J. Phys. Chem. B 2000, 104, 7993-8000
7993
Analytical Expressions for Feedback Currents at the Scanning Electrochemical Microscope Josep Galceran,*,† Joan Cecı´lia,‡ Encarnacio´ Companys,† Jose´ Salvador,† and Jaume Puy† Departament de Quı´mica and Departament de Matema` tica, UniVersitat de Lleida, RoVira Roure 177, 25198, Lleida, Spain ReceiVed: April 25, 2000; In Final Form: June 12, 2000
Steady-state currents of the scanning electrochemical microscope (SECM), which can be used to convert approach curves to topographical information, are considered in four cases, corresponding to the combination of positive and negative feedback on the substrate surface with either a finite or infinite glass insulator around the disc electrode. If the substrate is an insulator and both substrate and insulator extend to infinity (case I), steady-state cannot be attained. When the bulk concentrations can be assumed to be restored at a finite distance (the glass radius), both the negative (case II) and positive (case IV) feedback steady-state currents can be analytically found by using properties of the Bessel functions in the solution of a dual series equation. The strong influence of the glass radius on the current in case II can be taken into account in the exact computation and in simple approximate expressions. Positive feedback with an infinite domain (case III) can be solved with a dual integral equation. All solutions can be easily computed with modern software. Moreover, very simple and accurate approximate expressions (from the literature and new suggestions) are assessed and compared. A consistent way of linking the theoretical and experimental normalized currents for finite glass radii is suggested in terms of the values of the currents corresponding to an infinite distance between the electrode and the substrate. An expression to estimate the electrode radius from the insulator radius and the current with no feedback is suggested.
Introduction The scanning electrochemical microscope (SECM)1 has proven to be a powerful tool to study a wide variety of systems and phenomena.2,3 Topographical information can be gained from the use of approach curves (tip current vs. distance between the electrode and the surface known as substrate). Because the electrode is in many instances a disc microelectrode, steady state currents are usually recorded;4 thus, we make here no consideration of the transient current or the time needed to reach enough proximity to the steady state. The interpretation of steady-state experiments with disc electrodes can be based on numerical simulations (mainly with Finite Differences or Finite Elements),5-12 but analytical approaches, such as the dual integral method, are also feasible. Thus, exact and approximate expressions for the diffusion of an electroactive species through a multilayered medium13,14 are available to be used in a variety of situations such as coated electrodes15 or liquid interfaces.16,17 The (pseudo) first order EC′ scheme18,19 has also been tackled with the dual integral method.20,21 Here, we present 4 cases arising from considering positive feedback (conductive substrate) or negative feedback (insulating substrate) together with considering finite or infinite insulating glass (coplanar with the inlaid electrode). Diffusion of a redox couple is taken as the only transport phenomenon with no complications arising from chemical reaction or charge-transfer irreversibility. In all cases, we study a planar substrate surface parallel to that determined by the disc and its surrounding insulator. Other considerations about the geometry of the system have been analyzed elsewhere.11,12,22,23 * Corresponding author. Telephone: 34 973 70 28 26; e-mail: galceran@ quimica.UdL.es. † Departament de Quı´mica. ‡ Departament de Matema ` tica.
Figure 1. Scheme of case I: negative feedback mode with infinite substrate and infinite glass (insulators). The distance between the disk electrode acting as tip of the SECM (down, left, in dark grey) and the substrate (up, in light grey) is z1. Distances are normalized with respect to the electrode radius. This particular configuration cannot sustain steady state.
Both positive and negative feedback currents have been simulated with finite insulating glass.1 From the computed currents, simple analytical expressions were then fitted,24,25 but the dependence of the current on the insulator radius was lost. The aims of this article are to (1) describe the rigorous analytical solutions by using existing methodology from potential theory developments,26 (2) assess the influence of the finite insulator radius on the rigorous current, (3) assess the accuracy of approximate expressions found in the literature and of those newly derived, and (4) discuss the use of the normalized current. Mathematical Formulation Common to the 4 Cases The region where the diffusion of the species is considered can be seen as a cylinder whose axis is taken as the z coordinate. The origin of the cylindrical coordinate system (see Figure 1) is taken at the center of the disc electrode which, together with
10.1021/jp001564s CCC: $19.00 © 2000 American Chemical Society Published on Web 08/02/2000
7994 J. Phys. Chem. B, Vol. 104, No. 33, 2000
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the insulator around it, determines the bottom lid of the cylinder, while the substrate surface acts as the top lid. If cO denotes the concentration of an oxidized species O that is converted into the reduced form R at the disc electrode surface, steady state diffusion requires solving the Laplace equation
∂2cO
2 1 ∂cO ∂ cO + + 2 )0 r ∂r ∂r2 ∂z
(1)
where, for convenience, spatial variables r and z are normalized with respect to the electrode radius (rc). Now we prescribe the boundary conditions: 1) In this work we deal with limiting currents, so at the electrode surface the concentration of O vanishes:
z ) 0 r e 1 cO ) 0
(2)
The development done here is valid for any ratio of the diffusion coefficients and the extension to other currents under reversible electron transfer of the species is straightforward.2,21 2) The glass surface around the disc electrode is an insulator
z)0 r>1
( ) ∂cO ∂z
z)0
)0
(3)
3) Because of the axisymmetry:
r ) 0 ∀z
( ) ∂cO ∂r
r)0
)0
I π I ) ) / IT,∞ 4nFD c/ r 2cO O O c
∫01
( ) ∂cO ∂z
z)0
r dr
(5)
where n is the number of electrons, F is the Faraday constant, and DO is the diffusion coefficient of species O. Notice that the normalization factor is
IT,∞ ) 4nFDOc/Orc
(6)
corresponding to the current27 without substrate surface (z1 ) ∞) and implying an infinite insulator around the inlaid disk. Case I: Negative Feedback Mode Assuming Infinite Insulator Thickness A characteristic of case I is the negative feedback or insulating nature of the substrate surface, which is placed at a normalized distance z1 (also labeled L in some of the previous literature2) from the electrode-glass surface,
z ) z1 ∀r
( ) ∂cO ∂z
z)z1
)0
Case II: Negative Feedback Mode Assuming Finite Insulator Thickness Instead of considering that both substrate and glass extend up to r ) ∞ as in case I (see boundary condition eq 8), a wellknown approach,1,2,28-30 which we label case II, postulates that bulk conditions are gained at a finite (nondimensional) glass radius g (see Figure 2):
∀z r ) g cO ) c/O
(8)
where the superscript asterisk indicates the bulk conditions. It has been demonstrated elsewhere14 that steady-state cannot be attained when a disc probe embedded in an infinite insulator surface is placed at a finite distance of another infinite parallel insulating surface. This is an analytical result that indicates the
(9)
It is obvious that this boundary condition represents a first approximation that can be improved further,5,11,12,22,23 but, as shown below, the existence of an analytical solution provides some useful insight on the problem. The approximate boundary condition eq 9 is more realistic the higher g (according to the physical picture that the z-gradients of concentration, for r > 1, decrease when r increases, as the distorting impact of the electrode vanishes when we move away from the electrode). Thus, along this work, we restrict ourselves to g > 5 for the presented values of the current. Moreover, other transport phenomena apart from diffusion can be relevant in the restoration of bulk conditions in the proximity of the glass insulator edge (r ) g), especially if g is large and the corresponding time to reach a situation close to steady state is long. Thus, in the case of such forced restoration, the boundary condition (9) is also reasonable. Analytical Solution of Case II. It is shown in appendix A that the problem can be solved through separation of variables, which leads to a “dual series equation”.31 The normalized current (see appendix A) can be computed as
φg )
(7)
The other specific boundary condition of case I is the infinite domain of diffusion:
z < z1 r f ∞ cO ) c/O
need of taking into account other configurations of the system when dealing with insulating substrates.
(4)
For all the cases studied here, the analyzed response is the nondimensional current:2,14
φ≡
Figure 2. Scheme of case II: negative feedback mode with bulk conditions restored at a radial coordinate r equal to the insulator radius g. Dark grey indicates that the concentration of the electroactive species on the surface is prescribed. Light grey indicates that there is no flux across the corresponding surface.
x
πg2 b 22 0
(10)
where we have added a superscript minus to the normalized current symbol φ to indicate the negative feedback and a subscript g to indicate the finite domain. The coefficient b0 can be obtained from the linear system
{
∞
∑ Bm,0bm ) m)0 ∞
x
∑ Bm,sbm ) 0
m)0
2
π
(11) s ) 1,2,3..
Analytical Expressions for Feedback Currents
J. Phys. Chem. B, Vol. 104, No. 33, 2000 7995
being ∞
Bm,s )
∑
(coth(λnz1) - 1)J2m+1/2(λn)J2s+1/2(λn)
+
g2
2 λ2nJ21(λng) δms (-1)m+s2 ∞K0(t) t t I2m+1/2 I2s+1/2 dt (12) 0 4s + 1 π g g tI0(t)
[
() () ]
n)1
∫
where Jν are the Bessel functions of first kind and order ν, I0 and K0 are the zero order modified Bessel functions of first and second kind, respectively; λn are the ordered solutions of J0(λg) ) 0, and δms is the Kronecker delta. Approximate Expressions for Case II. A classical approximate expression for φg is
φg = 1 (13) 1.5385 0.15 + + 0.58 e-1.14/z1 + 0.0908e(z1-6.3)/1.017z1 z1
Figure 3. Continuous lines represent rigorously computed approach curves (nondimensional current φg using eqs 10 and 11 vs. electrodesubstrate distance z1) in negative feedback mode for different insulator radii g. The horizontal dashed line indicates the nondimensional current φg ) 0.6, which can be obtained with different electrode-substrate distances (intercept b for g ) 10; intercept 9 for g ) 100). Markers stand for the 2-domains approximation (36) with g′ ) 10: 4 for g ) 100, ) for g ) 1000 and × for g ) 10000.
which was suggested by Mirkin, Fan, and Bard24 from the fitting of the numerical values obtained for g ) 10 by using the finite element method.1 An accuracy of 0.5% in the range 0.05 < z1 < 20 was claimed for eq 13. The main drawback of this formula arises from the neglecting of the g-dependence of the current. This can be seen as reasonable in some systems because of the experimental difficulties in the actual assessment of g.2 Alternatively, using the analytical development given above, we can suggest a “zeroth order” approximate expression by drastically truncating the linear system eq 11 by using just B0,0 (i.e., m ) s ) 0) and an approximate value (given by eq A-18) for its integral term:
1
φg = nmax
4 πg
2
∑ n)1
(coth(λnz1) - 1)sin2(λn) λ3nJ21(gλn)
(14) +1-
0.5543 g
which is easily and accurately computed in a spreadsheet using32
λn ≈
[
1π 31 1 + (4n - 1) + g 4 2π(4n - 1) 6π3(4n - 1)3 3779 5 15π (4n - 1)5
]
λ1 ≈
2.40483 g
n>1 (15)
and nmax large enough (e.g., between 20 and 40). Results and Discussion on Case II. The current can be obtained analytically by means of eqs 10 and 11. Despite its formal complexity, the actual calculation can be straightforwardly performed with general-purpose computational software available in most laboratories. A version of the Mathematica33 code (for this and other cases presented in this article) can be obtained as Supporting Information, from the web page www.udl.es/usuaris/q4088428/, or from the authors upon request. Knowledge of the exact solution allows for the computation of the exact approach curve, the assessment of the effect of the parameter g, and the assessment of the accuracy of the approximate expressions.
Figure 4. Effect of the glass insulator radius g on the nondimensional current φg (rigorously computed using eqs 10 and 11) in negative feedback mode for different electrode-substrate distance z1. Dashed line stands for φg(g,∞) computed with eq 17.
We begin with an analysis of the effect of the electrodesubstrate distance z1 on the normalized current (using the rigorous expressions 10 and 11). In Figure 3 we have plotted 5 approach curves to an insulating substrate with 5 different g values. In all cases φg increases when the electrode moves away from the substrate (z1 increases), as expected from the increasing length of the source line at r ) g, where bulk conditions are fixed and from which the electroactive material is provided. One could alternatively use the interpretation that, while z1 increases, the shielding (or blocking effect generated by the insulator substrate) decreases, and thus the current is higher. Now, we turn our attention toward the impact of g on the normalized current. The dependence of φg on g can be seen in Figure 3. Comparing the 5 lines (each one with a different g), one concludes that the current decreases when g increases. This is consistent with the fact that, when g tends to infinity, we recover case I, which prescribes a null current. If we compute the exact values of φg for a fixed z1 (see any line in Figure 4) and change only g, the nonnegligible effect of g is also evidenced. It is seen in Figures 3 and 4 that a decrease of the insulator radius g enhances the dimensionless current. This enhancing effect can be physically explained from the fact that the source line (where Dirichlet bulk conditions are fixed) is closer to the electrode. We conclude that the dimensionless current φg is strongly dependent on g and we write φg (g,z1) to
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TABLE 1: Percentage of Error in the Determination of z1 (Computed As 100×(z1,apparent - z1,true)/z1,true) due to the Ignoring of the Dependence of g in the Normalized Current Obtained in Case II (Negative Feedback and Finite Insulator Thickness)a φg ) 0.6 φg ) 0.8
g)5
g ) 100
g ) 1000
g ) 10000
49.00% 59.64%
-56.27% -63.44%
-73.52% -79.83%
-81.41% -86.53%
a Two fixed φg values (0.6 and 0.8) are interpreted with the rigorous expressions 10 and 11 but erroneously taking g ) 10 (instead of the true value of g, which is detailed in the heading of each column).
Figure 5. Errors of the normalized current φg (negative feedback and bulk values restored at the insulator radius g) arising from the approximate expressions when the insulator radius g is 10. Marker × stands for Mirkin et al.’s expression 13 and marker 9 stands for the zeroth order approximation (eq 14).
explicitly take into account both parameters. On the other hand, as explained above, the dimensionless current is lower for lower values of z1. The enhancing effect (due to the finite g value) can overcome the shielding effect (due to the insulating substrate) and produce φg > 1 (which is possible due to the choice of the normalizing current 4nFDOc/Orc corresponding to g ) z1 ) ∞), as clearly seen in Figure 3 for g ) 5. We now illustrate that ignoring this dependence of the current on g can lead to errors in the determination of z1. A system with z1 ) 2.66 and g ) 100 produces φg ) 0.6 (see marker 9 in Figure 3). If one interprets the measured value φg ) 0.6 with the rigorous expressions 10 and 11 but using g ) 10, instead of its true value g ) 100 one would erroneously determine z1 ) 1.16 and not its real value z1 ) 2.66, thus underestimating z1 by 56.27%. This case can be seen in the horizontal dashed line in Figure 3 corresponding to the measure φg ) 0.6: the true distance z1 is the abscissa of the solid-square marker while the apparent z1 is the abscissa of the black bullet. Other cases are also collected in Table 1, where we have selected g ) 10 as the prescribed fixed insulator glass radius, because this is the value taken by Mirkin et al. to fit eq 13, while the true value of g was 5, 100, 1000 or 10 000. The percentage of error in the recovered z1 (see Table 1) ranges from +49.00% to -86.53%. As expected, the (absolute) percentage of error grows with the difference between the real g value and the erroneous g value. Note that the percentage of error increases with φg , as can also be predicted from Figure 3, where the approach curves increase their z1 separation when φg increases. We now focus on the accuracy of the approximate expressions. When g ) 10, Mirkin et al.’s expression 13 for the normalized current is practically more accurate than 1.5%, although it never is exact (see marker × in Figure 5) and for z1 f ∞ yields 2.8% of absolute error. For the same reference value,
g ) 10, the zeroth order approximation eq 14 is accurate to 0.6% for z1 > 0.5 and is accurate below 0.02% from 1.5 onwards (see Figure 5). If we compute now the errors of Mirkin et al., expression 13 with respect to the exact values for g ) 20 (i.e., we take also into account the problem of neglecting the g-dependence), we find that they are higher, as expected. For z1 > 0.5 and in the range 5 < g < 50, we have checked that the zeroth order approximation eq 14 is accurate to 0.8%. Let us work out an example of how the inaccuracy of eq (13) (mainly due to the neglecting of the g dependence) has effects on the measured distance. If the electrode radius is rc ) 5 µm, the actual g ) 20 and z1 ) 1 (so the dimensional distance between the electrode and the insulator is 5 µm), approximation eq 13 would yield a z1-value of 0.76, and hence it would represent a 24% error in the determination of the electrodesubstrate distance (3.8 µm). As pointed out above, the defined dimensionless current corresponds to dividing the current measured with the (finite sheath) electrode by the theoretical current IT,∞ ) 4nFDOc/Orc see ref 27 that arises for an infinite insulator with no substrate surface present (see eq 5). However, the practical experimental procedure will usually consist in dividing the measured current with the substrate surface present by the current obtained with the substrate surface absent or sufficiently far away from the tip, but in both cases the actual glass radius is finite. Then, only the shielding effect is practically encountered, so I(with substrate) < I(without substrate). This fact suggests that values of φg for z1 f ∞ allow a coherent estimation of the experimentally normalized current:
I(with substrate) φ4nFDOc/Or I(with substrate) g (g,z1) ) ) I(without substrate) I(without substrate) φg(g,∞) 4nFDOc/Or
(16)
The fact that the substrate is at an infinite distance renders the superscript minus in the denominator φ unnecessary and (as explained later when dealing with case IV) misleading, so we drop it. Some selected values for z1 ) ∞, which can be used for the experimental normalization purpose, are φg(5,∞) ) 1.126, φg(10,∞) ) 1.059, and φg(20,∞) ) 1.029. Instead of a rigorous computation of φg(g,∞) using eqs 10 and 11, one can use the zeroth order approximation eq 14, which for z1 ) ∞ collapses into
φg(g,∞) ≈
1 0.5543 1g
(17)
which is more accurate than 0.2% for g > 5. As seen in Figure 4, lines representing the dependence of φg on g exhibit a fast convergence (for z1 > 8) to φg(g,∞) (dashed line). Another application of eq 17 is the estimation of the electrode radius. Instead of the direct application of eq 6 to estimate rc from the measure of the current I (with the electrode away from any disturbance), a simple reorganization of eq 17 could also be useful:
rc ≈
1 / 0.5543 4nFDOcO + Rg I
where Rg ) rcg is the dimensional radius of the glass.
(18)
Analytical Expressions for Feedback Currents
J. Phys. Chem. B, Vol. 104, No. 33, 2000 7997
For extremely large values of g (say g > 100) and relatively low values of z1, one can consider that at a certain auxiliary distance r ) g′ the gradient of concentration in the z-direction is negligible (i.e., a common value c′O can be taken for any z). Thus, the original diffusion region consist of two domains: a 2-dimensional domain from r ) 0 to r ) g′ (where the solution / eq 11 can be used to find its flux as φg (g′,z1)c′O/cO) and a 1-dimensional region from r ) g′ to r ) g (where just radial diffusion applies and the flux of which is readily found to be πz1c′O/(2c/Oln(g/g′))). By matching the fluxes of both domains, one obtains
φg (g,z1) ≈
πz1φg (g′,z1) g 2φ+ πz1 g (g′,z1)ln g′
( )
Figure 6. Scheme of case III: the positive feedback mode at the substrate surface with bulk conditions restored at infinity. Color code as in Figure 2.
The response function is
(19) φ+ ∞ )
xπ2a
(24)
0
which allows the easy computation of the flux for those extremes cases when g > g′ . 1 using φg (g′,z1), which is the nondimensional current for g ) g′. This approximation has been applied in Figure 3 to compute φg (g,z1) for g ) 100 (markers ∆), g ) 1000 (markers )) and g ) 10 000 (markers ×) using the (rigorous) values obtained for g′ ) 10. It can be seen that the approximate current values perfectly agree with the rigorously computed ones (continuous lines). So the 2-domain approximation eq 19 could be a good choice for extremely large values of g if sufficiently accurate values of the flux for an intermediate g′ are available. For instance, any approximate expression (such as eq 13 yielding φg (g′,z1) for g′ ) 10 with reasonable accuracy) can be used to obtain φg (g,z1). As expected from the physical hypotheses on its derivation, eq 19 becomes less accurate when z1 increases, although this inaccuracy is practically imperceptible in Figure 3.
The subscript ∞ indicates that in this case the insulator around the electrode and the substrate surface extend up to infinity (i.e., there is no finite glass radius g to consider). The superscript + indicates a positive feedback. Approximate Expressions for Case III. Alternatively to the exact solution given by eqs 22-24, we may resort to analytically derived approximate expressions (see ref 14 for a discussion of them) such as the one found by Hale34 for short distances (z1 < 1):
Case III: Positive Feedback Mode Assuming Infinite Insulator Thickness
φ+ ∞ =1+
φ+ ∞ =
(20) φ+ ∞ =
and the assumption of an infinite insulator glass:
r f ∞ cR ) 0
{
∞
an +
being
Lm,n ≡ (4n + 1)
x
∑ Lm,nam ) 0
2
π
(22) n>0
m)0
tanh(λz1) - 1 J2m+1/2(λ)J2n+1/2(λ)dλ (23) λ
∫0∞
1 1 + L0,0
) 1
(21)
This problem is equivalent to that of a one-layer membrane electrode with a gas sample.14 The differential equation problem can be reformulated as a dual integral equation and solved through the linear system
∑ Lm,0am ) m)0
( ) ( )
2 ln2 2 2 ln2 3 ζ(3) 2 ln2 + + =1+ πz1 πz1 πz1 4πz31 0.4413 0.1947 0.0097 + (26) z1 z21 z31
We have recently suggested14 a “zeroth order approximation”
z ) z1 cR ) 0
∞
(25)
or the expansion for large distances (z1 > 1) found by Tranter:35
Apart from the general boundary condition eqs 2-4, as specific characteristics of case III (see Figure 6), we have a conducting substrate:
a0 +
π 0.7854 + ln 2 ≈ + 0.6931 4z1 z1
1+
2
∞
[ () ( 1
∑(-1)i 2arctan iz
π i)1
1
- iz1 ln 1 +
)]
1
(27)
(iz1)2
Following a quite different approach, Mirkin, Fan, and Bard24 fitted the numerical values for 0.02 < z1 < 20 claiming 0.7% error for their expression
φ+ ∞ =
0.78377 + 0.3315e-1.0672/z1 + 0.68 z1
(28)
One drawback of this formula is its prediction of a dimensionless current greater than 1 for an infinite z1.2 It is easy to identify the first and third terms in eq 28 with the first and second terms in eq 25. Introducing small corrections to expression 28, one can improve the accuracy of the approximation and succeed in
{
7998 J. Phys. Chem. B, Vol. 104, No. 33, 2000
Galceran et al. ∞
∑ Bm,0bm ) m)0 ∞
∑ Bm,nbm ) 0
x
2
π
(31) n>0
m)0
being ∞
Bm,s ) ∞
∑ n)1
∑ n)1
tanh(λnz1)J2m+1/2(λn)J2s+1/2(λn) λ2nJ21(λng)
(tanh(λnz1) - 1)J2m+1/2(λn)J2s+1/2(λn)
+
Figure 7. Errors in using some approximate expressions for the normalized current assuming positive feedback and infinite parallel planes for substrate and insulator. Markers: o for Hale expression (eq 25), 2 for Tranter expression (26), 9 for the zeroth order approximation (eq 27), × for Mirkin et al. expression (eq 28) and + for the new approximation (eq 29).
)
achieving a nondimensional current tending to 1 for z1 f ∞. We suggest
The normalized current, analogously to eq 10, can be computed with:
φ+ ∞ =
π + (1 - ln 2)e-1.11/(0.11+z1) + ln 2 4z1
δms
2 4s + 1
-
Case IV: Positive Feedback Mode Assuming Finite Insulator Thickness In this case, the common boundary condition eqs 2-4 are complemented by the bulk conditions restored at a finite g (eq 9) and by the positive feedback at the substrate surface
(30)
Following an analogous approach to that expounded in appendix A, one finds that the problem reduces to solve
(-1)m+s2
() () ]
K0(t) t t I2m+1/2 I2s+1/2 dt (32) g g tI0(t)
∫0∞
π
φ+ g )-
(29)
Results and Discussion of Case III. The rigorous value of the current can be obtained analytically by means of eqs 2224. Its computation is straightforward with software usually available to most laboratories. We can assess the accuracy of the different approximations for case III. The error of Hale’s approximate expression 25 is less than -0.5% for z1 < 0.18 (see marker o in Figure 7), but its inaccuracy increases rapidly with increasing z1. The error of Tranter’s approximate expression 26 is large for low z1 but falls under 0.5% for z1 > 1.54 (see marker 2 in Figure 7). The zeroth order approximation eq 27 underestimates the nondimensional current by less than 1.21% for z > 0.5 and less than 0.1% for z > 1 (see marker 9 in Figure 7). Mirkin et al.’s approximate expression 28 switches from underestimating the current to overestimating it around z1 ) 2 (see marker ×). Expression 28 is accurate to an absolute 0.5% in both regions z1 < 0.10 and 1.38 < z1 < 3.06 but reaches almost -1.89% error around z1 ) 0.5 and overcomes +1% when z1 > 10. Bearing in mind practical purposes, the previous results indicate that Mirkin et al.’s fitted expression is a reasonable solution for the problem, as the error always remains below 1.9% for any z1 value. A slight correction of Mirkin et al.’s expression yields a new approximation (29) which bounds the error to 0.75% everywhere (and over 0.5% just in the interval 0.21 < z1 < 0.75) and could be taken as the de facto solution (see marker + in Figure 7).
z ) z1 cR ) 0
[
g2
λ2nJ21(λng)
x
π g2 b 2 2 0
(33)
Approximate Expressions for Case IV. Mirkin et al.’s expression 28 can also be considered as an approximate expression for case IV, where, even though g ) 10 was taken in the actual fitting of the expression,2 the parameter g is not included in the final expression (in a fashion similar to the way g has also been disregarded in expression 13 for case II). From the exact solution eqs 31-33, we can suggest a zeroth order approximation:
1
φ+ g = 4
nmax(tanh(λ z ) n 1
∑
πg2 n)1
- 1)sin2(λn)
λ3nJ21(gλn)
(34) +1-
0.5543 g
where the integral term in eq 32 for B0,0 has also been approximate with eq A-18, as in expression 14. Expression 15 allows for the computation of eq 34 with a spreadsheet. Results and Discussion of Case IV. The solution of the problem, including the response function current, can be obtained exactly by means of eqs 31-33. + The normalized current φ+ g depends on g, i.e., φg ) + + φg (g,z1), but this dependence is mild.1 For instance: φg (5,5) + ) 1.133; φ+ g (10,5) ) 1.098; φg (20,5) ) 1.096. When z1 ) 2 and g > 5, one can consider the sheath as infinite since the difference between the rigorously computed response functions for case IV and case III is less than 0.5% so that both cases become practically indistinguishable. From a physical viewpoint, this corresponds to a negligible contribution of the radial flux (source line at r ) g) in front of the contribution from the feedback flux (source line at z ) z1). When z1 ) 5, the difference between the fluxes for cases III and IV is less than 0.5% for g > 8.1. The dependence of the normalized current on g and z1 can also be seen in Figure 8, where for a high enough value of g (see continuous line for g ) 100), case III (marker b) coincides perfectly with case IV for the z1 values considered. We now turn our attention to the assessment of the accuracy of the approximate expressions for case IV. When g ) 10, Mirkin et al.’s expression 28 is more accurate than 1.9% for
Analytical Expressions for Feedback Currents
J. Phys. Chem. B, Vol. 104, No. 33, 2000 7999 la Generalitat de Catalunya. Finally, we acknowledge financial support from the Ministerio de Educacio´n y Cultura (FPI grant to E. C.). Appendix A: Analytical Solution of the Steady State with Negative Feedback and Finite Domain for the Inlaid Microdisc Electrode Let us introduce the dimensionless concentration variable
θ(r,z) ≡ Figure 8. Rigorously computed approach curves (nondimensional current φ+ g using eqs 31-33 vs. electrode-substrate distance z1) in positive feedback mode for different glass radii g. Markers: b stands for the normalized current assuming positive feedback and infinite parallel planes for substrate and insulator (case III, eqs 22-24), and marker × stands for Mirkin et al. expression (eq 28).
c/0 - c0(r,z)
(A-1)
c/0
The diffusion eq 1 reads
∂2θ 1 ∂θ ∂2θ + + )0 ∂r2 r ∂r ∂z2
(A-2)
with boundary condition eqs 2, 3, 4, 7 and 9 being now
z)0 re1 θ)1 z)0 r>1
(∂θ∂z )
z)0
(∂θ∂r )
r ) 0 0 e z e z1 z ) z1 ∀r
Figure 9. Errors in using the approximate expressions for the approaching curve assuming negative feedback and bulk values restored at the insulator radius g ) 10. Markers: × stands for Mirkin et al. expression (eq 28) and 9 stands for the zeroth order approximation eq 34.
low and intermediate z1 (see marker × in Figure 9), while for z1 f ∞ it yields 4.4% error. Still, for the reference value g ) 10, the zeroth order approximation 34 is accurate to 1.2% for z1 > 0.5 and to less than 0.01% for z1 > 2 (see marker 9 in Figure 9). The error of the zeroth order approximation 34 decreases both with increasing z1 and with increasing g. In the less favorable case considered here (g ) 5), the error corresponding to z1 around 0.5 is 1.4% and falls quickly with increasing z1 (e.g., 0.08% error for z1 ) 5). Values of the nondimensional current for z1 f ∞ cannot be influenced by the conducting or insulator nature of the substrate surface. Therefore, we refer to its common value as φg(g,∞), without any superscript. φg(g,∞) can be used in the determination of the experimental normalized current, as in case II, with the parallel expression to eq 16:
φ+ g (g,z1)
I(with substrate) ) I(without substrate) φg(g,∞)
(35)
Again, expression 17 can serve as an excellent approximation for φg(g,∞). Acknowledgment. The authors gratefully acknowledge support of this research by the Spanish Ministry of Education and Science (DGICYT: Project PB96-0379), by the Ajuntament de Lleida, and by the Comissionat d’Universitats i Recerca de
(∂θ∂z )
(A-3)
)0
r)0
)0
(A-4) (A-5)
)0
(A-6)
r ) g 0 e z e z1 θ ) 0
(A-7)
z)z1
Through separation of variables and taking into account the boundary condition eqs A-5, A-6 and A-7, one can write ∞
θ)
∑ n)1
an coth(λnz1) [cosh(λnz) - tanh(λnz1)sinh(λnz)]J0(λnr) λn (A-8)
with λn being the roots J0(λg) ) 0. Imposition of the remaining two boundary condition eqs A-3 and A-4 leads to the formulation of the problem in terms of a “dual series equation”:26,31 ∞
∑ n)1
an coth(λnz1) J0(λnr) ) 1 r e 1 λn
(A-9)
∞
∑anJ0(λnr) ) 0
1
