Scanning electrochemistry microscopy (SECM) in the study of electron transfer kinetics at liquid/liquid interfaces: Beyond the constant composition approximation

Article abstract of DOI:10.1021/jp991414l

A numerical model is developed for the SECM feedback mode for the case of irreversible electron transfer (ET) processes at the interface between two immiscible electrolyte solutions (ITIES). In this application, a redox-active species is electrogenerated by the reduction/oxidation of the oxidized/reduced form of a couple at an ultramicroelectrode (UME) tip located in one liquid (phase 1). The tip is positioned close to the interface with a second immiscible liquid (phase 2), that contains the oxidized/reduced half of another redox couple. If ET occurs between the tip-generated species in phase 1 and the redox-active species in phase 2, then the original species in phase 1 is regenerated at the interface and undergoes positive feedback at the tip, enhancing the steady-state current. The feedback current, for a given separation between the tip and the interface, is shown to depend on the ratio of the concentrations of the redox-active species in the two phases, their relative diffusion coefficients, and the rate constant for the redox reaction. The results of the model are used to identify the conditions under which (i) diffusion in phase 2 has to be considered and; (ii) a simpler limiting (constant composition) model for phase 2, employed to analyze earlier SECM experiments, can be used. In addition to diversifying the range of conditions under which redox reactions at ITIES can be studied, the results of the model demonstrate that there are considerable advantages to lifting the constant composition restriction on phase 2 for the accurate characterization of rapid redox reactions. The theoretical predictions are examined through experimental studies of electron transfer between the electrogenerated, oxidized form of zinc-21H, 23H-tetraphenylporphine (ZnPor) in benzene or benzonitrile and the reductants Fe(CN)₆^4-, Ru(CN)₆^4-, Mo(CN)₈^4-, or FeEDTA^2- (where EDTA denotes ethylenediaminetetraacetic acid) in an aqueous solution. Bimolecular rate constants for each of these systems are reported, with the potential across the ITIES biased with either perchlorate or tetrafluoroborate ions in each phase.

Full text of DOI:10.1021/jp991414l

7260
J. Phys. Chem. B 1999, 103, 7260-7269
Scanning Electrochemistry Microscopy (SECM) in the Study of Electron Transfer Kinetics
at Liquid/Liquid Interfaces: Beyond the Constant Composition Approximation
Anna L. Barker and Patrick R. Unwin*,†
Department of Chemistry, UniVersity of Warwick, CoVentry CV4 7AL, U.K.
Shigeru Amemiya, Jungfen Zhou, and Allen J. Bard*,‡
Department of Chemistry and Biochemistry, UniVersity of Texas at Austin, Austin, Texas 78712
ReceiVed: April 29, 1999; In Final Form: June 19, 1999
A numerical model is developed for the SECM feedback mode for the case of irreversible electron transfer
(ET) processes at the interface between two immiscible electrolyte solutions (ITIES). In this application, a
redox-active species is electrogenerated by the reduction/oxidation of the oxidized/reduced form of a couple
at an ultramicroelectrode (UME) tip located in one liquid (phase 1). The tip is positioned close to the interface
with a second immiscible liquid (phase 2), that contains the oxidized/reduced half of another redox couple.
If ET occurs between the tip-generated species in phase 1 and the redox-active species in phase 2, then the
original species in phase 1 is regenerated at the interface and undergoes positive feedback at the tip, enhancing
the steady-state current. The feedback current, for a given separation between the tip and the interface, is
shown to depend on the ratio of the concentrations of the redox-active species in the two phases, their relative
diffusion coefficients, and the rate constant for the redox reaction. The results of the model are used to identify
the conditions under which (i) diffusion in phase 2 has to be considered and; (ii) a simpler limiting (constant
composition) model for phase 2, employed to analyze earlier SECM experiments, can be used. In addition to
diversifying the range of conditions under which redox reactions at ITIES can be studied, the results of the
model demonstrate that there are considerable advantages to lifting the constant composition restriction on
phase 2 for the accurate characterization of rapid redox reactions. The theoretical predictions are examined
through experimental studies of electron transfer between the electrogenerated, oxidized form of zinc-21H,
4-
23H-tetraphenylporphine (ZnPor) in benzene or benzonitrile and the reductants Fe(CN)64-, Ru(CN)6
,
Mo(CN)84-, or FeEDTA2- (where EDTA denotes ethylenediaminetetraacetic acid) in an aqueous solution.
Bimolecular rate constants for each of these systems are reported, with the potential across the ITIES biased
with either perchlorate or tetrafluoroborate ions in each phase.
Introduction
the tip and the ITIES, (ii) mediator diffusion in phase 2, (iii)
the rate of the reaction at the ITIES, and (iv) ion transport across
Scanning electrochemical microscopy (SECM) has emerged
as a powerful approach for studying the kinetics of electron,1-7
ion8-11 and molecular10,12 transfer across an ITIES. Studies of
ET have employed the feedback mode,13 in which a redox-active
species, e.g., Ox1 is electrogenerated from a reduced form, Red1,
at an UME tip located in one liquid (phase 1), close to the
interface with a second immiscible liquid (phase 2) that contains
the reduced form of a separate redox couple (Red2). As shown
schematically in Figure 1, if a redox reaction occurs between
Ox1 and Red2
the ITIES to balance charge. The kinetics of the interfacial
reaction can be evaluated from the tip current response, provided
the contributions from the other processes are either known or
nonlimiting.
The advantages of studying interfacial redox reactions at an
ITIES by SECM feedback are now well established.1,4 In
particular, (i) the SECM current allows discrimination between
electron and ion transfer processes at the interface; (ii) distortions
associated with iR drop and charging current are eliminated;
(iii) it is not necessary to externally bias the interface, thereby
avoiding complications due to the polarization window of the
ITIES and the variation of the interfacial properties with applied
potential. In addition, it is generally proven that SECM allows
the study of rapid interfacial processes, since the micrometer
and submicrometer dimensions of both the UME and the tip/
interface gap promote high rates of mass transport.
Ox1 + Red2 f Ox2 + Red1
then Red1 is regenerated at the interface and diffuses back to
the tip. The effect of this positive feedback of Red1 is to enhance
the current compared to the situation where there is no redox
reaction and Red1 simply reaches the tip by hindered diffusion.
There are four main processes that contribute to the magnitude
of the tip current:1 (i) mediator diffusion in phase 1 between
Previous SECM liquid/liquid feedback experiments often
employed a high concentration of reactant in phase 2 compared
to that of the mediator in phase 1 to avoid the complication of
diffusional effects in the second phase.1,4,7 While this approach
simplifies the model, it does limit the use of lower concentrations
† E-mail: p.r.unwin@warwick.ac.uk.
‡ E-mail: ajbard@mail.utexas.edu.
10.1021/jp991414l CCC: $18.00 © 1999 American Chemical Society
Published on Web 08/04/1999

SECM at Liquid/Liquid Interfaces
J. Phys. Chem. B, Vol. 103, No. 34, 1999 7261
allows the principle of mass conservation to be invoked in phase
1:
0 e r e rg, 0 < z < d: cOx (r,z) ) c/Red - cRed (r,z)
(2)
1
1
1
where cOx (r,z) and cRed (r,z) are the spatial-dependent concen-
1
1
trations of Ox1 and Red1, respectively, within the region of
interest (see Figure 1), defined in terms of r and z, which are
the coordinates in the directions radial and normal to the
electrode surface measured from the center of the electrode.
The parameters d and rg, respectively, denote the location of
the liquid/liquid interface and the edge of the glass sheath
surrounding the UME. The use of eq 2 simplifies the problem
to the consideration of species Red1 and Red2 alone. Although
the diffusion coefficients of Red1 and Ox1 may differ slightly
in real systems, it has been shown that under steady-state
conditions (of interest in this paper), the ratio DOx /DRed has
1
1
no effect on the positive feedback current characteristics.14
Time-dependent diffusion equations, appropriate to the axi-
symmetric SECM geometry, can be written for the species of
interest in each phase
Figure 1. Schematic of SECM feedback at an ITIES with the
coordinate system used for the theoretical model. The coordinates r
and z are measured from the center of the UME in the radial and normal
directions, respectively. The UME is characterized by an electrode
radius, a, and rg is the distance from the center of the electrode to the
edge of the surrounding insulating glass sheath. The ITIES is located
at distance, d. Species Red1 and Ox1 are confined to phase 1, while
species Red2 and Ox2 are present in phase 2.
∂c
∂2c
∂c
∂2c
Red1
Red1
Red1
Red1
1
r
Phase 1:
) DRed
+
+
+
+
(3)
(4)
[
[
]
]
∂r2
∂z2
1
∂t
∂r
∂c
∂2c
∂c
∂2c
Red2
Red2
Red2
Red2
1
r
Phase 2:
) DRed
∂r2
∂z2
2
∂t
∂r
in the second phase. Moreover, as shown in this paper, the use
of lower concentrations is particularly beneficial in determining
the rate constant when the electron transfer reaction is rapid.
Because of the advantages of SECM, compared to alternative
methods, we develop here a more general theory to both
considerably extend the applicability and to identify precisely
when the constant composition assumption is valid. We have
recently reported a theoretical model which includes depletion
and diffusion effects in the second phase for the process of
SECM induced transfer (SECMIT),10 and this paper serves to
extend the theory developed therein to the SECM feedback
mode.
where cRed and DRed are, respectively, the concentration and
2
2
diffusion coefficient of species Red2 in phase 2, and t is time.
In order to calculate the tip current response, the diffusion
equations must be solved subject to the boundary and initial
conditions of the system. Prior to the potential step, phase 1
and 2 contain only species Red1 and Red2, respectively. The
initial condition is thus
/
0 e r e rg,
0 e r e rg,
c
) c
t ) 0;
0 e z e d:
z > d:
Red1
Red1
cRed ) c/Red
(5)
2
2
A theoretical model is developed for the case of an irrevers-
ible, driven ET process at the interface, i.e. the situation where
the potentials of the redox couples in the two phases are widely
separated. The model would require only simple modifications
to be applicable to the cases of reversible and quasireversible
ET. Because of the number of parameters involved in the
problem, the aim of this paper is to give a general guide to the
conditions where depletion effects in the second phase become
important and to state the implications for the analysis of ET
kinetics. The predictions from the model are examined through
where c/Red denotes the initial bulk concentration of Red2 in
phase 2.
2
The potential of the UME tip is stepped from a value where
no electrode reaction occurs, to one sufficient to drive the
oxidation of Red1 at a diffusion-controlled rate. Species Red1
is assumed to be inert with respect to the insulating glass sheath
surrounding the electrode and to remain at bulk concentration
values beyond the radial edge of the tip (throughout phase 1).
In phase 2, species Red2 attains its bulk concentration for r >
rg and at a semi-infinite distance from the electrode. Conse-
quently, the exterior boundary conditions may be summarized
as follows:
experimental studies of ET between electrogenerated ZnPor+•
4-
in benzene or benzonitrile and the reductants Fe(CN)6
,
Ru(CN)64-, Mo(CN)84-, or FeEDTA2- in an aqueous solution.
These studies clearly demonstrate that lifting the constant
composition restriction on the second phase is particularly
advantageous for the accurate measurement of rapid redox
reactions.
cRed ) 0
(6)
(7)
(8)
z ) 0, 0 e r e a:
1
∂c
Red1
a < r e rg:
z ) 0,
DRed
) 0
1
∂z
/
Red2
Theory
0 e r e rg:
z f ∞,
c
) c
Red2
Formulation of the Problem. Consider the case where the
/
r > rg,
c
) c
0 < z < d:
Red1
Red1
diffusion coefficients of Red1 and Ox1 are equal, i.e., DRed
)
1
D
Ox , with only the reactant, Red1, initially present in phase 1
1
/
c
) c
at concentration c/Red . This is a useful assumption, since it
(9)
z > d:
Red2
Red2
1

7262 J. Phys. Chem. B, Vol. 103, No. 34, 1999
Barker et al.
This latter condition is valid provided that RG ) rg/a g 10,15
where a is the radius of the electrode.
The axisymmetric cylindrical geometry of the SECM implies
Method of Solution. Numerical solutions were achieved
using the alternating direction implicit finite-difference method
(ADIFDM).18 The application of this method to a variety of
SECM problems has been described elsewhere.10,12,14,16,19 The
following brief account of the modifications required to treat
the internal boundary condition in the above model should be
read with reference to more detailed descriptions of the
ADIFDM as applied to SECM.10,16,19
The ADIFDM involves the construction of implicit finite
difference equations at successive half-time intervals, respec-
tively, for the R and Z directions. The finite-difference grid
employed was as used previously for two-phase problems.10 For
the first half-time step, in which concentrations were calculated
in the radial direction from known values deduced in the
previous half-time step, the calculation proceeded from the
points j ) 1 to j ) NE + NG - 1, where j denotes the grid
points in the radial direction running from j ) 0 at R ) 0 to j
) NE at R ) 1 (over the tip electrode) and j )NE + 1 to j )
NE + NG over the insulating glass sheath. This results in NE
+ NG - 1 simultaneous equations and NE + NG - 1
unknowns for each grid point, k, in the Z coordinate. After
application of the boundary conditions above, these equations
may be expressed as a tridiagonal matrix, the solution of which
is found using the Thomas algorithm.20
∂c
Red1
r ) 0, 0 < z < d:
DRed
DRed
) 0
) 0
1
∂r
∂c
Red2
z > d:
(10)
2
∂r
The final internal boundary condition applies to the interface
and relates the flux of species Red1 and Red2, at the ITIES, to
the rate of the second-order redox reaction occurring at the
interface.
z ) d, all r:
∂c
∂c
Red2
Red1
DRed
) DRed
) k12(c/Red - cRed )cRed (11)
1 1 2
1
2
∂z
∂z
where k12 is the heterogeneous bimolecular rate constant (cm
s-1 M-1).
To formulate a general solution, the following dimensionless
terms are introduced:
The calculation was performed for each point on the finite-
difference grid, over phase 1 (from k ) 1 to k ) NZ1 - 1) and
then phase 2 (from k ) NZ1 + 1 to k ) NZ1 + NZ2 - 1). The
electrode was at k ) 0, the ITIES was located at k ) NZ1, and
k ) NZ1 + NZ2 corresponded to a sufficient distance from the
UME for a semi-infinite boundary condition to be applicable
(see ref 10). At the end of each first half-time step, the interior
boundary condition (eq 11), in finite-difference form, was used
to update the concentrations of species Red1 and Red2 at the
interface. Those remaining exterior boundary concentrations
involving no-flux boundary conditions were also updated.
During the second half-time step, concentrations in the Z
direction were calculated from the values of the concentrations
in the radial direction, evaluated in the first half-time step. For
k ) 1 to NZ1 - 1 and k ) NZ1 + 1 to NZ1 + NZ2 - 1, the
approach was similar to that employed previously.10 However,
for k ) NZ1 the finite-difference equations derived from the
interior boundary condition are
tD
Red1
τ )
(12)
a2
r
R )
(13)
(14)
a
z
Z )
a
D
Red2
γ )
(15)
(16)
D
Red1
cRed
CRed
)
i where i denotes either 1 or 2.
/
Redi
i
c
/
Red2
k12ac
D
K )
(17)
(18)
Red1
Red1 //
Red1 //
//
2
Cj,NZ1-1
-
Cj,NZ1(1 + K∆ZRed Cj,NZ1) +
/
c
c
Red2
K∆ZRed Cj,NZ1 ) 0 (21)
//
2
Kr )
/
Red1
K∆ZRed
Krγ
K∆Z
Krγ
Red1 //
Red2 //
Cj,NZ1
2C/j,/NZ1
-
Cj,NZ1 1 +
+
(
)
(
)
It should be clear how these terms affect the diffusion equations
(eqs 3 and 4) and associated boundary and initial conditions.
The tip current response is calculated as a function of time
and tip/interface separation, for particular Kr, K, and γ values.
The UME current is related to the flux of Red1 at the electrode
surface and hence the dimensionless current ratio is given by16
Red2 //
Cj,NZ1+1 ) 0 (22)
Redi //
In eqs 21 and 22,
Cj,NZ1+m is the new (normalized) concen-
tration of Redi (where i is either 1 or 2) at the loci j, NZ1 + m
(where m is -1, 0 or +1). ∆Z is the grid spacing in the Z
direction (which is uniform in phase 1, expanding with
increasing distance in phase 2, and matched at the interface, as
∂CRed
1
1
i
π
2
)
RdR
(19)
described in ref 10). It is apparent from these equations that
∫
(
)
0
∂Z Z)0
i(∞)
Redi //
the calculation of
Cj,k for k ) 1 to k ) NZ1 + NZ2, for
//
each value of j, requires a prior knowledge of Red Cj,NZ1 for the
2
where i(∞) is the steady-state diffusion-limited current at an
inlaid disk electrode positioned at an effectively infinite distance
from the interface.17
matrix equations to reduce to tridiagonal form. This problem
was overcome by adopting an iterative procedure in which, for
Red2 //
a particular value of j,
Cj,NZ1 was initially approximated by
C/j,NZ1 (calculated in the first-half time step). Following the
Red2
i(∞) ) 4nFaDRed cR/ ed
(20)
Redi //
1
1
calculation of
Cj,k (for i ) 1 and 2), the procedure was

SECM at Liquid/Liquid Interfaces
J. Phys. Chem. B, Vol. 103, No. 34, 1999 7263
Red2 //
repeated using the newly determined value for
values of
Cj,NZ1 until
Red2 //
Cj,NZ1 were unchanged (precision 10-10) upon
further iteration. At the end of each second half-time step all
matrix elements were checked to verify convergence within the
required precision (10-8).
Theoretical Results and Discussion
The numerical model developed above involves the param-
eters K, γ, Kr, and the normalized tip/interface distance, L )
d/a. The aim of this section is to analyze the effect of these
parameters on the tip current. Although the ADIFDM calculates
the chronoamperometric response of the UME probe, the results
described here are for the steady-state characteristics, determined
from the long-time limit, which are of primary interest in SECM
feedback experiments on liquid/liquid interfaces.1-7 The simula-
tions reported in this section were performed for an UME
characterized by RG ) rg/a ) 10, which is a typical value used
in SECM experiments and theoretical treatments.
The effect on the normalized approach curves of allowing
Kr to take finite values is illustrated in Figure 2, which shows
simulated data for three rate constants for redox couples
characterized by γ ) 1. The rate parameters considered are
typical of the upper, medium, and lower constants that might
be encountered in feedback measurements at liquid/liquid
interfaces. In each case, values of Kr ) 1000 or 100 yield
approach curves which are identical to the earlier solution for
no depletion in phase 2.1,19b This behavior was predicted earlier1
and is expected, given the relatively high concentration of Red2
compared to Red1. This ensures that the concentration of Red2
adjacent to the liquid/liquid interface is maintained close to the
bulk solution value, even when the redox process, eq 1, is driven
at a fast rate [Figure 2(i)].
For Kr e 10, the calculated normalized currents are lower
than predicted by the constant composition model, particularly
at close tip/interface separations, where the diffusion rate of
the Red1/Ox1 couple in phase 1 is highest and competes
effectively with the diffusion of Red2 in phase 2. As expected,
the deviation of the current from that predicted, assuming
constant composition conditions, becomes increasingly signifi-
cant as Kr is reduced, resulting in increasing diffusional
limitations from Red2 in phase 2. On the basis of the data in
Figure 2, the constant composition assumption is seen to break
down for Kr e 10, irrespective of the size of the rate constant
characterizing the interfacial process. The conditions under
which the approximation becomes invalid are discussed further
below.
The reasons for the deviation of the full model from the
constant composition model are apparent when the concentra-
tions of Red1 and Red2 are examined. Because of the axisym-
metric SECM geometry, the concentration profiles of Red1 and
Red2 are shown as cross-sections over the domain R g 0, Z g
0, as illustrated schematically in Figure 3. Note that in this figure
the tip position has been inverted compared to that in Figure 1.
Figures 4, 5 and 6 show profiles for K ) 100, 10, and 1,
respectively, at a normalized tip/interface separation, d/a ) 0.1
(which is typical of the smallest d/a attainable in SECM
measurements). In each case, normalized concentration distribu-
Figure 2. Simulated normalized steady-state current as a function of
tip/interface distance for (i) K ) 100, (ii) K ) 10, and (iii) K ) 1. In
each case γ ) 1 and Kr takes the values (a) 1000, (b) 100, (c) 10, (d)
5, (e) 2, and (f) 1.
tions are shown in terms of CRed (i ) 1 or 2) for Kr ) 1000,
i
10, and 5. With the former value of Kr, phase 2 is maintained
at constant composition for the range of rate constants consid-
ered, and profiles for this case are thus provided for phase 1
only. These latter profiles show the expected changes in the
concentration of Red1 as the kinetics change from the fast limit,
where positive feedback is observed [Figure 4(i)], to a situation
where the turnover of Red1 at the interface becomes increasingly
slow, so that hindered diffusion (negative feedback) becomes
important [Figures 5(i) and 6(i)].
In the fast kinetic regime (K ) 100), the turnover of Red1 at
the portion of the interface directly under the tip is sufficiently
fast that Red2 is depleted in this interfacial region, even when

7264 J. Phys. Chem. B, Vol. 103, No. 34, 1999
Barker et al.
Figure 3. Schematic of the region (cross-hatched) represented by the
concentration profiles in Figures 4 to 6. The ITIES is located at Z )
L.
Figure 5. Steady-state concentration profiles for species Red1 and Red2
in phases 1 and 2, respectively, for K ) 10 and γ ) 1, with Kr taking
the values (i) 1000, (ii) 10, and (iii) 5.
Figure 4. Steady-state concentration profiles for species Red1 and Red2
in phases 1 and 2, respectively, for K ) 100 and γ ) 1, with Kr taking
the values (i) 1000, (ii) 10, and (iii) 5.
Kr ) 10, and consequently a quasihemispherical diffusion field
of Red2 is established in phase 2 [Figure 4(ii)]. The effect is to
diminish the extent of positive feedback, as reflected by the
change in the phase 1 profile between Figure 4(i) and 4(ii) and
evident by the lower current observed at close tip/interface
separations, compared to the constant composition case [Figure
2(i)]. For Kr ) 5, the depletion of Red2 in phase 2 becomes
very significant, such that the Red1 profile in phase 1 has a
considerable hindered diffusion component [Figure 4(iii)].
Although the tendency for depletion in phase 2 becomes less
significant as the interfacial rate constant is decreased, there
are still considerable diffusional limitations from Red2 in phase
2 for K ) 10, with Kr ) 10 and 5 [Figure 5(ii) and (iii)]. Even
for K ) 1, Red2 is depleted at the interface when Kr ) 10 and
5 [Figure 6(ii) and (iii)].
Figure 6. Steady-state concentration profiles for species Red1 and Red2
in phases 1 and 2, respectively, for K ) 1 and γ ) 1, with Kr taking
the values (i) 1000, (ii) 10, and (iii) 5.
herein, as a function of log Kr for a range of rate constants
typically encountered in SECM feedback studies of ITIES. The
results of this exercise are shown in Figure 7 for γ ) 1, log-
(d/a) ) -0.5, -0.8, and -1.0, and K values from 2 to 1000.
At low Kr (log Kr ) -2), the concentration of Red2 in phase 2
The precise conditions under which the constant composition
model is valid can be identified by plotting the calculated
normalized current ratio derived from the full model, outlined

SECM at Liquid/Liquid Interfaces
J. Phys. Chem. B, Vol. 103, No. 34, 1999 7265
Figure 8. Working curves of normalized tip current versus log Kr for
(i) γ ) 2 and (ii) γ ) 0.5, with log(d/a) ) -1.0. K takes the values (a)
1000, (b) 100, (c) 50, (d) 20, (e) 10, (f) 5, and (g) 2. The region to the
left of the dashed line is where the constant composition model tends
to be invalid.
negligible (generally log Kr > 2). Between these two limits,
the current ratio is sensitive to the value of Kr.
If we consider that an experimental measurement of i/i(∞),
under arbitrary conditions of Kr, can be distinguished from the
corresponding constant composition value, provided the former
is at most 90% of the latter (which should generally be possible),
the data in Figure 7 can be divided into zones to indicate the
conditions under which the constant composition model tends
to be valid and invalid. The dashed lines bisecting the working
curves in Figure 7 indicate the borders between these two
domains for different distances between the tip and the ITIES.
As expected, the boundary shifts to higher Kr as d/a decreases,
as a result of the higher mass transport rates promoted at close
tip/interface separations. The boundaries also show a slight
dependence on K, but, at the smallest d/a, the constant
composition model is generally not a good approximation when
Kr < 10 for all kinetic cases considered.
The constant composition approximation is both Kr- and
γ-dependent. When the diffusion coefficient of Red2 is greater
than that of Red1, the constant composition model is valid to
lower Kr, as indicated by the data in Figure 8(i), simulated for
γ ) 2. Conversely, if Red2 is characterized by a low diffusion
coefficient compared to Red1, the approximation breaks down
at higher Kr, as illustrated in Figure 8(ii) for γ ) 0.5.
Figure 7. Working curves of normalized tip current versus log Kr for
γ ) 1, with K taking the values (a) 1000, (b) 100, (c) 50, (d) 20, (e)
10, (f) 5 and (g) 2. The data are for (i) log(d/a) ) -0.5, (ii) log(d/a)
) -0.8, (iii) log(d/a) ) -1.0. The region to the left of the dashed line
is where the constant composition model tends to be invalid.
is so low that effectively negative feedback is observed at each
separation considered (irrespective of the value of K). As Kr is
increased the current ratio is enhanced to a K-dependent limiting
value which holds when diffusional effects in phase 2 are

7266 J. Phys. Chem. B, Vol. 103, No. 34, 1999
Barker et al.
Figure 10. Contour plot of percentage error in the rate constant k12
that results from analyzing data in terms of the constant composition
model, rather than the full model presented in this paper. The data are
for a tip/ITIES separation, d/a ) 0.1, with a range of K and Kr values.
previous experimental investigations of ITIES by SECM),1,4,7
this corresponds to an upper limit on k12 of 8 cm s-1 M-1. In
contrast, for Kr ) 3, the approach curves are such that ready
kinetic discrimination should be possible for K e 100 [Figure
9(ii)]. For the same values of DRed and a, with a bulk
1
concentration of Red2 in phase 2, c/Red ) 1.5 mM, this
2
corresponds to a bimolecular rate constant, k12, of ca. 500 cm
s-1 M-1. It is also interesting to note that since the approach
curves in Figure 9(ii) have peaks (at least for the fast kinetic
cases that are most of interest), it is not actually necessary to
know the tip to ITIES separation absolutely. Rather, the rate
constant for the process of interest could be deduced simply by
measuring the peak current value of a tip approach curve.
Figure 9. Simulated approach curves of i/i(∞) versus normalized tip/
interface separation, d/a, for (i) constant composition conditions with
K ) (a) 100, (b) 50, (c) 20, (d) 10, (e) 5, (f) 2, and (g) 1; (ii) full
model conditions with Kr ) 3 and K ) (a) 1000, (b) 100, (c) 50, (d)
20, (e) 10, (f) 5, (g) 2, and (h) 1.
A final assessment of the applicability of the constant
composition assumption, compared to the full model, is shown
in Figure 10. This is a plot of the percentage error in the
interfacial rate constant that would result from analyzing
experimental data at log(d/a) ) -1.0, obtained with arbitrary
Kr, using the earlier constant composition theory.1,19(b) To use
the constant composition model to analyze data obtained under
conditions of finite Kr is to underestimate the rate constant;
therefore, the errors in Figure 10 are the relative percentage
underestimates of the rate constant. It can be seen that for Kr >
40, the constant composition model is a good one, resulting in
errors of less than 10-15% over the full range of rate constants.
For Kr < 20, however, sizeable errors result, particularly in the
fast kinetic regime. This diagram can be used as a guide as to
the likely errors involved in analyzing data for particular
conditions with the constant composition model.
In addition to extending the range of conditions under which
SECM feedback measurements can be made at liquid/liquid
interfaces, lifting the restriction on the composition of phase 2
is particularly beneficial for enhancing both the range and
precision with which fast kinetics can be investigated. This is
not only due to the fact that decreasing the bulk concentration
of Red2 in phase 2 lowers the dimensionless rate constant for
the system (eq 17) but also because the approach curves in the
fast kinetic limit are more readily distinguished from one another
when Kr < 10. This point is well illustrated by the data presented
in Figure 9, which shows simulated tip approach curves for a
range of normalized rate constants (i) under the constant
composition approximation and (ii) with Kr ) 3 and γ ) 1.
Although there are differences in the approach curves with
the constant composition model, it would be extremely difficult
to distinguish between any of the K cases practically, unless K
was below 10. Even for K ) 10, an uncertainty in the tip
position from the interface of ( 0.1d/a would not allow the
experimental behavior for this rate constant to be distinguished
Experimental
Chemicals. Na4Fe(CN)6 from Johnson Matthey (Ward Hill,
MA), Na2EDTA, FeCl2‚4H2O, ZnPor, and benzonitrile from
Aldrich (Milwaukee, WI), and benzene from Fluka Chemika
(Switzerland) were used as received. Tetrahexylammonium
perchlorate (THAClO4; Fluka Chemika) was recrystallized twice
from an ethyl acetate/ether (9:1) mixture and dried under
vacuum overnight at room temperature. Tetrabutylammonium
from the diffusion-controlled case. For a typical value of DRed
1
) 10-5 cm2 s-1 and electrode radius a ) 12.5 µm, this
corresponds to an effective first-order heterogeneous rate
constant of just 0.08 cm s-1. Assuming one needs Kr g 20 to
ensure constant composition conditions and at least 0.5 mM of
mediator in phase 1 for reasonable measurements (as in all

SECM at Liquid/Liquid Interfaces
J. Phys. Chem. B, Vol. 103, No. 34, 1999 7267
tetrafluoroborate (TBABF4; Southwestern Analytical Chemicals,
Austin, TX) was recrystallized from acetone/ether. Na4Ru(CN)6
and Na4Mo(CN)8 were synthesized as reported previously.7
Aqueous solutions with varying concentrations of reductants
were generally prepared with 0.1 M NaCl and 0.1 M NaClO4
as supporting electrolytes (for exceptions, see below). Aqueous
solutions of FeEDTA2- were prepared by adding a stoichio-
metric amount of FeCl2‚4H2O to deaerated solutions of Na2-
EDTA buffered with 0.01 M citrate at pH 4.8. Benzonitrile
solutions, containing 0.25 M THAClO4 and 0.3-1 mM ZnPor,
were mixed with at least twice their volume of water by vigorous
shaking and centrifuged to separate the organic solution from
the aqueous phase. This procedure was repeated three to five
times to remove trace amounts of surfactants from the organic
phase that might adsorb on the benzonitrile/water interface. For
experiments with Na4Fe(CN)6, benzene was used instead of
benzonitrile. For experiments with Na4Ru(CN)6, 0.2 M NaBF4
and 0.2 M TBABF4 were used instead of 0.1 M NaClO4 and
0.25 M THAClO4, respectively. All aqueous solutions were
prepared from deionized water (Milli-Q, Millipore Corp).
Figure 11. Experimental approach curves (9) for the oxidation of
ZnPor at a tip UME in benzene approaching a benzene/aqueous
interface, with the aqueous phase containing Fe(CN)64-. The bulk
concentration conditions in the organic and aqueous phases, respec-
tively, were as follows: (a) [ZnPor]* ) 0.500 mM, [Fe(CN)64-]* )
7.00 mM; (b) [ZnPor]* ) 0.500 mM, [Fe(CN)64-]* ) 3.50 mM; (c)
[ZnPor]* ) 0.380 mM, [Fe(CN)64-]* ) 0.700 mM; (d) [ZnPor]* )
0.380 mM, [Fe(CN)64-]* ) 0.350 mM; (e) [ZnPor]* ) 0.380 mM,
[Fe(CN)64-]* ) 0.255 mM. The solid lines show the behavior predicted
Electrodes and Electrochemical Cells. SECM tips were
prepared by heat-sealing Pt wires (25-µm diameter, Goodfellow,
Cambridge, U.K.) in glass capillaries as described previously.21
The tip electrode was rinsed with ethanol and water and then
polished and dried before each measurement. A three-electrode
configuration (tip, auxiliary, and reference) was used in all
experiments, and all electrodes were placed in the top (organic)
phase. The SECM cell was described previously.4 An ionic
bridge containing 0.1 M NaCl and 0.1 M NaClO4 was placed
between the SCE reference electrode and organic solution,
except for experiments with Na4Ru(CN)6, for which a Ag/AgCl
wire was used as a reference electrode. Experiments with
FeEDTA2- were carried out in a glovebag (Aldrich) under
nitrogen.
for γ ) 1.7 and a bimolecular rate constant, k12 ) 91 cm s-1 M-1
,
while the dashed lines show the behavior for a diffusion-controlled
process for each of the five cases considered, simulated using the model
described herein. The diffusion-controlled characteristics for the two
cases with the highest Kr are indistinguishable.
Experimental Results and Discussion
A set of steady-state approach curves of i/i(∞) versus tip/
interface separation, d, is shown in Figure 11 for the oxidation
of ZnPor to ZnPor+•, in a benzene phase, at an UME tip
translated toward the interface with an aqueous solution,
containing various concentrations of Fe(CN)64-. This is a useful
system to begin the assessment of the new model, since the
driving force for the reaction is large, with a measured ∆E1/2
of 585 mV. Previous SECM investigations of the kinetics of
this process, under constant composition conditions, found only
that the rate constant was too high to be measured, with the
process deduced to be diffusion-controlled, i.e., showing an
approach curve indistinguishable from that characteristic of total
positive feedback.7
SECM Apparatus and Procedure. A home-built SECM
instrument22 was employed for experiments with Na4Fe(CN)6,
while a commercially available SECM (model CHI-900; CH
Instruments, Cordova, TN) was used for measurements on other
systems. In all SECM experiments, the tip electrode was biased
at a potential corresponding to the plateau current of the
oxidation wave of ZnPor to ZnPor+•. Approach curves were
obtained by moving the tip toward the ITIES and recording the
tip current, i, as a function of the distance, d.
Measurement of the Driving Force for Interfacial ET. For
quantitative comparison of the ET rate constant, k12, obtained
for the different redox species, the driving force for the
interfacial ET was measured as reported previously.7 The formal
potentials for most of the aqueous redox species and ZnPor in
benzene were evaluated as the half-wave potential, E1/2, of the
nernstian steady-state voltammograms. The formal potential for
the FeEDTA2- was found from the quasireversible steady-state
voltammogram, using a procedure reported previously.23 These
voltammograms were measured with respect to the SCE
reference electrode with the same ionic bridge as used for SECM
measurements. The difference of half-wave potentials for an
aqueous redox species and ZnPor is
The experimental data in Figure 11 are shown alongside the
best fits to the model presented herein. These simulations were
carried out for γ ) 1.7, given the measured values of DZnPor
)
4.0 × 10-6 cm2 s-1 and DFe(CN) ) 6.7 × 10-6 cm2 s-1 in the
4-
6
two solvents of interest. For the highest two concentration ratios
considered (Kr ) 14 and 7), the interfacial redox reaction
appears to be diffusion controlled and the rate constant cannot
be determined. However, as the value of Kr is decreased, by
employing smaller concentrations of Fe(CN)64- in the aqueous
phase, it becomes possible to distinguish the measured behavior
from the predictions of the diffusion-controlled cases. For the
lowest three Kr values investigated, the experimental current-
distance data lie below the simulated characteristics for a
diffusion-controlled process and can be analyzed well in terms
of a unique rate constant, k12 ) 91 cm s-1 M-1
.
∆E1/2 ) ∆E° + ∆φ
(23)
In addition to expanding the range of rate constants that can
be studied, lifting the constant composition restriction on phase
2 also enhances the precision and certainty with which rate
measurements can be made. As outlined in the theoretical results
and discussion section, the measurement of rapid rate constants,
where ∆E° is the difference of formal potentials of the aqueous
redox species and ZnPor, and ∆φ is the relative value of the
potential drop across the ITIES. ∆E1/2 gives the absolute value
of the driving force according to eqs 5 and 7 in ref 7.

7268 J. Phys. Chem. B, Vol. 103, No. 34, 1999
Barker et al.
Figure 13. Experimental approach curves (9) for the oxidation of
ZnPor at a tip UME in benzonitrile approaching a benzonitrile/aqueous
interface, with the aqueous phase containing Ru(CN)64-. For all
experiments, ZnPor was present at a concentration of 1.00 mM, while
the concentration of Ru(CN)64- was such that the following values of
Kr resulted: (a) 9.8, (b) 8.2, (c) 7.7, (d) 6.6, (e) 5.3, (f) 4.5, (g) 3.7, (h)
2.8, (i) 1.7, (j) 1.3, (k) 0.8, (l) 0.3, and (m) 0.1. The solid lines are the
best fits of the experimental data to theory with k12 ) 0.25 cm s-1 M-1
and γ ) 2.8.
Figure 12. Experimental approach curves (9) for the oxidation of
ZnPor at a tip UME in benzonitrile approaching a benzonitrile/aqueous
interface, with the aqueous phase containing Mo(CN)84-. For all
experiments, ZnPor was present at a concentration of 0.500 mM, while
the concentration of Mo(CN)84- was such that the following values of
Kr resulted: (a) 14, (b) 10, (c) 6, (d) 4, (e) 3, (f) 2, (g) 1.2, (h) 0.8, and
(i) 0.4. The solid lines are the best fits of the experimental data to
theory for γ ) 3.2 with k12/cm s-1 M-1 ) (a) 2.75, (b) 2.75, (c) 2.75,
(d) 1.75, (e) 2.1, (f) 3.0, (g) 3.5, (h) 3.25, and (i) 3.0. The dashed line
shows the predicted behavior for a diffusion-controlled process under
constant composition conditions.
this case, ∆E1/2 ) 70 mV and the rate constant k12 ) 0.25 cm
s-1 M-1. For this system, it is particularly satisfying that there
is an excellent fit of the data to the new model over 2 orders of
magnitude in Kr, from a value of 9.8 down to 0.1. In analyzing
these data, a value of γ ) 2.8 was used, based on the measured
diffusion coefficient of Ru(CN)64- in the aqueous phase of 6.7
under constant composition conditions, requires that similar
approach curves involving enhanced currents at close distances
can be distinguished from one another, placing considerable
importance on the accurate and reproducible attainment of close
tip/interface separations. In contrast, by working under condi-
tions of lower Kr the approach curves flatten and indeed may
peak as the tip/interface separation is decreased from high to
low values. In the latter situation, uncertainties in the separation
between the tip and the interface become less important, as the
magnitude of the peak current alone is the unequivocal pointer
to the kinetics of the interfacial process. This effect is found in
practice in the two approach curves with the lowest Kr in Figure
11.
× 10-6 cm2 s-1
.
As a final example, a system was investigated where the
4-
driving force was larger than that of the Fe(CN)6 case
considered in Figure 11. Using FeEDTA2- as the reductant in
the aqueous phase, with ZnPor again oxidized at the tip in
benzonitrile solution, a driving force, ∆E1/2 ) 776 mV, was
established. Steady-state current distance approach curves for
this system are shown in Figure 14 for Kr over the range 2.4 to
8.2. A value of γ ) 2.4 was employed to fit the experimental
data to theory, on the basis of the measured diffusion coefficient
4-
The key points introduced above with Fe(CN)6 as the
reductant are further emphasized in Figure 12, which shows an
for FeEDTA2- in the aqueous phase of 5.7 × 10-6 cm2 s-1
.
4-
extensive set of approach curves obtained with Mo(CN)8 as
The data can be analyzed in terms of a fairly consistent value
of k12 in the range 7.5-10 cm s-1 M-1. These data show clearly
again that it becomes increasingly easier to identify an accurate
value of k12, distinct from the diffusion-controlled behavior, by
decreasing the Kr value.
the reductant in the aqueous phase. These measurements were
made with benzonitrile, rather than benzene, as the organic
solvent. The experimental data can be analyzed very well in
terms of a value of k12 of 2.5 ( 0.75 cm s-1 M-1, given γ )
3.2, based on measured diffusion coefficients for ZnPor and
Mo(CN)84-, respectively, of 2.4 × 10-6 cm2 s-1 and 7.2 × 10-6
cm2 s-1 in the two solvents of interest. However, it can be seen
again that even with this relatively low rate constant, tip/interface
separations have to be known with high precision to make a
kinetic assignment of the data obtained under conditions where
constant composition of the second phase can be assumed (case
a, with Kr ) 14). In contrast, by lifting the restriction on the
composition of phase 2, an assignment of k12 is readily achieved.
Although it is difficult to make a precise comparison of the
rate constants obtained from the data in Figures 11 and 12,
because of the different organic solvents used, the decrease in
rate constant between Fe(CN)64- and Mo(CN)84- is at least, in
part, due to a lower driving force in the latter case, with ∆E1/2
) 140 mV. As the driving force is reduced further, the rate
constant decreases, as evidenced by the data in Figure 13, for
The rate constant derived from the data in Figure 14, with
FeEDTA2- is lower than with Fe(CN)64- as the reductant, even
though the driving force, based on ∆E1/2, is apparently higher
in the former case. Although this paper is concerned with a
new general model for SECM feedback at ITIES with a focus
on the role of diffusion of the redox-active species in phase 2,
some comment on the rate constants derived in the four systems
is warranted. A plot of log k12 vs driving force is shown in
4-
Figure 15. Although the solvent used for the Fe(CN)6 study
was different than that of the other three reductants, the overall
trend is consistent with the predictions of Marcus theory,24
showing first an increase in the reaction rate with increased
driving force, followed by a decrease in the inverted region.
The Marcus λ values probably differ for the various couples,
but a general plot according to Marcus theory with λ ) 0.55
eV (and an arbitrary maximum rate constant of about 50 cm
s-1 M-1) shows a trend consistent with the data. While more
detailed studies with more couples are needed to confirm these
4-
experiments with Ru(CN)6 as the reductant in the aqueous
phase, with ZnPor again the initial species in benzonitrile. In

SECM at Liquid/Liquid Interfaces
J. Phys. Chem. B, Vol. 103, No. 34, 1999 7269
the concentration of the redox couple in the second phase be
15 to 20 times that in the phase containing the probe, if the
constant composition theory is to be used.
Lifting the restriction on the composition of the second phase
to allow the use of a relatively low concentration of redox-
active mediator has been shown to result in considerable
advantages for the study of rapid kinetics. In particular, rate
processes can be characterized that would appear purely
diffusion-controlled under constant composition conditions.
Thus, it has been possible to measure the kinetics of the reaction
between ZnPor+• and Fe(CN)64-, which has appeared diffusion-
controlled in previous SECM studies.
The rate constants obtained for the four systems investigated
generally follow a trend with driving force consistent with
Marcus theory, in particular suggesting inverted region behavior
for interfacial electron transfer. Further studies of ET at ITIES
are underway using the methodology and theory outlined in this
paper.
Figure 14. Experimental approach curves (9) for the oxidation of
ZnPor at a tip UME in benzonitrile approaching a benzonitrile/aqueous
interface, with the aqueous phase containing FeEDTA2-. For all
experiments, ZnPor was present at a concentration of 0.500 mM, while
the concentration of FeEDTA2- was such that the following values of
Kr resulted: (a) 9.8, (b) 5.4, (c) 3.8, and (d) 2.4. The solid lines are the
best fits of the experimental data to theory with γ ) 2.4 and k12/cm
s-1 M-1 values of (a-c) 10 and (d) 7.5. The dashed line shows the
predicted behavior for a diffusion-controlled process under constant
composition conditions.
Acknowledgment. The support of this research by the
EPSRC (P.R.U. and A.L.B.), the National Science Foundation
(CHE9870762), and the Robert A. Welch Foundation (A.J.B.)
is gratefully acknowledged. S.A. expresses thanks for a post-
doctoral fellowship from the Japan Society for the Promotion
of Science.
References and Notes
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Phys. Chem. B 1998, 102, 1586.
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243.
Figure 15. Plot of log k12 versus ∆E1/2 derived from the data for the
four systems in Figures 11-14. The solid line is expected behavior
based on Marcus theory for λ ) 0.55 eV and a maximum rate constant
of 50 cm s-1 M-1
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results, the proposed existence of interfacial inverted region
behavior suggested by these studies is consistent with a previous
study of the electron transfer at an ITIES modified with a
monolayer of surfactant.7 The methodology outlined in this
paper should be useful in enhancing the range of kinetics open
to quantitative investigation at the ITIES by the SECM
technique.
(19) (a) Unwin, P. R.; Bard, A. J. J. Phys. Chem. 1991, 95, 7814. (b)
Bard, A. J.; Mirkin, M. V.; Unwin, P. R.; Wipf, D. O. J. Phys. Chem.
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(24) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265.
Conclusions
By developing a full treatment for SECM feedback for the
case of a redox reaction at an ITIES, a fuller range of reaction
conditions may now be explored than could with earlier limiting
models based on the constant composition approximation. The
conditions under which the latter approximation is valid have
been identified. For a given tip/substrate separation and tip
geometry, the key parameters are the relative diffusion coef-
ficients and concentrations of the redox reactants in the two
phases. When the diffusion coefficients of the redox-active
species in the two phases are similar, it is recommended that