10410
J. Phys. Chem. 1993,97, 10410-10415
Hydrodynamic Voltammetry with Microelectrodes. Channel Microband Electrodes: Theory and Experiment Richard G. Compton,' Adrian C. Fisher, and R. Geoffrey Wellington Physical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom
Peter J. Dobson and Peter A. Leigh Department of Engineering Science, Oxford University, Parks Road, Oxford OX1 3PJ. United Kingdom Received: May 5, 1993; In Final Form: July 22, 1993"
Theory is reported that predicts the mass-transport limited current flowing at a microband electrode located in a rectangular channel through which solution is pumped under laminar flow conditions. The effect of electrode size and solution flow rate is quantified and the relative roles of axial diffusion and convective mass transport are identified. Theory is found to be in excellent agreement with experiments conducted on the reduction of p-chloranil and the oxidation of ferrocene by using gold channel microband electrodes.
Introduction
Y-
The use of microelectrodes has revolutionized voltammetric practice within a period of just a few Of their many advantages the greatly enhanced rates of mass transport as compared to electrodes of conventional dimensions offers the mechanistic electrochemist opportunity for studying very fast chemical processes coupled with heterogeneous electron-transfer events that were previously too rapid to be probedvoltammetrically and were simply masked by diffusional control. Notably, various workers3.4 have used rapid-scan cyclic voltammetry at microdisc electrodes to access kinetics that approach the nanosecond time scale and this has transformed the scope of mechanistic voltammetry. It is of interest to inquire as to the fastest processes amenable to study via electrochemical microelectrode measurements. For voltammetric visibility the time scale of the chemical event to be studied must a t least be comparable to the rate of mass transport to the electrode surface. If the process is much slower, a standard voltammetric response free of kinetic information will be measured; if it is too fast, the absence of a reverse peak in cyclic voltamnetry, a t least, may leave no "handle" with which to interrogate the electrode reaction mechanism or kinetics. The success of microelectrodes in probing ultrafast homogeneous kinetic events coupled to electrochemistry derives from the much greater rates of diffusion to these electrodes. In principle the electrochemical time domain might be extended yet further if the rate of mass transport to a microelectrode could be augmented. The aim of this paper is to examine the response of microband electrodes under conditions where mass transport occurs predominantly by convection rather than diffusion. We anticipate that the use of such hydrodynamic microelectrodes may offer advantages in the study of electrode reaction mechanisms containing rapid chemical, or particularly photochemical,sv6 steps. Specifically we develop a theoretical treatment for the flow-rate dependence of the transport-limited currents flowing at microband electrodes of different sizes located in a channel flow cell (Figure 1) and report correspondingexperimental results, which are found to be in excellent agreement with theory. The latter considersaxial convection through the flow cell and diffusion both axially and normal to the electrode surface. It is shown that in a limit of sufficiently fast flow rates approximate theory is quantitatively adequate and the electrode response can be predicted by a model that neglects axial diffusion. Abstract published in Advance ACS Abstracts, September 1, 1993.
0022-3654/93/2097- 10410%04.00/0
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/I
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Figure 1. Schematic diagram showing a microband channel electrode.
Theory
In this section we describe a theoretical model for mass transport to a microband channel electrode and consider the transportlimited current due to the discharge of a species A at the electrode. A*e--.B We assume the presence of enough supporting electrolyte that migration effects may be neglected and that the width, w, of the microband electrode is much larger than its length, x, (w >> xe), so that the calculation reduces to a two-dimensional problem. Note that the flow through the cell is parallel to the electrode length. Under these conditions the steady-state convective diffusion equation descibing the spatial distribution of A is
where x and y are defined in Figure 1, a is the concentration of the species of interest, D is its diffusion coefficient, and vx represents the solution velocity profile in the x direction. The latter is parabolic, provided one is considering laminar flow and that a sufficiently long lead in section is present for the flow to become fully developed.' Quantitatively,
where uo is the velocity at the center of the channel and 2h is the channel depth (height). The boundary conditions pertinent to the problem of interest are
y=O
OCxCx,
y =0 y=o
X>X,
y=2h
x 0 the boundary conditions previously identified are applied. Time increments At are used with a counter T so that
t* = 0, 1, 2, ..., T t = t*At We use the symbol '*aj,&to denote the concentration of A at the coordinate G,k). Equation 1 becomes
This is solved by using the hopscotch algorithm applied in the standard manners1 and the current evaluated from the following expression
I=
I
I
k= Ks+Kt J
F i p e 3. Finite-difference grid, note that there are J box= across the cell depth, K1 boxes along the electrode length, KZboxes upstream, and K3 box- downstream of the electrode.
x
10.0
wFD[A](,,&Xk-" AY g'al,&
for different flow rates and cell/electrode geometries. For the purposes of interpreting our computational results presented below, we next seek to identify analytically the independent combinations of the cell parameters that control the
magnitude of the current flowing. To this end we first define the normalized coordinates
x= x/x,
Y =y/x,
(3)
Equation 2 becomes
v, = - 1 -h
(4)
so that eq 1 reduces to
where V = Vf/hd, d is the channel width, and 5 is the volume flow rate (cm3 s-'). We thus infer that a is a solely a function of the parameters X,Y, (x,/h),and (Vx,/D). It follows that the current will depend on D, (x,/h),and (Vx./D) and this dictates the way in which we have chosen to present some of our results below.
Theoretical Results and Discussion First the numerical strategy outlined above was verified by calculating the transport-limited current flowing at an electrode of conventional dimensions, as follows: x, = 0.4 cm, 2h = 0.4 mm, w = 0.4 cm. A diffusion coefficient of 10-5 cm2 s-1 and a concentrationof 10-6mol~ m wereassumed. - ~ For such electrodes it is well established12 that the current is given by the following expression
where F is the Faraday constant and Vf the volume flow rate. Figure 4 shows the results of our computations (made with At = 2 X l(r s, T = 25 000,J = 500, Kr = 20, K2 = 20, and K3 = 20, these minimum values generated "converged" currents; increasing the grid size beyond thesevalues produced no significant change,
