Experimental Validation of MarcusTheory for Outer-SphereHeterogeneous Electron-TransferReactions: The Oxidation ofSubstituted 1,4-Phenylenediamines

Antony D. Clegg, Neil V. Rees, Oleksiy V. Klymenko,Barry A. Coles, and Richard G. Compton*[a]

Introduction

In this Communication, we report a quantitative investigationof Marcus theory[1±6] using high-precision measurements of fastouter-sphere heterogeneous electron-transfer rate constants.Theoretical developments since Marcus' initial work have con-sidered the outer-sphere electron-transfer process to involvethe formation of a precursor complex between the reactantmolecule and the electrode surface,[7] leading to the commonlyfound expression for the standard electrochemical rate con-stant [Eq. (1)]

k0 ¼ kelKpnnexp

��DG�

RT

�ð1Þ

where Kp is the equilibrium constant for the precursor complexformation, DG� is the free energy of activation for the electrontransfer, nn is the frequency of crossing the free energy barrier,and kel is the probability of electron tunnelling in the transitionstate.[6±8] The pre-exponential terms are sensitive to the system,and in particular the nuclear frequency and electronic trans-mission factors have been considered in detail.[6, 7, 9, 10] It hasbeen shown that if the reaction free energy is zero, and the

[a] A. D. Clegg, N. V. Rees, O. V. Klymenko, Dr. B. A. Coles, Prof. R. G. ComptonPhysical and Theoretical Chemistry LaboratoryOxford University, South Parks RoadOxford OX1 3QZ, (UK)Fax: (+44)1865-275410E-mail : richard.compton@chemistry.ox.ac.uk

1234 ¹ 2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/cphc.200400128 ChemPhysChem 2004, 5, 1234 ±1240

™weak overlap∫ limit is assumed, that is the electronic couplingis small, then Equation (2) holds,[8]

nn ¼ t�1L

�DG�

4pRT

�1=2

ð2Þ

where tL is defined as tL=tDe¥/es, tD is the experimentalDebye relaxation time and e¥ and es are the high-frequencyand static dielectric permittivities, respectively.[7] In order to ac-count for any nonadiabaticity in the electron transfer, which isoften observed in outer-sphere reactions,[7] the electronictransmission probability, kel, can be stated as in Equation (3),

kel ¼ k0el exp½�Bðr0�sÞ� ð3Þ

where r’ is the molecule±electrode separation, s is the distanceof closest approach of the molecule and electrode, and B is aconstant.[7] If the free energy of activation is solely due to theouter-sphere reorganisation energy then Equation (4) applies:

lo ¼NAe

2

8pe0

�1r� 1

2d

��1eop� 1es

�ð4Þ

where e is the electronic charge, r is the radius of the mole-cule, and d is the distance from the reactant to the metal sur-face[11] which is usually set to infinity following Hale,[12] thenEquation (1) can be expressed as Equation (5),

k0 ¼ Q

�y

4p

�1=2�

1r

�1=2

exp

���Br þ y

r

��ð5Þ

where Q=Kpk0elt�1L exp[�B(d�s)] , y= (NAe

2/32pe0RT)[(1/eop)�(1/es)] , and r’= r+d.

There have been few attempts to interpret experimentallydetermined rates of electron transfer using expressions such as(1) and (5), and most comparisons with Marcus theory remainin a qualitative sense only. One reason for this may lie in thenecessity to obtain high-precision measurements of k0, togeth-er with the need to be able to measure very fast as well asslow values of k0, to perform a satisfactory test of the theory.In most cases, the method of choice for the kineticist wouldbe cyclic voltammetry including the recently developed fieldof fast-scan voltammetry.[13±15] However, the experimental diffi-culties of obtaining precise and accurate data via these meth-ods are documented.[16,17] The exacting requirements for preci-sion and reliability of the experimental methodology lendthemselves to steady-state methods, since there is negligibledistortion from iR or capacitative effects. We have therefore se-lected the high-speed channel electrode (HSChE) for this studybecause of its proven ability to unambiguously measure kineticinformation for the fastest electrode processes,[18±21] althoughwe note that small amplitude a.c. voltammetry offers somemerits in this context since capacitance contributions are re-duced (but not eliminated) compared to fast-scan cyclic vol-tammetry.[22±24]

In a recent publication, we developed a methodology formeasuring k0 for fast electron transfers with a view to testing

Marcus Theory expressions such as (5), using the HSChE tomake the necessary high-precision measurements of electrontransfer rates.[25] We also considered what measures of the mo-lecular radius r might be used. Of the several methods availa-ble, such as crystallographic or computed values from themean spherical or ellipsoidal approximations,[8,26, 27] it was sug-gested that the hydrodynamic radius would provide the mostphysically meaningful value, despite having been rarely usedas such in the literature.[8] It is recognised that this method as-sumes that the electron transfer is orientation-independent, al-though in practice some weighted average of molecular orien-tations and their respective ™radii∫ will probably be relevant.Nevertheless, in the absence of specific orientational require-ments or adsorption, the hydrodynamic radius should be relat-ed to the true effective radius by some factor, and thereforeshould show the correct trend for our interpretation. The hy-drodynamic radius is also conveniently measurable from exper-imental voltammetry, since it can be simply calculated fromthe Stokes±Einstein equation [Eq. (6)]:[28]

r ¼ kTPphD

ð6Þ

in which h is the viscosity, D is the diffusion coefficient, and Pis either 4 or 6 depending whether the ™stick∫ or ™slip∫ limit isassumed for Equation (6).[28] To further assist the fitting of theo-retical results to the experimental data, Equation (5) can be lin-earised, and also rendered dimensionless for convenience bymaking the substitutions y= r/y, b=yB, K=k0/k’ and q=Q/k',where k’=1 cms�1; it then becomes Equation (7).

lnðK ffiffiffiyp Þ þ 1

y¼ �by þ ln

�q

2ffiffiffipp

�ð7Þ

Herein we present experimental results for comparison withEquations (2) and (7) to investigate the validity of the applica-tion of these theoretical results to heterogeneous electrontransfer over a range of molecular radii from 3 to 9 ä in alkylcyanide solvents. This range of molecular sizes is significantlywider than previously attempted[25] and hence a more rigoroustest of the validity of the above equations. First, the solventeffect on the rate of electron transfer is studied for the oxida-tion of N,N,N’,N’-tetramethyl-1,4-phenylenediamine dihydro-chloride (TMPD) in the solvents acetonitrile (MeCN), propioni-trile (EtCN), butyronitrile (PrCN), and valeronitrile (BuCN) withreference to Equation (2). Then, the variation of k0 for a seriesof related compounds are investigated by considering the firstoxidations of the ten tetraalkyl-1,4-phenylenediamines (TRPDs)shown in Table 1, in MeCN.

Experimental Section

Reagents: The chemical reagents used for these experiments thatwere available from commercial sources were TMPD and N,N’-di-ethyl-1,4-phenylenediamine (DEPD), and N,N’-diphenyl-1,4-phenyl-enediamine (DPPD; Aldrich, 98%), tetrabutylammonium perchlo-

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rate (TBAP; Fluka, Puriss.>99%), and the solvents MeCN (Fisherscientific, >99.99%), EtCN (Fluka, purum>99%), PrCN (Aldrich,98%), and BuCN (Fluka, puriss>99%). These were the highestgrades available, and were used without further purification. TBPD,THxPD, THpPD, TOPD, DEDB, and DEDHx (see Table 1 for abbrevia-tions) were synthesised by Prof. D. J. Walton.[29±31] Solvents werestored over molecular sieves (Linde 5 ä, Aldrich) for several hoursprior to use and thoroughly degassed with argon (PureshieldArgon, BOC Gases Ltd, UK) before and after solution preparation.All solutions contained 0.10m TBAP as supporting electrolyte andexperiments were conducted at a temperature of 2942 K.

Instrumentation: The high-speed channel electrode (HSChE) andpressurised apparatus have been described previously[18,19, 21] (seeFigure 1). High flow rates across a microband electrode are ach-ieved by pressurising a chamber containing the solution and elec-trode assembly up to 1.5 atm. The solution passes through theflow-cell (width, d=0.200 cm and height, 2 h=126 mm) and outthrough one of three capillaries of varying internal bore size, tothe exit at atmospheric pressure. This achieves volume flow ratesof between 0.10±3.2 cm3 s�1 (corresponding to linear flow velocities

close to the electrode of 0.7 to20 ms�1), and the Reynoldsnumber, Re, given by Equa-tion (8),[32]

Re ¼3Vf

2hdnð8Þ

can attain maximum values of9000 under well-defined laminarconditions,[18,19] since the channelflow-cell has been designed toensure these Reynolds numbersare present for less than 2 mmbefore the electrode, whilst a™lead-in∫ length of approximately

4 mm is needed for the development of turbulent flow.[19] Voltam-mograms are measured by means of an in-built potentiostat at ascan rate of 400 mVs�1 with a platinum microband electrode oflength (xe) 40.5 mm.The potentiostat used for microdisk voltammetry was a mAutolabType II (Eco Chemie BV, Utrecht, Netherlands) controlled by a DellOptiplex GX110 Pentium III computer using General Purpose Elec-trochemical System v4.8 software (Eco Chemie BV, Utrecht, Nether-lands).

Microelectrodes: Microband and microdisk electrodes were fabri-cated by fusing platinum (99.95%, Johnson Matthey plc, London,UK) into soda glass according to literature methods,[19,25] and theirworking surfaces ground and polished to a mirror finish. The mi-croband electrode width (w), as measured by microscope, was0.096 cm. For both types of electrode, the dimensions were con-firmed by electrochemical calibration,[33] and the radius (rd) of themicrodisk found to be 14.1mm. The electrodes were cleaned withultrapure water and polished using 0.25 mm alumina slurry on softlapping pads, then finally rinsed in ultrapure water and dried care-fully before use. The counter electrode was a smooth, bright, plati-num mesh, and a silver wire (99.95%, Johnson Matthey plc,London, UK) was used as a quasi-reference electrode. Microdiskvoltammetry was performed within a Faraday cage to reduce elec-trical noise.

Analysis of hydrodynamic voltammetry: The analysis of steady-state voltammograms recorded at the HSChE has been reported.[20]

The current response for each voltammogram was normalised bydivision by the respective limiting current, Ilim, and the middle 60%of the wave was plotted against the potential. These data werethen input into a program and compared against a calculated vol-tammogram of (I/Ilim) versus E for a selected range of a, k0 and E0

f

values. The quasi-reversible electron transfer to be simulated canbe treated as the Reaction (9):

A� ekf

kb

�! �B ð9Þ

where

kf ¼ k0 exp

�ð1�aÞF

RTðE�E0

f Þ�¼ k0 exp½ð1�aÞq� ð10Þ

kb ¼ k0 exp

��aFRTðE�E0

f Þ�¼ k0 expð�aqÞ ð11Þ

and its simulation was achieved by using the following analyticalresults [Equations (12) to (14)]:[18,20, 34,35]

Table 1. The tetraalkyl-1,4-phenylenediamines under investigation.

R1 R2 R3 R4 Abbreviation

H H H H PPDethyl ethyl H H DEPDmethyl methyl methyl methyl TMPDn-butyl n-butyl n-butyl n-butyl TBPDn-hexyl n-hexyl n-hexyl n-hexyl THxPDn-heptyl n-heptyl n-heptyl n-heptyl THpPDn-octyl n-octyl n-octyl n-octyl TOPDethyl ethyl n-butyl n-butyl DEDBethyl ethyl n-hexyl n-hexyl DEDHxH phenyl H phenyl DPPD

Figure 1. Schematic diagram of the channel cell and microband electrodeshowing the geometrical parameters.

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iirev¼ 1�2uþ 2u2 lnð1þ u�1Þ ð12Þ

where

u ¼ 0:6783D2=3B ð3Vf=4dxeh

2Þ1=3k0fexp½�ð1�aÞq� þ ðDA=DBÞ

2=3 expðaqÞgð13Þ

and

irev ¼0:925nFw½A�bulkðxeDAÞ

2=3ðh2dÞ�1=3V1=3f

1þ ðDA=DBÞ2=3 expð�qÞ

ð14Þ

Vf is the volume flow rate, [A]bulk is the bulk concentration of theelectroactive species, Di is the diffusion coefficient of species i, andthe geometrical parameters are as given in Figure 1. The quantityirev is the current that would flow if the electrode kinetics were re-versible. These approximate results have been verified against nu-merical simulations using the backward implicit finite differencemethod,[18] and found to be valid over the range of flow rates ach-ievable in the HSChE.For each simulated voltammogram, a mean scaled absolute devia-tion (MSAD) given by Equation (15)

MSADða,k0,E0f Þ ¼

XN

k¼1

jIexpðEkÞ�Ithða,k0,Ek�E0f Þj

IexpðEkÞð15Þ

was calculated as the sum of the differences between each simu-lated point (Ith) and each experimental point (Iexp). Here, Ek, k=1,º,N are the potentials of the experimental data points underanalysis, and N is typically above 20. Each simulated voltammo-gram was therefore awarded its own MSAD value and this enableda minimum to be found (by a Direct Search Method) correspond-ing to the optimum values of a, k0 and E0

f . Contour plots of MSADas a function of a and k0, and as a function of k0 and E0

f were pro-duced showing the existence of a single minimum in every case.

Results and Discussion

First, a solution containing 1.12 mm TMPD and 0.10m TBAP inMeCN was introduced into the pressure chamber of the HSChEapparatus fitted with the 40.5 mm Pt microband electrode. Alinear sweep voltammogram was then recorded at an arbitraryflow rate (1.14 cm3 s�1) which yielded an effectively steady-state response, enabling a limiting current (Ilim) to be measuredfor the first oxidation wave as shown in the inset of Figure 2.This was repeated for a range of volume flow rates (Vf) from0.15 to 3.10 cm3 s�1. Figure 2 also shows a ™Levich plot∫ ofmeasured limiting currents against V

1=3f , according to the Levich

equation[36] for channel flow cells [Eq. (16)]:

Ilim ¼ 0:925nFw½A�bulkðxeDÞ2=3

�Vf

h2d

�1=3 ð16Þ

where n, F, Vf, and [A]bulk have their usual meaning, and theother geometrical terms are shown in Figure 1. The gradient ofthis plot was used to obtain a value for the diffusion coeffi-cient of TMPD in MeCN of (2.200.35)î10�5 cm2 s�1, whichcompares with a literature value of 2.1î10�5 cm2 s�1.[37] Thesedata were then put into the computer program described

above and optimum values of k0, a and E0f were found simulta-

neously for each voltammogram and the mean values taken.Figure 3 shows the three-dimensional surface plots for k0,E

0f

and k0,a as they vary independently with MSAD. The plots

Figure 2. Levich plot for 1.12 mm TMPD in MeCN/0.1m TBAP (gradient=10.03 -mAcm�1 s1/3 ; R2=0.984). The inset shows a typical steady-state linear-sweepvoltammogram (Vf=1.14 cm3s�1).

Figure 3. Contour plots for TMPD in MeCN, with Vf=1.87 cm3s�1, showing a) k0vs. Ef

0, and b) k0 vs. a. Numbers shown on contours are the MSAD values.

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show single minima, which demonstrates that there is onlyone set of optimised values.

The same procedure was repeated for solutions of TMPDand 0.10m TBAP in EtCN (1.16 mm), PrCN (1.16 mm), and BuCN(1.32 mm). In all cases, excellent Levich plots were obtained(with R2 � 0.990), and Table 2 lists the diffusion coefficientsand kinetic parameters derived from analysis.

A logarithmic plot of the determined vales of k0 against thereported longitudinal dielectric relaxation times (MeCN=

0.20 ps, EtCN=0.31 ps, PrCN=0.52 ps, BuCN=0.74 ps)[38] ac-cording to Equation (17), yields a linear plot with gradient�0.92 as shown in Figure 4. This result compares with the the-

oretically expected inverse relationship with tL. However, thereare several studies that have investigated the electron transferrates in a variety of different solvents[39±44] using an empiricalform of Equation (1) to accountfor observed departures fromthe inverse dependence on tL,namely Equation (17).

lnk0 ¼ �q lntL þ lnA ð17Þ

These reports have confirmeda linear relationship betweenlnk0 and lntL for various valuesof the constant q ranging from1±0.2, and also indicate thepresence of an outer-sphererather than an inner-sphere

pathway for TMPD.[39±41,43] In these papers, q=1 indicates anadiabatic, purely outer-sphere electron transfer, and thereforeq<1 indicates a degree of either nonadiabaticity or inner-sphere contribution to the reorganisation energy.

In particular, our results for TMPD are consistent with report-ed findings which considered the relationship between the for-ward rate constant kf [see Eq. (9)] and tL for 12 solvents, which

gave an average value of q=

0.53, although the value recordedin MeCN reflects a higher valueof q.[39] Considering these results,we infer that the departure ofthe gradient from unity is due toa failure of the solvent continu-um model, perhaps attributableto the solvation of the aminogroups by RCN molecules caus-

ing additional solvent inertial effects.Next, the oxidation of different TRPDs was investigated by

the HSChE method using solutions of TBPD (1.16 mm), DEDB(1.16 mm), DEDHx (1.84 mm) and DPPD (1.00 mm) in 0.10mTBAP and MeCN. In all cases, excellent Levich plots were ob-tained (with R2�0.980) which yielded values for the respectivediffusion coefficients that were all in good agreement (seeTable 3) with those predicted by the Wilke±Chang expression[Eq. (18)] .[45]

D � ð7:4� 10�8ÞTffiffiffiffiffiffiffiffiffi�ms

p

hs0:6ð18Þ

where T is the absolute temperature, f is the solvent affinityfactor (unity for aprotic solvents), ms is the molecular mass ofthe solvent, h is the solvent viscosity, and s the molecularvolume.[45] The analysis confirmed that these compounds ex-hibit a simple one-electron oxidation, and the kinetic parame-ters derived are shown in Table 3 (mean standard deviation).Previously published results for the kinetics of oxidation ofPPD and DEPD have been included for completeness.[20]

In the case of THxPD, THpPD, and TOPD, initial HSChE ex-periments appeared to indicate some adsorption effects, aswell as having k0 values apparently approaching the lowerlimit of reliability for the HSChE analysis method.[46] It wastherefore decided that quantitative kinetic measurements were

Table 2. Diffusion coefficients and kinetic parameters obtained for the oxidation of TMPD in alkyl cyanides.

Solvent Dî105 [cm�2 s�1] k0 [cm�1 s�1] a Ef0 [V�1 (vs. Ag)]

MeCN 2.200.15 0.550.08 0.500.01 0.3740.002EtCN 1.200.10 0.450.03 0.530.02 0.4360.025PrCN 1.110.10 0.330.03 0.540.02 0.2430.003BuCN 0.840.06 0.180.01 0.520.01 0.2040.002

Figure 4. Plot of lnk0 vs. lntL for TMPD in alkyl cyanides (gradient=�0.92,R2=0.944).

Table 3. Diffusion coefficients and kinetic parameters obtained for the oxidation of substituted 1,4-phenylenedi-amines in MeCN. DWC is the Wilke±Chang estimate for the diffusion coefficient.

Compound Dî105 [cm�2 s�1] DWC î105 [cm�2 s�1] k0 [cm�1 s�1] a Ef0 [V�1 (vs. Ag)]

PPD[20] 2.09 2.1 0.840.16 0.50 0.2870.002DEPD[20] 1.88 1.6 1.640.25 0.50 0.2080.003TMPD 2.200.35 1.6 0.550.08 0.500.01 0.3740.002TBPD 1.240.12 1.0 0.300.05 0.440.05 0.1630.007THxPD 0.820.08 0.8 0.0080.001 0.400.02 0.2240.005THpPD 0.720.07 0.7 0.0140.003 0.330.07 0.1810.007TOPD 0.770.08 0.7 0.0070.001 0.310.02 0.2320.010DEDB 1.050.12 1.1 0.360.03 0.500.01 0.2090.009DEDHx 0.960.10 1.0 0.0370.006 0.400.06 0.2440.017DPPD 1.700.10 1.3 1.100.15 0.530.03 0.4070.008

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best obtained using steady-state voltammetry at a microdiskelectrode, due to the ease of mechanically cleaning the elec-trode surface. Solutions of 1.82 mm THxPD, 0.92 mm THpPDand 0.85 mm TOPD were investigated in this way, with carefulpolishing of the electrode between voltage scans. In each case,several steady-state voltammograms were recorded and ana-lysed according to the method of Mirkin and Bard[47] to extractvalues for k0 and a. The results are also presented in Table 3.

These results can be interpreted by mean of Equation (7),and a graph of this function is shown in Figure 5, yielding a

linear plot with a gradient of �86.7 and R2=0.970. From thisplot, values of Q and B can be extracted using the knownvalue for y of 37.3 ä. This gives B=2.31 ä�1 and Q=1.79î108 cms�1. These values compare reasonably with other valuesfor B of 1.37 ä�1[25] and within the range 1±2 ä�1.[7] Thesevalues can then be put into Equation (5) to visualise the directrelationship between k0 and r, as shown in Figure 6, showing avery good correlation between the values determined for k0

and Equation (5). This justifies the use of the hydrodynamicradius in this context, as well as providing a rigorous test ofMarcus theory over a range of k0 that spans more than twoorders of magnitude.

In using Equations (5) and (7), we assume that the contribu-tion to the activation energy due to inner-sphere reorganisa-tion is not significant. The basis of this assumption is that lihas been calculated to be no more than 10% of the value oflo for PPD and TMPD,[27] and would be expected to be less forthe larger TRPDs. In choosing the hydrodynamic radius as ameasure of molecular size, it is noted that Kapturkiewicz hasdemonstrated the usefulness of the ellipsoidal model, using anellipsoidal ™radius∫ calculated from the lengths of the molecu-lar semi-axes in the crystallographic literature.[27] Due to the ab-sence of published structural data for most of the compoundsin the present study (including those which are liquid at ambi-ent temperatures), a thorough comparison of ellipsoidal andhydrodynamic radii is not appropriate. However, for the com-

pounds PPD and TMPD the calculated ellipsoidal radii are0.28 nm and 0.40 nm respectively,[27] which compare favourablywith those determined for the hydrodynamic radii (under the™slip limit∫) of 0.29 nm and 0.37 nm.

Conclusions

Hydrodynamic and stationary electrodes have been used tomeasure the heterogeneous electron-transfer rates for the 1,4-phenylenediamines under steady-state conditions, therebyavoiding the problems of charging currents and iR losses whichare associated with alternative conventional methodologybased on transient measurements such as cyclic voltammetry.The dependence of k0 on the hydrodynamic radius (r) has beeninvestigated over the significantly wide range 3 ä� r�9 ä andthe experimental results found to be in excellent agreementwith the values determined for k0, which range from 7î10�3 to1.64 cms�1, and Equation (7). This justifies the use of the hydro-dynamic radius in this context, as well as providing a rigoroustest of Marcus theory. The dependence of k0 on the solvent lon-gitudinal dielectric relaxation (tL) has been investigated forTMPD and a minor departure from the theoretical tL

�1 relation-ship in Equation (2) was found, and these have been ascribedto the solvation effects of the amino groups.

Acknowledgements

We thank the Clarendon Fund for partial funding for O.V.K. , andboth the EPSRC for a studentship and Avecia Ltd for CASE sup-port for N.V.R.

Keywords: electrochemistry ¥ electron transfer ¥ kinetics ¥Marcus theory ¥ oxidations ¥ phenylenediamines

[1] R. A. Marcus, J. Chem. Phys. 1956, 24, 966.[2] R. A. Marcus, J. Chem. Phys. 1956, 24, 979.

Figure 5. Best fit plot of [ln(Kffiffiffiyp

)+1/y] vs. y for the 1,4-phenylenediamines in

MeCN with gradient=�86.46 and intercept=17.74 (R2=0.970). Concentrationsare as given in the text and the compounds are as follows: a) PPD, b) DEPD, c)TMPD, d) TBPD, e) THxPD, f) THpPD, g) TOPD, h) DEDB, i) DEDHx, and j) DPPD.

Figure 6. Plot of k0 vs. r for 1,4-phenylenediamines in MeCN using the valuesB=2.3 ä and Q=1.79î108cms�1 derived above from the linear plot inFigure 5 for the optimum fit of the data. Concentrations are as given in thetext and the compounds are as follows: a) PPD, b) DEPD, c) TMPD, d) TBPD, e)THxPD, f) THpPD, g) TOPD, h) DEDB, i) DEDHx, and j) DPPD.

ChemPhysChem 2004, 5, 1234 ±1240 www.chemphyschem.org ¹ 2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1239

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