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Geometrical Insights of Transient Diffusion Layers ´ ngela Molina,*,† Joaquı´n Gonza´lez,† Francisco Martı´nez-Ortiz,† and Richard G. Compton*,‡ A Departamento de Quı´mica Fı´sica, UniVersidad de Murcia, Espinardo 30100, Murcia, Spain, and Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, United Kingdom ReceiVed: December 4, 2009; ReVised Manuscript ReceiVed: January 21, 2010
Very simple explicit expressions for the diffusion layer thicknesses (δl) under transient conditions have been deduced for different geometries of the diffusion field. From these expressions we have completely quantified and characterized the variation of δl in normal pulse voltammetry, staircase voltammetry, and linear sweep voltammetry at planar, cylindrical, and spherical electrodes of any size. The influence of the time duration, the potential, the sweep rate, and the electrode radius has been analyzed, and interesting limit expressions for δl corresponding to E . E0 and E , E0 have also been deduced. The conditions required to attain stationary state are given. 1. Introduction The Nernst diffusion layer thickness (δl) provides an estimation of the spatial variation of concentration profile of species participating in electrochemical reactions and also its dependence upon experimental control variables.1-5 δl provides valuable information related to the nature of the electrode and to the diffusion field. So, it can be compared to the diameter of a planar electrode or the maximal diffusion length in an electrochemical cell to discuss the validity conditions for the approximation of semi-infinite linear diffusion of electroactive species. The comparison between δl and the thickness of the convection-free domain, δconv, which in usual solvents and electrochemical working conditions ranges from a few tenths of micrometers to less than a few hundred micrometers, will also allow valuable information to be obtained on convection-free experiments.2 At electrodes of millimetric dimensions, δconv is necessarily negligible versus the dimension of the electrode, and therefore, only planar diffusion can be observed, with the electrochemical data being independent of the electrode shape. Conversely, if at least one dimension of the electrode is made smaller than δconv, diffusion may extend without significant interference of convection over distances that exceed this particular dimension of the electrode. Under these conditions, nonplanar diffusion may be observed and electrochemical data become affected by the electrode shape. White et al. first quantified the influence of natural convection in ultramicroelectrodes.6 Amatore et al. have theoretically studied and experimentally measured the conditions under which the diffusion domain is not affected by natural convection, and therefore, convection-free diffusion can be observed, both in conventional electrodes and ultramicroelectrodes.7-9 The knowledge of δl has proved to be as very important under a wide variety of electrochemical applications. For example, the time dependence of diffusion layer thickness can act as an electrochemical probe, which is very useful in the voltammetric sizing, shaping, positioning, and tracking of drops and small inert particles located in the proximity of an electrode or electrode array.10-14 Similarly, δl is a very valuable tool in the
design of arrays of microelectrodes when various electrochemical techniques are used.1,15-17 The diffusion layer thickness has been extensively studied under steady state1-3,18 and limiting current conditions1,2 but has received much less attention under potential-dependent transient conditions. Recently, Prasad and Sangaranarayanan have studied δl for a reversible charge transfer reaction at planar electrodes in linear sweep voltammetry (LSV).5 However, Montella’s group19 has clearly stated that the analysis carried out in ref 5 (based on the use of Pade´ approximants) fails to predict the potential dependence of diffusion layer thickness over the whole potential domain. Conversely, in ref 19, this problem is appropriately treated and the correct dependence of δl on potential was obtained under planar LSV conditions. In this paper we study the behavior of the diffusion layer thickness for a reversible electrochemical process under transient conditions by using some of the more usually used electrochemical techniques, such as, normal pulse voltammetry, staircase voltammetry (SCV), LSV, and cyclic voltammetry (CV). In this study we consider planar electrodes and also spherical and cylindrical electrodes of any radius. In all the cases, we analyze the variation of δl with the time duration, the potential, the sweep rate, and the electrode radius to deduce the necessary conditions for attaining a stationary I/E response. In all the geometries above considered, our study was carried out under conditions in which natural convection does not play any role in the mass transport;2,6-8 therefore, our results are of interest from both the theoretical and the experimental points of view. The new physical insights from this paper result from the derivation of very simple expressions for the diffusion layer thickness from which we have completely quantified and characterized the variation of δl in LSV for planar electrodes and also for spherical and cylindrical ones of any size. From these results, we have deduced interesting limiting expressions for δl and also examine if the steady state has been reached or not. 2. Diffusion Layer Thickness under Transient Conditions
* To whom correspondence should be addressed. † Universidad de Murcia. ‡ Oxford University.
The measured current I, obtained when a constant or time variable potential perturbation is applied to a simple charge
10.1021/jp9115172 2010 American Chemical Society Published on Web 02/15/2010
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transfer taking place in an homogeneous solution, can be defined from the equivalence between the gradient at the electrode surface and the difference between bulk (c*O) and surface (cO,surf) and concentrations divided by the Nernst diffusion layer thickness, δl, of the concentration profile of oxidized species,
(
∂cO(q, t) I ) nFAD ∂q
)
)
qsurf
∆cO δl
cO,surf )
∆cO )
- cO,surf
∆cO(E, t, c*, D, qsurf, ks, R) (∂cO(q, t)/∂q)qsurf
(6)
1 + eη
in all the cases considered (planar, spherical, and cylindrical), f(t) is given by
(2)
(3)
In the case of a reversible electrochemical process, the surface concentration cO,surf is only dependent on the applied potential,20-28 that is, ∆cO ) ∆cO(E), and is independent of the electrode geometry and time. Therefore, the determination of the diffusion layer thickness δl becomes much simpler in this case. In the following, we will consider only nerstian electrochemical charge transfer processes to know the behavior of the diffusion layer thickness in the more usual voltammetric techniques for electrodes and microelectrodes of different geometries. 3. Results and Discussion 3.1. Potential Step. In this case, we consider the application of a constant potential to an electrode of planar, spherical, or cylindrical geometry when both electroactive species are initially present in concentrations c*O and c*R and the diffusion coefficients of both species can be considered as equal. The area of the spherical electrode is 4πr02 with r0 being the electrode radius, whereas for the cylindrical electrode, the area is 2πr0l, (with l . r0). Under these conditions, the dependence of the measured current I with any applied potential E at any time is given by20,21
I ) ∆cO(E)f(t) nFAD
cO* - cR*eη
(1)
q is the coordinate on which the concentrations depends and qsurf is its value at any electrode surface, D is the diffusion coefficient of oxidized and reduced species, n is the number of electrons transferred, and A is the electrode area. cO,surf is, in general, a function of time, potential applied, diffusion coefficient of electroactive species, bulk concentrations of all species (c*), electrode geometry, and kinetic parameters (i.e., the heterogeneous rate constant and the charge transfer coefficient). When the expressions corresponding either to surface concentration of the electroactive species or to those of its surface gradient are known, it is possible to determine δl in any electrochemical technique in the following way:
δl )
(5)
1 + eη
∆cO )
with
cO*
(cO* + cR*)eη
(4)
Equation 4 shows that, for the three above electrode geometries, the I-E-t dependence can be expressed as the product of a potential function (∆c(E)) by a time function f(t) being20,21
{
1
planar1,3
√πDt 1 1 spherical20,21 + r √πDt 0 f(t) ) r0 Dt + 0.422 + 0.0675 log 2 ( r0 √πDt 2 Dt 0.0058 log 2 - 1.47 cylindrical29-31 r0
(
with
)
( () η)
()
)
(7)
nF(E - E0) RT
(8)
The upper sign in the expression of f(t) for a cylindrical electrode applies for positive values of (log (Dt/r02) - 1.47), whereas the lower one applies for negative values. In agreement with eqs 4 and 7, the diffusion layer thickness is, in the three electrode geometries, independent of the value of the potential applied, and it is given by (see eqs 1-3):
δl,planar ) √πDt 1
δl,spherical )
1
δl,cylindrical )
√πDt r0
(
+
√πDt
1 r0
()
+ 0.422 + 0.0675 log
( ()
0.0058 log
))
Dt - 1.47 r20
2 -1
Dt ( r20
(9)
In Figure 1 we have plotted the transient accurate concentration profiles cO(q,t)/c*O and the linear concentration profiles cO(q,t) ) (∆cO/δl)q + cO(qsurf,t), with q ) x for planar electrodes and q ) (r - r0) for spherical and cylindrical electrodes, for the application of different potential steps to planar (1a), cylindrical (1b), and spherical (1c) electrodes at a fixed time value. The profiles corresponding to planar and spherical electrodes have been calculated from eqs 11, 12, 22, and 23 of ref 27, whereas those of the cylindrical electrode have been calculated using numerical simulation in the way discussed in ref 27. From all these curves we can observe that the Nernst diffusion layers are independent of the potential in all the cases in spite of their having been obtained under transient conditions. This is in accordance with eqs 4 and 9, which show that the potential dependence is only present in the surface concentrations.
Geometrical Insights of Transient Diffusion Layers
Figure 1. Concentration profiles of a planar (1a), cylindrical (1b), and spherical (1c) electrode for the application of a potential pulse. Dashed lines correspond to their linear concentration profiles: t ) 0.1 s. Profiles corresponding to planar and spherical electrodes have been calculated from eqs 11, 12, 22, and 23 of ref 27, whereas those of the cylindrical electrode have been calculated by using a numerical simulation in the way discussed in ref 27. The electrode radius for the cylindrical and spherical electrodes is 10 µm. D ) 10-5 cm2 s-1. The values of the applied potential (E - E0; in mV) are on the curves.
Figure 2 shows the time dependence of the concentration profiles for the three electrode geometries mentioned above. As can be seen, at shorter times the behavior of the three electrodes is very similar, as corresponds to the prevalence of linear diffusion for small time values. However, at larger times, the perturbed spatial region is greater in planar diffusion (Figure 2a) than in the cylindrical one (Figure 2b), becoming minimum and constant ()r0) for spherical electrodes (Figure 2c). This
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Figure 2. Concentration profiles of a planar (2a), cylindrical (2b), and spherical (2c) electrode for the application of a potential pulse. Dashed lines correspond to their linear concentration profiles. (E - E0) ) -500 mV. The values of time (in s) are in the curves. Other conditions as in Figure 1.
behavior is in accordance with the greater ease in reaching a steady state at spherical electrodes versus planar and cylindrical ones. It can also be observed that the difference between the linear diffusion layer and the accurate diffusion layer (determined from the upper limit of these profiles) is greater for spherical electrodes than for cylindrical ones and both are greater than that of planar electrodes. In Figure 3b we can observe how for a spherical electrode of r0 ) 1 µm the Nernst diffusion layer is practically equal to the electrode radius r0 (i.e., a stationary current is obtained),
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Molina et al. the three electrode geometries discussed above, as has been reported in refs 20 and 21. Under these conditions, the diffusion layer thickness for any potential pulse p of the above sequence is given by
δlp(Ep, t) )
∆cOp
(10)
(∂cOp (q, t)/∂q)qsurf
p ∆cOp ) cO* - cO,surf (Ep)
(11)
where the superindex p refers to the pth potential pulse and ∆cOp is given by eq 5, according to the above discussion. The Ip - Ep - t response corresponding to the application of any potential Ep can be written in the following general way, by supposing cR* ) 019,20 p Ip 1 m-1 m (cO,surf (Em-1) - cO,surf (Em))f(tmp) ) nFAD √πD m)1
∑
(12) 0 with cO,surf (E0) ) c*O. Introducing eq 5 into eq 12 obtains
Ip cO* p Zmf(tmp) ) nFAD √πD m)1
∑
(13)
with Figure 3. Concentration profiles of cylindrical (3a) and spherical (3b) electrodes for the application of a potential pulse. Dashed lines correspond to their linear concentration profiles. (E - E0) ) -500 mV; r0 ) 1 µm. Other conditions as in Figure 2.
whereas for a cylindrical electrode of the same radius, it is heavily time-dependent (Figure 3a). We can also conclude from these two last figures that, as has been indicated in ref 2, the attainment of a steady state concentration profile requires much greater duration times than those needed for achieving a steady state current (i.e., independent of a time Nernst diffusion layer) because the latter only j t f 0 (see eqs 1-3), whereas an independent requires that r0/D j t f 0. of time profiles requires that (r - r0)/D 3.2. Staircase Potential and Linear Sweep Potential. Now we will consider the application of a staircase of potential steps E1, E2, ..., Ep, of pulse amplitude ∆E and duration τ (SCV), and also a linear ramp of potentials (which corresponds to the limit of pulse amplitude tending to zero for a given sweep rate, V ) ∆E/τ as corresponds to LSV). We demonstrated in previous papers that, under these conditions, the surface concentration of both species for any potential pulse can also be written in the way indicated in eq 5 because, for a reversible charge transfer, surface concentrations do not depend on the previous history of the process.20-28 However, the current (or the surface concentration gradient) is given in a more complex way than that of eq 4 (i.e., it cannot be expressed as the product of two functions dependent on the potential and time), although it presents an explicit form for
{
1 if m ) 1 1 + eη1 Zm ) 1 1 1 + eηm 1 + eηm-1
if
m>1
(14)
with ηm given by eq 8 with E ) Em. Equation 13 is a general expression for the current corresponding to SCV, LSV, and CV, applicable to planar, spherical, and cylindrical electrodes by substituting f(tmp) in eq 13 with
tmp ) (p - m + 1)τ
(15)
by the expressions given in eq 7 for the tree electrode geometries considered. Therefore, the Nernst diffusion layer thickness can be immediately calculated from eqs 3, 5 and 13-15,
√πDτ
p δl,planar )
p
(1 + e ) ηp
(16)
Zm
∑ √p - m + 1
m)1
1
p δl,spherical )
p
(1 + e ) ηp
1 + r0
Zm
∑ √p - m + 1
m)1
√πDτ
(17)
Geometrical Insights of Transient Diffusion Layers
(
p δl,cylindrical ) r0 (1 + eηp)
p r0 √πD Zm + ∑ r0 m)1 √πDtmp
( ) (( )
0.422 + 0.0675 log 0.0058 log
Dtmp r20
Dtmp r20
[ ) ])
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(
-1
- 1.47
2
(18)
In Figures 4a and 5a we have plotted the variation of δpl with the potential for different values of the electrode radius obtained in LSV with spherical (Figure 4a) and cylindrical (Figure 5a) electrodes, both including the limit case of a planar electrode (r0 f ∞). These curves have been calculated by using eqs 17 and 18 for a sweep rate V ) 0.1 V s-1. From these figures it can be observed that the diffusion layer thickness decreases with the electrode radius in both spherical and cylindrical electrodes, although this decrease is more important in the former, for which δpl , a constant value δpl ) r0, is reached when r0 e 5 × 10-4 cm (i.e., a stationary I-E response is obtained), whereas δpl continues to be potential-dependent for the cylindrical geometry under these conditions. From these curves it can also be observed that, for any electrode radius
Figure 5. Nernst diffusion layer thickness δpl obtained in linear sweep voltammetry (a) and cyclic voltammograms (4b) corresponding to a cylindrical electrode. These curves have been calculated from eqs 13 and 18 for ∆E ) 10-5 mV and V ) 100 mV s-1. The values of the electrode radii appear on the curves.
and for anodic potential values E . E0, δpl takes a constant value in both spherical and cylindrical geometries given by
(δlp)E>>E0 )
1 1 + r0
(19) a D
for spherical electrodes and
(δlp)E>>E0 )
1 0.5 + r0
(20) a D
for cylindrical ones, with
a) Figure 4. Nernst diffusion layer thickness δpl obtained in linear sweep voltammetry (a) and cyclic voltammograms (b) corresponding to a spherical electrode. These curves have been calculated from eqs 13 and 17 for ∆E ) 10-5 mV and V ) 100 mV s-1. The values of the electrode radii appear on the curves.
nFV RT
(21)
This anodic limit behavior of eqs 17 and 18 is observed for E - E0 g 95 mV with errors below 1%. Moreover, for cathodic potential values E , E0, δpl takes the following limiting expressions:
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(δlp)E
