Measurement of the free-energy dependence of interfacial charge-transfer rate constants using ZnO/H2O semiconductor/liquid contacts

Article abstract of DOI:10.1021/ja0436188

The dependence of electron-transfer rate constants on the driving force for interfacial charge transfer has been investigated using n-type ZnO electrodes in aqueous solutions. Differential capacitance versus potential and current density versus potential measurements were used to determine the energetics and kinetics, respectively, of the interfacial electron-transfer processes. A series of nonadsorbing, one-electron, outer-sphere redox couples with formal reduction potentials that spanned approximately 900 mV allowed evaluation of both the normal and Marcus inverted regions of interfacial electron-transfer processes. All rate processes were observed to be kinetically first-order in the concentration of surface electrons and first-order in the concentration of dissolved redox acceptors. The band-edge positions of the ZnO were essentially independent of the Nernstian potential of the solution over the range 0.106-1.001 V vs SCE. The rate constant at optimal exoergicity was observed to be approximately 10^-16 cm⁴ s^-1. The rate constant versus driving force dependence at n-type ZnO electrodes exhibited both normal and inverted regions, and the data were well-fit by a parabola generated using classical Marcus theory with a reorganization energy of 0.67 eV. NMR line broadening measurements of the self-exchange rate constants indicated that the redox couples had reorganization energies of 0.64-0.69 eV. The agreement between the reorganization energy of the ions in solution and the reorganization energy for the interfacial electron-transfer processes indicated that the reorganization energy was dominated by the redox species in the electrolyte, as expected from an application of Marcus theory to semiconductor electrodes.

Full text of DOI:10.1021/ja0436188

Published on Web 05/07/2005
Measurement of the Free-Energy Dependence of Interfacial
Charge-Transfer Rate Constants using ZnO/H
2
O
Semiconductor/Liquid Contacts
Thomas W. Hamann, Florian Gstrein, Bruce S. Brunschwig, and Nathan S. Lewis*
Contribution from the DiVision of Chemistry and Chemical Engineering, Noyes Laboratory,
1
27-72, California Institute of Technology, Pasadena, California 91125
Received October 20, 2004; E-mail: nslewis@caltech.edu
Abstract: The dependence of electron-transfer rate constants on the driving force for interfacial charge
transfer has been investigated using n-type ZnO electrodes in aqueous solutions. Differential capacitance
versus potential and current density versus potential measurements were used to determine the energetics
and kinetics, respectively, of the interfacial electron-transfer processes. A series of nonadsorbing, one-
electron, outer-sphere redox couples with formal reduction potentials that spanned approximately 900 mV
allowed evaluation of both the normal and Marcus inverted regions of interfacial electron-transfer processes.
All rate processes were observed to be kinetically first-order in the concentration of surface electrons and
first-order in the concentration of dissolved redox acceptors. The band-edge positions of the ZnO were
essentially independent of the Nernstian potential of the solution over the range 0.106-1.001 V vs SCE.
-16
4
-1
The rate constant at optimal exoergicity was observed to be approximately 10 cm s . The rate constant
versus driving force dependence at n-type ZnO electrodes exhibited both normal and inverted regions,
and the data were well-fit by a parabola generated using classical Marcus theory with a reorganization
energy of 0.67 eV. NMR line broadening measurements of the self-exchange rate constants indicated that
the redox couples had reorganization energies of 0.64-0.69 eV. The agreement between the reorganization
energy of the ions in solution and the reorganization energy for the interfacial electron-transfer processes
indicated that the reorganization energy was dominated by the redox species in the electrolyte, as expected
from an application of Marcus theory to semiconductor electrodes.
Carefully prepared (100)-oriented n-type Si/CH OH viologen2
+/+
I. Introduction
3
contacts have shown the predicted dependence of interfacial
A. Background. The Marcus semiclassical description of
outer-sphere electron-transfer reactions has been studied exten-
sively for electron-transfer processes between molecular donor
charge-transfer rate constants, ket, on changes in standard
1
0
interfacial free energies, ∆G°′, for driving forces up to, and
slightly beyond, that of optimal exoergicity. However, measure-
ments at higher driving forces were precluded because redox
couples having more positive potentials than the valence band
edge of Si oxidize the Si surface and/or induce carrier inversion
processes that prevent changes in interfacial driving force as
the Nernstian potential of the electrolyte is increased.11 Similar
considerations limit the experimentally accessible range of
1
and acceptor species. Agreement between theory and experi-
2-5
ment has been demonstrated in numerous cases. In contrast,
there have been relatively few studies of electron transfer at
semiconductor/liquid contacts. Current versus potential data for
electron transfer from ZnO to deuterated and normal thianthrene
acceptors were interpreted in terms of the normal and inverted
6
regions of Marcus theory. Transient absorption data for a series
1
2
driving forces for InP electrodes. Such considerations are
expected to complicate kinetics measurements at high exoer-
gicity for other small band gap (<2 eV) semiconducting
electrodes as well.
of TiO2-Fe(CN)5L contacts that showed increasing excited-
state lifetimes with increasing relative driving force were
presented as evidence of the inverted region.7
For nonadsorbed, outer-sphere redox species, extraordinarily
low defect densities at the semiconductor/liquid interface are
required to prevent adsorption and surface-state-related reactions
The metal oxide semiconductor, ZnO, is an attractive material
to mitigate these drawbacks and, thereby, allow direct investiga-
tion of the behavior of ket at large interfacial exoergicities. The
wide band gap of ZnO (3.3 eV) allows a large variation in the
driving force, and ZnO is not susceptible to the oxidation or
passivation processes that are prevalent in small band gap
semiconductors.
8
,9
from dominating the observed interfacial kinetics processes.
(
(
(
(
1) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265-322.
2) Chou, M.; Creutz, C.; Sutin, N. J. Am. Chem. Soc. 1977, 99, 5615-5623.
3) Gray, H. B.; Winkler, J. R. Annu. ReV. Biochem. 1996, 65, 537-561.
4) Fox, L. S.; Kozik, M.; Winkler, J. R.; Gray, H. B. Science 1990, 247, 1069-
1
071.
(
(
5) Closs, G. L.; Miller, J. R. Science 1988, 240, 440-447.
(8) Lewis, N. S. J. Phys. Chem. B 1998, 102, 4843-4855.
6) Nakabayashi, S.; Itoh, K.; Fujishima, A.; Honda, K. J. Phys. Chem. 1983,
(9) Lewis, N. S. Annu. ReV. Phys. Chem. 1991, 42, 543-580.
(10) Fajardo, A. M.; Lewis, N. S. Science 1996, 274, 969-972.
(11) Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. B 1997, 101, 11136-11151.
(12) Pomykal, K. E.; Lewis, N. S. J. Phys. Chem. B 1997, 101, 2476-2484.
8
7, 5301-5303.
(
7) Lu, H.; Prieskorn, J. N.; Hupp, J. T. J. Am. Chem. Soc. 1993, 115, 4927-
4
928.
10.1021/ja0436188 CCC: $30.25 © 2005 American Chemical Society
J. AM. CHEM. SOC. 2005, 127, 7815-7824
9
7815

A R T I C L E S
Hamann et al.
3
+/2+
Figure 1. Numerical symbols and chemical names of the redox compounds used in this study, given in the order of decreasing potential. I [Ru(bpy)3]
,
3
+/2+
3+/2+
3+/2+
3+/2+
3+/2+
, VII [Os(Me2bpy)2(MeIm)2]3+/2+,
II [Os(terpy)2]
, III [Os(bpy)3]
, IV [Os(Me2bpy)3]
, V [Os(bpy)2(MeIm)2]
, VI [Os(bpy)2(Im)2]
3
+/2+
VIII [Os(Me2bpy)2(Im)2]
.
Several studies using ZnO electrodes in the 1960s noted a
linear relationship between the logarithm of the current density,
J, and the applied potential, E, with a slope of E versus J of 60
mV/decade, indicating that ZnO/liquid contacts exhibit an
Scheme 1. Thermodynamic and Kinetics Parameters of a
Semiconductor/Liquid Junctiona
“
ideal” first-order dependence on the electron concentration at
13-16
the surface of the semiconductor.
the redox couples studied, notably Fe(CN)63
However, only a few of
-/4-
, showed a first-
order dependence of the interfacial rate on the concentration of
the redox acceptor species in solution. This latter condition is
also required to allow a straightforward interpretation of the
observed interfacial current density. Another important result
of these earlier ZnO studies is that a relatively small frequency
dispersion was observed in the differential capacitance measure-
ments, allowing a reliable determination of the surface electron
concentration and, additionally, a reliable measurement of the
position of the band edges versus a fixed reference electrode.
In addition, corrosion or passivation of the surface of the ZnO
electrodes was minimal. Subsequent attempts to extract rate
constants from steady-state J versus E data were thwarted
because most of the simple metal-ion-based redox systems that
a Energy versus distance for an n-type semiconductor in contact with a
redox species (A/A ). Formation of a space-charge region produces a
spatially dependent electric potential drop in the solid. Ecb and Evb are the
-
energies of the conduction and valence band edges, respectively, and
-
-
E(A/A ) is the Nernstian potential of the redox species (A/A ). The value
of ∆G°′, the standard free-energy change for interfacial charge transfer, is
-
-
given by ∆G°′ ) Ecb - qE°′(A/A ), where E°′(A/A ) is the formal
reduction potential of the (A/A ) redox system. The surface and bulk
-
electron concentrations are denoted as ns and nb, respectively. (a) At
equilibrium, the electrode potential, E, is such that the Fermi level of the
4+
2-
were investigated, such as Ce (in HNO3 and H2SO4), IrCl6
,
3
+
+
-
V
(in HCl), and Ag(NH3)2 , are known to adsorb onto
semiconductor, E , equals qE(A/A ), and n ) n , where n is the surface
F
s
so
so
electron concentration at equilibrium of the solid/liquid interface. (b) If a
potential is applied such that the junction is biased away from equilibrium,
ns is not equal to nso, and a nonzero net current, J, flows across the
semiconductor/liquid interface.
hydroxylated surfaces or to involve inner-sphere electron-
transfer pathways. Little follow-up of these early experiments
has apparently occurred over the subsequent four decades.
In this work, the interfacial electron-transfer kinetics of a ZnO
semiconductor/liquid interface has been studied at low and high
driving forces. To achieve this goal, we have synthesized a series
of one-electron outer-sphere redox couples having potentials
that span approximately 900 mV (Figure 1) in the band gap
region of ZnO. Rate constants have been measured for such
systems in contact with n-type ZnO electrodes. NMR line
broadening experiments were performed to determine the
reorganization energies of the compounds in the same medium
as that used to determine the interfacial kinetics at ZnO. Such
measurements have allowed a detailed investigation of the
dependence of the rate constant for electron transfer on driving
force in both the normal and Marcus inverted region, for a
homologous series of outer-sphere redox-active compounds at
an “ideally” behaving semiconductor/electrode interface.
B. Rate Laws and Models for Charge-Transfer Processes.
Scheme 1 depicts the thermodynamic and kinetics parameters
that describe electron-transfer processes at a nondegenerately
doped n-type semiconductor/liquid interface. No electronic states
are present in the band gap region of an ideal semiconductor,
so only electrons that are thermally excited to the conduction
band for an n-type material can participate in majority carrier
based electron-transfer events. At forward bias, the net flux of
electrons from the conduction band to randomly dissolved
1
7
acceptors in solution is given by
(
(
(
(
13) Dewald, J. F. Bell Technol. J. 1960, 39, 615-639.
14) Dewald, J. F. J. Phys. Chem. Solids 1960, 14, 155-161.
15) Morrison, S. R. Surf. Sci. 1969, 15, 363-379.
J(E) ) -qk [A]ns
(1)
where J is the current density (A cm ); E is the applied
et
-
2
16) Morrison, S. R.; Freund, T. J. Chem. Phys. 1967, 47, 1543-1551.
7816 J. AM. CHEM. SOC.
9
VOL. 127, NO. 21, 2005

Free-Energy Dependence of Charge-Transfer Rate Constants
A R T I C L E S
potential (V) relative to a saturated calomel electrode, SCE; q
Inc., San Clemente, CA). Ga-In eutectic was used as an ohmic contact,
and silver print (GC electronics 22-201, Rockford, IL) was used to
connect the Ga-In to a tinned copper wire. White epoxy was used to
seal the ZnO electrode assembly in a glass tube. The resulting electrode
area was determined by digitizing photographs of a microruler and of
-
19
is the charge of an electron (1.6022 × 10
C); ket is the
4
-1
electron-transfer rate constant (cm s ); [A] is the acceptor
-3
-3
concentration (cm ), and ns is the electron concentration (cm )
at the surface of the semiconductor. The concentrations of the
acceptor, [A], and of the electrons in the conduction band at
the surface of the semiconductor, ns, appear explicitly in the
expression for the current density, thus yielding a second-order
rate law for the charge-transfer process.
2
the exposed ZnO surface. An area of 0.51 cm was determined with
2
an estimated error of 0.03 cm . Before use, the electrode was etched
for 7 min in concentrated phosphoric acid (Aldrich), rinsed with 18
MΩ cm resistivity water (Barnstead NANOPure), and blown dry with
N
2
(g).
The value of ns is related to the potential difference between
E and the potential of the conduction band edge, Ecb/q, through
a Boltzmann-type relationship:8
B. Electrolyte Solutions. Electrochemical experiments were carried
out in a 55 mM phthalate buffer prepared by adding 12 mL of 1.0 M
KOH(aq) to 250 mL of 0.11 M potassium hydrogen phthalate (5.76 g)
solution, followed by dilution to 500 mL. The pH was then adjusted to
pH 4.99 using 1 M KOH(aq). The ionic strength was adjusted to 1.0
M by adding 37.4 g of KCl (Aldrich, 99+%) to provide the supporting
electrolyte for the electrochemical measurements.
(Ecb-qE)/kBT
n ) N e
(2)
s
c
where kB is Boltzmann’s constant; T is the temperature; Ecb is
the energy of the conduction band edge, and Nc is the effective
density of states in the conduction band of the semiconductor.
Hence, application of a potential to an ideally behaving
semiconductor electrode interface effects a change in the
observed current density (i.e., the charge-transfer rate) by
changing the value of the electron concentration at the surface
of the solid, as opposed to changing the rate constant, or the
energetics, of the interfacial charge-transfer process.
If J is shown to follow eq 1, with knowledge of ns and [A],
the value of ket can be calculated from the observed steady-
state J versus E data. Unlike the situation for metallic electrodes,
the relatively small, and controllable, value of the electron
concentration at the semiconductor surface affords the ability
to avoid redox-coupled mass-transport limitations on the charge-
transfer flux even for reactions at optimal exoergicity. Hence,
rate measurements at semiconductor electrodes can be performed
using simple steady-state methods with dissolved redox species,
even for relatively large values of the interfacial charge-transfer
rate constant.
C. Redox Compounds. Ammonium hexachloroosmate(IV),
2
2,2′-bipyridine (bpy), 4,4′-dimethyl-2,2′-bipyridine (Me bpy), terpyri-
dine (terpy), imidazole (Im), 1-methylimidazole (MeIm), NH
(n-C NCl (TBACl) were purchased from Aldrich and used as
received. All solvents were reagent grade and were used as received.
Ru(bpy) Cl ‚6H O, I (Figure 1), was purchased from Strem Chemicals
4 6
PF , and
4 9 4
H )
3
2
2
and used as received. All other compounds were made following
1
8,19
modified literature procedures, as described briefly below.
. Synthesis of [Os(terpy) ](PF (II), [Os(bpy) ](PF
Os(Me bpy) ](PF
of bpy or Me bpy or 2.5 equiv of terpy was added to (NH
0.25 g, 0.56 mmol) dissolved in 25 mL of ethylene glycol. The solution
1
2
)
6 2
3
)
6 2
(III), and
[
2
3
)
6 2
(IV). In a 50 mL round-bottom flask, 3.5 equiv
[OsCl
2
4
)
2
6
]
(
was heated to reflux for 1 h with rapid stirring under Ar and was then
cooled to room temperature. Then, 2-3 equiv of NH PF (aq) was
4
6
-
added, and the resulting PF
6
salt precipitate of the desired compound
was filtered, yielding a dark-green product that was washed with cold
water and diethyl ether.
2
. Synthesis of [Os(bpy)
2
(MeIm)
(MeIm) ](PF
(VIII). In a 50 mL round-bottom flask, 2 equiv of bpy or
[OsCl ] (1.1 g, 2.3 mmol) in 30 mL of
2
](PF
6
)
2
(V), [Os(bpy)
2
(Im)
2
]-
2
-
(
(
PF
6
)
2
(VI), [Os(Me
](PF
2
bpy)
2
2
)
6 2
(VII), and [Os(Me
2
bpy)
Im)
2
6 2
)
Me
2
bpy was added to (NH
)
4 2
6
ethylene glycol. The solution was heated to reflux for 1 h with rapid
stirring under Ar and was then cooled to room temperature. To reduce
any Os(III) species that may have formed, approximately 100 mL of
II. Experimental Section
A. Electrodes. Hydrothermally grown, n-type, 〈0001〉 oriented, ZnO
single crystals having dimensions of approximately 10 × 10 × 0.5
mm were purchased from Commercial Crystal Laboratories, Inc.
2 2 4
cold 1 M aqueous Na S O was slowly added, and the solution was
cooled for 1 h in an ice bath. The dark precipitate was collected by
vacuum filtration, washed with cold water and diethyl ether, and used
in further reactions without additional purification.
(
Naples, FL). The resistivity of the crystals was reported by the
1
4
manufacturer to be between 10 and 10 Ω cm.
To make the imidazole complexes, in a 100 mL round-bottom flask,
Electrochemical experiments reported in this work were confined
to the Zn-rich surface of such electrodes. Due to the limited number
of high-quality ZnO single crystals available, a meaningful statistical
approach was not feasible. For consistency, all of the data reported
herein were collected using a single electrode to evaluate the trends in
energetic and kinetics behavior for the entire series of redox couples
of interest. At least two other electrodes with nominally identical
capacitance versus E behavior to the one reported herein exhibited
essentially identical J versus E behavior, for several different contacts,
to the electrode reported herein. Other electrodes produced similar trends
in the rate constant versus driving force behavior, except that the
calculated -∆G°′ and ket values for these other electrodes were
somewhat smaller for every redox system in the series because these
electrodes exhibited more positive flat-band potentials than the electrode
reported herein.
5
-20 equiv of Im or MeIm was then added to the dried product, [OsCl
bpy ] or [OsCl Me bpy ], in 50 mL of ethylene glycol. The solution
was heated to reflux for 2-3 h with rapid stirring under Ar and was
2
-
2
2
2
2
then cooled to room temperature. Then, 2-3 equiv of NH
added, and the resulting PF
dark-brown product that was washed with cold water and diethyl ether.
. Purification and Characterization of the Os Complexes. Metal
complexes II-VIII were purified on an activated neutral alumina
column using acetonitrile as the eluent. For II-IV, a dark-green band
was collected, whereas V-VIII yielded a brown band. The solvent
was removed in vacuo. The residue was dissolved in a minimal amount
of acetone, and the complex was precipitated by addition of diethyl
ether. The product was then filtered and dried under vacuum. Yields
in excess of 80% were obtained for II-IV, and yields were in excess
of 30% for V-VIII. Elemental analysis yielded the following
4
PF
6
(aq) was
-
6
salt precipitate was filtered, yielding a
3
The crystal was first polished with water-based diamond suspensions
of grain size 6, 3, and 1 µm. The crystal was then chemically polished
in a silica/KOH suspension (0.05 µm, pH > 10, South Bay Technology
(calculated). II: C 37.99 (38.06), H 2.41 (2.56), N 8.89 (8.88). III: C
(18) Nakabayashi, Y.; Omayu, A.; Yagi, S.; Nakamura, K. Anal. Sci. 2001, 17,
945-950.
(
17) Morrison, S. R. Electrochemistry at Semiconductor and Oxidized Metal
(19) Nakabayashi, Y.; Nakamura, K.; Kawachi, M.; Motoyama, T.; Yamauchi,
Electrodes; Plenum: New York, 1980.
O. J. Biol. Inorg. Chem. 2003, 8, 45-52.
J. AM. CHEM. SOC.
9
VOL. 127, NO. 21, 2005 7817

A R T I C L E S
Hamann et al.
M KCl electrolyte in an ice bath. A graphite disk electrode was used
as the working electrode; a platinum mesh was employed as the counter
electrode, and an SCE, in a separate cell at room temperature, was
used as the reference electrode. Scans were taken from -0.1 to 1.1 V
-
1
vs SCE at a scan rate of 50-75 mV s . An error of (5 mV was
-
estimated for E°′(A/A ) of each redox couple, as determined from the
cyclic voltammetry data.
D. NMR Line Broadening Experiments. NMR spectra were
measured on a Varian 500 MHz spectrometer. Since these results were
used in the interpretation of electrochemical measurements, the
conditions of the two experiments were matched as closely as possible.
Figure 2. (A) Electrospray mass spectrum of [Os(Me2bpy)3]+ from
[
Os(Me2bpy)3]Cl2. (B) Simulated isotopic peak distribution for the molecular
+
Measurements were carried out at 3-4 °C in buffered D
that contained 1 M KCl. The deuterated buffer was prepared as follows.
NaOD in D O was added dropwise, until the pH reached 4.50, to a
2
O solutions
ion [OsC36H36N6] .
Table 1. Mass Spectroscopic and Cyclic Voltammetric Results
2
calcd
mass
observed
m/z
E
°′
solution of 0.134 g of deutero-phthalic acid dissolved in 10 mL of D
(78 mM).
2
O
ion
(V vs SCE)
Diamagnetic Os(II) samples of IV were prepared by dissolving
sufficient IV, as the Os(II) chloride salt, to make a 10 mM solution in
I
II
III
IV
V
VI
VII
VIII
n/a
328
330
372
333
319
361
347
n/a
329
330
372
334
320
362
348
1.001
0.673
0.572
0.385
0.248
0.248
0.111
0.106
2
+
2
+
D
2
O that contained 1 M KCl. An aliquot of this solution was oxidized
by exposure to Cl (g) until the solution turned bright red. To remove
dissolved Cl (g), the solution was purged with Ar(g), and the resulting
2
+
2
2
+
2
2
+
sample was used as the pure paramagnetic Os(III) sample of IV. Both
the diamagnetic and paramagnetic solutions were then diluted by 50%
with the buffer solution. Aliquots of the paramagnetic solution were
added to the diamagnetic solution, and the resulting solutions were used
as mixed samples. Several solutions with different concentrations of
paramagnetic species were prepared in this fashion.
2
+
2
+
a
+
Mass peaks of singly charged ions are given for 2 ions that were
reduced from the matrix; 1 ions were also observed in all spectra with a
m/z 1/2 that of the calculated mass.
+
Diamagnetic samples of V and VIII were prepared from the Os(II)
chloride salt by making a 5 mM solution in a deuterated buffer (2.5
mM, 1 M KCl). Bulk electrolysis was employed to oxidize controlled
amounts of the complex of interest, and aliquots were taken after each
bulk electrolysis step. The mass-transport-limited currents recorded with
a platinum microelectrode were used to verify the mole fractions of
diamagnetic and paramagnetic species of each aliquot. The pure
paramagnetic sample was prepared by bulk electrolysis carried out to
completion, as verified by the lack of significant anodic faradaic current
in the resulting solution.
For data analysis, NMR free-induction decays were imported to
Mestrec software and a Fourier transform was performed. Spectra were
then exported to Sigma Plot software for analysis. All reported peak
positions and line widths were derived from nonlinear least-squares
fits of the peak of interest to a Lorentzian line shape. Errors given in
the spectral parameters are the standard errors from the peak fitting
routine, propagated for the determination of the error in the self-
3
3
7.93 (37.98), H 2.73 (2.55), N 8.64 (8.86). IV: C 41.81 (41.86), H
.49 (3.51), N 7.89 (8.14). V: C 35.15 (35.15), H 3.07 (2.95), N 11.34
(11.71). VI: C 33.56 (33.63), H 2.46 (2.60), N 11.77 (12.07). VII: C
3
9.22 (37.95), H 3.67 (3.58), N 11.23 (11.06). VIII: C 36.65 (36.59),
H 3.31 (3.28), N 11.13 (11.37).
Compounds were converted to the chloride salt by dissolving in
acetone followed by addition, while stirring, of a concentrated solution
of TBACl in acetone. The chloride salt precipitated out of solution,
was filtered, washed with acetone and ether, and then dried in vacuo.
Elemental analysis yielded the following (calculated). II‚2H
47.18), H 3.18 (3.17), N 11.12 (11.00). III‚4H O: C 45.04 (44.95),
H 4.02 (4.02), N 10.48 (10.48). IV‚4H O: C 44.57 (48.81), H 4.26
5.01), N 10.10 (9.49). V‚4H O: C 39.63 (41.53), H 3.72 (4.48), N
3.15 (13.84). VI‚2H O: C 41.41 (41.88), H 3.66 (3.78), N 14.24
15.03). VII‚2H O: C 47.33 (46.32), H 4.94 (4.86), N 12.20 (13.50).
O: C 45.01 (44.94), H 4.48 (4.53), N 12.33 (13.98). Since
2
O: C 47.42
(
2
2
(
1
(
2
2
2
VIII‚2H
2
chloride salts can adsorb water, water was added to the molecular
formula to account for the analysis results.
NMR, mass spectroscopic, and cyclic voltammetric data were
collected on all of the compounds of interest. Analytic NMR measure-
ments were made on a Varian 300 MHz spectrometer. Spectra were
exchange rate constant, kse
.
The line-width method20 was used for proton signals that were not
complicated by coupling. Because the osmium polypyridyl compounds
of interest in this study are in the fast-exchange regime, the chemical
shifts of mixed species were assumed to vary linearly with the mole
fraction of species in the solution.20 Accurate mole fractions, therefore,
can be determined by the ratio of the frequency shift to the contact
shift21
-
-
obtained of the PF
6
salts in CD CN and of the Cl salts in D O. The
3
2
expected integration and shifts of the proton peaks of the ligands were
observed for both types of salts. NMR spectra of the chloride salts
indicated the presence of the expected ligands and the lack of hydrogen-
containing impurities, other than water.
Electrospray mass spectroscopic data (Table 1) were collected for
compounds II-VIII and confirmed the composition of the materials.
The major peaks were also isotopically resolved and compared to theo-
retical spectra for the compounds of interest. Figure 2 displays the peaks
observed for [Os(Me
spectrum of this species. The [Os(Me
reduction of [Os(Me
X ) |ν - ν |/∆ν
(3)
p
d
where X
species, respectively (with X
proton peak in the mixed sample; ν
the pure diamagnetic sample, and ∆ν is the shift difference between the
peaks of the pure paramagnetic and diamagnetic samples (∆ν ) ν - ν ).
p
and X
d
are the mole fractions of paramagnetic and diamagnetic
) 1 - X ); ν is the frequency shift of a
is the corresponding peak shift in
d
p
d
+
2
bpy)
3
]
along with the theoretically predicted
+
2
bpy)
3
]
ion was produced by
p
d
2
+
2
bpy)
3
]
by the matrix; peaks corresponding to
E. Electrochemical Measurements. All buffers were deaerated prior
to use by purging with Ar, and electrochemical measurements with
ZnO were performed under an Ar atmosphere. ZnO electrodes were
etched and then immersed in the supporting electrolyte for at least 5
2
+
2 3
[Os(Me bpy) ]
were also observed. Differences in mass between the
predicted and observed spectra are due to the loss of a hydrogen atom.
The agreement between the observed and calculated spectra and isotopic
patterns confirmed the composition of each compound.
-
The formal reduction potential, E°′(A/A ), of each compound (Table
(
20) Chan, M. S.; Wahl, A. C. J. Phys. Chem. 1978, 82, 2542-2549.
1
) was determined using cyclic voltammetry in buffered H
2
O with 1
(21) Jameson, D. L.; Anand, R. J. Chem. Educ. 2000, 77, 88-89.
7
818 J. AM. CHEM. SOC. VOL. 127, NO. 21, 2005
9

Free-Energy Dependence of Charge-Transfer Rate Constants
A R T I C L E S
min before being transferred to the electrochemical cell. All experiments
were carried out in an ice bath with a solution temperature of ∼3 °C.
The Os(III) compounds were created in situ via bulk electrolysis using
a carbon mesh working electrode to create a 5 mM concentration; a
k T
Jl,a
Jl,a - J
B
ηconc
)
ln
- ln
(6)
( ){ (
)
(J - J )}
nq
-Jl,c
l,c
E
) E - iR - η
(7)
corr
cell
conc
5
% error in [A] was estimated. The concentration of acceptor, [A],
was varied by diluting an aliquot of the redox solution with buffer.
The Nernstian potential of the solution, E(A/A ), changed by less than
with Ecorr the corrected potential of the semiconductor electrode.
-
III. Results
3
mV during each measurement and by less than 15 mV following
A. Measurements of Self-Exchange Rate Constants using
NMR Line Broadening. Figure 3 displays the fit of the methyl
dilution. The J versus E data and the open-circuit potential of each
electrode were recorded before and after each set of differential
capacitance measurements.
Differential capacitance measurements were performed with a
Schlumberger Instruments Model 1260 Impedance Gain-Phase Analyzer
interfaced to a Model SI1287 potentiostat. Measurement parameters
were adjusted to minimize the exposure of the electrode to acidic media,
as it is known that ZnO can dissolve in acidic solutions.22 The
differential capacitance versus E behavior of the semiconductor/liquid
contact was recorded for DC biases that were stepped in 100 mV
increments over the potential range of 0.1-0.8 V vs SCE. A 10 mV
AC signal was superimposed on the DC bias. Each capacitance
2
4
measurement consisted of frequency sweeps from 10 to 10 Hz in
equally spaced logarithmic steps.
To deduce the space-charge capacitance, the impedance spectra were
fitted to an equivalent circuit that consisted of the cell resistance, R
in series with two parallel components: the resistance to charge transfer,
sc, and the space-charge capacitance, Csc. Because Csc is much less
than the differential capacitance, Cdiff, of either the Helmholtz layer or
s
,
R
1
2+
Figure 3. H NMR spectra of the methyl proton peak of Os(Me2bpy)3
1
1,17
2+
3+
3
the double layer, Cdiff was set equal to Csc
.
A linear regression was
2 2
(circles) and of a mixture of Os(Me bpy)3 and Os(Me bpy) (squares).
The lines indicate the results of nonlinear least-squares fitting of each
spectrum.
-
2
used to fit the Cdiff versus E data in accordance with the Mott-
1
7
Schottky equation:
k T
peak of IV to a Lorentzian line shape for a solution of the
diamagnetic Os(II) complex and for a solution containing both
the Os(II) and Os(III) forms of the complex. The self-exchange
-
2
2
B
Csc
)
E - E -
fb
(4)
(
)
2
q
qꢀꢀ N A
D s
o
2
0
where ꢀ is the static dielectric constant of the electrode (8.65 for ZnO);23
rate constant, k , was calculated from
ex
ꢀ
o
is the permittivity of free space; N
semiconductor; A is the surface area of the semiconductor electrode,
and Efb is the flat-band potential of the semiconductor/liquid contact.
Values for N and Efb were obtained from the slope and from the
x-intercept adjusted by k
D
is the dopant density of the
2
s
4πX X (∆ν)
d p
k )
(8)
ex
(
W - X W - X W )C
dp d d p p
D
B
T/q, respectively. With knowledge of N
D
and
where Wdp is the line width (full width at half-maximum) of
the mixed species resonance peak; Wp and Wd are the line widths
of the paramagnetic and diamagnetic peaks, respectively, and
C is the total concentration of exchanging species. The line width
of the diamagnetic species was ∼3-5 Hz. The reported values
of kse are averages of measurements on at least three compo-
sitionally different mixed samples. A value of kse was also
determined from the line broadening of the methyl and of the
E
fb, the energy of the conduction band edge, Ecb, was determined using
the expression
ND
Ecb ) qE + k T ln
(5)
fb
B
( )
N
c
) 3.5 × 10 cm-3 for ZnO.24 The interfacial free energy for
18
where N
c
charge transfer under standard conditions, ∆G°′, was then computed
for each redox system by subtracting qE ′(A/A ) from Ecb
o
-
.
3
,3′ proton peak, as both are uncoupled. This analysis produced
The J versus E data were obtained with a Schlumberger Instruments
8
-1 -1
a value of kse ) 1.1 × 10 M
s
(Table 2) for IV in buffered
Electrochemical Interface Model SI1287 potentiostat. Two scans at a
D2O (pH ) 4.5) at 3 °C and 1 M ionic strength.
-
1
rate of 20 mV s were measured. IR corrections were applied to all
of the J versus E data, despite the negligible effect due to the low
currents in the region of interest. The cell resistance, Rcell, was obtained
from the fit of the impedance spectra to the equivalent circuit. The
data for the low driving force redox couples, VII and VIII, were also
Table 2. Relevant Parameters, Self-Exchange Rate Constants,
1
and Reorganization Energies from H NMR Line-Broadening
Measurements in Buffered D2O (pH ) 4.5) with an Ionic Strength
of 1 M at 3 °C
2
5
corrected for the concentration overpotential, ncone
,
calculated from
νd
a
∆ν
b
Wp
c
kse
×
10
-
7
λse
(eV)
(M- s-1)
1
the mass-transport-limited cathodic and anodic current densities, Jl,c
and Jl,a, respectively, that were measured with a one-sided Pt-foil
electrode of known area, according to eqs 6 and 7:
(Hz)
(Hz)
(Hz)
IVMed
IV3,3′d
V
1161
4064
1643
1122
8297
4892
5079
3857
194
155
101
141
11.3 ( 0.7
12 ( 3
7.3 ( 0.8
7.1 ( 0.7
0.64 ( 0.01
0.64 ( 0.02
0.68 ( 0.01
0.69 ( 0.01
(
22) Gerischer, H.; Sorg, N. Electrochim. Acta 1992, 37, 827-835.
23) Bhargava, R. Properties of Wide Bandgap II-VI Semiconductors; Inspec:
London, 1997; Series No. 17.
VIII
(
a
Quantity νd is the shift of the pure diamagnetic species. b ∆ν is the
(
24) Sze, S. M. The Physics of Semiconductor DeVices, 2nd ed.; Wiley: New
c
shift difference between the diamagnetic and paramagnetic species. Wp is
York, 1981.
the paramagnetic line width. d Subscripts Me and 3,3′ designate results from
the methyl protons and 3,3′ protons, respectively, for compound IV.
(25) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and
Applications; John Wiley & Sons: New York, 2001.
J. AM. CHEM. SOC.
9
VOL. 127, NO. 21, 2005 7819

A R T I C L E S
Hamann et al.
Figure 5. Mott-Schottky plots of compounds I (hexagon w/cross), II (open
circle), III (open square), IV (open triangle up), V (open triangle down),
VI (open diamond), VII (triangle w/cross up), and VIII (square w/cross).
The line indicates the least-squares fit of all of the data.
To obtain values for Csc, which is taken to be equal to Cdiff,
the impedance spectra were fitted over the frequency range of
2
4
1
0 -10 Hz to the equivalent circuit described above. No
frequency dependence of the capacitance was observed, resulting
in very small errors (<1%) for each fit. The series resistance
of the system, Rs, was essentially constant for all measurements,
with a value of 850 Ω. This series resistance can be accounted
for by the ohmic resistance of the sample because the measured
impedance of other electrodes prepared from crystals from the
same batch was ∼1 kΩ.
-
2
Figure 5 displays Mott-Schottky plots in the form of Cdiff
versus E data for all contacts grouped and weighted by their
respective (negligible) errors. All of the plots were linear, as
predicted by eq 4. The resulting slope and intercept of a linear
least-squares fit were used to extract values for Nd and Efb,
respectively. The standard errors resulting from the fit were used
to calculate the errors in Nd and Efb, producing values of Efb )
14
-3
-
0.10 ( 0.01 V vs SCE and Nd ) (5.8 ( 0.6) × 10 cm .
Equation 5 was then used to calculate a value for Ecb/q ) -0.31
0.01 V vs SCE. The invariance of the capacitance data at a
(
fixed electrode potential for all of the compounds, despite the
large variation in the Nernstian potential of the cells, confirms
that the band-edge position is invariant (the “ideal” model) rather
than changing with the solution potential (the Fermi level
pinning situation).
Figure 4. Bode plots of the magnitude of the impedance versus frequency
for I (a), II (b), and VII (c) biased at 0.8 V vs SCE (solid circles) and 0.2
V vs SCE (open square).
-
2
Cdiff
versus E measurements were also performed in
solutions that had varying concentrations of oxidized and
reduced species of the same redox couple. For all of the
junctions studied, the band-edge positions remained constant,
to within 15 mV, after a 10-fold reduction in acceptor
concentration.
2. Current Density versus Applied Potential Measure-
ments. All of the junctions showed rectifying behavior, produc-
ing a limiting anodic current density and an exponentially
increasing cathodic current density, in accord with the diode
equation:
Self-exchange rate constants were also determined for
compounds V and VIII. Both of these compounds have
uncoupled methyl protons, facilitating a straightforward analysis
7
of the NMR data. Values of kse were calculated to be 7 × 10
-1
-1
M
s
for V and VIII (Table 2).
B. Electrochemical Data. 1. Differential Capacitance
versus Applied Potential Measurements. Figure 4 presents
representative Bode plots for some of the semiconductor/liquid
junctions studied in this work. Data are shown for two applied
potentials, 0.8 and 0.2 V vs SCE. The Bode plots of the
impedance magnitude, |Z|, were linear over at least 2 orders of
magnitude variation in ac signal frequency, f, having slopes of
about -1 and phase angles of the current versus ac voltage of
about -90°. The observed impedance of these systems was thus
dominated by a single capacitive circuit element, with Zim ≈
-
qEcorr/γkBT
J ) - J (e
- 1)
(9)
0
where J0 is the exchange current density, γ is the diode quality
factor, and Ecorr is the applied potential corrected for both
concentration overpotential and for series resistance losses in
the electrochemical cell (eq 7).
-
1 11
(
2πfCdiff) .
7820 J. AM. CHEM. SOC.
9
VOL. 127, NO. 21, 2005

Free-Energy Dependence of Charge-Transfer Rate Constants
A R T I C L E S
5
). The value of ket was then readily calculated in accordance
with the rate law given in eq 1, by dividing J by the quantity
-qns[A]} at a given potential. The J versus E data collected
{
at the largest acceptor concentration were used to minimize the
error in the concentration and because the diode quality factors
were close to 1 (typically 1 < γ < 1.1) under such conditions.
The quoted ket for each contact represents the average of values
calculated using the same high cathodic current density portion
-6
-2
of the J versus E curve (-(1-2) × 10 A cm ) for each redox
couple. A standard error analysis was performed in conjunction
with calculation of the rate constants by propagating the errors
of all measured parameters used in the calculation of ket. The
error in Ecb dominated the error in ket, due to the exponential
dependence of ns on (Ecb - qE). Table 3 summarizes the values
of ket determined for each of the ZnO/H2O-redox couple
junctions evaluated in this study.
Figure 6. Plots of the dark current density versus Ecorr for compound III
at high concentration ([A] ) 5 mM; filled circles) and low concentration
(
[A] ≈ 0.5 mM; open squares). As noted in the text, a decrease in [A]
should result in a shift of the J-Ecorr curve according to (kBT/q)ln([A]high/
A]low). Potentials are referenced to SCE.
IV. Discussion
[
A. Reorganization Energies. Self-exchange rate constants
Table 3. Results of Current Density versus Applied Potential
Measurements, Barrier Height, and Rate Constant Determinations
7
-1 -1
7
of kse ) 2 × 10 M
s
for III and IV and kse ) 1 × 10
-
1
-1
M
s
for I have been measured in CH3CN at 31 °C with
E (theor)b
(mV)
qE(A/A-
(eV)
)
−
E
c
k
et
d
∆
cb
-
20
a
a
(cm4 s-1
PF6 as the counterion and at an ionic strength of ∼0.1 M.
compound
γ
high
γ
low
)
Rate constants were found to be larger in water and to increase
-
-
-
-
-
-
-
-
19
17
16
17
17
17
18
18
I
II
1.33 1.53
1.05 1.15
1.03 1.02
1.04 1.06
1.06 1.14
1.02 1.19
1.06 1.19
1.19 1.29
55 (60)
44 (54)
42 (51)
50 (54)
46 (48)
35 (45)
50 (54)
25 (34)
1.30 ( 0.01 (1.4 ( 0.6) × 10
0.96 ( 0.01
(2 ( 1) × 10
9
-1 -1
with ionic strength, with a value of kse ≈ 2 × 10 M
s
-
III
IV
V
VI
VII
VIII
0.86 ( 0.01 (1.6 ( 0.7) × 10
estimated for the ClO4 salt of I at 25 °C in a 1 M HClO4
0.67 ( 0.01
0.54 ( 0.01
0.54 ( 0.01
0.40 ( 0.01
0.40 ( 0.01
(6 ( 2) × 10
(2 ( 1) × 10
(3 ( 1) × 10
(5 ( 2) × 10
(6 ( 2) × 10
26
solution. Comparison of these results highlights the effects
that solvent, counterion, and ionic strength have on kse. Thus,
in our study, the self-exchange rate constants were measured
-
under conditions (buffered D2O with Cl counterion and 1 M
KCl at 3 °C) that were as close as possible to those used in the
electrochemical measurements on ZnO.
a
The quantities γhigh and γlow are the diode quality factors at high acceptor
concentration, [A]high, and low acceptor concentration, [A]low, respectively.
b
The value of [A]high was always 5.0 mM, and [A]low ≈ 0.5 mM. The
The self-exchange rate constants for the redox couples
employed were all very similar. A mean value of (8.6 ( 0.7)
theoretical shift of the J versus Ecorr data was calculated by ∆E(theor) )
c
-
(
kBT/q)ln(Jl,c,high/Jl,c,low). The quantity qE(A/A ) - Ecb was calculated using
7
-1 -1
d
× 10 M
s
(D2O, 1 M KCl, 3 °C) was measured here for
the high acceptor concentration data. As noted in the text, ket was calculated
using the high acceptor concentration data for all redox systems investigated.
complexes IV, V, and VIII. The self-exchange rate constants
of a series of osmium tetramethyl, dimethyl, and unsubstituted
phenanthroline complexes have been previously observed to
Figure 6 displays plots of ln(-J) versus Ecorr for compound
III (plots for all other compounds investigated in this work are
given as Supporting Information). The diode quality factors were
close to 1.0, in accord with expectations for a process that is
kinetically first-order in the concentration of electrons at the
surface of the semiconductor. The dependence of the rate on
the concentration of acceptor species in the solution was
determined by reducing [A], which produced shifts of the J
versus Ecorr data, ∆E ) (kBT/q)ln([A]high/[A]low), as expected
for a first-order process in [A]. The magnitude of the change in
27
vary only by a factor of 2. Therefore kse is assumed to be
similar for I, III, and IV under our conditions. In addition, kse
for II is also assumed to be similar to that for III-IV since II
has the same metal center and very similar ligands. The rate
constants for V and VIII were taken to be representative of
values for compounds V-VIII. No diffusion corrections were
made to the measured self-exchange rate constants since the
values are significantly below the diffusion-limited value of ∼3
9
-1 -1 28
×
10 M s .
The reorganization energy, λse, for a given species in a self-
[
A] was verified by measuring the limiting cathodic current
exchange electron-transfer process can be related to the self-
densities at both acceptor concentrations, Jl,c,high and Jl,c,low, with
a Pt microelectrode, and the theoretical shift of the J versus
Ecorr data was calculated by ∆E(theor) ) (kBT/q)ln(Jl,c,high/Jl,c,low).
Values of γ, ∆E, and ∆E(theor) are given in Table 3 for the
systems of interest in this work.
1,28-31
exchange rate constant by the expression
-λse/4kBT
k ) K κ ν Γe
(10)
ex
A el n
where KA is the equilibrium constant for the formation of the
3
. Rate Constants for Interfacial Charge Transfer, ket.
Since nondegenerately doped semiconductor electrodes have
relatively little Frumkin correction associated with the liquid
part of the solid/liquid double layer, the acceptor concentration
can be assumed to be equal to the bulk value. The surface
electron concentration at each applied potential, ns(E), was
computed according to eq 2 using the conduction band-edge
energy extracted from the flat-band potential determinations (eq
(26) Young, R. C.; Keene, F. R.; Meyer, T. J. J. Am. Chem. Soc. 1977, 99,
2468-2473.
(
27) Triegaardt, D. M.; Wahl, A. C. J. Phys. Chem. 1986, 90, 1957-1963.
8
(28) Sutin, N. Acc. Chem. Res. 1982, 15, 275-282.
(
29) Meyer, T. J.; Taube, H. In ComprehensiVe Coordination Chemistry;
Wilkinson, S. G., Gilliard, R. D., McCleverty, J. A., Eds.; Pergamon
Press: New York, 1987; Vol. 1, p 331.
(
30) Sutin, N. Prog. Inorg. Chem. 1983, 30, 441-498.
31) Brunschwig, B. S.; Logan, J.; Newton, M. D.; Sutin, N. J. Am. Chem. Soc.
1980, 102, 5798-5809.
(
J. AM. CHEM. SOC.
9
VOL. 127, NO. 21, 2005 7821

A R T I C L E S
Hamann et al.
1
4
-3
precursor complex of the reactants; κel is the electronic transmis-
sion coefficient; νn is the effective nuclear vibration frequency
of the activated complex, and Γ is a correction for nuclear
tunneling. For an ionic strength of 1 M, the work to bring two
positively charged species together is less than kBT, which
V vs SCE, Nd ) (5.8 ( 0.6) × 10 cm , and T ) 276 K) at
pH 5 is therefore in good agreement with prior results on ZnO
in H2O.
C. Current Density versus Applied Potential Measure-
ments. A notable feature of the ZnO/H O contacts reported
herein is their nearly ideal J versus E behavior. Diode quality
factors for data collected at high acceptor concentrations were
generally found to be between 1 and 1.1, and the J versus E
curves shifted by the theoretically expected amounts as the
acceptor concentration was varied. These observations imply
that the interfacial kinetics follows the rate law of eq 1 and
indicate that surface-state effects do not dominate the charge-
2
32,33
11
implies that KA ≈ 1.
Given the similarity of the complexes
studied, they are assumed to have similar values of κel ≈ 1 (i.e.,
the reactions are adiabatic).31
The total reorganization energy in self-exchange reactions is
comprised of changes in the bond lengths and angles in the inner
coordination sphere of the complexes, λse,in, and changes in the
polarization of the solvent in the outer coordination sphere, λse,out
1
1
(λse ) λse,in + λse,out). For Os and Ru complexes that involve
transfer processes of the systems investigated.
bipyridyl ligands, the inner-sphere does not undergo significant
changes upon electron transfer.34 The reorganization energy is,
therefore, dominated by the solvent reorganization energy which
D. Dependence of Interfacial Charge-Transfer Rate Con-
stants on Driving Force: Comparison between Theory and
Experiment. One of the most interesting predictions of Marcus
theory is that of the inverted region, in which the electron-trans-
fer rate constant decreases with increasing driving force. The max-
imal rate for a molecular donor-acceptor system is observed
when the standard free-energy change for the reaction, ∆G°′, ex-
actly cancels the reorganization energy, λ. When the driving force
is either increased or decreased relative to this point, an ener-
gy barrier is present for the reaction, and the rate decreases.4
The Marcus description of electron transfer has been general-
ized for interfacial electron-transfer reactions for both metal and
semiconductor electrodes. Metal electrodes, in contrast with
molecules, have many closely spaced electronic levels. The rate
of electron transfer from a metal electrode to a redox molecule
in solution consists of a summation of rates from each of the
occupied metal levels, with each level having a rate that is
characterized by the Marcus free-energy relationship. Thus,
when the driving force for the reaction is made high enough
31
has a significant effect on νn. For these reactions, νn was taken
to be 1011 s .
-1 29,35
The values of λse obtained using eq 9 with
the measured values of kse are 0.67 ( 0.04 eV (Table 2).
The outer-sphere reorganization energy for two identical
1
spherical reactants in solution can be calculated as
2
,5,38-40
(
∆zq)
1
a
1 1
1
ꢀ)
λout
)
-
-
(11)
(
)
(
2
4πꢀ0
[
R
]
n
1
7
where ∆z is the change in charge of each reactant species to
form products, a is the radius of the reactant; R is the reactant
center-to-center separation; n is the refractive index of the
solvent (1.3438 for 0.98 M KCl in H2O at 20 °C, with a very
small temperature dependence ), and ꢀ is the static dielectric
constant of the solvent (86.5 for H2O at 3 °C ). Using a )
.60 nm and R ) 1.2 nm produces a value of λse,out ) 0.65 eV,
36
3
6
0
(the electrode potential is made negative enough), an inverted
in excellent agreement with the value derived from the self-
region barrier and a decreased rate of electron transfer will be
produced from the highest occupied energy levels. Since there
are many closely spaced levels of lower energy, however, there
will always be occupied electronic states at optimal exoergicity
that dominate the interfacial current. Thus, for metal electrodes,
the total rate will not decrease with driving force, which
precludes the direct observation of the inverted region.
An ideal semiconductor has no electronic levels in the band
gap region, so only electrons with energies near the conduction
band, for an n-type material, can contribute to the current flow.
This situation is then analogous to a donor-acceptor system.
Thus, as the driving force of the reaction increases, the electron-
transfer rate of the conduction band electrons should increase,
reach a maximal value, and then decrease. It should, therefore,
be possible to observe directly the inverted region for semi-
conductor/liquid contacts.
exchange rate-constant measurements (Table 2).
B. Differential Capacitance versus Applied Potential
Measurements. The nearly ideal behavior of the Mott-Schottky
plots allowed accurate determination of the flat-band potentials
for the ZnO/H2O interfaces of interest. The Efb values for a given
ZnO/liquid contact did not vary significantly as the measurement
frequency was changed. Furthermore, Efb values were essentially
constant vs SCE as the Nernstian potential of the cell was changed
by ∼0.9 V by varying the redox species in the homologous
series of compounds investigated in this work (Figure 5).
In prior work on ZnO, the pH dependence of Efb was studied
37
and the predicted 2.3kBT/q shift of Efb per pH unit was verified.
A value for Efb ≈ -0.26 V vs SCE was determined at 25 °C at
16
-3
pH 5 with Nd ) 5 × 10 cm , which corresponds to a value of
Ecb/q ≈ -0.37 V vs SCE. In another report, a value for Efb of
1
4
-3
-
0.33 V vs SCE was found for Nd ) 7 × 10 cm at 25 °C
14
at pH 8.5, which corresponds to a value of Ecb/q ≈ -0.34 V
Several different levels of theory have been used to derive
an expression for the rate constant at optimal exoergicity of an
interfacial electron transfer for a semiconductor electrode in
vs SCE at pH 5. Our experimental value of Ecb/q ) -0.31 (
0
.01 V vs SCE (calculated using eq 5 with Efb ) -0.10 ( 0.01
contact with an acceptor species in solution. These approaches
(
32) Sutin, N.; Brunschwig, B. S. ACS Symposium Series 198, 1982, 105-
35.
8-11,41
include a simple collisional model
and an electronic
1
(
33) Sutin, N. In Tunneling in Biological Systems; Chance, B., Marcus, R.,
DeVault, D. C., Schrieffer, J. R., Frauenfelder, H., Sutin, N., Eds.; Academic
Press: New York, 1979; pp 201-228.
coupling model based on Marcus’ treatment of electron transfer
(34) Biner, M.; Burgi, H.-B.; Ludi, A.; Rohr, C. J. Am. Chem. Soc. 1992, 114,
(38) McCleskey, T. M.; Winkler, J. R.; Gray, H. B. Inorg. Chim. Acta 1994,
225, 319-322.
5
197-5203.
(
35) Marcus, R. A. J. Phys. Chem. 1963, 67, 853-857.
36) Lide, D. R. CRC Handbook of Chemistry and Physics, 81st ed.; CRC
Press: Boca Raton, FL, 2001.
(39) Creutz, C.; Sutin, N. J. Am. Chem. Soc. 1977, 99, 241-243.
(40) Closs, G. L.; Calcaterra, L. T.; Green, N. J.; Penfield, K. W.; Miller, J. R.
J. Phys. Chem. 1986, 90, 3673-3683.
(
(
37) Lohmann, F. Ber. Bunsen-Ges. Phys. Chem. 1966, 70, 428-434.
(41) Gerischer, H. J. Phys. Chem. 1991, 95, 1356-1359.
7822 J. AM. CHEM. SOC.
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VOL. 127, NO. 21, 2005

Free-Energy Dependence of Charge-Transfer Rate Constants
A R T I C L E S
at the interface of two immiscible liquids.42-44 An electronic
coupling model based on the Fermi Golden Rule applied to the
case of a semiconductor electrode in contact with a random
distribution of acceptor species in solution has produced the
following expression for the rate constant:45
where âsc is the coupling attenuation factor; lsc is the effective
coupling length in the semiconductor; λsc is the reorganization
energy of the acceptor species near the semiconductor electrode,
2
and dsc is the atomic density of the solid. The quantity H
AB,sc
represents the square of the matrix element that couples reactant
and product states at energy E, averaged over all degenerate
states in the semiconductor in a plane parallel to the electrode
surface. This value is assumed to be independent of energy over
the range of interest.45 The subscript “sc” indicates parameters
for a semiconductor electrode. Equation 12 can be rewritten as
2
-(∆G°′ + λsc)
ket ) ket,max exp
(13)
{
}
4
λ k T
sc B
where the prefactor has been combined into ket,max, which is
the rate constant at optimal exoergicity. While ket,max is not
independent of the reorganization energy, and thus will vary
for different redox couples, this variation is modest.
The similarity in the values of the self-exchange rate constants
for the complexes studied indicates that both the electronic
coupling in homogeneous solution and the reorganization
energies are essentially constant. For the electrode reactions,
since all compounds are coupling to the same electrode, the
electronic coupling coefficient, HAB, is also likely to be similar
for the redox species of concern.
Figure 7. Plots of the electron-transfer rate constant for compounds I-VIII
-
as a function of the standard driving force, -∆G°′ ) qE°′(A/A ) - Ecb,
for the redox systems investigated. (a) The solid line represents the predicted
-
16
4
-1
ket versus ∆G°′ behavior for ket,max ) 1 × 10
cm s and λsc ) 0.67
eV. (b) The dotted line represents the predicted ket versus ∆G°′ behavior
-
16
4 -1
for ket,max ) 1 × 10
cm s and λsc ) 0.8 eV, and the dashed line is the
4
-16
-1
predicted ket versus ∆G°′ behavior for ket,max ) 1 × 10
) 0.53 eV.
cm s and λsc
Figure 7 shows a semilogarithmic plot of ket versus the
plexes.20 These results, therefore, support the theoretical ex-
pectation of the Marcus inverted region for these interfacial
electron-transfer processes.
standard driving force for interfacial electron transfer, -∆G°′
-
)
qE°′(A/A ) -Ecb. The rate constant at optimal exoergicity,
-
16
ket,max, is obtained when ∆G°′ ) -λsc, and has a value of ∼10
4
-1.45
The value of the reorganization energy for an electron-transfer
reaction between a redox couple and a ZnO electrode, where
both the redox couple in solution and the image charge in the
semiconductor contribute to the total reorganization energy, is
expected to be less than that for the self-exchange reaction of
the couple in homogeneous solution. A theoretical value for
the outer-sphere reorganization energy, λsc,out, of a redox couple
cm s
A ket versus ∆G°′ curve calculated according to eq
-
16
4
-1
1
3 with ket,max ) 10
cm s and λsc ) 0.67 eV (the value
derived from self-exchange measurements; see discussion of
the reorganization energy below) has been superimposed on the
plot of Figure 7a (solid line). The agreement between the
experimental data and theory is very good.
An uncertainty in the absolute value of Efb would introduce
a constant offset in the standard driving force, and resulting
rate constants for all compounds, but such an error in Efb and/
or Nd would not change the observed trend of the rate constant
with increasing driving force. The interfacial rate constant for
I was observed to be 2 orders of magnitude smaller than that
for II, despite >300 mV greater driving force. Although
complex I has a metal center different than those of complexes
II-VIII, the self-exchange rate constants have been shown to
be very similar for analogous Os and Ru polypyridyl com-
4
6,47
at a ZnO electrode can be calculated by eq 14:
where nZnO and nSol are the refractive index of ZnO (1.923,48
and the aqueous solution, respectively; ꢀZnO and ꢀSol are the static
)
2
3
dielectric constants of ZnO (8.65 ) and the solution, respec-
tively, and Re is the distance from the acceptor to the electrode.
(
42) Marcus, R. A. J. Phys. Chem. 1990, 94, 1050-1055.
(46) Kuciauskas, D.; Freund, M. S.; Gray, H. B.; Winkler, J. R.; Lewis, N. S.
J. Phys. Chem. B 2001, 105, 392-403.
(47) Marcus, R. A. J. Phys. Chem. 1991, 95, 2010-2013.
(48) Ashkenov, N.; Mbenkum, B. N.; Bundesmann, C.; Riede, V.; Lorenz, M.;
Spemann, D.; Kaidashev, E. M.; Kasic, A.; Schubert, M.; Grundmann, M.;
Wagner, G.; Neumann, H.; Darakchieva, V.; Arwin, H.; Monemar, B. J.
Appl. Phys. 2003, 93, 126-133.
(
43) Pomykal, K. E.; Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. 1996, 100,
3
652-3664.
(
44) Gao, Y. Q.; Georgievskii, Y.; Marcus, R. A. J. Chem. Phys. 2000, 112,
358-3369.
45) Royea, W. J.; Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. B 1997, 101,
1152-11159.
3
(
1
J. AM. CHEM. SOC.
9
VOL. 127, NO. 21, 2005 7823

A R T I C L E S
Hamann et al.
Using a ) 0.60 nm and Re ) 0.60 nm produces a value of λsc,out
polypyridyl redox couples. Differential capacitance measure-
ments showed that the band edges of ZnO were fixed to within
)
0.53 eV, which is approximately 80% of the outer-sphere
reorganization energy of this system in a self-exchange reaction.
1
0 mV with respect to SCE when the solution potential was
A ket versus ∆G°′ curve calculated according to eq 13 with
changed by ∼900 mV. Current density versus potential mea-
surements displayed a first-order dependence on acceptor and
surface electron concentrations. This behavior allowed the
experimental determination of interfacial electron-transfer rate
constants for such systems. The driving force was changed by
varying the formal reduction potential of the redox couple in
solution. The reaction with the highest driving force had the
smallest rate constant. This observation of decreasing rate
constant with increasing driving force indicates that semicon-
ductor/liquid contacts can operate in the inverted regime with
nonadsorbed outer-sphere redox systems.
-
16
4
-1
ket,max ) 10
cm s and λsc,out ) 0.53 eV is superimposed
on the plot of Figure 7b (dashed line). Such a curve yields a
poor fit to the experimental data, with a λsc that is too low.
Inspection of the plot indicates that the maximum rate constant
occurs at -∆G°′ ≈ 0.8 eV, which implies a λsc for our series
of ∼0.8 eV. A ket versus ∆G°′ curve calculated according to
-
16
4
-1
eq 13 with ket,max ) 10
cm
s
and λsc ) 0.8 eV is
superimposed on the plot of Figure 7b (dotted line). This
theoretical curve fits the low driving force regime well; however,
it deviates significantly for the high driving force compounds.
The reorganization energy derived from the self-exchange
reactions, and calculated according to eq 11, appears to be the
best estimate for the reorganization energy of the redox
compounds investigated herein at the ZnO/water interface. The
slight increase in reorganization energy over that calculated from
eq 14 may indicate some contribution to the reorganization
energy arising from the electrode and is consistent with similar
Acknowledgment. We acknowledge the Department of
Energy, Office of Basic Energy Sciences, for support of this
work. Dr. Mona Shahgholi provided helpful assistance in
acquiring mass spectroscopic data, and Drs. Nagarajan Srivatsan
and Norman Sutin are acknowledged for helpful conversations.
observations for interfacial charge-transfer processes of silicon
electrodes, as reported previously.1
0,11
However, clearly most
Supporting Information Available: Additional plots of dark
current density versus Ecorr for compounds I, II, and IV-VIII.
This material is available free of charge via the Internet at
http://pubs.acs.org.
of the reorganization energy for ZnO/redox systems investigated
is well described by expectations based on eq 14.
V. Conclusions
The ZnO/H2O junctions displayed nearly ideal energetic and
kinetics behavior in contact with a homologous series of osmium
JA0436188
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VOL. 127, NO. 21, 2005