Scanning Electrochemical Microscopy (SECM) as a Probe of Transfer

1586

J. Phys. Chem. B 1998, 102, 1586-1598

Scanning Electrochemical Microscopy (SECM) as a Probe of Transfer Processes in Two-Phase Systems: Theory and Experimental Applications of SECM-Induced Transfer with Arbitrary Partition Coefficients, Diffusion Coefficients, and Interfacial Kinetics Anna L. Barker, Julie V. Macpherson, Christopher J. Slevin, and Patrick R. Unwin* Department of Chemistry, UniVersity of Warwick, CoVentry CV4 7AL, U.K. ReceiVed: October 16, 1997

Scanning electrochemical microscopy induced transfer (SECMIT) is introduced as a new approach for probing the dynamics of partitioning of electroactive solutes between two immiscible phases. The partitioning process, initially at equilibrium, is perturbed by electrolysis of the target solute at an ultramicroelectrode (UME) tip positioned in one phase, close to the interface with a second phase. For a particular separation between the tip and interface, the flux of solutesand hence current flowingsat the UME is governed primarily by the partition coefficient of the solute, Ke, the relative diffusion coefficients of the solute in the two phases, γ, and the interfacial transfer kinetics. A numerical model is developed for SECMIT under potential step chronoamperometric conditions, where the target solute is electrolyzed at the UME at a diffusion-controlled rate. When the interfacial kinetics are nonlimiting, the steady-state current is strongly dependent on the product Keγ, particularly for Keγ > 1, while for constant Keγ the corresponding chronoamperometric response depends on the individual values of Ke and γ. In principle, this methodology provides a route to determining the diffusion coefficient and/or concentration of a target solute in a medium, without contact from the UME. This aspect of SECMIT is illustrated, under steady-state conditions, through studies on the transfer of (i) several types of electroactive ions between hydrogels and aqueous solutions and (ii) oxygen transfer across the interface formed between water and either 1,2-dichloroethane (DCE) or nitrobenzene (NB). The application of SECMIT to probe interfacial kinetics is illustrated through studies of cupric ion extraction and stripping between an aqueous phase and DCE containing the oxime ligand, Acorga P50 (LH).

Introduction A greater understanding of interfacial kinetics and mechanisms in condensed-phase systems has been achieved, over the past decade, through the use of scanning electrochemical microscopy (SECM) techniques.1-13 The SECM response depends on the rate of mass transfer of a target species to a tip ultramicroelectrode (UME), positioned in a solution (phase 1), close to the interface with a second phase (phase 2), which may be a solid,1-13 liquid14-20 or gas.19,20 The mass transfer rate, in turn, is governed by physicochemical processes at the part of the interface directly under the tip.1 The small characteristic dimension of the SECM probe, typically of the order of micrometers1 to tens of nanometers,1,21 enables high rates of diffusional mass transfer to be attained in the gap between the UME and the interface. This facilitates the measurement of fast interfacial processes, with high spatial resolution, under well-defined and calculable conditions. SECM has traditionally been applied to study processes at liquid/solid interfaces.2-13 For example, it has found particular use in: (i) measuring fast electron-transfer processes at electrodes2-6 and immobilized redox enzymes;7,8 (ii) characterizing dissolution and growth processes at a variety of materials including metals,9 semiconductors,10 ionic crystals,11 and polymers;12 (iii) determining adsorption/desorption kinetics and surface diffusion rates.13 More recently, SECM has proved a powerful kinetic probe of reactions that occur at the interface between two immiscible liquids.14-20 This class of reactions is of widespread importance,22-27 encompassing the physical sciences (e.g., electron and ion transfer), biology (e.g., models for biological

membranes), and industrial processes (e.g., solvent extraction in hydrometallurgy). SECM studies, hitherto, include the use of the equilibrium perturbation mode to probe molecular and ion transfer processes,14,19 and the feedback mode to measure the rates of interfacial redox reactions.15,16 Part of the success of the SECM, over more conventional methodologies,27 stems from the ability to measure reaction rates under conditions where the contributions from transport and interfacial processes can readily be separated. Additionally, the use of a direct, rather than supported, liquid/liquid contact of well-defined area, facilitates the precise determination of interfacial fluxes.14 SECM liquid/liquid studies are currently limited to situations where phase 2 is maintained at constant composition during a measurement. Although this simplifies the supporting theoretical treatments considerably, through the elimination of diffusional and depletion effects in phase 2, it places restrictions on the conditions under which SECM can be used quantitatively. The primary aim of this paper is therefore to develop the first general model for SECM that considers interfacial kinetics, coupled with arbitrary diffusion coefficients and concentrations for the species of interest in both phases on either side of the target interface, thereby extending the range of conditions and systems to which SECM may be applied. The model is formulated with specific reference to SECM-induced transfer (SECMIT), part of the family of SECM equilibrium perturbation techniques,11,13,14,19 the principles of which are outlined below. It should be noted, however, that the numerical approach adopted will be readily applicableswith only minor modificationssto the host of other SECM modes that are available.1

S1089-5647(97)03370-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 02/06/1998

SECM as a Probe in Two-Phase Systems

J. Phys. Chem. B, Vol. 102, No. 9, 1998 1587

A number of interesting new experimental applications of SECMIT arise under the conditions considered herein. It will first be shown that this approach allows the diffusion coefficient and/or concentration of a target species in phase 2 to be determined without the UME tip having to contact or enter that phase. This aspect of SECMIT is illustrated with studies of the electrochemically induced transfer of (i) Ru(bipy)32+, Ru(NH3)63+, and Fe(CN)64- across the interface between an aqueous solution and the hydrogel, poly(hydroxyethyl acrylate), and (ii) oxygen across the interfaces formed by water in contact with (a) dichloroethane (DCE) and (b) nitrobenzene (NB). Second, building on recent work,14 the validity of the constant composition approximation for phase 2, for kinetic applications of SECMIT, will be assessed with studies of the extraction/ stripping of cupric ions from aqueous solution by LH in DCE. Principles of SECMIT SECMIT is a development of the SECM equilibrium perturbation mode, employed originally to induce and monitor reversible phase-transfer processes, such as adsorption/desorption13 crystal dissolution/growth,11 and more recently transfer processes across liquid/liquid interfaces.14 In SECMIT, the UME tip is positioned (in phase 1) close to the interface between two phases, such as two immiscible liquids, each containing a common electroactive species, A. The partitioning of this species between the two phases (denoted by the subscript 1 or 2) can be represented as follows k1

A1 y\ z A2 k

(1)

2

where k1 and k2 are first-order interfacial rate constants for the transfer of A from phases 1 and 2, respectively. The target interface is assumed to be mechanically stable and also sharply defined so that, initially at equilibrium, each phase has a uniform bulk concentration of A, ci* (i ) 1 or 2). With eq 1 at dynamic equilibrium, there is zero net flux of species A across the interface. A potential step is subsequently applied to the UME, in phase 1, sufficient to electrolyze A at the tip, at a diffusion-controlled rate. This perturbs the interfacial equilibrium, inducing the transfer of the target species across the interface, from phase 2 to phase 1. The flux of species A to the UME tip, and hence the resulting current-time behavior, following the potential step, is governed by both the interfacial kinetics and the rates of mass transfer in each phase. In principle, the tip current response could therefore be used to provide information about the kinetics of the interfacial process, the diffusion coefficients of A in the two phases and the partition coefficient, while only one of the phases is probed directly by the UME. Theory Formulation of the Problem. Time-dependent diffusion equations for A, appropriate to the axisymmetric SECM geometry, shown in Figure 1, can be written for each phase.

[

]

∂2ci 1 ∂ci ∂2ci ∂ci ) Di 2 + + 2 ∂t r ∂r ∂r ∂z

(i ) 1 for 0 e r e rg, 0 < z < d; i ) 2 for 0 e r e rg, z > d) (2)

where ci and Di are the concentration and diffusion coefficient

Figure 1. Principles of SECMIT, along with the coordinate system used to define the two-phase model. The coordinates r and z are, respectively, those in the radial and normal directions to the electrode surface. The electrode radius is denoted by a, and rg is the distance from the center of the electrode to the outermost edge of the insulating sheath surrounding the UME. The interface is located at a distance, d, from the end of the tip, while phase 2 is considered to extend a semiinfinite distance in the z-direction. Species A is the target electroactive solute, while B denotes the product of the electrode reaction.

of the electroactive species, A, in phase i, r and z are the coordinates radial and normal to the electrode surface, respectively, measured from the center of the electrode, and t is time. As highlighted in Figure 1, rg denotes the radius of the probe (electrode plus insulating sheath), while d represents the separation between the end of the probe and the target interface. Phase 2 is assumed to be semi-infinite in the z-direction, but the model could readily be modified to consider the situation where phase 2 has a finite thickness. To calculate the tip current response, the diffusion equations must be solved subject to the boundary conditions of the system. Prior to the potential step, the initial condition is

t ) 0: ci ) ci*

(i ) 1 for 0 e r e rg, 0 < z < d; i ) 2 for 0 e r e rg, z > d)

(3)

The tip UME potential is subsequently stepped from a value where no electrode reaction occurs to one sufficient to drive the electrolysis of A at a diffusion-controlled rate. Species A 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-substrate domain. The following exterior boundary conditions therefore apply:

z ) 0, 0 e r e a: c1 ) 0 z ) 0, a < r e rg: D1

∂c1 )0 ∂z

(4)

(5)

r > rg, all z: ci ) ci* (i ) 1 for 0 < z < d; i ) 2 for z > d) (6) This latter condition is valid11 provided that RG ) rg/a g 10, where a is the radius of the electrode.

1588 J. Phys. Chem. B, Vol. 102, No. 9, 1998

Barker et al.

The axisymmetric geometry of the SECM implies

∂ci r ) 0, all z: Di ) 0 (i ) 1 for 0 < z < d; i ) 2 ∂r for z > d) (7) A further boundary condition is imposed that reflects that at a semi-infinite distance from the electrode, in phase 2, the electroactive species attains its bulk concentration, c2*.

z f ∞, 0 e r e rg: c2 ) c2*

(8)

There is a final internal boundary condition to take account of the presence of the interface. For a first-order process, the net diffusive flux of A across the interface from phase 2 to phase 1, as the system attempts to reattain equilibriumsfollowing the electrolytic depletion of A in phase 1sis given by

∂c1 ∂c2 ) k2(c2 - Kec1) z ) d, all r: D1 ) D2 ∂z ∂z

(9)

Ke ) (c2*/c1*) ) (k1/k2)

(10)

where

The internal boundary condition (eq 9) couples the timedependent diffusion equations for the two phases (eq 2). For the formulation of a general solution it is convenient to use dimensionless quantities:

τ ) (tD1)/a2

(11)

R ) r/a

(12)

Z ) z/a

(13)

γ ) D2/D1

(14)

Ci ) ci/ci*

(15)

K ) (k2a)/D2

(16)

The effect of each of these terms on the diffusion equations and boundary conditions, highlighted above, should be evident. The aim of the calculation is to determine the tip current response as a function of time and tip/interface separation for particular Ke, γ, and K values. The UME current is related to the flux of c1 at the electrode surface,

∫01

i ) 2πnFaD1c1*

( ) ∂C1 ∂Z

Z)0

R dR

(17)

where n is the number of electrons transferred per redox event and F is Faraday’s constant. The normalized current ratio is given by:

π i ) i(∞) 2

∫01

( ) ∂C1 ∂Z

Z)0

R dR

(18)

where i(∞) is the steady-state diffusion-limited current when the tip is positioned at an effectively infinite distance from the interface28

i(∞) ) 4nFaD1c1*

(19)

Method of Solution. Numerical solutions were obtained using the alternating direction implicit finite-difference method

Figure 2. SECMIT chronoamperometric characteristics for log(d/a) ) -0.5, K ) 105 and γ ) 1. The curves correspond to the equilibrium partition coefficient, Ke, taking the values: (a) 1000, (b) 50, (c) 10, (d) 5, (e) 2, (f) 1, (g) 0.5, and (h) 0.1. The lower dashed curve represents the behavior for an inert interface (i.e., no transfer from phase 2 to phase 1).

(ADIFDM), which has previously been applied to a variety of SECM problems.11,13,14,29 The Appendix provides a brief outline of the modifications required to the algorithm described previously, focusing particularly on the treatment of the internal boundary condition within the numerical method. This account should be read in conjunction with the original description of the ADIFDM as applied to SECM.29 Theoretical Results and Discussion The tip current response for SECMIT is governed primarily by K, Ke, γ, and the dimensionless tip-substrate distance, d/a. For long times, and interfaces that approach the chemically inert limit, the value of RG may also influence the current value,11 but this is of relatively minor importance. As with previous theoretical treatments of the SECM current response,11,13,14,29 all simulations were performed for an UME with RG ) 10: a similar value to that for tip UMEs employed in the experiments described later. The aim of this section is to examine the effects of the abovementioned parameters on the chronoamperometric and steadystate SECMIT characteristics. To limit the length of this paper, we focus the discussion of the current-time characteristics to a typical tip/interface distance, defined by log(d/a) ) -0.5; decreasing or increasing the separation between the tip and the interface primarily accentuates or diminishes the chronoamperometric features described below, as found for previous SECM problems.11,13 All chronoamperometric data are presented as normalized current ratio versus τ-1/2 in order to emphasize the short-time characteristics, for the reasons outlined previously.11 Steady-state characteristics, derived from the chronoamperometric data in the long-time limit, are examined over the full range of tip/substrate separations generally encountered in SECM. Effect of Ke. The effect on the current-time behavior of varying Ke, while keeping the kinetics of the interfacial process high and nonlimiting, is shown in Figure 2. For the range of Ke values of interest, a value of K ) 105 was found to be sufficiently high to ensure that interfacial kinetics were effectively nonlimiting over the tip/interface separations examined, with γ ) 1.



Figure 3. Simulated normalized steady-state current as a function of normalized tip-interface distance for K ) 105, γ ) 1, and Ke taking the values: (a) 1000, (b) 50, (c) 10, (d) 5, (e) 2, (f) 1, (g) 0.5, and (h) 0.1. The lower dashed line represents the approach curve for an inert interface (i.e., no transfer from phase 2 to phase 1).

The general trends in Figure 2 can be explained as follows. At short times, the diffusion field at the UME tip is not of sufficient size to intercept the interface, and there is thus no perturbation of the interfacial equilibrium. In this time regime, τ-1/2 > 8, the tip current is similar for all of the cases considered. This is a common feature of SECM potential step chronoamperometry.11,13,29,30 At longer times, the diffusion field of the UME intercepts the interface and the flux of A to the UME (and hence current) is then governed by the value of Ke. Given that, under the defined conditions, there is no interfacial kinetic barrier to transfer from phase 2 to phase 1, the concentrations immediately adjacent to each side of the interface may be considered to be in dynamic equilibrium throughout the course of a chronoamperometric measurement. For high values of Ke, the target species in phase 2 is in considerable excess so that the interfacial concentration in phase 1 is maintained at a value close to the initial bulk value, with minimal depletion of A in phase 2. Under these conditions, the response of the tip (Figure 2, case a) is in agreement with that predicted for other SECM diffusion-controlled processes with no interfacial kinetic barrier, such as induced dissolution11 and positive feedback.29,30 A feature of this response is that the current rapidly attains a steady-state, the value of which increases dramatically as the separation between the tip and interface decreases, as illustrated by case a in Figure 3. At the other extreme, for low Ke, the relatively small concentration of A in phase 2scompared to that in phase 1sis insufficient to maintain the concentrations of A on both sides of the interface at their initial bulk values, during SECMIT measurements. Consequently there is extensive depletion of A in both phases. The smaller the value of Ke, the more extensive the depletion of A, as illustrated by Figure 4, which depicts the steady-state concentration profiles of A for Ke ) 0.1 (a), 1 (b), and 10 (c). Notice, however, that although A is depleted in both phases, equilibrium conditions prevail at the interface, i.e., there is no abrupt discontinuity in the normalized concentrations of A on crossing the interface (for any value of R). The main result of the preceding observations for SECMIT measurements is that as Ke decreases, the current tends to increasingly smaller values (Figures 2 and 3). In the limit Ke f 0, the chronoamperometric (Figure 2) and steady-state (Figure 3) current responses approach the characteristics predicted previously for an inert interface,29-31 as expected.

Figure 4. Steady-state profiles of the distribution of A in the two phases during SECMIT, for K ) 105, γ ) 1, and Ke taking the values: (a) 0.1, (b) 1, and (c) 10. The concentration values relate to C1 and C2 for phases 1 and 2, respectively.

The plot of normalized steady-state current vs tip-interface distance, shown in Figure 3, demonstrates that as the tipinterface distance decreases the steady-state current becomes more sensitive to the value of Ke. Under the defined conditions, it is interesting to note that the shape of the approach curve is highly dependent on the concentration in the second phase, for Ke values over a very wide range, with a lower limit less than 0.1 and upper limit greater than 50. The case of Ke ) 1, with γ ) 1, provides an additional check on the method adopted for the simulation. Since the concentration and diffusion coefficient of A are the same for the two phases, and interfacial kinetics are nonlimiting, the two phases effectively act together as a single bulk phase. The tip current response is thus expected to resemble that predicted for a conventional UME in bulk solution,28,32 as found for the chronoamperometric (Figure 2, case f) and steady-state (Figure 3, case f) current responses, as well as the steady-state concentration profile (Figure 4b). Effect of γ. As might be expected, similar trends to those identified above are observed as γ is varied, while maintaining Ke ) 1 and K ) 105. The transient and steady-state current responses, shown in Figures 5 and 6, respectively, vary between a lower limit, that is close to the response for an inert interface when γ < 0.01, and an upper limit (when γ g 1000), which is characteristic of SECM diffusion control in phase 1 with no resistance from interfacial kinetics or transport in phase 2.


Barker et al.

Figure 5. SECMIT chronoamperometric characteristics for log(d/a) ) -0.5, K ) 105, and Ke ) 1. The curves correspond to the ratio of the diffusion coefficients, γ, taking the values: (a) 1000, (b) 100, (c) 10, (d) 2, (e) 1, (f) 0.5, (g) 0.1 and (h) 0.01. The lower dashed curve represents transient behavior for an inert interface. Figure 7. Steady-state profile of the distribution of A in the two phases during SECMIT, for K ) 105, Ke ) 1, and γ values of 0.1 (a) and 10 (b).

Figure 6. Simulated normalized steady-state current as a function of normalized tip-interface distance for K ) 105, Ke ) 1, and γ taking the values: (a) 1000, (b) 100, (c) 10, (d) 2, (e) 1, (f) 0.5, (g) 0.1, and (h) 0.01.

The effect of increasing γ is to increase the diffusion coefficient of the solute in phase 2 compared to that in phase 1. For a given value of Ke this means that when a SECMIT measurement is made, the higher the value of γ, the less significant the depletion effects in phase 2 and the concentrations at the target interface are maintained closer to the initial bulk values, as illustrated by the profiles in Figures 7 and 4b, for γ ) 0.1, 1, and 10. Consequently, as γ increases, the chronoamperometric and steady-state currents increase from a lower limit, characteristic of an inert interface, to an upper limit defined above. For comparison with Figures 5 and 6, Figures 8 and 9, respectively, show the dependences of the chronoamperometric and steady-state currents on γ when Ke is increased from 1 (Figures 5 and 6) to 10 (Figures 8 and 9). As expected, on the basis of the results presented hitherto, the chronoamperometric and steady-state currents increase, within the defined range, when Ke is increased. More striking is the observation that the normalized steady-state currents depend strongly on the value of the product Keγ (related to the permeability of the solute in phase 2), but for constant Keγ are largely independent of the individual values of Ke and γ, especially for Keγ > 1. This

Figure 8. SECMIT chronoamperometric characteristics for log(d/a) ) -0.5, K ) 105, and Ke ) 10. The curves correspond to the ratio of the diffusion coefficients γ taking the values: (a) 1000, (b) 100, (c) 10, (d) 2, (e) 1, (f) 0.5, (g) 0.1, and (h) 0.01. The lower dashed curve represents transient behavior for an inert interface.

can be seen clearly by comparing the data for Keγ ) 100, 10, 1, and 0.1 in Figures 6 and 9. In contrast, the current-time behavior is highly dependent on the individual Ke and γ values, as can be seen by comparing the data for Keγ ) 10, 1, and 0.1 in Figures 5 and 8. This difference in the steady-state and chronoamperometric characteristics holds true when the data for Keγ ) 10, 1, and 0.1, in Figures 2 and 3, are also considered. A comparison of the concentration profiles in Figures 4c and 7b demonstrates that the close agreement in the steady-state currents for a particular value of the product Keγ is due to a similarity in the steadystate distribution of the target solute across the two phases. Effect of Interfacial Kinetics. The influence of an interfacial kinetic barrier on the transfer process is readily illustrated by fixing the concentrations and the diffusion coefficients of A for the two phases and examining the current response of the UME as K is varied. For the purposes of this discussion, we



Figure 9. Simulated normalized steady-state current as a function of normalized tip-interface distance for K ) 105, Ke ) 10, and γ taking the values: (a) 1000, (b) 100, (c) 10, (d) 2, (e) 1, (f) 0.5, (g) 0.1 and (h) 0.01.

Figure 11. Steady-state profile of the distribution of A in the two phases during SECMIT, for Ke ) 1, γ ) 1, and K ) 0.1 (a), 1 (b), and 10 (c). Figure 10. SECMIT chronoamperometric characteristics for log(d/a) ) -0.5, γ ) 1, and Ke ) 1. The curves correspond to K taking the values: (a) 1000, (b) 20, (c) 5, (d) 2, (e) 1, (f) 0.5, and (g) 0.0.

arbitrarily set Ke ) 1 and γ ) 1; i.e., initially the equilibrium conditions are such that there are equal concentrations of the target solute in the two phases, and the solute diffusion coefficient is phase-independent. Figure 10 shows the chronoamperometric characteristics for K values of 1000, 20, 5, 2, 1, 0.5, and 0. Under the defined conditions, these values of K reflect the ease with which the transfer process can respond to a perturbation of the local concentration of A in phase 1, due to electrolytic depletion. For small K, i.e., K ) 0.5 in Figure 10, the response of the equilibrium to the depletion of species A in phase 1 is slow compared to diffusional mass transport, and consequently the current-time response and mass transport characteristics are close to those predicted for hindered diffusion with an inert interface. The steady-state concentration profile, shown in Figure 11 a, illustrates this point, demonstrating that there is a large discontinuity in the concentration distribution at the interface, with little perturbation of the solute in phase 2. As K is increased, the interfacial equilibrium responds more rapidly to the electrochemically induced depletion process in phase 1. The transfer of the species across the interface generates an enhanced flux to the UME, causing the long-time current to be larger than that predicted for an inert interface. The higher the value of K, the greater the extent of interfacial

transfer, resulting in a greater depletion of material in phase 2 at steady-state (Figure 11). Moreover, as K increases, the discontinuity in the concentration distribution at the interface becomes less marked, as the interfacial process moves closer to equilibrium. For large K (K ) 1000 in Figure 10) an upper limit is reached, where the interfacial kinetics are sufficiently fastson the time scale of the SECM measurementssuch that the concentrations of A in the two phases, adjacent to the interface, are always in equilibrium even though A is generally depleted. The tip current response is then dependent on the rate of mass transfer within both phases and for Ke ) 1 and γ ) 1 resembles that predicted for a conventional UME in bulk solution.32 The plot of the normalized steady-state current against tipinterface distance, shown in Figure 12, can be explained by a similar rationale. For large K the steady-state current is controlled by diffusion of the solute in the two phases and for the specific Ke and γ values considered is thus independent of the separation between the tip and the interface. For K ) 0, the current-distance relationship is identical with that predicted for the approach to an inert substrate. Within these two limits, the steady-state current increases as K increases. Experimental Implications. The theoretical results presented above have implications for the design of experimental approaches for the study of transfer processes across the interface between two immiscible phases. It has clearly been shown above that the current response in SECMIT is sensitive


Figure 12. Simulated normalized steady-state current as a function of normalized tip-interface distance for γ ) 1, and Ke ) 1 with K taking the values: (a) 1000, (b) 20, (c) 5, (d) 2, (e) 1, (f) 0.5, and (g) 0.0.

to the relative diffusion coefficients and concentrations of a solute in the two phases and the kinetics of the interfacial transfer over a wide range of values of these parameters. Previous investigations of liquid/liquid interfaces14,15,17,18 have employed a relatively high concentration in phase 2 compared to that in phase 1, in order to exclude the possibility of depletion effects and diffusion limitations in phase 2. The above results show that for comparable diffusion coefficients in the two phases and nonlimiting transfer kinetics, systems characterized by different Ke should be resolvable on the basis of transient and steady-state current responses to a value of Ke > 50 at practical tip/interface separations. If the diffusion coefficient in phase 2 becomes lower than that in phase 1, diffusion in phase 2 will be partly limiting at even higher values of Ke. On the other hand, as the value of γ increases or interfacial kinetics become increasingly limiting, lower values of Ke suffice for the constant composition assumption for phase 2 to be valid. Another practical consequence relates to the different dependences of the steady-state and transient currents on Ke and γ. The results above suggest that it may be possible to determine independently both Ke and γ, by correlating measurements of the steady-state current, as a function of distance from the interface, with chronoamperometric measurements (if there is no interfacial kinetic barrier). Alternatively, steady-state measurements alone should provide a powerful approach to determining the product Keγ. These observations are of considerable practical importance in opening up a new route for measuring concentrations and diffusion coefficients in phases that have, hitherto, been difficult to study with dynamic electrochemistry. As with previous kinetic applications of SECM, it should be noted that experimental measurements can be tuned to the kinetic region of interest by varying the radius of the electrode (eq 16) and separation between the tip and interface. In essence, the smaller the UME, and/or tip-interface separation, the higher the diffusion rates that may be generated and, consequently, the greater the tendency for interfacial kinetic limitations. Experimental Section Apparatus and Instrumentation. All electrochemical measurements were made using a two-electrode arrangement. A silver wire served as a quasi-reference electrode (AgQRE), and a 25-µm diameter Pt disk UME functioned as the working

Barker et al. electrode tip. The UME, characterized by RG ≈ 10, was fabricated and polished as described previously.11 The potential was controlled with a purpose built triangular wave/pulse generator (Colburn Electronics, Coventry, UK) and the current was measured using a home-built current follower (gains of 10-5-10-9 A V-1). Voltammetric and chronoamperometric measurements were mainly recorded using a PC equipped with a data acquisition card (Lab PC+ card, National Instruments, Austin, TX). Alternatively, for some experiments, current-time transients were recorded using a NIC310 (Nicolet, Coventry, UK) digital oscilloscope equipped with disk storage capabilities. To facilitate movement of the tip UME in the z-direction, normal to the target interface, two types of instrumentation were employed. For all experiments involving measurements of interfacial oxygen transfer, the tip position was controlled using a TSE-75 stage from Burleigh Instruments (Fischer), which provided spatial resolution in the position of the tip of 0.05 µm. In these measurements the tip was scanned at a velocity of 2.2 µm s-1 toward the target interface, while simultaneously recording the current as a function of position. For all other experiments, the UME was initially positioned close to the target interface (within 30-60 µm) using 431-2 stages (Newport Corp., Fountain Valley, CA). Scanning of the tip position in the z-direction, at a rate of 0.8 µm s-1, was then effected with a piezoelectric translator, incorporating a strain gauge sensor (translator model P173 or P178 and controller P273, Physik Instrumente, Waldbronn, Germany). The voltage waveform used to control the positioner was delivered via one of the D/A channels from a Lab PC+ card. This card was also used to record the current as a function of the tip position. For investigations of liquid/liquid interfaces, a one-piece cylindrical glass cell (40-mm diameter and 50-mm depth) was employed. The cell incorporated an optical glass window (diameter 30 mm) so that a zoom microscope with a CCD camera attachment (maximum resolution 2.2 µm per screen pixel) could be used to aid positioning of the electrode near the interface. The distance of closest approach of a particular tip to the liquid/liquid interface of interest was established through steady-state hindered diffusion mode experiments, as employed successfully in earlier work,11,29 for the reduction of hexaammineruthenium(III) [Ru(NH3)63+] in aqueous solution as a function of tip/interface separation. The composition of the aqueous solution was as for the hydrogel experiments described below. For hydrogel experiments, fully detachable cells were used, comprising a Teflon base to which the hydrogel was fixed, a cylindrical glass body, and a Teflon lid. As with our previous SECM studies,11,14,20 all measurements were made at ambient temperature (24 ( 1 °C). Materials and Solutions. All aqueous solutions were prepared from Milli-Q reagent water (Millipore Corp.). The organic solvents used were DCE (HPLC grade, Sigma-Aldrich, Gillingham, UK) and NB (ACS Reagent, Sigma-Aldrich). Hydrogels had a typical composition of 3.0 g of 2-hydroxyethyl acrylate (96%, ACS Reagent, Sigma-Aldrich), 0.6 g of acrylic acid (ACS Reagent, Sigma-Aldrich), 9.25 g of poly(ethylene glycol) (Sigma-Aldrich), and 7.15 g of water. Sodium persulfate (98+%, Sigma-Aldrich) was used as the polymerization initiator at a level of 0.5%. The above components were mixed together thoroughly and baked in a level receptacle at 70 °C for 60 min until set. The hydrogel was then soaked in a large excess of Milli-Q water to remove unreacted monomer and ensure hydration.

SECM as a Probe in Two-Phase Systems A small section of the hydrogel (typically a square of 5-mm length and 1-2-mm depth) was doped and equilibrated with the mediator solution before use by soaking with a large excess of the electrolyte solution of interest for at least 24 h. The sample was fixed in the base of the cell with two Teflon anchors positioned over the edges of the sample and secured with small screws. Three mediator solutions were investigated: 0.012 mol dm-3 tris(2,2′-bipyridyl)ruthenium(II) chloride hexahydrate (Ru(bipy)32+) (Sigma-Aldrich); 0.010 mol dm-3 hexaammineruthenium(III) chloride (99%, Strem Chemicals, Newburyport, MA), both with 0.2 mol dm-3 potassium nitrate (Analytical Reagent, Fisher, Loughborough, UK) as background electrolyte; and 0.050 mol dm-3 potassium ferrocyanide with 2 mol dm-3 potassium chloride (both Analytical Reagent, Fisher) as background electrolyte. When employing ferrocyanide as the mediator, the solution was deaerated with Ar during doping to prevent air oxidation of ferrocyanide. For investigations of cupric ion extraction and stripping processes, equilibrium conditions were established by shaking together (for a period of ca. 15 min) equal volumes of an aqueous solution comprising 0.02 mol dm-3 copper sulfate pentahydrate (Analytical Reagent, Fisons), 0.5 mol dm-3 potassium sulfate (Analytical Reagent, Fisons), and 0.4 mol dm-3 sulfuric acid (Analytical Reagent, Sigma-Aldrich) with DCE. The latter contained 0.2 mol dm-3 LH, which was kindly supplied by Zeneca Specialties (UK). The DCE solution employed for the chronoamperometric determination of the diffusion coefficient of the complexed cupric ion, CuL2, comprised 0.05 mol dm-3 tetra-n-hexylammonium perchlorate as a supporting electrolyte, together with 10-3 mol dm-3 LH. Cu2+ was extracted into this solution from a 1.0 mol dm-3 aqueous copper sulfate solution (unbuffered) by shaking the two phases together as before. For all measurements on this system, solutions were deaerated with Ar for 15 min prior to running experiments. Experimental Results and Discussion The aim of the experiments reported herein was to assess several applications of SECMIT that derive from the theoretical results above, under steady-state conditions. The application of transient SECMIT methods will be considered in more detail in subsequent papers. SECMIT of Solutes across Hydrogel/Aqueous Interfaces. As an initial system for examining the theoretical model, the use of a hydrogel as phase 2, in contact with an aqueous solution as phase 1, was considered to be ideally suitable for several reasons. First, hydrogels are insoluble in water but are extremely hydrophilic. Thus, although a hydrogel will effectively act as a separate phase in contact with an aqueous solution, any species soluble in aqueous medium is likely to partition into the hydrogel network. This provides a large number of possibilities for model solutes for study. Second, the diffusion coefficient and concentration of the solute can be determined in each phase, with high precision, through the use of UME chronoamperometry,33-35 providing an independent method for checking the values of Ke and γ derived from SECMIT measurements. Third, the properties of hydrogels have been exploited practically in a variety of pharmaceutical and biomedical applications,36 including controlled drug delivery devices and soft contact lenses. To optimize such applications, an understanding of the diffusive properties of solutes in hydrogel systems is of considerable importance. The aqueous/hydrogel system was investigated with the three different electroactive solutes in separate experiments. As

J. Phys. Chem. B, Vol. 102, No. 9, 1998 1593 discussed above, the concentration and diffusion coefficient were obtained for each phase from the diffusion-limited chronoamperometric responses with a UME first positioned in bulk aqueous solution and then inserted ca. 100 µm into the hydrogel. This was achieved, for each of the three solutes, by stepping the potential of the UME from a value at which there were no Faradaic processes to one at which the electrolytic process of interest occurred at a diffusion-controlled rate. For the oxidation of Fe(CN)64- the potential (vs AgQRE) was stepped from 0.00 to 0.65 V, for Ru(bipy)32+ oxidation the potential range was 0.75 to 1.05 V, and for Ru(NH3)63+ reduction the potential was stepped from 0.00 to -0.50 V. In each case the current response, normalized with respect to the steady-state current, was plotted vs t-1/2 and compared with the theoretically predicted response for a diffusion-controlled electrolysis process32 at a disk UME:

i/i(∞) ) 0.7854 + 0.4431(tD/a2)-1/2 + 0.2146 exp[-0.3911(tD/a2)-1/2] (20) where D is the diffusion coefficient of the target solute. Equation 20 was found to provide excellent descriptions of the experimental data obtained with all three solutes in both aqueous solutions and the hydrogels, over the experimental time range employed (ca. 2.5 ms and longer), as shown in Figure 13. The fitting procedure involved D as the only adjustable parameter, allowing the extraction of its value. Armed with a knowledge of the diffusion coefficient, the solute concentration was then readily determined using eq 19. The concentrations and diffusion coefficients of the solutes in the two phases, deduced from the data in Figure 13, are shown in Table 1, together with the calculated values of Ke and γ. It can be seen that the hydrogel medium is permselective to the cations studied (Ke > 1), while for the ferrocyanide anion Ke < 1. The diffusion coefficients are lower in the hydrogel than in aqueous solution, as found in previous studies.35 Together, these three systems provide a good range of Ke and γ values to examine with SECMIT measurements. Approach curves of steady-state current vs the distance between the tip (in aqueous solution) and the hydrogel surface, for the three solutes, are shown in Figure 14. For all solutes, the current decreases as the separation between the tip and the target interface is decreased, but the currents are larger than expected for the approach to an inert interface. These effects may be attributed to the electrochemically induced transfer of the target solute across the interface from the hydrogel phase to the aqueous bathing solution. The current eventually reaches a plateau value when the UME contacts and then pushes into the hydrogel. For all three solutes, the steady-state currents at this point are consistent with the values of D2 and c2* measured directly with chronoamperometry. In each case, the best fits of the experimental approach curves to the theoretical model were obtained using the values of Ke and γ determined via chronoamperometry, as described above, assuming that there is no interfacial kinetic barrier to transfer at the interface. This analysis is seen to provide an excellent description of the SECMIT data (Figure 14). The lack of any significant interfacial kinetic barrier is expected, since the target solutes are effectively transferred between two discrete aqueous phases. The consistency between the values of Keγ derived from SECMIT measurements and those obtained directly provides good evidence for the validity of the theoretical model described above. SECMIT of Oxygen across the Interfaces between Water and Organic Solvents. Reduction of Oxygen at an UME. A


Barker et al.

Figure 14. Steady-state diffusion-limited current vs tip/interface separation for the approach of a tip to an aqueous/hydrogel interface with the tip held at a potential to electrolyze the following solutes that partition between the two phases: oxidation of Ru(bipy)32+ (∆); reduction of Ru(NH3)63+ (0); oxidation of Fe(CN)64- (O). The solid lines through each of these data are the best theoretical fits using the SECMIT model with the values of Ke and γ defined in Table 1, assuming no interfacial kinetic barrier (K ) 105). The dashed line is the response expected if the interface was inert with respect to the transfer of the target solute.

Figure 13. Chronoamperometric characteristicssplotted as normalized current ratio versus t-1/2sfor: (a) the oxidation of Fe(CN)64-, (b) the oxidation of Ru(bpy)32+, and (c) the reduction of Ru(NH3)63+ in hydrogels (∆) and aqueous solutions (]). The dashed lines show the best fits of eq 20 to the experimental data using the diffusion coefficients indicated.

well-defined steady-state cathodic voltammogram with a halfwave potential of -0.58 V (vs AgQRE) and height of 10.7 nA was measured with an UME positioned in bulk aerated water, which may be attributed to the reduction of oxygen.37-39 Taking a value of 2.5 × 10-4 mol dm-3 for the concentration of oxygen

in air-saturated water,37 a diffusion coefficient of 2.2 × 10-5 cm2 s-1 was calculated, assuming a four-electron process for the reduction of oxygen.40 This value for the diffusion coefficient agrees closely with that reported elsewhere.37 The assumption of a four-electron process for the cathodic reduction of oxygen in water at a Pt UME of this size is consistent with earlier studies,37 which suggest that oxygen reduction involves two 2e- transfer steps occurring at similar potentials, coupled by a fast chemical step.37,38 The apparent number of electrons transferred therefore varies from four to two with increasing rates of mass transport. The steady-state diffusion rate to a 25 µm diameter UME is sufficiently low that a process involving close to four electrons is observed.37 In the SECMIT investigations reported below of oxygen transfer across the interfaces between aqueous and (i) DCE and (ii) NB, the diffusion rate remained sufficiently low (as determined by the relative diffusion coefficients for both phases), for the assumption of a four-electron process to be valid for all tip/ interface separations employed. This assumption was found to be valid in recent SECMIT studies of oxygen between biological tissues and aqueous solutions.39 Oxygen Transfer across Liquid/Liquid Interfaces. In light of some of our recent work20 which has demonstrated that Br2 is transferred across aqueous/organic interfaces at a diffusioncontrolled rate on the time scale of SECM measurements in this paper, it was anticipatedsand confirmed (vide infra)sthat there would be no apparent measurable kinetic barrier for oxygen transfer in the systems of interest with the UME size and tip/interface separations, employed. This assumption could typically be verified by using a range of tip sizes, but, for oxygen

TABLE 1: Measured Values for the Diffusion Coefficients, Concentrations, Ke, and γ for Various Electroactive Solutes in Aqueous/Hydrogel Systems species 4-

Fe(CN)6 Ru(bipy)32+ Ru(NH3)63+

c*aq/10-3 mol dm-3

c*hydrogel/10-3 mol dm-3

Ke

Daq/10-6 cm2 s-1

Dhydrogel/10-6 cm2 s-1

γ

50.1 11.7 9.9

22.2 26.8 13.0

0.44 2.3 1.3

6.7 4.8 8.6

1.9 1.8 4.7

0.28 0.38 0.55


Figure 15. Steady-state diffusion-limited current for the reduction of oxygen in water at a tip approaching a water/DCE interface (O). The solid line is the behavior predicted theoretically for γ ) 1.2 and Ke ) 5.5, with no interfacial kinetic barrier. The lower and upper dashed lines denote the current-distance characteristics for the situations where there is no interfacial transfer and where transfer occurs without limitations from diffusion in the DCE phase.

reduction at Pt, this would introduce complications associated with a variation in the apparent number of electrons transferred in the electrode reaction,37 as discussed above. With the above assumption, the aim of the measurements was to demonstrate that the concentration and/or diffusion coefficient of oxygen could be probed indirectly in a phase, where a direct voltammetric measurement would be impossible due to high resistivity effects or a limited voltammetric window. For these measurements, the UME was placed in the aqueous phase, and neither this nor the organic phase contained any deliberately added electrolyte. Approach curves were recorded by holding the UME potential at -0.8 Vsto effect the diffusionlimited reduction of oxygenswhile the tip was translated toward the interface, from an initial tip/interface distance of ca. 40 µm. A typical approach curve of diffusion-limited current vs tip separation from an aqueous/DCE interface is shown in Figure 15. As the tip/interface separation decreases, the current increases consistent with the electrolysis process inducing the transfer of oxygen from the DCE phase into the aqueous phase, thereby increasing the flux of oxygen at the tip, compared to the case where the interface is inert. There are, however, diffusional limitations in the DCE phase, since the measured currents are lower than predicted theoretically for the situation KKeγ f ∞. The best fit of the experimental data to theory was obtained with a value of the product Keγ ) 6.6 ((0.1). This is in excellent agreement with the value expected, on the basis of the following known parameters for this system: caq ) 0.25 × 10-3 mol dm-3,38 cDCE ) 1.39 × 10-3 mol dm-3,40 Daq ) 2.2 × 10-5 cm2 s-1,38 and DDCE ) 2.7 × 10-5 cm2 s-1.41 Thus γ ) 1.2 and Ke ) 5.6. The excellent level of agreement between the experimental data and the theoretically predicted characteristics, with a well-found value of Keγ, reconfirms that, under the defined conditions, there are no interfacial kinetic limitations to oxygen transfer between DCE and water on the SECM time scale. NB is an example of a solvent in which oxygen cannot be determined directly by voltammetric methods, even with supporting electrolyte present, due to the limited cathodic window available. Previous attempts to determine the diffusion coefficient of oxygen in nitrobenzene by alternative electrochemical


Figure 16. Steady-state diffusion-limited current for the reduction of oxygen in water at a tip approaching a water/NB interface (O). The solid lines are the characteristics predicted theoretically for γ ) 0.58 (middle line) ( 0.053 (upper and lower lines) with Ke ) 3.8 and no interfacial kinetic barrier. The lower and upper dashed lines denote the current-distance characteristics for the situations where there is no interfacial transfer and where transfer occurs without limitations from diffusion in the NB phase.

methods, such as the employment of a polarographic diffusion cell, have met with little success.41 A SECMIT approach curve for oxygen reduction in the water/ NB system, obtained by translating the tip in water toward a water/NB interface, is shown in Figure 16. Theoretically simulated characteristics for no transfer and transfer without diffusional limitations in the NB phase are also shown for comparison. As with the case above, it can be seen that the UME process induces the transfer of oxygen from the NB phase, across the water/NB interface, enhancing the current compared to the case where there is no interfacial transfer. There are, however, diffusional limitations in the NB phase, since the measured currents are lower than predicted theoretically for the situation KKeγ f ∞. For this two-phase system, a value of Ke ) 3.8 may be calculated.41 Assuming there are no interfacial kinetic limitations, which is reasonable given the arguments above, the experimental data may then be fitted to theory by varying γ until an optimal fit is achieved. The best fit occurred for γ ) 0.58 ((0.05), giving the diffusion coefficient for oxygen in NB as 1.3 ((0.1) × 10-5 cm2 s-1. This result agrees with the value of 1.35 × 10-5 cm2 s-1, which may be estimated by considering that oxygen diffusivities in pure liquids are empirically correlated with µ-2/3, where µ is the solvent viscosity.41 Kinetics of Cu2+ Extraction/Stripping between Aqueous/ DCE Phases Promoted by Oxime Ligands. The kinetics of this process have previously been studied rigorously using the rotating diffusion cell.42 Several general drawbacks to this approach, including a mass transfer resistance from the membrane used to support the interface, led us to develop SECMIT as an alternative approach for determining the interfacial kinetics under conditions of well-defined mass transport with direct contact between the two phases.14 The latter studies were, however, carried out under conditions where there was a considerable excess of CuL2 in the organic phase (phase 2) compared to cupric ion in the aqueous phase. Here, we consider the extent to which the constant composition model is valid when both phases have approximately equal concentrations of the target solute. The overall extraction/stripping process can be expressed as follows:

1596 J. Phys. Chem. B, Vol. 102, No. 9, 1998 extraction

Cu2+(aq) + 2LH(org) {\ } CuL2(org) + 2H+(aq) stripping

Barker et al.

(21)

Under the conditions of this study, LH in the DCE phase and H+ in the aqueous phase were in large excess, and the process may be written more simply in terms of the cupric species alone ke

} CuL2(org) Cu2+(aq) {\ k

(22)

s

where ke and ks are the rate constants for the extraction and stripping reactions. Under the (relatively low) concentration conditions employed, the extraction and stripping processes are expected to be first order in Cu2+and CuL2 respectively.42 At equilibrium there is no net flux of copper species in either direction, and the following equation applies

ke[Cu2+]* ) ks[CuL2]*

(23)

where [Cu2+]* and [CuL2]* are the bulk concentrations of the two cupric species in the aqueous and DCE phases, respectively. For this system, the equilibrium constant, Ke, can be expressed as

Ke )

ke [CuL2]* ) ks [Cu2+]*

(24)

With a UME located in the aqueous phase to effect SECMIT of cupric ions from the DCE into the aqueous phase, the flux of Cu2+ across the interface in the stripping direction, jCu2+, is given by

jCu2+ ) ks([CuL2]i - Ke[Cu2+]i)

Figure 17. Steady-state diffusion-limited current for the two-electron reduction of Cu2+ in aqueous solution at a tip approaching the water/ DCE interface (O), where the latter phase contains CuL2, as defined in the text. The solid line is the best fit of the experimental data to the SECMIT model in this paper (with the parameters defined in the text). The dotted line shows the best fit for the constant composition model with the parameters in the text. The dashed line shows the behavior predicted for the situation where there is no transfer of cupric ions from phase 2 to phase 1 (hindered diffusion only).

(25)

where [Cu2+]i and [CuL2]i are the interfacial concentrations of the two copper species in their individual phases. This equation has the same form as eq 9, which formed the interfacial (internal) boundary condition in the model described in this paper. To determine the interfacial kinetics from steady-state measurements alone, a knowledge of Ke and γ is imperative. The equilibrium concentrations of the cupric species in each phase, determined by UV/visible spectroscopy, were 10.4 × 10-3 mol dm-3 CuL2 for the DCE phase (λmax ) 677 nm; ) 102 dm3 mol-1 cm-1) and 9.8 × 10-3 mol dm-3 Cu2+(aq) (λmax ) 804 nm; ) 11.8 dm3 mol-1 cm-1). The latter value was confirmed by steady-state UME voltammetry (eq 19), using a measured value (eq 19) of DCu2+ ) 6.0 × 10-6 cm2 s-1 (in good agreement with previous values).43 Thus, under the experimental conditions, Ke ) 1.06. By difference, at equilibrium, [HL] in the DCE phase was 0.180 mol dm-3 and the measured pH of the aqueous phase was 0.97. The chronoamperometric determination of the diffusion coefficient of CuL2 in DCE involved stepping the potential from 0.0 to -1.4 V vs AgQRE, where the reduction process was found to be diffusion controlled. The chronoamperometric data were in excellent agreement with eq 20 for D ) 5.0 × 10-6 cm2 s-1 at times greater than 2.5 ms, indicating γ ) 0.83 for the system under the defined conditions. Figure 17 shows a typical steady-state approach curve for Cu2+ reduction at a tip UME translated in aqueous solution toward the aqueous/DCE interface, while held at a potential of -0.75 V vs AgQRE. The electrodeposition of Cu on the UME during such measurements was not sufficiently extensive to alter the area (or geometry) of the electrode, evident from the observation that the bulk (steady-state) currents for Cu2+

reduction before and after running an approach curve were in close agreement. Cu was anodically stripped from the UME in the period between the acquisition of successive approach curves by taking the potential to +1.0 V (vs AgQRE). The experimental data in Figure 17 indicate that there is a gradual decline in the current on approaching the interface, but the measured currents are larger than those predicted for the approach to an inert interface. The data are seen to be in good agreement with both the theoretical model developed in this paper and the constant composition model outlined earlier.14 However, the rate constants derived from the two models are different. The full model described herein produces a value of ks ) 2.6 ((0.2) × 10-4 cm s-1, while the constant composition assumption for phase 2 leads to ks ) 1.8 ((0.2) × 10-4 cm s-1. The higher value for the stripping rate constant in the former case is because the depletion of [CuL2] in the DCE phase (particularly at the interface) is taken into account when modeling the interfacial fluxes. In contrast, the constant composition assumption requires a lower first-order rate constant to predict interfacial fluxes of the same magnitude. Even though this system is characterized by low interfacial fluxes, an error of ca. 30% in the rate constant results from assuming constant composition in phase 2. For faster processes, the constant composition assumption would lead to more serious errors. Conclusions The first model for SECM that allows for diffusion in the phase containing the UME and the second phaseswhich forms the target interfaceshas been developed, with specific reference to the induced transfer mode. SECMIT shows considerable promise for determining interfacial transfer kinetics of electroactive solutes, with a wide range of values of the partition coefficient, interfacial kinetics, and diffusion coefficients for the two phases. Under conditions where there is no detectable interfacial resistance to the induced transfer process, the approach allows the measurement of the diffusion coefficient and/or concentration of a solute in a target phase without the UME having to enter or contact that phase. This extends the use of electrochemistry to analyze phases where direct volta-

SECM as a Probe in Two-Phase Systems mmetric measurements are prohibited, for example by a limited solvent window or high electrical resistance, or where the physical presence of the UME could damage the structural integrity of the phase. The advantages of SECMIT in this latter case are well-illustrated by preliminary investigations of oxygen diffusion in biological tissues, reported elsewhere,39 which complement the studies in this paper. The numerical approach described herein for treating diffusion in two phases could readily be extended to other modes of SECM such as the positive feedback mode1,15-18 and doublepotential step chronoamperometry,20 thereby significantly diversifying the range of processes and conditions that may be studied with SECM techniques. Acknowledgment. We thank the EPSRC, Zeneca, and the Royal Society Paul Instrument Fund for support of this work. Helpful discussions with Dr. David Haddleton (University of Warwick) and Dr. John Atherton and John Umbers (Zeneca Huddersfield Works) are much appreciated. Appendix For the problem under consideration, the R-coordinate was transformed with exponential functions, as defined by eqs A1 and A2 in ref 29. This served to effectively match the radial grid to the predicted steady-state concentration profiles, thereby increasing the efficiency of the simulation. For the Z-coordinate, a linear grid was constructed over the domain of phase 1 (0 < Z < d/a), whereas for phase 2 the Z-coordinate was transformed with an exponential function, similar in form to that used for R-space over the glass insulator, which served to effectively concentrate the grid points close to the interface (in real dimensionless Z-space). For the Z-direction, a distance of 20 electrode radii from the electrode surface was found to be sufficient to represent this semi-infinite domain, on the time scale of the simulations. The ADIFDM solves the time-dependent diffusion equations, subject to the boundary conditions, through the formulation of implicit finite-difference equations at successive half-time intervals, respectively, for the R and Z-directions. For the first half-time step, in which concentrations are calculated along the R-coordinate from known values (at the previous half-time step) in the Z-direction, the methodology was similar to that described previously.29 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 ) a (over the tip electrode) and j ) NE + 1 to j ) NE + NG over the insulating glass sheath. This approach produces a tridiagonal matrix of NE + NG - 1 simultaneous equations in NE + NG - 1 unknowns for each grid point, k, in the Z-coordinate, that can be solved by the Thomas algorithm.44 The calculation was carried out for phase 1 from k ) 1 to k ) NZ1 - 1 and then phase 2 from k ) NZ1 + 1 to k ) NZ1 + NZ2 - 1, where the points 0 e k e NZ1 covered phase 1 and NZ1 e k e NZ1 + NZ2 covered phase 2, with the target interface located at k ) NZ1. After this set of calculations, the exterior and interior boundary values were updated using the boundary conditions above, in standard finite-difference form. For the second half-time step, the concentrations in the Z-direction are unknown and are calculated from the values of the concentrations in the radial direction, evaluated in the first half-time step. After points at the exterior boundaries in the Z-direction (at Z ) 0 and Z ) 20) are eliminated, the result is a set of simultaneous equations, involving NZ1 + NZ2 unknowns, for each value of j in the internal domain 1 e j e

J. Phys. Chem. B, Vol. 102, No. 9, 1998 1597 NE + NG - 1. For the grid points 1 e k e NZ1 - 1 and NZ1 + 1 e k e NZ1 + NZ2 - 1, the equations are simply derived from the diffusion equations in ADIFDM form, as outlined previously.29 The two additional equations, which provide the link between these two sets of equations, relate to the interior boundary and may be derived by writing eq 9, after adimensionalization, in ADIFDM form, viz:

C1**(j, NZ1 - 1) - [1 + KKeγ∆Z]C1**(j, NZ1) + KKeγ∆Z C2**(j, NZ1) ) 0 (A1) KC1**(j, NZ1) - [1 + K]C2**(j, NZ1) + C2**(j, NZ1+1) ) 0 (A2) The double asterisk superscript signifies that the concentrations are those to be calculated at the second half-time step. The indices in the brackets, following the concentration terms, identify a specific point on the finite-difference grid. The forms of eqs A1 and A2 means that the NZ1 + NZ2 set of simultaneous equations, at each j value, are in tridiagonal matrix form, allowing a direct solution by application of the Thomas algorithm.44 References and Notes (1) (a) Bard, A. J.; Fan, F.-R.; Pierce, D. T.; Unwin, P. R.; Wipf, D. O.; Zhou. F. Science 1991, 254, 68. (b) Bard, A. J.; F.-R. Fan. Faraday Discuss. 1992, 94, 1. (c) Bard, A. J.; Mirkin, M. V.; Fan, F.-R. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1993; Vol. 18, p 243. (d) Arca, M.; Bard, A. J.; Horrocks, B. R.; Richards, T. C.; Treichel, D. A. Analyst 1994, 119, 719. (e) Mirkin, M. V. Anal. Chem. 1996, 68, 177A. (2) Wipf, D. O.; Bard, A. J. J. Electrochem. Soc. 1991, 138, 469. (3) (a) Bard, A. J.; Mirkin, M. V.; Unwin, P. R.; Wipf, D. O. J. Phys. Chem. 1992, 96, 1861. (b) Engstrom, R. C.; Small, B.; Kattan, L. Anal. Chem. 1992, 64, 241. (4) Wipf, D. O.; Bard, A. J. J. Electrochem. Soc. 1991, 138, L4. (5) Mirkin, M. V.; Bulhoes, L. O. S.; Bard, A. J. J. Am. Chem. Soc. 1993, 115, 201. (6) Engstrom, R. C.; Meaney, T.; Tople, R.; Wightman, R. M. Anal. Chem. 1987, 59, 2005. (7) Pierce, D. T.; Unwin, P. R.; Bard, A. J. Anal. Chem. 1992, 64, 1795. (8) Pierce, D. T.; Bard, A. J. Anal. Chem. 1993, 65, 3598. (9) (a) Mandler, D.; Bard, A. J. J. Electrochem. Soc. 1989, 136, 3143. (b) Mandler, D.; Bard, A. J. J. Electrochem. Soc. 1990, 137, 1079. (c) Meltzer, S.; Mandler, D. J. Electrochem. Soc. 1995, 142, L82. (d) Wipf, D. O. Colloids Surf. A 1994, 93, 251. (e) Macpherson, J. V.; Slevin, C. J.; Unwin, P. R. J. Chem. Soc., Faraday Trans. 1996, 92, 3799. (10) (a) Mandler, D.; Bard, A. J. J. Electrochem. Soc. 1990, 137, 2468 (b) Mandler, D.; Bard, A. J. Langmuir 1990, 6, 1489. (11) (a) Macpherson, J. V.; Unwin, P. R. J. Phys. Chem. 1994, 98, 1704. (b) Macpherson, J. V.; Unwin, P. R. J. Phys. Chem. 1994, 98, 3109. (c) Macpherson, J. V.; Unwin, P. R. J. Phys. Chem. 1995, 99, 3338. (d) Macpherson, J. V.; Unwin, P. R. J. Phys. Chem. 1995, 99, 14824. (e) Macpherson, J. V.; Unwin, P. R. J. Phys. Chem. 1996, 100, 19475. (f) Macpherson, J. V.; Hillier, A. C.; Unwin, P. R.; Bard, A. J. J. Am. Chem. Soc. 1996, 118, 6445. (12) Kranz, C.; Gaub, H. E.; Schuhmann, W. AdV. Mater. 1996, 8, 634. (13) Unwin, P. R.; Bard, A. J. J. Phys. Chem. 1992, 96, 5035. (14) Slevin, C. J.; Umbers, J. A.; Atherton, J. H.; Unwin, P. R. J. Chem. Soc., Faraday Trans. 1996, 92, 5177. (15) Wei, C.; Bard, A. J.; Mirkin, M. V. J. Phys. Chem. 1995, 99, 16033. (16) Solomon, T.; Bard, A. J. J. Phys. Chem. 1995, 99, 17487. (17) Tsionsky, M.; Bard, A. J.; Mirkin, M. V. J. Phys. Chem. 1996, 100, 17881. (18) Shao, Y.; Mirkin, M. V.; Rusling, J. F. J. Phys. Chem. B 1997, 101, 3202. (19) Unwin, P. R.; Slevin, C. J. Abstracts of the Fall Meeting of the Electrochemical Society, 1996; Vol. 96-2, p 810. (20) Slevin, C. J.; Macpherson, J. V.; Unwin, P. R. J. Phys. Chem. B 1997, 101, 10851. (21) (a) Fan, F. R. F.; Bard, A. J. Science 1995, 267, 871. (b) Fan, F. R. F.; Bard, A. J. Science 1995, 270, 1849. (22) Freiser, H. Chem. ReV. (Washington, D.C.) 1988, 88, 611.

1598 J. Phys. Chem. B, Vol. 102, No. 9, 1998 (23) Danesi, P. R.; Chiarizia, R. CRC Crit. ReV. Anal. Chem. 1980, 10, 1. (24) Atherton, J. H. Res. Chem. Kinet. 1994, 2, 193. (25) Girault, H. H.; Schiffrin, D. J. In Charge and Field Effects in Biosystems; Allen, M. J., Usherwood, P. N. R., Eds.; Abacus Press: Tunbridge Wells, 1984; p 171. (26) Ohkouchi, T.; Kakutani, T.; Senda, S. Bioelectrochem. Bioenerg. 1991, 25, 71, 81. (27) For a review, see, for example: Hanna, G. J.; Noble, R. D. Chem. ReV. (Washington, D.C.) 1985, 85, 583. (28) Saito, Y. ReV. Polarogr. 1968, 15, 177. (29) Unwin, P. R.; Bard, A. J. J. Phys. Chem. 1991, 95, 7814. (30) Bard, A. J.; Denuault, G.; Dornblaser, B. C.; Friesner, R. A.; Tuckerman, L. S. Anal. Chem. 1991, 63, 1282. (31) Kwak, J.; Bard, A. J. Anal. Chem. 1989, 61, 1221. (32) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1982, 140, 237. (33) Winlove, C. P.; Parker, K. H.; Oxenham, R. C. J. Electroanal. Chem. 1984, 170, 293. (34) Denuault, G.; Mirkin, M. V.; Bard, A. J. J. Electroanal. Chem. 1991, 86, 27.

Barker et al. (35) (a) Pyo, M.; Bard, A. J. Electrochim. Acta 1997, 42, 3077. (b) Zhou, H., Che, G.; Dong, S. Electroanalysis 1997, 9, 41. (36) Kroschwitz, J. I. Encyclopedia of Polymer Science and Engineering, 2nd ed.; Wiley-Interscience: New York, 1985; Vol. 7, pp 783-807. (37) Pletcher, D.; Sotiropoulos, S. J. Electroanal. Chem. 1993, 356, 109. (38) Pletcher, D.; Sotiropoulos, S. J. Chem. Soc., Faraday Trans. 1995, 91, 457. (39) Macpherson, J. V.; O’Hare, D.; Unwin, P. R.; Winlove, C. P. Biophys. J. 1997, 73, 2771. (40) Luhring, P.; Schumpe, A. J Chem. Eng. 1989, 34, 250. (41) Schumpe, A.; Luhring, P. J. Chem. Eng. Data 1990, 35, 24. (42) (a) Albery, W. J.; Choudhery, R. A.; Fisk, P. R. Faraday Discuss. Chem. Soc. 1984, 77, 53. (b) Albery, W. J.; Choudhery, R. A. J. Phys. Chem. 1988, 92, 1142. (43) Quickenden, T. I.; Qingzhong, X. J. Electrochem. Soc. 1996, 143, 1248. (44) (a) Lapidus, L.; Pinder, G. F. Numerical Solutions of Partial Differential Equations in Science and Engineering, Wiley: New York, 1982. (b) Ames, W. F. Numerical Methods of Partial Differential Equations; Wiley: New York, 1977.

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