF I ,
(3)
where tz - t , represents the time taken for R to diffuse through the reaction layer and react instantaneously at the electrode, and the product to diffuse out. Since eq 1 predicts an essentially steady-state current for electrodes of radius > z , and for reasonable values of E, eq 7 reduces to CRz/CRb= 0, which is the boundary condition used to solve the diffusion equation yielding eq 1. For instance, if ro = 0.1 mm and z = 10 A, then CR'/CRb is less than 0.001 for any value of 5 smaller than 250 at times longer than 1 s. If r and z have comparable dimensions, then eq 7 predicts a nonzero concentration of R in the reaction zone layer adjacent to the electrode surface. The behavior predicted here is represented qualitatively in Figure 4. Qualitative concentration profiles are sketched for different electrode radii ro. To a first-order approximation, the current should scale with the difference between the bulk and surface concentrations, CRb - CRz,yielding a deviation from the diffusion controlled flux, eq 1 , as follows:
T .1 'C
42.9 nA
and after substitution of eq 7
I
For large electrodes, eq 9 is equivalent to eq 1 , as expected. The corrected limiting current given by eq 9 is plotted in Figure 5 with 6 = 3 and for several values of z, and in Figure 6 with z = 20 8, for several values of [. Inspection of Figures 5 and 6 indicates that the critical dimension at which the current deviates from eq 1 is primarily a function of the dimensions of R, while the magnitude of this deviation is more strongly effected by the fluid properties near the electrode surface. However, for z = 10 A, eq 9 predicts a significant decrease in the quasi-steady-state limiting current (at t = 25 s) beginning a t ca. 2000 A, regardless of the value of t. Comparison of Figure 3 with Figures 5 and 6 demonstrates qualitative agreement between the experimental values and eq 9 for 10 < z < 20 A and 3 < $. < 5. Because of the qualitative arguments leading to eq 9 and the assumptions made concerning the electrode geometry, adjustment of z and f to obtain a more exact fit is unwarranted. The arguments leading to eq 9 contain the assumption that electron transfer occurs with R in intimate contact with the electrode. A nonzero electron-transfer distance would have the effect of increasing the apparent radius of the electrode, resulting in larger currents than predicted by eq 9. Because the diffusion current depends on In ro, this effect is very small. In order to maintain equivalent surface concentrations for different electrode reactions, it can be argued from the preceding
- 0.4
I
0.0
I
I
0.4
0.8
V vs. SSCE Figure 7. Voltammetric response of 500-, loo-, and 20-A-wide band electrodes in nitrobenzene (0.2 M TBAP) containing 9.99 mM decamethylferrocene ( E l l 2= 0.03 V) and 6.02 mM ferrocene (Ell2= OSV). Scan rate = 20 mV/s. Dashed line indicate base lines to measure limiting currents for ferrocene oxidation.
discussion that the size of the reaction zone, z, should scale with the molecular dimensions of R. Comparison of predicted current values for z = 10 A and z = 20 A in Figure 5 at values of In (4Dtlr:) = 20 and In (4Dtlr:) = 25 (corresponding to electrode widths of approximately 500 and 30 A, respectively) indicates that the ratio of limiting currents, i ( z = 10 A ) / i ( z = 20 A), increases from 1.1 at 500-A wide electrodes to 1.7 at 30-Awide electrodes. To test this prediction, we measured the limiting currents corresponding to the oxidation of ferrocene (Fc) and decamethylferrocene (DMFc) in nitrobenzene (NB) solutions at 500-, loo-, and 20-A-wide electrodes. The choice of this system was based on the similarity of the redox chemistry, the solubility of both compounds in nitrobenzene, and the substantial difference in the radii of the two molecules. In addition, the half-wave potentials of DMFc and Fc are separated by 0.6 V in NB, allowing simultaneous measurement of limiting currents for both oxidations in a single voltammetric experiment. Figure 7 shows one set of
3564 The Journal of Physical Chemistry, Vol. 91, No. 13, 1987 TABLE I: Limiting Currents for Ferrocene and Decamethylferrocene Oxidation in Nitrobenzene’ electrode
run
500A
A B
ilim(Fc)X A 220 260 137 370
C D
IOOA
A B
C D E 20A
A B
C D “0.1
ilim(DMFc)X lo4, A 320 530 259 660
ilim(Fc)/ ili,(DMFc) 0.68 0.49 0.52 0.56 av = 0.56 f 0.08
27.6 22.5 24.0 40.0 32.5
41.5 34.0 27.0 68.5 52.5
0.58 0.66 0.88 0.58 0.62 av = 0.67 f 0.11
7.0 8.5 6.7 10.1
8.2 10.0 8.7 11.0
0.85 0.85 0.78 0.92 av = 0.85 f 0.05
M TBAP as supporting electrolyte.
04-
, 100
C
200
300
400
500
r., A
Figure 8. Ratio of limiting currents obtained for ferrocene (i(Fc)) and decamethylferrocene (i(DMFc)) as a function of Pt band width. Same solution conditions as in Figure 7.
measurements obtained in N B containing 0.1 M TBAP at a scan rate of 20 mV/s. The voltammetric curves corresponding to the different size electrodes were obtained in the same solution so that direct comparison of the ratio of limiting currents can be made without correction for bulk concentrations of Fc and DMFc. For similar reasons, comparison of limiting currents at the same electrode minimizes the possible effects of any uncertainty in the total active electrode area or geometry. The results of these experiments and others are given in Table I. Two trends are apparent from the data in Table I. First, the absolute limiting currents for both Fc and DMFc oxidation decrease rapidly with decreasing electrode width, consistent with the results obtained for Fe(CN)6” (in H20) and Fc (in MeCN), Figure 3. Second, as the electrode width is made smaller, the ratio of limiting currents, i(Fc)/i(DMFc), increases from 0.56 at 500 8,to 0.85 at 20 8,,Figure 8. The trend in limiting current ratios is predicted by the size effect theory and yields a value of z(Fc)/z(DMFc)
-
0.65. Since t h e reaction layer thickness z is
roughly proportional to diameter of R, the size effect should scale with molecular dimensions, resulting in lower diffusion currents
Morris et al. for larger molecules. A z ratio of 0.65 is qualitatively expected based on the relative diameters of the two redox molecules. Wave-Shape Analysis and Electron- Transfer Kinetics. A key element of the proposed model is that the current is determined by local fluid properties, e.g., viscosity, diffusivity, at the electrode surface (within a distance of ca. 22). These properties, in turn, depend on the potential and charge distribution which vary as the applied voltage is scanned during a voltammetric experiment. For this reason, complete waveshape analysisi6of the voltammograms obtained at electrodes of molecular dimensions will require knowledge of the local fluid behavior as a function of the electrode potential. The effects of electrode kinetics on the voltammetric response are expected to become increasingly more important as the electrode width is reduced. However, a slow electron-transfer rate is compensated, in part, by the decrease in molecular transport as the electrode width is reduced. Thus, waveshape characteristics useful in determining kinetic parameters, e.g., half-wave potential, are expected to be less dependent on the electrode radius than previous predictions have s ~ g g e s t e d . ~ ~ , ~ ’
Conclusion We have demonstrated that band electrodes of width approaching the dimensions of small molecules can be fabricated by simple evaporation techniques. Qualitatively, the electrochemical behavior of these electrodes in voltammetric experiments is similar to larger microelectrodes (micron dimensions). The magnitude of voltammetric limiting currents deviates strongly from theoretical predictions at electrodes of width
