J. Phys. Chem. 1996, 100, 14067-14073
14067
Hydrodynamic Microelectrodes. The Microstrip Electrode: Theory and Experiment W. J. Aixill, A. C. Fisher,* and Q. Fulian School of Chemistry, UniVersity of Bath, ClaVerton Down, Bath BA2 7AY, United Kingdom ReceiVed: January 16, 1996; In Final Form: May 6, 1996X
Theory is presented to predict the mass transport limited current flowing at a hydrodynamic microstrip electrode located in a rectangular flow cell under steady state and chronoamperometric conditions. The influence of electrode size is quantified with the relative effects of lateral diffusion, normal diffusion, and axial convection identified. Theory is found to be in good agreement with experimental measurements conducted using the oxidation of tris(4-bromophenyl)amine at a platinum microstrip electrode. The microstrip electrode is seen to exhibit considerably enhanced mass transport characteristics over its macroelectrode equivalent and consequently is proposed as a new tool for the study of electrode reaction mechanisms.
Introduction The introduction of microelectrodes has effectively revolutionized electrochemical measurements over a relatively short period of time.1-5 Of their many advantages these tiny electrodes offer substantially enhanced rates of mass transport over electrodes of conventional dimensions. The microdisc electrode has been widely adopted by many workers6-8 in conjunction with rapid scan techniques as a method with which to study rapid chemical reactions coupled to electron transfer processes. These microdisc electrodes are operated under conditions of diffusion-controlled mass transport, and experimentally they have enabled first-order reactions with rate constants approaching 104 s-1 to be studied. More recently it has been proposed that the introduction of forced convection might extend the accessible time scale of reaction yet further.9,10 The aim of this paper is to examine the response of a new electrochemical device, the hydrodynamic microstrip electrode, where material is brought to the microelectrode via convection and diffusion. In particular a theoretical treatment is employed to calculate the flow rate dependence of the current flowing under steady state and potential step conditions. Experimental investigations are also reported and found to be in good agreement with those predicted using the computational model. Figure 1 shows a schematic of the cell employed with the microstrip electrode embedded in one wall of a rectangular duct. Solution is pumped through the cell under laminar flow conditions to facilitate ease of computation. It is demonstrated under fast flow and no flow conditions that the electrode response is in good agreement with that predicted by relevant analytical theory. Theory In this section the theoretical model of the mass transport to the hydrodynamic microstrip electrode is developed. In particular the transport-limited current flowing at the electrode due to the one-electron electrolysis of a reactant A is considered.
A ( e- f B It is assumed that sufficient supporting electrolyte is present so that migration may be neglected and that the length of the electrode (xe) is much larger than the width (w). In this situation X
Abstract published in AdVance ACS Abstracts, August 1, 1996.
S0022-3654(96)00167-0 CCC: $12.00
the diffusional transport to the electrode reduces to a twodimensional problem, and the convective diffusion equation describing the distribution of the reactant a is given by
∂[A] ∂2[A] ∂2[A] ∂[A] ) DA 2 + DA 2 - Vx ∂t ∂x ∂y ∂z
(1)
where x, y, and z are defined in Figure 1, DA is the diffusion coefficient of species A, and Vx is the solution velocity in the x direction. Under conditions of laminar flow and semi-infinite cell width, the solution velocity is parabolic (subject to a sufficient lead in length to enable the flow to become fully developed12).
(
Vx ) V0 1 -
)
(h - y)2 h2
(2)
where V0 is the velocity at the center of the channel and 2h is the height of the flow cell. The boundary conditions relevant to the problem are all x all x all x all x all x
y ) 2h y)0 y)0 all y all y
all z z < w/2 z > w/2 zf∞ z)0
∂[A]/∂y ) 0 [A] ) 0 ∂[A]/∂y ) 0 [A] ) [A]bulk ∂[A]/∂y ) 0
To solve eq 1, the ADI algorithm is adopted.13 This has been applied to a number of electrochemical problems and more recently to the simulation of flow cell techniques.14,15 This approach utilizes a finite difference grid as depicted in Figure 2 and covers x, y, and z directions using increments of ∆x, ∆y, and ∆z, respectively, so that x ) k∆x y ) j∆y z ) i∆z
k ) 1, K j ) 1, J z ) 1, I + I1
where ∆x ) xe/K where ∆y ) 2h/J where ∆z ) Nw/(I + I1)
where I and I1 are defined in Figure 2 and N is an integer that enables the number of boxes between the edge of the electrode (z ) I) and bulk solution to be varied according to the solution flow rate concentrating the numerical calculations within the region of concentration change. Since the problem is symmetrical about the center axis of the electrode (Figure 2), only half of the cell is used for calculation purposes. The computational approach proceeds by solving the time dependent form © 1996 American Chemical Society
14068 J. Phys. Chem., Vol. 100, No. 33, 1996
Aixill et al.
Figure 1. Schematic diagram of the microstrip flow cell electrode.
Figure 2. Finite difference grid covering the microstrip electrode.
Figure 4. Current transients for microband electrodes of length (a) 4 and (b) 40 µm using the analytical expression (O) found in ref 17 and the microstrip electrode of width (a) 4 and (b) 40 µm computed using the ADI algorithm (s).
Figure 3. Current transients calculated for a macroelectrode using (O) the analytical expression found in ref 16 and computed via the ADI algorithm (s).
of eq 1 starting from some initial boundary conditions. In this case the cell is filled uniformly with a reactant A.
t)0
[A] ) [A]bulk
all x, and y, and all z
The calculation then proceeds in a manner identical to that described previously for a disc arrangement.13 The simulation generates values of concentration tai,j,j, and these are then employed to evaluate the current flowing at time t for any particular flow rate using the following expression: t
I ) FDA[A]bulk
∆x∆z ∆y
K
I
t ai,1,k ∑ ∑ k)1 z)1
(3)
Theoretical Results and Discussion Initially the computational strategy was assessed by calculating the mass transport limited current flowing at a macroelectrode of dimensions xe ) 0.4 cm, 2h ) 0.04 mm, w ) 0.4 cm, and d ) 0.6 cm. The bulk concentration of A was 10-6 mol cm-3, and the diffusion coefficient was 10-5 cm2 s-1. Under these conditions it is established12 that the mass transport limited current varies as predicted by the Levich equation:
( )
ilevich ) 0.925nF[A]bulk(xeDA)2/3
Vf
h2d
1/3
(4)
Figure 5. Ratio of microstrip current to that calculated using the Levichn equation, for electrode widths 4, 11, 40, and 400 µm. The data were calculated for a cell height of 0.04 mm and with a diffusion coefficient of 10-5 cm2 s-1.
where F is the Faraday constant; Vf, the volume flow rate; and n, the number of electrons transferred. Simulations were performed for the above electrode geometry using the ADI approach with the following parameters: ∆t ) 0.1, I ) 20, I1 ) 100, J ) 100, and K ) 40 with volume flow rates 10-1, 10-2, and 10-3 cm3 s-1. Increasing the grid parameters employed was found to produce no significant difference (
