Solid State Electron Self-Exchange Dynamics in Mixed Valent Poly

Chem. , 1994, 98 (16), pp 4343–4351. DOI: 10.1021/j100067a022. Publication Date: April 1994. ACS Legacy Archive. Cite this:J. Phys. Chem. 98, 16, 43...
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J . Phys. Chem. 1994,98, 4343-4351

4343

Solid State Electron Self-Exchange Dynamics in Mixed Valent Poly(viny1ferrocene) Films Melani G. Sullivan+and Royce W. Murray' Kenan Laboratories of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290 Received: October 21, 1993; In Final Form: February 4, 1994"

The redox conductivity of mixed valent poly(viny1ferrocene) films is measured by a steady state method using interdigitated array electrodes. Spectroelectrochemistry shows that the Fc/Fc+ site concentration ratio has a simple Nernstian dependence on applied potential. Measurements of the bimolecular Fc/Fc+ electron selfexchange rate constant using concentration gradient- and electrical gradient-driven electron transfers are in agreement and give an average value of 2 X lo5 M-l s-l for dry Nz bathed films. The solid state dispersive transport model of Scher, Montroll, and Pfister is used in interpretation of the electrical potential gradierttdriven electron transport. The rate constant kEX remains roughly constant as the concentration ratio of Fc+ and Fc sites is varied from 0.04 to 3.0.

The dynamics of electron self-exchange (or hopping) reactions in solid media between diffusively immobile, electronically welldefined molecular electron donor/acceptor pairs has been an active area of study.' Glassy insulating materials containing mixtures of electrondonor and acceptor molecules,' usually transition metal complexes or conjugated non-metal systems, have conductivities in the range lP9-10 Q-I cm-l. Our laboratory has pursued2 versatile experimental routes to measure bimolecular electron transfer rate constants in mixed valent solid media, notably polymeric phases containing metal complexesand organometallic species. These methods involve preparation of ultrathin, mixed valent films sandwiched between metal electrodes in a manner permitting (i) exposure to and equilibration with a variety of liquid and gaseousbathing environments2cJvgJand (ii) application of a potential bias under conditions that allow, or do not allow, interfacial electrode/film electrolysis and development of concentration gradients of donor and acceptor sites in the film's bulk. In this way we are able to inspect the dependency of electron transfer rates on the state of solvation of the film and to extract rate data for transfers driven by gradients of electrical potential aloneZi-l or by gradients of donor/acceptor concentration.%-hThe work aims both at reliable procedures and theory for measuring solid state electron transfer processes and at a more detailed understanding of their behavior than exists presently. This paper presents a study of electron transport through mixed valent films of poly(vinylferrocene), PVF, in which electron transport occurs by ferrocene/ferrocenium bimolecular selfexchange reactions

Fc,

+ Fc;

kex

Fc,'

+ Ec,

where the subscripts denote position. In the experiments described, thin films of PVF on interdigitated electrode arrays (IDA'S) are made mixed valent in Fc and Fc+,and their currentpotential responses are measured with the films immersed in either acetonitrile liquid, acetonitrile vapor, or dry Nz. When in liquid or vapor acetonitrile, the film's perchlorate counterions are sufficiently diffusive that application of a potential bias causes electrolysis at the electrode/polymer interfaces, converting Fc+ to Fc at the negative terminal and the reverse at the positive terminal. Currents arising from electron hopping down the resulting crossed concentration gradients are described as beforeh4 by a Fickian model as equivalent electron diffusion coefficients De, which are related to the electron self-exchange Present address: Paul Scherrer Institut, CH-5232Villigen PSI, Switzerland. * Abstract published in Aduance ACS Abstracts, April 1, 1994.

rate constant of eq 1 byla.3

where C, is the total Fc site concentration and 6 is the average intersite distance, (CTN*)-II~,with NA = Avogadro's number. This experimental situation, termed concentrationgradient-driven electron transport, is characterized by sigmoidallyshaped currentpotential waves with limiting currents corresponding to limiting concentration gradients of electron donor and acceptor. In a dry N2 bath, the film's perchlorate counterions are, on the other hand, diffusively immobile on useful experimental time scales, so that application of a potential bias creates an electrical gradient across the bulk of the film, supplying impetus for electron hopping by an intersite potential 4 and free energy AG = -nF& We have described2i-Isuch currents, which rise exponentiallywith potential bias, in terms of a (modified) Marcus ratefree energy r e l a t i ~ nand , ~ refer to them as electrical gradient-driven. The films are capable of sustaining gradients of ca. 105 V/cm. On a molecular level this gradient, through large, amounts to only a modest solid state reaction free energy (a few tens of millielectronvolts),a value small in comparison to that of crossreactions familiar in homogeneous fluid solutions. This paper will address the following points regarding electron transport in mixed valent PVF films. I. In contrast to the well-known5 nonideal cyclic voltammetry of PVF films, spectroelectrochemical results show that long electrode equilibration times produce film behavior that is in accord with a simple Nernstian concentration-potential relationship. This is important, since it allows us to reliably establish the relative concentrations of Fc and Fc+ present in the film before an electron transport measurement. 11. Electron self-exchangerate constants for (1) are measured by concentration gradients in liquid and vapor acetonitrile and under dry Nz and by electrical gradients in dry Nz to evaluate the effects of bathing medium and free energy source on the electron transfer dynamics, as we have done before with other mixed valent materials.2 Additionally, we examine the bimolecular rate law by varying Cox/Crdfrom 0.04 to 3.0. 111. Our previous application of Marcus' theoryzi-Is6employed an empirical fitting parameter ( p ) because currents rise too steeply with potential. We have found that this fitting parameter tends to vary with the value of the electrical gradient, prompting examination of an alternative theory, that of Pfister, Scher, and Montroll,7which accounts for dispersion of electron hopping rates in disordered solid media.

0022-3654/94/2098-4343$04.50~0 0 1994 American Chemical Society

4344 The Journal of Physical Chemistry, Vol. 98, No. 16, 1994

Experimental Section Materials. Poly(viny1ferrocene) (PVF, Polysciences,nominal MW 50 000) dissolved in toluene (Fisher) at ca. 10 mM (in Fe) gave, after separation of a small amount of insoluble material, a clear solution which was used to cast films. Gel permeation chromatographyindicates that the molecular weight of dissolved polymer is somewhat less than the nominal, but larger than that of a second sample tested with nominal MW of 26 000. Spectroelectrochemistry. PVFfilms (60Ck1200nm thick) were cast from the toluene solution onto transparent tin oxide electrodes and warmed for 10 min at 70 OC to improve adhesion. The polymer-coated working electrode was immersed in an optical cuvette filled with 0.1 M Bu4NC104/CH3CNa The SSCE and Pt auxiliary electrodes were outsidethe light beam of the HewlettPackard 8452A diode array spectrophotometer. Potentials were applied to the film (at ca. 25-mV increments) until the current decayed to background levels (ca. 2 min at each potential), using a potentiostat and voltage sweep generator of local design.*Spectra were recorded at each applied potential, starting with a reduced film and moving to more positive potentials until the spectrum no longer changed (at ca. +700 mV us SSCE) and then incrementally returning to the original potential in the same manner. The absorbance at 636 nm was used to calculate the ratios of oxidized and reduced ferrocene sites, assuming the film to be completely oxidized at +800 mV and completely reduced at 0 mV us SSCE. Since films are slightly soluble in acetonitrile, film thickness was determined with a Tencor Alpha Step 100 surfaceprofiiometer before and after the spectroelectrochemistry. The ca. 25% decrease in film thickness was incorporated into conversions of absorbanceto quantities of sites, assuming a linear film loss with time. InterdigitatedArray Electrodes with Pt Fingers? The following four IDA geometries used were: IDA 11 (100 electrode finger pairs, 1-pmfinger width, 1-pmgap), IDA 32 (50 electrodefinger pairs, 3-pm finger width, 2-pm gap), IDA 52 (50 electrodefinger pairs, 5-pm finger width, 2-pm gap), and IDA 102 (25 electrode finger pairs, 10-pm finger width, 2-pm gap). All IDA'S have an oxidizedsilicon wafer substrate, a Si02 coating over theconnecting leads, and Pt fingers nominally 0.1 pm high and 2 mm long. To calculate the area of the electrodes, the finger walls facing one another across the gap are taken as parallel plate electrodes. For 2N fingers, finger length A, and finger height h, the parallel area is

A = ( 2 N - 1)Xh

(3)

The width of the fingers is ignored in this area calculation, which leads to a small dependence of the determined k E X on IDA finger width (see Tables 1-3, vide infra).l0 Preparation of Dry,Mixed Valent Films. PVF films (ca. 200 nm thick) were cast from toluene over the exposed Pt IDA finger region, with any film spreading beyond the finger region being removed with a solvent-soaked wipe. Films were warmed at 70 OC for 10 min to improve adhesion and electrolyzed at the desired potential us SSCE or Ag/Ag+ in 0.1 M Bu4NC104/CH$N with the IDA electrode finger sets shorted together, followed by rinsing with acetonitrile and drying under N2. Some films were made 1:l Fc:Fc+ mixed valent with a Pine Instruments RDE4 bipotentiostat, by potentiostatingone electrodefinger set negative of Eo and slowly (1 mV/s) sweeping the other to more positive potentials. The concentrationgradient-establishedlimiting currents can be used to calculate electron diffusion coefficients DE for Bu4NClO4/CH3CN-bathed films. Following removal from the electrolyte solution, the Fc/Fc+ concentration gradient is allowed to relax by removing potential control, and the film is rinsed and dried as above. ElectronTransport Measurements. Transport measurements were made using mixed valent films prepared as above with the

Sullivan and Murray film-coated IDA bathed in either CH3CN (saturated) vapor or in dry Nz. With the film bathed in acetonitrile vapor, the film's perchlorate counterions are relatively mobile, and Fc/Fc+ concentration gradients are readily established by applying a potential biasacrosstheIDA. The fiimcounterion mobilityduring fast *lo-V sweeps was, however, too large to achieve a pure electrical potentialgradient, without the signatureof i-E hysteresis indicating development of concentration gradients around the electrodes. When the Fc/Fc+ films are bathed in dry N2, on the other hand, counterion motion is thoroughly quenched, and f lO-V potential sweepsgeneratecurrent-voltage responses characteristic, by the absence of hysteresis in the i-E plot, of electron transport driven by a bulk phase electric field. Voltage sweeps at 50-0.1 Hz produce identically shaped responses. Triangular waves (k10 V) were supplied (Wavetek function generator) to a locally-built potentiostat with reference and auxiliary connections shorted together and connected to one set of IDA fingers and the working electrode connection made to the other finger set. Data were transferred from a Nicolet 310 oscilloscope to a microcomputer for Levenberg-Marquardt fitting" to eqs 7 and 8. Concentration gradient-based electron transport data in a N2 environment were obtained by applying the desired potential bias across the IDA fingers with the film bathed in acetonitrile vapor and with the bias still applied, switching the bathing gas to N2 to evaporate the CH3CN, and then recording the current. This point-by-point method produces a voltammogram of shapesimilar to that seen in CH3CN vapor, but with smaller currents.

Results and Discussion Determinationof @' and the Dependence of Concentration on Film Potential. In films of poly(viny1ferrocene) drop cast onto interdigitated array electrodes, mixed valency is most easily attained by electrolysis of the film in acetonitrile/Bu4NClO4 solution. Electrolysis is ideally conducted by applying an electrolysis potential which by application of the Nernst relation gives a known Fc/Fc+ site concentration ratio. Alternatively,to prepare a film with equal numbers of Fc+ and Fc sites (e.g., 1:l mixed valency), potentials on opposite sides and sufficiently removed from Eo' can be applied to establish limiting, crossed concentration gradients of Fc and Fc+, the film removed from the solution, and the potentials disconnected to allow the gradients to relax while the film is still solvent-wetted. The above procedures rely on knowledge of Eo' and on having Nernstian control of the concentration ratios. Films of poly(vinylferrocene) on electrodes typically, however, display cyclic voltammograms with nonideal shapes, as illustrated in Figure 1. The anodic and cathodic branches are both dissimilar and asymmetric, and there is a break-in effect in which currents in the first cyclical scan differ from those on subsequent scans. A variety of explanations have been offered for the nonideal voltammetry of PVF films. Peerce and BardM observed multiple peaks and used digitally simulated voltammograms to support a square scheme model with ferrocene sites of two different formal potentials. LavironSfand Daumsg proposed site interactions (activity effects) to account for the narrow anodic and broad cathodic peak. Another model,l2 supported by QCM experim e n t ~ , ~proposes ~ , ' ~ an oxidation state-dependentfilm resistance to counterion transport, to account for wave-shape and break-in effects. The significance of these voltammetric problems to electron transport studies lies in theaccurate experimental setting of mixed valent states and the appropriatenessof transport relations that connect rate and free energy. We accordingly undertook a spectrophotometric inspection of the PVF voltammetry, in experiments allowing longer film equilibration times than feasible in transient electrolysis like cyclic voltammetry. Oxidized PVF films are blue-green in color and, when reduced, amber yellow. Both forms absorb in the 400-500-nm range, but

The Journal of Physical Chemistry, Vol. 98, No. 16, I994 4345

Mixed Valent Poly(viny1ferrocene) Films

TABLE 1: Concentration Gradient-Driven Electron Transport Results bathing medium IDA B~NClO4/acetonitrile IDA 52/IDA 32 acetonitrile vapor IDA 11 IDA 32

IDA 11 1 pm finger width 1-pm gap width

IDA 32 3-pmfinger width 2-pm gap width

CFC* (M) 1.25 1.25 1.275 2.725 1.36 2.725 2.725 2.725 2.725 1.108 1.252 1.252 2.20 2.72 3.50 0.633 0.858 1.18 1.53 1.108

from ili, (eqs 4,5) DE (cma/s) km (M-I s-I) av valuesa (6.4 5 ) x 10-9 (1.6 k 1) X lo6 2.1 x 10-9 5.1 x 105 2.3 x 10-9 5.6 x 105 4.0 x 105 1.6 x 10-9 2.0 x 10-9 4.9 x 105 3.2 x 10-9 7.7 x 105 4.0 x 10-9 9.8 x 105 1.6 x 10-9 3.9 x 105 1.0 x l e 2.5 X lo6 2.1 x 10-9 5.1 x 105 av values (3.2 k 2) x 10-9 (7.9 f 4) x 105 4.9 x 104 2.0 x 10-10 1.4 X lo5 5.9 x 10-10 4.3 x 104 1.8 x 10-10

*

2.5 X 10-Io 2.1 x 10-10 1.9 X 10-I0 2.7 X 10-Io 2.8 X 10-Io 7.7 x 10-11 3.8 X 10-" 3.8 X 10-Io IDA 11 av (2.4 f 1) X 10-I0

6.1 X 5.1 x 4.5 x 6.5 X 6.8 X 1.9 x 9.3 x 9.3 x

104 104 104 104 104 104 103 104

from slope (eq 6) DE (cm2/s) km (M-I s-1) (6.3 k 5 ) x 10-9 (1.5 X 1) X 106 1.2 x 2.0 x 2.2 x 2.1 x 3.8 x 1.2 x 9.3 x 2.0 x (4.3

10-9 10-9 10-9 10-9 10-9 1 P i t 9

10-9 4) x 10-9

6.0 X 10-10

(5.8 2) x 104 6.0 X 9.6 X 10-l0 2.725 2.3 X lo5 2.725 9.8 X 10-Io 2.4 x 105 2.725 2.6 x 10-9 6.2 X lo5 2.725 8.5 x 105 3.5 x 10-9 2.725 6.7 X l 0 - I o 1.6 x 105 4.4 x 10-10 1.1 x 105 2.725 2.7 X 10-l0 6.7 x 104 1.36 2.4 X lo5 1.0 x 10-9 2.725 1.8 x 10-9 2.1 x 10-9 5.1 X lo5 2.725 4.0 x 104 1.9 X 10-l0 1.6 X 10-l0 1.108 6.1 X 10-Io 1.5 x 105 2.0 x 10-10 1.108 1.2 x 10s 4.3 x 10-10 4.8 X 2.20 IDA 32 av (1.1 0.8) x 10-9 (2.8 k 2) X los (6.6 f 8) X 2.8 x 10-9 0.886 6.8 X lo5 4.8 X lo5 1.25 2.0 x 10-9 2.725 2.4 X los 9.9 x 10-10 1.6 X 10-l0 3.8 x 104 1.108 2.7 x 105 3.35 1.1 x 10-9 3.8 x 104 1.6 X 1.8 x 10-10 0.830 IDA 102 av (1.2 f 1) x 10-9 (2.9 2) x 105 1.8 X 5.7 x 10-10 2.0 x 105 av of all dry NZdata 7.9 x 10-10

*

IDA 102 10-pm finger width 2-pm gap width

*

The average of 5 IDA 2 measurements and 16 IDA 32 measurements. only the oxidized form absorbs at 636 nm. Figure 2A shows spectra of PVF films during oxidation at incremented potentials; Figure 2B shows the ensuing re-reduction (see Experimental Section). Since the PVF site concentration (CT = 5.45 M) is known14 and the total number of sites in the film is measured from voltammograms like Figure 1, film concentrations of Fc and of Fc+ can be calculated from the absorbance at 636 nm. With some deviations at extreme potentials, the logarithm of their ratio, plotted us electrode potential in Figure 2C, gives a 59-mV slope. Assuming a completely oxidized film at +800 mV US SSCE, the ratio CF~+/CT, given by AE/AE.~W, is plotted against potential in Figure 2D along with a curve of ideal Nernstian shape and a formal potential of +440 mV. This potential agrees with the E112 of the electron transport voltammogram in Bu4NClO,/CH$N in Figure 3A (vide infra) and lies in Figure 1 at a point where ca. 50% of the charge has been passed in the positive potential sweep. These results show that PVF films can behave in a relatively ideal manner. There is no evidence for multiple redox states or strong site-site interactions that would cause nonlinear activityconcentration relations. The present conditions are somewhat different from previous work;$the spectroelectrochemical condi-

2.8 x 4.9 x 5.5 x 5.1 x 9.1 X 2.9 X 2.3 X 4.8 X (1.0

105 105 105 105 105 106 106 105 1) x 106

1.4 X los 1.4 X 10s

4.5 x 4.6 x 4.8 X 1.0 x (1.6

105 104 104 105 2) X los

4.3 x 104 4.3 x 104 1.4 X lo5

tions allow ample time for film equilibration, and the films have been deposited by casting as opposed to electrochemical depositionsasdor plasma polymeri~ation.5~.C~C.B The absorbance in Figure 2B at 400-500 nm did not return to exactly its original value (Figure 3A) upon re-reduction of the film. This may be an optical scattering effect, since the films are visually somewhat roughened after the lengthy oxidationreduction cycle. Also, thus cycled films are more adherent than freshly cast ones. A fresh film appears smooth and glassy and is easily rinsed away with toluene or CH2C12, whereas removal of a cycled film requires scrubbing with a solvent soaked wipe. Electron Transport in PVF Driven by ConcentrationGradients of Fc and Fc+Sites. Figure 3 shows experimentsin various bathing environments in which concentration gradients were established in PVF films on IDA'S. In Figure 3A, in Bu4NC104/acetonitrile bathing medium, the voltammogram's limiting current is related to the electron diffusion coefficient in the solvent-wettedpolymer byla,2a-h,15

CT

ilim= oonFADE-;T

(4)

Sullivan and Murray

4346 The Journal of Physical Chemistry, Vol. 98, No. 16, 1994

TABLE 2

Electric Gradient-Driven Electron Transport Results, in Dry Ni

IDA 11 1-pm fingers 1-pm gaps

IDA 32 3-pm fingers 2-pm gaps

IDA 102

10-pm fingers 2-pm gaps

E

2.05 2.41 2.725 0.976 1.33 2.0

3.6 3.3 2.1 4.3 3.5 2.2

2.725 2.725 2.725 2.725 2.20 1.71 0.191 1.106

3.7 3.2 2.2 3.8 3.4 2.7 4.6 7.3

1.37 1.85 2.30 3.15 2.67 2.10 2.62 2.88 2.725 2.79 2.30 3.25

5.7 5.2 4.7 8.1 6.8 7.1 9.1 6.6 6.4 5.9 5.5 3.6

5.18 5.02 5.40 5.32 5.35 6.10

1.8 1.7 1.1 2.3 1.9 1.3

1.73 1.42 2.30 1.78 2.02 7.02

1.7 f 0.4 (av) 3.4 3.1 2.2 3.9 3.8 3.6 3.7 6.0 3.7 f 1 (av)

9.1 9.7 10.1 10.2 11.5 13.3 8.0 8.2 8.57 8.33 7.93 7.86 8.05 8.1 1 8.01 8.25 8.54 9.01 9.58 9.76

12.4 15.6 48.2 9.8 20.4 32.4 196.4 2.1 2.99 3.04 2.88 1.74 3.06 2.22 1.71 6.16 2.99 2.66 3.50 7.07

4.9 4.4 3.7 6.4 5.5 5.8 7.3 5.4 5.4 5.3 5.3 3.5 5.2 & 1 (av)

1.2 x 1.2 x 7.4 x 1.5 x 1.2 x 7.5 x

105 105 104 105 105 104

(1.1 & 0.3) X 105/(av) 2.7 X 105 2.4 X lo5 1.6 x 105 2.9 X 105 2.6 x 105 2.1 x 105 3.1 x 105 5.0 x 105 (2.8 f 1) X 105 4.0 x 105 3.6 x 105 3.2 X 105 5.4 x 105 4.6 x 105 4.8 x 105 6.1 x 105 4.5 x 105 4.4 x 105 4.3 x 105 4.1 X lo5 2.7 X lo5

0.946 0.949 0.941 0.944 0.942 0.923

2.12 1.79 2.43 2.63 2.40 4.03

0.960 0.954 0.942 0.945 0.934 0.910 0.970 0.967

13.57 17.91 40.86 8.14 19.81 25.56 196.10 3.35

0.964 0.967 0.970 0.969 0.968 0.968 0.968 0.966 0.963 0.960 0.954 0.952

5.18 5.85 4.90 3.31 4.68 4.68 3.44 8.31 3.97 5.10 5.70 9.64

(4.31 f 0.9) X lo5

From eq 7. xz is a measure of goodness-of-fit;the smaller x2, the better the fit: x2

=

E(

Yi,mear -Yi,fit ‘i

>’

where ui is uncertainty in the measured y value. From eq 8.

TABLE 3: Comparison. of Concentration Gradient-Driven with Electrical Gradient-DrivenElectron Self-Exchange Rate Constants, in Dry N2 electrical gradient eq 7 1 ~ - 5 ~( ~k - ~1S-1)~ eq 8 1 P 5 k w (M-I s-I)

concn gradient

i/AE slope l P 5 &EX (M-I 1.4 1.6 0.4 a Electrical gradient data are from Table 2; concentration gradient data are from Table 1; all are averaged for each IDA type.10 IDA 11 IDA 32 IDA 102

1.7 & 0.4 3.7 f 1.0 5.2 f 1.0

1.1 & 0.3 2.8 i 1.0 4.3 f 0.9

A

i b &EX

(M-I s-I)

s-l)

(5.8 & 2) X 104 (2.8 x 2) x 105 (2.9 f 2) X lo5

according to the Nernst equation, produces Fc and Fc+ concentrations in the film of 4.2 and 1.25 M, respectively. The film was then subjected to a slow potential bias scan in acetonitrile vapor bathing gas to produce a voltammogram with a shape characteristic of “ion budgeted” mixed valent films as described in which the average limiting current is related to electron diffusion by

2Gim ilim= nFAD,-;7-

1

Figure 1. Typicalcyclicvoltammogramof a PVFmodified IDAelectrode in 0.1 M Bu,NC104/acetonitrile us SSCE at 5 mV/s: (1) first scan; (2) all subsequent scans. The film thickness is approximately 200 nm. The cathodic current is plotted as positive. where CT = 5.45 M Fc sites,14 d is interelectrode gap distance, A is area, and coo is a small migration correction15 which is neglected.16 Figure 3B presents an experiment in which the P V F had been electrolyzed in Bu4NC104/acetonitrile a t a potential which,

where Climis the lesser of the two concentrations CF,+and CF,. Results for thus obtained values of & and of k a (eq 2) in acetonitrile liquid and vapor-bathed films are given in Table 1. Figure 3C shows a concentration gradient voltammogram with the P V F film in dry Nz where Clod- is diffusively immobile. Concentration gradients were established a t selected potential biases in acetonitrilevapor (e.g., as in Figure 3B), and the bathing gas was then changed to dry Nz. T h e voltammogram shows hysteresis presumably because the trans-gap concentration gradient a t small potential biases had not completely come to steady state. ilim was taken on the flattened plateaus a t higher potential biases and used in eqs 2 and 5 t o calculate the DEand EX results in Table 1.

The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4341

Mixed Valent Poly(viny1ferrocene) Films

1

7

0.1

b

1

0.564

b

0.1

"I

m

W

o

C

S O

UJ 0

360100

-

0

b

Figure 3. Concentration gradient voltammogramsof PVF films on IDA electrodes. (a) Four-electrode voltammogram in 0.1 M BmNC104/ acetonitrile, WE1 held at +200 mV us SSCE, WE2 scanned to +800 mV at 1 mV/s. The current shown is at WE2. WE = working electrode; in the IDA these have a 5-pm finger width and a 2-pm gap width. & = 2.0 X lO+'cm2/s, k u =: 4.9 X lo5M-1 s-1. (b) Mixed valent film ( C F ~ = 1.25M, CF,= 4.20 M) in acetonitrilevapor,1mV/s, maximum potential bias 350 mV. The IDA has a I-pm finger width and a 1-pmgap width. & = 2.1 X 10-9 cm2/s, km = 5.1 X lo5 M-l s-l. (c) Pointwise voltammogram of a mixed valent film (CF,+= 0.83 M, CF,= 4.62 M) in dry N2, concentration gradient set in acetonitrilevapor at each point; maximum potential bias 350 mV. The IDA has a IO-pm finger width and a 2-pm gap width. & = 3.2 X I&lo cm2/s, km = 7.7 X 104 M-l S-1.

la

-

0,

Ea

W

I "

-

J 8 A

a-

/Gd

:; :> Qo-

-

numu"

that measured in mixed valent PVF immersed in acetonitrile electrolyte solution (Table 1) and 45-fold larger than that in dry N2 bathing gas. The differences between polymer and solution phase electron transfer dynamics parallel our previous observations of bathing medium effects, which in all cases2CpfaJ save one2b display acetonitrile solvent-bathed polymer phase kex values slightly smaller than that for the related acetonitrile-dissolvedmonomer, and much smaller kEX in dry N2. Presumably, acetonitrilevaporbathed films contain somewhat less imbibed acetonitrile and, of course, none at all in a dry N2 bath. That is, the Fc and Fc+ sites are "desolvated" of acetonitrile and their environment is replaced with neighbor Fc and Fc+ sites and counterions, and this results in slower electron transfers. A previous solvent dynamics study'' of the km of C P ~ C O + / ~ and Cp2Fe+/0monomer couples in various solvents showed that the former exhibited nearly ideal k ~ us x q - 1 behavior and the latter a relatively small dependence of kEX on the longitudinal relaxation time TL of the solvent employed. If one assumes that the ferrocene sites in PVF films exhibit a similarly small sensitivity toward the dynamics of local dipole motions, then one tends to ascribe the changes in EX of monomer ferrocene us PVF polymer bathed in acetonitrile liquid us acetonitrile vapor us dry N2 to some other factor@). Those factors diminishing km in vapor and dry N2-bathed PVF films may be (i) increased reorganizational energetics of the local dipoles comprising the "outer sphere solvent shell" of the ferrocene sites, (ii) enhanced "inner sphere" reorganizational energy barrier terms associated with ion-pairing

4348

The Journal of Physical Chemistry, Vol. 98, No. 16, 1994

f o r d distortionsof the Fc+nuclear positions or with the energetics of local microscopicoscillationsof Fc and Fc+ toward one another as part of the barrier path, or with some other factor, and (iii) decreases in intersite distances, and thus electron transfer distances, caused by shrinkage of the polymer. The last factor can be excluded as a dominant effect since a decreased intersite distance should cause an increase in kEx, the opposite of what is observed. We are able to raise, but not resolve, the distinctions of (i) us (ii) by the present data. We can also compare the kEx results in dry N2 to the older conductivity results in mixed valent PVF pellets by Pittman.16 Those electrical gradient (200 V over 0.6-1 .O-mm-thick pressed pellets) measurements gave a maximum conductivity ca. 5 X f2-I cm-l, which corresponds to kEX = 237 M-I s-l, over 100-fold smaller than the present results in dry N2. At small potential biases in Figure 3B,C and near El/zin Figure 3A, linearization of the Nernst equation leads to another relation for D Ebased on the current us potential slope6J9

Inasmuch as the spectrophotometric results (Figure 2) confirm Nernstian behavior of the PVF film, D Eand kEx obtained in this way should be comparable to those obtained from limiting currents. Table 1 shows that theselow-bias electron conductivity data indeed agree rather well with the limiting current values. Electron Transport in PVF Using an Electrical Potential Gradient. A mixed valent film with diffusively immobile counterions,and no initial concentration gradient, cannot develop one owing to electroneutrality constraints on eiectrolysis at the film/electrode interfaces. A signature for this condition is a current-potential response which exhibits no hysteresis and is independent of the rate of potential variation (aside from capacitative charging). The currents rise exponentially with increasing potential bias and have, for electropolymerized metal bipyridine films, been modeled with a modified Marcus relation: 2i- I ,4

where the empirical (non-Marcussian) fitting factor p was introduced2'-l because observed currents rise more steeply than anticipated from simple Marcus theory. Analogous effects (unusually steep electrical field dependence) are known in timeof-flight measurements of electron hopping mobility in xerographic materials20 and are commonly associated with dispersive transport. Of the models and explanations that have been offered,2l most in some way include a dispersion of electron hopping rate constants associated with the microstructural disordercharacteristicof amorphous solid state materials. Related rate constant dispersions occurZ2 in static quenching of excited states in solutions but on time scales sufficiently short so as to mimic the pseudosolid state. Because of the empirical nature of the p-modification in eq 7, and a further observation as described below, we have sought alternative modes of analysis of current-potential curves like that obtained under N2 in Figure 4 and apply here the model of Scher and MontrolP and Pfister'b for anomalous dispersive transport where variations in rate constants arise from a distribution in nearest-neighbor reactant distances,

where e measures the rate constant dispersion (an ordered material has e = 1)

Sullivan and Murray

(9) wheref is the potential sweep frequency and 0is the wavefunction overlap parameter in the rate-distance relation23

in which k, is the contact rate, r, is the reactant diameter, and 6 is the center-center separation at electron transfer. In eq 8, Lis introduced as film thickness in the original theory' as a time-scaling factor for time-of-flight experiments. In the present experiments, we translate L as the product of the average electron transport distanceduring theexperiment (the time which for simplicity we define as one-fourth of a potential cycle, f/4) with the average electron velocity (hop frequency, CTkEX, traversing the hop distance, 6, modified by the linearized dependence on the average voltage bias, AE,,/2), thus

The L term disappears where B = 1, and eq 8 reduces to the classical Marcus r e l a t i ~ n . ~ Figure 4 shows an electrical gradient current-potential (*10V) response for a 1:1 mixed valent PVF film under N2 on an IDA. When such responses are analyzed according to eq 7 with LEX and p as adjustable parameters and to eq 8 with kEx and t as fitting parameters, parts A and B of Figure 4 show typical comparisons of experimental and calculated i-E curves. Figure 4C shows residuals for the fits in Figure 4A,B, which are small and similar. Numerical results are given in Table 2. Examination of Table 2 and of the summary Table 3 reveals that (a) kEx from thep-modified Marcus relation (eq 7)is smaller than that from the Scher-Montroll-Pfister (SMP) theory but that (b) the product pkex agrees quite well with results from the SMP analysis and that (c) kex from the SMP analysis and pkEX from the p-modified Marcus analysis agree rather well with kex values from concentration gradient measurements (Table 1). The only difference between mixed valent films used in concentration gradient and electricalgradient measurements is that in the former the populations of electron donor and acceptor sites, and the counterions, are linearly polarized (eq 6), whereas in the latter they are uniform across the film. Assuming that changes in the local densities of ionic sites do not provoke major changes in the electron transfer dynamics,one expectsthat electron self-exchange rate constants should be the same whether derived from concentration gradient or electrical gradient experiments. Accordingly, agreement between the results of the concentration gradient and electrical gradient experiments can be taken as a significant success of the data analysis.10 Considering further the p-modified Marcus relation eq 7, note that Table 2 results for p using the 1-pm IDA (IDA 11) differ from the 2-pm IDA'S (IDA 32 and 102) by about 2-fold. An apparent dependence of p on the size of the electrical gradient noticed6 in [O~(vbpy)3]~+/~+/[Zn(vbpy)#+ copolymers and in earlier data" was confirmed by analyzing the mixed valent PVF current-potential responsesover different incrementsof potential. The results for p (Figure 5 ) , while scattered at low gradients where i-E responses show only slight curvature, reveal a systematic variation of p with the size of the electrical gradient. The results for kEX also vary with gradient, but in an almost exactly opposite manner so that, for the examples in Figure 5 , the product pkex remained constant to within * 5 % with no trends. Considering the observation of Figure 5 , and the fact that from analysis using eq 7, pkm, not km, agrees with the SMP and concentration gradient results, it appears that the empirical eq 7 does produce reasonable self-exchange rate constant data, but

The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4349

Mixed Valent Poly(viny1ferrocene) Films

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potential bias (V) Figure 4. Mixed valent PVF film in dry Nz, linear voltage bias sweep to f10 V at 5 Hz. The IDA has I-pm finger width and a 1-pm gap width. (a) A fit to cq 7 yields k a = 2.2 X 104 M-l s-] and p = 6.10. (b) A fit to eq 8 yields &EX = 7.5 X 104 M-1 s-1 and c 5 0.923. (c) Residuals, the difference between measured and calculated current: (open circles) residuals for the fit to cq 7; (dark squares) residuals for the fit to eq 8.

as pkm. Indeed, in previous metal bipyridine studies?'.' we observed that k a results using eq 7 were smaller than those obtained using concentration gradient experiments. This dis-

agreement is reconciled by multiplying the previous electric potential gradient km values by the p values obtained there. Nonetheless, analysisof electrical gradient experimentsby a more

Sullivan and Murray

4350 The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 70-4

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4E OEO 1E4 2E4 3E4 424 5E4 6E4 1E4 8E4 9E4 1E5 1 applied field (V/cm) Figure5. Variation of p with applied electrical field. Error bars (shifted slightly for clarity) represent &1 u, averaging all experimental results (6-10) at each electric field interval on each IDA type. The best-fit value of &ex decreases in inverse proportion as p increases, so that the product pkm remains constant to within 5%.

physically understandable foundation is desirable, hence our investigation of the Scher-Montroll-Pfister model.’ The Table 2 results from the SMP analysis (eq 8), in good agreement with theconcentration gradient data (Table l), further show that the rate constant dispersion factor e is near unity in PVF, meaning that the PVF rate constant dispersion is not very broad. That is, the SMP model indicates that the distances intervening (or the barrier sizes) between Fc and Fc+ sites undergoing electron transfers are rather uniform throughout the mixed valent film. Importantly, the value of e does not seem to depend on the electrical gradient employed. The REX values represent, according to the model, the average electron transfer rate constant probed on the experimental time scale. A “slow” site might not participate in electron transport at high measurement frequency (such sites would be “traps”); potentially, the obtained REX could thus depend on the experimental time scale (potential sweep frequency). However, no frequency dependence was observed in the present data. That e is near unity also causes the pre-exponential Ll-I/t term in eq 8 to be near unity and thus has little effect on the fitting process. A preliminary estimate of L can be obtained from the electron hopping conductivity near zero potential bias and the linearized form of eq 7,

Values of L ranged from 0.3 to 0.6 pm, about the same as the IDA gap width. As noted above, we take L in eq 8 as an average electron transit distance during the experiment. A calculation of the wavefunction overlap term, j3, using eq 9 and REX = 2 X 105 M-1 s-1 and t = 50 ms, gives j3 = 0.84 cm-1, a value close to many estimates of the overlap term in the electron transfer literat~re.~3 Lastly, a variety of electrolysis potentials were employed in preparing mixed valent PVF films, in order to evaluate the bimolecular rate dependence implicit in the transport eqs 7 and 8. In principle, electron hopping conductivity as expressed by e~

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should be parabolic with changing proportions of electron donor and acceptor sites in the mixed valent film, with a maximum exchange current at CF,= CF,+. The results from Table 2 are plotted in Figure 6 against the concentration of Fc+ sites, along with curves of theoretically anticipated shape. The scatter in the experimental results is unfortunately too substantial to affirm the bimolecular rate law formulation with any great confidence.

Acknowledgment. This research was supported in part by grants from the National Science Foundation. We gratefully acknowledge gifts of Pt interdigitated array electrodes from the scientific staff of Nippon Telephone and Telegraph Co., Tobei, Japan.

References and Notes (1) (a) Murray, R. W. Molecular Design of Electrode Sutfaces; John Wiley and Sons, Inc.: New York, 1992. (b) Facci, J. S.; Schmehl, R.H.; Murray, R. W. J. Am. Chem.Soc. 1982,104,4959.(c) Buttry, D. A,; Anson, F. C. J. Electroanal. Chem. Interfacial Electrochem. 1983,130,333. (d) Conti, A. J.; Kaji, K.; Nagano, Y.;Sena, K. M.; Yumoto, Y.; Chadha, R.K.; Rheingold, A. L.; Sorai, M.; Hendrickson, D. N. Inorg. Chem. 1993,32,2681. (e) Forster, R. J.; Vos, J. G. J. Electrochem. Soc. 1992, 139, 1503. ( f ) Gaudiello, J. G.; Ghosh, P. K.; Bard, A. J. J. Am. Chem.Soc. 1985,107,3027. (9) Shu, C.-F.; Wrighton, M. S . J. Phys. Chem. 1988,92,5221. (h) Murray, R.W. Annu. Rev. Mater. Sci. 1984,14,145. (i) Chidsey, C. E. D.; Murray, R. W. Science 1986,231,25.(j) Wrighton, M. S . Science 1986,231,32.(k) Abruna, H. D. In Electroresponsive Molecular and Polymeric Systems; Skotheim, T., Ed.; Marcel Dekker: New York, 1988. (2) (a) Pickup,P. G.; Murray, R.W. J. Am. Chem.Soc. 1983,105,4510. (b) Pickup, P.; Kutner, W.; Leidner, C. R.;Murray, R.W. J. Am. Chem.Soc. 1984,106,1991. (c) Jernigan, J. C.; Chidsey, C. E. D.; Murray, R. W. J. Am. Chem. SOC.1985,107,2824. (d) Chidsey, C. E. D.; Feldman, B. J.; Lundgren, C.; Murray, R. W. Anal. Chem. 1986,58,601.(e) Chidsey, C. E. D.; Murray, R. W. J. Phys. Chem. 1986,90,1479. (f’) Jemigan, J. C.; Murray, R. W. J. Am. Chem.SOC.1987,109,1738.(8)Dalton, E.F.; Murray, R. W. J. Phys. Chem. 1991,95,6383.(h) Nishihara, H.; Dalton, F.; Murray, R. W. Anal. Chem. 1991, 63,2955. (i) Jernigan, J. C.; Murray, R.W. J. Phys. Chem. 1987,91,2030.(j) Jemigan, J. C.; Surridge, N.; Zvanut, M. E.; Silver, M.; Murray, R. W. J. Phys. Chem. 1989,93,4620. (k) Dalton, E.F.; Surridge, N. A.; Jernigan, J. C.; Wilbourn, K. 0.;Facci, J. S.; Murray, R. W. Chem. Phys. 1990, 141, 143. (1) Surridge, N. A,; Zvanut, M. E.; Keene, F. R.; Sosnoff, C. S.; Silver, M.; Murray, R.W. J. Phys. Chem. 1992, 96,962. (3) (a) Andrieux, C. P.; Saveant,J. M. J. Electroanal. Chem. Interfacial Electrochem.1980,111,311.(b) Laviron,E.J. Electroanal.Chem.Interfm‘al Electrochem. 1980,112, 1. (4) (a) Marcus, R.A. Discuss. Faraday Soc. 1960,29,21.(b) Marcus, R. A. Annu. Rev. Phys. Chem. 1964,15,155. (c) Marcus, R. A. J . Chem. Phys. 1965,43,679.(d) Sutin, N. Acc. Chem.Res. 1982,15,275.(e) Levich, V.G. In Advances in Electrochemistry andElectrochemica1Engineering, 4th ed.;Delahay, P., Ed.; Interscience: New York, 1966;p 314. Bagley, B. G . Solid State Commun. 1970,8,345.

Mixed Valent Poly(viny1ferrocene) Films

The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4351

( 5 ) (a) Merz, A.; Bard, A. J. J. Am. Chem. Soc. 1978, 100, 3222. (b) Daum, P.; Murray, R. W. J. Electrwnal. Chem. Interfacial Electrochem. 1979, 103, 289. (c) Daum, P.; Lcnhard, J. R.; Rolison, D. R.; Murray, R. W. J. Am. Chem. Soc. 1980, 102, 4649. (d) Pccrce, P.; Bard, A. J. J. Electrwnal. Chem. Interfacial Electrochem. 1980,114,89. (e) Nowak, R. J.; Schultz, F. A.; Umana, M.; Lam, R.;Murray, R. W. AMI. Chem. 1980, 52, 315. (0Laviron, E.; Roullier, L. J. Electrwnal. Chem. Interfacial Electrochem. 1980,115,65. (e) Daum, P.; Murray, R. W. J. Phys. Chem. 1981,85,389. (h) Varineau, P. T.; Buttry, D. A. J. Phys. Chem. 1987, 91, 1292. (6) Sosnoff, C. S. Master's Thesis, Universityof North Carolina, Chapel Hill. NC. - -___, - -, 1990. - - - -. (7) (a) Scher, H.; Montroll, E. W. Phys. Rev. E 1975, 12, 2455. (b) Pfister, G. Phys. Rev. E 1977, 16, 3676. (8) Woodward, W. S.;Rocklin, R. D.; Murray, R. W. Chem. Biomed. Environ. Instrum. 1979, 9, 95. (9) Aoki, K.; Moritz, M.;Niwa, 0.;Tabei, H.J. Electrwnal. Chem. Interfacial Electrochem. 1988, 256, 269. (10) The rate comparisons of concentration and electrical gradient experimentsare best made using the same IDA geometry. Comparison of km results in Tables 1 and 2 at IDA's 11,32, and 102 (seeTable 3) suggests that the electron transport r a t a increase with IDA finger width. This effect arises from our simplifying treatment of the IDA as a parallel plate cell, ignoring concentration gradient or electrical gradient transport that originates from the film that is present on the finger tops. The finger widths are 1-10 pm whereas the finger and film heights are 0.1 and ea. 0.2 pm, respectively. The effect is small; participation of only about 10%of the finger width could cause a 4-fold variation in km between IDA's 11 and 102. The effect complicated the intended scrutiny of possible contact resistances which the use of different ppsizeswasdesigned toexplore. Thattheconcentrati~andel~calgradient experiments produced the same rate constants, in spite of the very different electrical gradients that exist near the electrode/film interfaces in the two experiments, dots provide circumstantial support for the absence of contact resistances. (11) Press, W. H.; Flannery, B. P.;Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in Pascal;Cambridge University Prcss: New York, 1986; pp 572-580. ~~

(12) Gottesfeld,S.;Redondo, A.; Rubenstein,I.; Feldberg, S.J.Electrwnal. Chem. Interfacial Electrochem. 1989, 265, 15. (13) Hillman, A. R.; Hughes, N. A,; Bruckenstein, S.J. Electrochem. SOC.1992, 139, 74. (14) Carlin, C. M.; Kepley, L. J.; Bard, A. J. J. Electrochem.Soc. 1985, 132, 353. (15) Saveant, J. M. J.Electroanal. Chem. InrerfacialEIectrochem.1988, 242, 1. (16) (a)wo iscalculated*5tobe 1.5fora 1:l perfectlypermselectiveO/l+ film. Since poly(viny1ferrocene) is not very permselective,'6b-c especially in theneutralstate,andsincethe filmareemployed hereat other mixedvalences, we expect wo to be even smaller and have assumed wo = 1.0. (b) Hillman, A. R.; Loveday, D. C.; Bruckenstein, S . J. Electroanal. Chem. Interfacial Electrochem. 1989,274, 157. (c) Ikeda, S.;Oyama, N. Anal. Chem. 1993, 65, 1910. (17) McManis, G. E.; Nielson, R. M.; Gochev, A.; Weaver, M. J. J. Am. Chem. Soc. 1989, 1 1 1 , 5533. (18) Pittman, C. U.;Surynarayanan, B.;Sasaki, Y. Adv. Chem.Ser. 1975, No. 150, 46. (19) Dalton, E. F. Doctoral Dissertation, University of North Carolina, Chapel Hill, NC, 1990. (20) (a) Facci, J. S.;Stolka, M. Philos. Mug.E 1986,54, 1. (b) Pai, D. M.; Yanus, J. F.; Stolka, M.; Renfer, D.; Limburg, W. W. Philos. Mag. E 1983,48,505. (c) Stolka, M.; Yanus, J. F.; Pai, D. M. J. Phys. Chem. 1984, 88, 4707. (21) Bottger, H.; Bryksin, V. V. Hopping Conduction in Solids; VCH: Berlin, 1985; p 218. (22) Miller, J. R.; Beitz, J. V.;Huddleston, R. K . J. Am. Chem.Soc. 1984, 106, 5057. (23) Newton, M. D.; Sutin, N. Annu. Rev. Phys. Chem. 1984, 35,437. (24) Jernigan, J. C. Doctoral Dissertation, University of North Carolina, Chapel Hill, NC, 1983.