2170
J. Phys. Chem. 1996, 100, 2170-2177
Steady-State Currents at Sphere-Cap Microelectrodes and Electrodes of Related Geometry L. C. Roland Alfred and Keith B. Oldham* Departments of Physics and Chemistry, Trent UniVersity, Peterborough, Ontario K9J 7B8, Canada ReceiVed: May 26, 1995X
Mercury microelectrodes, fabricated by electrodeposition on an inlaid disk, adopt the shape of a sphere cap. This article derives the steady-state diffusion-limited voltammetric current at such electrodes and correlates the current with the radius, area, and volume of the sphere cap. Also predicted are the steady-state currents at an entire sphere emerging from an insulating plane.
1. Geometry of the Sphere Cap By “sphere cap” we mean either of the two bodies formed when a solid sphere (of radius a) is cut by an infinite plane. Generally, one of these sphere caps will be smaller, and one larger, than a hemisphere; we use the terms “subhemisphere” and “superhemisphere” to distinguish these two varieties of cap. The sphere cap is bounded by two surfaces: (i) a portion of the original spherical surface with a radius of curvature a and area A; and (ii) a flat circular disk of radius λ (λ e a) and area πλ2 (πλ2 < A), which we refer to as the “basal disk”. Because of the method, described below, usually used to prepare sphere-cap microelectrodes, important questions are how the area A and radius a of the spherically curved portion of the sphere cap are related to the volume V of the sphere cap and its basal radius λ. The relationships are quite complicated:1
A ) πλ2[{1 + V2 + xV4 + 2V2}1/3 + {1 + V2 - xV4 + 2V2}1/3 - 1] (1.1) where V is the volume of the sphere cap normalized by division by the volume Vhs of a hemisphere and multiplication by a convenient numerical factor:
V)
x18V x8V ) Vhs πλ3
(1.2)
Knowing A, the radius of curvature is then calculable from
a)
A 2xπA - π2λ2
(1.3)
These formulas apply equally to subhemispherical, hemispherical, and superhemispherical sphere caps. Note that specifying a and λ does not uniquely identify the sphere cap. For each a in the range λ < a < ∞ there are two sphere caps, one subhemisphere and one superhemisphere, that share the same a value. These two caps are, of course, complementary in the sense that they may be united into a single whole sphere. 2. Sphere-Cap Microelectrodes The standard method1-17 of preparing a mercury microelectrode is to cathodically electrodeposit liquid mercury onto a circular metal (or sometimes carbon) disk which is inlaid into an insulating planar support. Usually the mercury wets the disk, and the large surface energy of the solution/mercury interface X
Abstract published in AdVance ACS Abstracts, January 1, 1996.
0022-3654/96/20100-2170$12.00/0
Figure 1. Geometry of (a) a subhemispherical and (b) a superhemispherical sphere-cap electrode.
then ensures that this interface adopts a minimal area, giving it the shape of a sphere cap with the basal plane coincident with the original inlaid disk.18 The electrode area and its radius of curvature depend on the mercury volume, which is calculable coulometrically, in accordance with (1.1)-(1.3). Let us erect a cylindrical coordinate system with its origin at the center of the electrode’s basal disk, as in Figure 1. This disk and the surface of the surrounding insulator occupy the z ) 0 plane. The r-coordinate is directed radially from an r ) 0 axis perpendicular to this plane and through the origin. Then the surface of the electrode is given by
r2 + z2 - 2zxa2 - λ2 ) λ2
zg0
(2.1)
where the upper sign applies to a superhemispherical and the lower to a subhemispherical sphere cap. Later in this article we shall encounter the supplementary coordinate R. This is defined by
R ) xr2 + z2
(2.2)
and represents the distance from the origin to the point (r,z) in the vicinity of the sphere cap, as illustrated in Figure 1. These diagrams also identify the angle Ξ that will play an important role in what follows. Our interest is in the steady-state current that ultimately flows when a sphere-cap microelectrode is immersed in a solution containing a bulk concentration cb of some electroactive species, following the imposition of a totally concentration polarizing potential to the electrode. We shall assume that transport is by diffusion only (with a diffusivity D) and that the electrode reaction is an n-electron oxidation. Thus, we seek to solve the steady-state version of Fick’s second law:
D∇2c ) ∂c/∂t ) 0
(2.3)
where c denotes the local concentration of electroactive species, © 1996 American Chemical Society
Steady-State Currents at Sphere-Cap Microelectrodes
J. Phys. Chem., Vol. 100, No. 6, 1996 2171
subject to the boundary conditions
∂c/∂z ) 0 c f cb c f cb
at z ) 0 and all r > λ
(2.4)
as r f ∞ and all z g 0
(2.5)
as z f ∞ and all r
(2.6)
cs ) 0
(2.7)
and
The current density i at each point on the surface of the spherecap electrode could then be found by use of Faraday’s and Fick’s first laws:
i ) -nFjs ) nFD(∂c/∂N)s
(2.8)
In the last two equations a superscript “s” is used to indicate conditions on the electrode surface, i.e. when r and z are interrelated according to (2.1). Other new symbols in (2.8) are F, Faraday’s constant; j, the local flux density of electroactive species; and N, the outward directed normal to the electrode surface. The final step in the evaluation of the current I would be the integration of i over the entire electrode surface. However, it is inconvenient to attempt this exercise in cylindrical coordinates. Difficulties are introduced because the electrode surface is delineated by the complicated equation (2.1), involving two spatial coordinates. This complication does not arise with an alternative orthogonal coordinate system, which we now describe.
Figure 2. Toroidal coordinate system.
radius a of curvature by
Ξ)
ξ ) arccot
{
∇ 2c )
1 ∂ ∂c csch{η} ∂ ∂c h + h sinh{η} h ∂η ∂η h3 ∂ξ ∂ξ
}
(3.4)
where h, defined by
λ cosh{η} - cos{ξ}
h)
r2 + z2 - λ2 2λz
(3.3)
where, as before, the upper sign applies to a superhemispherical sphere cap and the lower sign to a subhemispherical sphere cap. The Laplacian operator in toroidal coordinates, when there is axial symmetry, is20,21
3. The Toroidal Coordinate System In cases where, as here, there is complete rotational symmetry about an axis through the origin, there are two nontrivial coordinates, ξ and η, in the toroidal system.19 These are related by
{}
λ π - arccos 2 a
(3.5)
is the scale factor. The supplementary coordinate R is related to the toroidal coordinates by
R ) λxW ( xW2 - 1
(3.1)
(3.6)
where
and
{
}
r2 + z2 + λ2 η ) arcoth 2λr
W)
(3.2)
to the cylindrical coordinates. Figure 2 portrays the toroidal coordinate system. In this system, the z ) 0 plane is replaced by two coplanar regions: a disk ξ ) π; and a surrounding perforated infinite plane, ξ ) 0. Every other surface of constant ξ, 0 < ξ < π, is a portion of a sphere of radius λ csc{ξ} centered on the symmetry axis at a distance λ cot{ξ} from the origin. As Figure 2 illustrates, all these spheres intersect on the hoop r ) λ, z ) 0. This hoop corresponds to η ) ∞, whereas the symmetry axis z ) 0 corresponds to η ) 0. Each intermediate value of η, 0 < η < ∞, corresponds to a toroid centered at the origin with minor and major radii of λ csch{η} and λ coth{η}, respectively. The toroidal coordinate system is felicitous for our present problem because the basal disk of a sphere cap can be associated with the ξ ) π disk. The electrode surface then corresponds to some constant value of ξ in the range 0 < ξ < π. The symbol Ξ will be used to denote this value. The shape of the sphere cap reflects the value of Ξ as follows: for a subhemispherical sphere cap, Ξ is an obtuse angle, π/2 < Ξ < π; for a hemisphere, Ξ is a right angle, Ξ ) π/2; and for a superhemispherical sphere cap, Ξ is an acute angle, 0 < Ξ < π/2. Ξ is related to the
sech2{η} + sec2{ξ} tanh2{η} + tan2{ξ}
(3.7)
This relationship reduces to
R)
2λ
xη2 + ξ2
when η f 0 and ξ f 0
(3.8)
when both coordinates are small, i.e. remote from the origin. 4. Steady-State Concentrations In the toroidal system, the boundary conditions equivalent to (2.4-2.7) are
∂c/∂ξ ) 0
at ξ ) 0 and all η
(4.1)
c f cb
as ξ f 0 and η ) 0
(4.2)
c f cb
as η f 0 and ξ ) 0
(4.3)
and
c ) cs ) 0
at ξ ) Ξ and all η
(4.4)
These are the conditions under which we seek to solve the
2172 J. Phys. Chem., Vol. 100, No. 6, 1996
Alfred and Oldham
steady-state version of Fick’s second law. From (3.4), this law becomes
∂ ∂c ∂ ∂c h ) -csch{η} h sinh{η} ∂ξ ∂ξ ∂η ∂η
(4.5)
where ∇2c is set to 0. In (4.5), the local concentration c is a function of both ξ and η, but we shall now assume21 that c may be written as
[ x
c(ξ,η) ) cb 1 -
]
λ f(ξ) g(η) h
(4.6)
where f is a function of ξ only and g is a function of η only. Substitution of this assumption into (4.5) leads eventually to
[
]
1 d2f 1 d2g dg g ) + coth{η} + f dξ2 g dη2 dη 4
cb - c(ξ,η) cb
cosh{µξ}PV{cosh η} (4.13)
Note that replacement of µ by -µ leaves this equation unchanged, so that negative values of µ need not be considered. Because any positive value of µ is admissible as the squareroot of the separation constant, the specific solution that we seek may incorporate all values in the range 0 e µ < ∞. This can be accomplished by integration:
cb - c(ξ,η)
(4.8)
cb
d2g dg 1 + coth{η} + g µ2 + ) 0 dη 4 dη2
∫0∞β cosh{µΞ}PV{cosh η} dµ
(4.9)
(
)
(4.10)
That this is a version of Legendre’s differential equation22a may be shown by setting x ) cosh{η}, which converts (4.10) into the standard form,
d2g dg + 2x - V(1 + V)g ) 0 2 dx dx
(x2 - 1)
g ) bPV{cosh η} + b′QV{cosh η}
1
x2[cosh{η} - cos{Ξ}]
(4.12)
where b and b′ are independent of the toroidal coordinates. However, QV{1} ) ∞, whereas the concentration must remain bounded along the symmetry axis η ) 0 as boundary condition (4.2) implies; therefore b′ must be 0. It follows that g is proportional to the Legendre P function of argument cosh{η} and degree (-1/2 + iµ); such functions are also known as conical functions.23 Some properties of conical functions are presented in the Appendix.
(4.15)
) ∫0 sech{πµ} cosh{(π-Ξ)µ} PV{cosh η} dµ ∞
where the second equality is a consequence of the result (11.4) from the Appendix. Accordingly
β ) sech{µΞ} sech{πµ} cosh{(π-Ξ)µ} ) 1 - tanh{πµ} tanh{Ξµ}
(4.16)
The final result, giving the steady-state concentrations in the vicinity of the sphere-cap electrode, is therefore
cb - c(ξ,η) cb
) x2[cosh{η} - cos{ξ}] ×
∫0∞cosh{µξ}[1 - tanh{πµ} tanh{Ξµ}]PV{cosh η} dµ (4.17)
(4.11)
where V is the complex number (-1/2 + iµ). The general solution is therefore22
(4.14)
Access to the term β is provided by boundary condition (4.4), which shows that
and has the general solution
where B and B′ are terms that are independent of the toroidal coordinates. However, boundary condition (4.1) requires that df/dξ be 0 at ξ ) 0, which demands that B′ be 0. Notice that if the separation constant had been taken as negative, then f would have been a sinusoidal function of ξ, which is physically inadmissible. The right-hand expression in (4.7) also equals µ2, which leads to the equation
) x2[cosh{η} - cos{ξ}] ×
∫0∞β cosh{µξ}PV{cosh η} dµ
) f ) B cosh{µξ} + B′ sinh{µξ}
) βx2[cosh{η} - cos{ξ}] ×
(4.7)
The left-hand expression in this equation is a function of ξ only, whereas the right-hand expression is a function of η only. It follows that each expression must be a term independent of both ξ and η. This separation constant will be denoted µ2. We assume µ to be real and positive and justify this assumption later. The ξ portion of (4.7) can be rewritten
d2f ) µ2f dξ2
The terms B and b remain unassigned; we combine them into a single term defined by β ) Bb/x2. With B′ and b′ replaced by 0, (3.5), (4.6), (4.9), and (4.12) now may be reassembled into
This equation describes, in toroidal coordinates, the solution to the diffusion problem. Its correctness can be verified by showing that it satisfies all of (4.1)-(4.5). 5. The Steady-State Current With c(ξ,η) treated as a constant, (4.17) prescribes a relationship between ξ and η that delineates a steady-state equiconcentration surface. In the steady state, the same total flux J of electroactive species must cross all such equiconcentration surfaces. Let us examine the shape of those equiconcentration surfaces which lie very far from the electrode, i.e. when c(ξ,η) is close to cb. As Figure 2 demonstrates, this corresponds to the coordinates ξ and η having acquired very small values. In
Steady-State Currents at Sphere-Cap Microelectrodes
J. Phys. Chem., Vol. 100, No. 6, 1996 2173
TABLE 1: Steady-State Current I at a Sphere-Cap Microelectrode, Calculated by Numerical Integration of (7.1), Presented as a Function of the Volume V, Area A, and Radius a, of the Sphere Cap
a
sphere-cap shape
V/Vhs
A/Ahs
a/λ
Ξ/π
I/Ihs ) I/2πnFDcbλ
disk subhemisphere subhemisphere subhemisphere subhemisphere one-quarter-sphere subhemisphere one-third-sphere subhemisphere subhemisphere subhemisphere hemisphere superhemisphere superhemisphere superhemisphere two-thirds-sphere superhemisphere three-quarters-sphere superhemisphere superhemisphere superhemisphere superhemisphere very large cap resting sphere
0 0.059148 0.119782 0.183518 0.252265 0.328427 0.415214 0.481125 0.517131 0.640786 0.796314 1.000000 1.279413 1.684150 2.310270 2.598076 3.361873 5.328427 9.596322 21.190673 92.819586 522.376342 2/Ξ3 ∞
0.500000 0.503097 0.512543 0.528819 0.552786 0.585786 0.629808 0.666667 0.687762 0.763932 0.864727 1.000000 1.185444 1.447214 1.831470 2.000000 2.425920 3.414214 5.236068 9.174861 20.43173 81.22382 2/Ξ2 ∞
∞ 6.392453 3.236068 2.202689 1.701302 1.414214 1.236068 1.154701 1.122326 1.051462 1.012465 1.000000 1.012465 1.051462 1.122326 1.154701 1.236068 1.414214 1.701302 2.202689 3.236068 6.392453 1/Ξ ∞
1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.666... 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.333... 0.30 0.25 0.20 0.15 0.10 0.05 Ξ/π 0
0.6366198 ) 2/π 0.6535576 0.6727876 0.6947489 0.7199938 0.7492257 ) x8 + 1 - 16/x27 0.7833339 0.8094011 ) 4/x3 - 3/2 0.8235728 0.8714813 0.9292668 1.0000000 ) 1 1.088126 1.200321 1.347090 1.406420 ) 2/π + 4/x27 1.545947 1.828427 ) x8 - 1 2.257597 2.980591 4.438744 8.838495 1.386294/Ξ ) ln{4}/Ξ ∞
All data have been normalized with respect to the hemisphere. λ is the radius of the basal disk and is treated as a constant.
this limit, since cosh{η} - cos{ξ} f (η2 + ξ2)/2, cosh{ξµ} f 1, and PV{cosh η} f 1, (4.17) becomes
cb - c(ξf0,ηf0) cb
)
xξ2 + η2∫0∞[1 - tanh{πµ} tanh{Ξµ}] dµ
(5.1)
This limit, however, corresponds to (3.8), so that these remote equiconcentration surfaces may be reformulated as
c(ξf0,ηf0) ) cb -
2cbλ ∞ ∫ [1 - tanh{πµ} tanh{Ξµ}] dµ R 0 (5.2)
Equation (5.2) demonstrates, as indeed would be expected, that each equiconcentration surface remote from a sphere-cap microelectrode is a hemisphere centered at the origin. The flux density across such a surface is easily found from Fick’s first law, j ) -D(dc/dR), and the previous equation shows this to be
2Dcbλ ∞ jRf∞ ) - 2 ∫0 [1 - tanh{πµ} tanh{Ξµ}] dµ (5.3) R Multiplying this by 2πR2 will give the total flux across these hemispherical equiconcentration surfaces:
J ) -4πDcbλ∫0 [1 - tanh{πµ} tanh{Ξµ}] dµ (5.4) ∞
However, as noted above, the total flux across all equiconcentration surfaces, hemispherical or not, must be of this same magnitude in the steady state. One such equiconcentration surface is the microelectrode surface itself. Recalling that we are assuming an n-electron oxidation, the steady-state current is found to be
{ }]
I ) 4nFDcbλ∫0 1 - tanh{u} tanh ∞
[
Ξu du π
after the integration variable is changed to u ) πµ.
(5.5)
The integral in (5.5) may be evaluated numerically for any value of the Ξ/π parameter, and this method was used in compiling Table 1. Alternatively, the integral may be recast in the form of a rapidly convergent infinite series, as we demonstrate in section 7, and the latter may be summed exactly in many cases. Moreover, direct analytical evaluation of the integral in (5.5) is possible for certain special values of Ξ, as discussed in the following section. 6. Special Cases It is the Ξ parameter in (5.5) that reflects the shape of the sphere-cap microelectrode. A hemispherical cap corresponds to Ξ ) π/2; the integrand in this case simplifies to sech(u), and therefore the current is
IΞ)π/2 ) 4nFDcbλ∫0 sech{u} du ) 2πnFDcbλ (6.1) ∞
This is the well-known result24 for the hemisphere. The limit Ξ f π corresponds experimentally to plating a negligible volume of mercury onto the basal disk. Thus, the electrode remains disk shaped. The integrand in this case becomes sech2{u} so that the current becomes
IΞ)π ) 4nFDcbλ∫0 sech2{u} du ) 4nFDcbλ ∞
(6.2)
This well-known result was first derived by Saito.25 When Ξ is sufficiently small, tanh{Ξu/π} is effectively 0 at all u values at which tanh{u} is not effectively unity. In these circumstances, the integrand in (5.5) becomes 1 - tanh{Ξµ/ π}, and therefore the current is
{Ξuπ }] du
IΞf0) 4nFDcbλ∫0 1 - tanh ∞
[
)
4πnFDcbλ ln{2} Ξ
(6.3)
Moreover, when Ξ is small, then (3.3) reduces to Ξ ) λ/a, so that
2174 J. Phys. Chem., Vol. 100, No. 6, 1996
IΞf0 ) 4π ln{2}nFDcba
Alfred and Oldham
The geometry to which this corresponds is that of a sphere cap so large that it is virtually a complete sphere resting on an insulating plane. The steady-state current for this geometry was reported by Bobbert et al.,26 and their formula agrees with (6.4). The congruence of these three results with established formulas lends credence to (5.5). Another special case occurs when Ξ ) π/4. In this case the integrand may be reformulated as [2 csch{u/2} - coth{u/2} + tanh{u/2}]/[coth{u/2} + tanh{u/2}], and thereby (5.5) may be converted to
IΞ)π/4 ) 8nFDcbλ × exp{-u/2} + exp{-3u/2} - exp{-u} du (6.5) ∫0∞ 1 + exp{-2u} A standard integral (see (11.7) of the Appendix) may now be employed to evaluate the current in terms of three instances of Bateman’s G function
IΞ)π/4) 2nFDcbλ[G(1/4) + G(3/4) - G(1/2)] ) 2(x8 - 1)πnFDcbλ
(6.6)
Similarly, easy integrations (change the integration variable in (5.5) in tanh{u/3}) permit the steady-state current to be found for sphere caps with Ξ angles equal to 2π/3 or π/3 and lead to the exact results shown at the extreme right of Table 1. These special cases are named the “one-third-sphere” and “two-thirdssphere”, respectively, in the table. 7. Series Solutions It is convenient to normalize the steady-state current at a sphere-cap microelectrode by division by the current Ihs ) 2πnFDcbλ at a hemisphere of identical basal disk. The resulting dimensionless quantity
{ }]
Ξu 2 ∞ I ) ∫ 1 - tanh{u} tanh du Ihs π 0 π
[
(7.1)
was evaluated for numerous values of Ξ/π, and the results of some of these calculations are displayed in Table 1. A change of integration variable converts integral (7.1) into
{ }]
I πu 2 ∞ ) ∫0 1 - tanh{u} tanh du Ihs Ξ Ξ
[
I
(6.4)
[{} ] [ ] ∑ {}
ln{4} Ξ
Ihs
) ln{4}
1
2
∞
jπ
∑(-1)j G Ξ j)1 -
Ξ
1
-
π
-
Ξ
jπ
∞
2
Ξ j)1
Ξ
(-1)jG
jπ
(7.4)
Ξ
where the second step follows from the rule ∑(-)j (1/j) ) -ln{2}. By making use of the πF{Ξ/π} ) ΞF{π/Ξ} property, an alternative representation is
I Ihs
[ ]
) ln{4}
1
-
π
1
-
Ξ
2
{}
∞
(-1)jG ∑ π j)1
jξ
(7.5)
π
a result that may also be derived more directly from (7.1). Truncated versions of the infinite series (7.4) can be used to compute I/Ihs for any value of the Ξ/π ratio, but because numerical values of Bateman’s G function are not readily accessible computationally, it is doubtful whether this procedure is greatly superior to numerical integration of (7.1). However, for many interesting values of the Ξ/π ratio it is possible to sum (7.5) exactly. This possibility arises whenever Ξ/π equals the ratio m/n of two integers, with either m or n being even. In this circumstance, as proved in section 11, (7.5) may be replaced by the remarkably simple formula
I Ihs
)-
1 n-1
{}
∑(-1)jG
π j)1
jm
-
n
n
m-1
{}
(-1)kG ∑ mπ k)1
kn m
(7.6)
which involves only finite sums. This formula is not valid if both m and n are odd. Interestingly, implementing the nF{m/n} ) mF{n/m} conversion leaves formula (7.6) unchanged. 8. Discussion of Sphere-Cap Electrodes In compiling Table 1, it was convenient to use the Ξ/π parameter as the independent variable. However, this parameter is not experimentally relevant, and therefore three other columns have been added to Table 1. These relate the steady-state current at the sphere-cap microelectrode to the geometric features of the cap: its radius, its area, and its volume. These are expressed in normalized form, where normalization is achieved by division by the corresponding feature of the hemispherical special case. The formulas
(7.2) a ) csc{Ξ} λ
The equivalence of the last two expressions shows that the steady-state current has an interesting property. Representing the I/Ihs ratio by F{Ξ/π}, then πF{Ξ/π} ) ΞF{π/Ξ}. We shall make use of this equivalence in section 9 and to transform (7.4) below into (7.5). Integral (7.2) may be split into two:
I 2 ∞ ) ∫0 [1 - tanh{u}] du + Ihs Ξ 2 ∞ tanh{u} du (7.3) ∫ 1 - tanh πu Ξ 0 Ξ
[
)
{ }]
The first integral evaluates to (1/Ξ) ln{4} by analogy with (6.3). In the Appendix we discuss some of the properties of Bateman’s G function. One of these, (11.8), in conjunction with the expansion 1 - tanh{πu/Ξ} ) -2∑(-)j exp{-2jπu/Ξ}, with the summation index j running from 1 to ∞, may be applied to the second integral. The result is
(8.1)
[ x ] a2 -1 λ2
A aa A ) ) ( Ahs 2πλ2 λ λ
) csc{Ξ} cot{Ξ/2}
(8.2)
and
3V V ) Vhs 2πλ3 ) )
[ ]x
a2 1 a3 ( + λ3 λ2 2
a2 -1 λ2
1 + 3 tan2{Ξ/2} 4 tan3{Ξ/2}
indicate how this normalization is accomplished.
(8.3)
Steady-State Currents at Sphere-Cap Microelectrodes
J. Phys. Chem., Vol. 100, No. 6, 1996 2175
Figure 3. Cross sections of a selection of sphere caps. In all diagrams the heavy line indicates the position of the metal/solution interface and the shaded region represents the electrolyte solution. The basal disk has the same size in all cases, and the angle Ξ takes the following values: (a) π/6, (b) π/3, (c) π/2, (d) 2π/3, (e) 5π/6, and (f) π.
Several of the numbers in Table 1 duplicate those obtained by Bobbert et al.26 by a different, more elaborate method. That method, moreover, did not permit the current to be calculated for the range π e Ξ e 2π/3. Table 1 shows that, as expected, the current increases as one descends through the tabular entries, corresponding to an increasing sphere-cap size. However, the current density, I/A, decreases. Interestingly, the I/xA ratio remains almost constant, a feature that has been noted previously.27 Notice the marked insensitivity of the steady-state current to the volume of mercury for sphere-cap microelectrodes. Going from a one-quarter-sphere to a hemisphere requires a tripling of the mercury volume but produces only a 33% increase in the current. Likewise, the 5-fold volume increase that converts a hemisphere into a three-quarters-sphere yields only an 83% current increase. This insensitivity could have beneficial experimental consequences; for example it means that a microelectrode will closely resemble a hemisphere voltammetrically even if an inexact volume of mercury has been electrodeposited. The other side of the coin is that steady-state voltammetry is a poor method of assessing the shape of a spherecap electrode. The first five diagrams in Figure 3 will help clarify our terminology. Diagrams (a) and (b) are superhemispherical caps; specifically, (b) is a two-thirds cap. Diagram (c) is a hemispherical sphere cap, whereas (d) and (e) are subhemispherical, (d) being a one-third-sphere cap. Diagram (f) is a disk electrode.
Figure 4. Sphere-cap electrodes (a-c) and entire spheres (d-f) which share a common curvature radius.
9. Caps of Common Curvature Up to this point, as Figure 3 makes abundantly clear, we have compared the behavior of sphere caps which share a common basal plane. There is also interest in comparing the voltammetric properties of sphere caps that share a common radius of curvature. Several such sphere caps are shown in Figure 4. Those labeled (a), (b), and (c) correspond respectively to a quarter-sphere, a hemisphere, and a three-quarters-sphere. However, unlike the situation considered earlier in which a mercury cap was built upon a metal disk, in Figure 4 these sphere-cap electrodes are considered to arise because an entire conducting sphere is gradually emerging from burial within an insulating slab. When state (d) is reached, emergence is complete, the basal plane has shrunk to a point, and the electrode sphere rests on an insulating plane. Of course, the emergence process eventually leads to the sphere breaking contact with the plane and occupying a position similar to that illustrated by (e) and (f) in Figure 4. The steady-state currents for the electrodes shown as (a), (b), (c), and (d) in Figure 4, and from similar emergent sphere caps, are easily found from Table 1 and from the accompanying discussion in sections 5-7. To take (c), the three-quarterssphere, as an example I ) 1.828427(2πnFDcbλ) ) 1.292893 × (2πnFcba). These currents have been plotted in Figure 5 as
Figure 5. Steady-state current I, normalized by division by the current 2πnFDcba at a hemispherical sphere-cap electrode, plotted as a function of l /a. l is the distance of the center of the sphere above the insulating plane. The letters (a), (b), ..., (f) identify the points with the diagrams in Figure 4. Note that, as λ f ∞, the situation of a totally isolated sphere is approached, for which I ) 2πnFDcba ) 2Ihs, so that the curve is asymptotic to a horizontal line at an ordinate of 2.0. The full curve was drawn with the aid of (7.1); the dashed curve used (9.3).
a function of l /a, where l is the distance that the center of the sphere lies above the insulating plane, as indicated in Figure 1b and given by
l ) xa2 - λ2 ) a sin{Ξ}
(9.1)
Of course, l is negative for subhemispheric sphere-cap electrodes. The present paper is concerned with sphere caps and has no apparent relevance to conditions beyond point (d) on Figure 5, where the electrode has fully emerged from the insulating slab. It has recently been demonstrated28 that the steady-state current at such an electrode fits the equation
2176 J. Phys. Chem., Vol. 100, No. 6, 1996
xl 2 - a2 ∞ b I ) 8πnFDc ∑ a j)1
1+
(
1
Alfred and Oldham TABLE 2: Some Values of Bateman’s G Function
)
l + xl 2 - a2 a
x 2j+1
(9.2)
exactly, and the empirical equation
I ) Ihs
[2l 4l+ a + exp{- 2.9la }]
(9.3)
with an error less than 0.15%. The broken line in Figure 5 was drawn with the help of (9.3). Gratifyingly, it displays perfect continuity with the full line, showing that similar factors determine the voltammetry of postemergent and preemergent spheres. It also suggests a liaison between (7.1) and (9.2), but we have not pursued this.
G(x) ∞ 2/x - (π + ln{4} - 2) + ... x2π + x8 ln{x2+1} 2π/x3 + ln{4} π 2π/x3 - ln{4} x2π - x8 ln{x2+1} ln{4} 4-π 2 - ln{4} ln{4} - 1
0 small 1/4 1/3 1/2 2/3 3/4 1 3/2 2 3 m ) 1, 2, 3, ...
m-1
(-1)m-1[ln{4} + 2
k
k)1 (m-1)/2
m/2 ) 1/2, 3/2, 5/2, ...
(-1)(m-1)/2[π + 4
∑
(-1)j/(2j - 1)]
j)1
1/x + 1/2x2 + ...
large
10. Appendix: Conical Functions
∑(-1) /k]
The conical function P-1/2+iµ(x) is one solution of the differential equation
Table 2; others may be evaluated via the theorem of Gauss22c or by applying the recursion
d2g dg (x2 - 1) 2 + 2x + (1/2 + µ2)g ) 0 dx dx
2 G(1 + x) ) - G(x) x
(11.2)
G(1 - x) ) 2π csc(πx) - G(x)
(11.3)
(10.1)
Despite the fact that the nomenclature of conical functions makes use of a complex degree, these functions are real, as the series expansion
(1 - x)/8 + P-1/2+iµ(x) ) 1 + (1 + 4µ ) 1! [(1 - x)/8]2 (1 + 4µ2)(9 + 4µ2) + ... (10.2) 1!2!
and/or reflection
formulas. Equation (11.2) generalizes to
2
confirms. In physical applications, conical functions often occur with arguments that are trigonometric or hyperbolic cosines. The latter circumstance is illustrated by the present article. One of several integral representations of the conical function is
P-1/2+iµ(cosh η) ) sin{θµ} ∞ x2 coth{πµ}∫η dθ (10.3) π xcosh{θ} - cosh{η} from which it is possible to
demonstrate29
} {}
(11.4)
where m is an odd integer and to m-1(-1)k
G(x) - G(m + x) ) 2 ∑
k)0 k
+x
for m ) 2, 4, 6, ... (11.5)
when m is even. Either (11.4) or (11.5) reduces to
1
∞
∞ (-1)k 1 ) +∑ x k)1 k + x k)0 k + x
G(x) ) ∑
(-1)k
exp{-xt}
∫0∞1 + exp{-t} dt ) 21G(x)
Bateman’s G function30,22b is defined in terms of the digamma (or psi) function
{
for m ) 1, 3, 5, ...
(11.6)
(10.4)
11. Appendix: Bateman’s G Function
x+1 x -ψ G(x) ) ψ 2 2
+x
when m is allowed to approach infinity. There are a number of integral representations of Bateman’s G function; one is
∫0∞sech{πµ} cosh{(π-ξ)µ} P-1/2+iµ{cosh η} dµ ) x2[cosh{η} - cos{ξ}]
k)0 k
2
that
1
m-1(-1)k
G(x) + G(m + x) ) 2 ∑
(11.1)
It is denoted 2β(x) in one popular mathematical compendium.31 A remarkable property of this function is its expressibility, for all integer or rational values of its argument, in terms of constants and simple functions. Some examples are listed in
(11.7)
and this leads easily to
∫0∞exp{-2xu} tanh{u} du ) 21[G(x) - 1x]
(11.8)
Many infinite alternating series of Bateman functions of arguments that form an arithmetic progression of rational numbers can be expressed as finite sums. We shall investigate the sum 2∑(-)j G(jm/n), where m and n are integers. We can group the summands as follows:
Steady-State Currents at Sphere-Cap Microelectrodes
()
∞
jm
2∑(-1)jG j)1
n
n
()
j)1
3n
()
(-1)jG ∑ j)n+1
jm
() ()
2n
jm
) ∑(-1)jG
n
j)1
4n
+
n
(-1)jG ∑ j)2n+1
n
Acknowledgment. The assistance of Jan Myland and the financial support of the Natural Sciences and Engineering Research Council of Canada are greatly appreciated.
+ ... (11.9)
References and Notes
jm
+ ∑(-1)jG
jm n
J. Phys. Chem., Vol. 100, No. 6, 1996 2177
+
Other than the first, each of the right-hand summations contains 2n Bateman functions of alternating signs. The first function in each summation may be paired with the (n+1)th, the second with the (n+2)th, and so on. The argument of the second member of every pair will exceed that of the first member by m. If n is even, each member of the pair will have the same sign, whereas the signs will differ if n is odd. Consulting (11.4) and (11.5), we learn that these equations can be used to sum the pair either if n is even and m odd or if n is odd and m is even. The ultimate result is
()
∞
2∑(-1) G j
j)1
jm n
()
n
) ∑(-1) G j
j)1
jm n
∞
m-1
(-1)j
+ 2n ∑ (-1) ∑ k)0 j)1 kn + jm (11.10) k
in either of these cases. The k ) 0 summand gives (-n/m) ln{4}, and the others may be summed via (11.6). One finds ∞
()
2∑(-1) G j
j)1
jm n
()
n
jm
) ∑(-1)jG j)1
n
n m-1 m k)1
m
()
∑ (-1) G k
n
-
kn m
ln{4} + m-1(-1)k
- 2∑
k
k)1
(11.11)
if either m or n is even. As Table 2 demonstrates, the last summand in the first summation equals (-1)n+m-1[ln{4} + 2∑(-1)k/k], and since only one of n and m is even, the sign is positive. The final summation in (11.11) is thereby canceled, leaving the result ∞
()
2∑(-1)jG j)1
jm n
n-1
()
) ∑(-1)jG j)1
jm n
-
n-m m
n m-1
ln{4} +
()
∑ (-1)kG m k)1
kn m
(11.12)
If n is even, further simplification is possible from applying the reflection formula (11.3) to the first right-hand summation. However, since this option is not available if m is even, we shall be content with (11.12) as it stands. Note that this formula does not apply if both m and n are odd.
(1) Colyer, C. L.; Luscombe, D.; Oldham, K. B. J. Electroanal. Chem. 1990, 283, 379-387. (2) Gunasingham, H.; Ang, K. P.; Ngo, C. C. Anal. Chem. 1985, 57, 505. (3) Ciszkowska, M.; Stojek, Z. J. Electroanal. Chem. 1985, 191, 101. (4) Baranski, A. S. Anal. Chem. 1987, 59, 662. (5) Stulikova, M. J. Electroanal. Chem. 1973, 48, 33. (6) Baranski, A. S.; Quon, H. Anal. Chem. 1986, 58, 407. (7) Batley, G. E.; Florence, T. M. J. Electroanal. Chem. 1974, 55, 23. (8) Bindra, P.; Brown, A. P.; Fleischmann, M.; Pletcher, D. J. Electroanal. Chem. 1975, 58, 31. (9) Wehmeyer, K. R.; Wightman, R. M. Anal. Chem. 1985, 57, 1989. (10) Harman, A. R.; Baranski, A. S. Can. J. Chem. 1988, 66, 1036. (11) Stojek, Z.; Osteryoung, J. Anal. Chem. 1988, 60, 131. (12) Wightman, R. M.; Wipf, D. O. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1989; Vol. 15, p 268. (13) Golas, J.; Galus, Z.; Osteryoung, J. Anal. Chem. 1987, 59, 389. (14) Bowyer, W. J.; Evans, D. H. J. Org. Chem. 1988, 53, 5234. (15) Bowyer, W. J.; Engelman, E. E.; Evans, D. H. J. Electroanal. Chem. 1989, 262, 67. (16) Howell, J. O.; Kuhr, W. G.; Ensman, R. E.; Wightman, R. M. J. Electroanal. Chem. 1986, 209, 77. (17) Stojek, Z.; Osteryoung, J. Anal. Chem. 1989, 61, 1305. (18) Colyer, C. L.; Oldham, K. B.; Fletcher, S. J. Electroanal. Chem. 1990, 290, 33-48. (19) Arfken, G. Mathematical Methods for Physicists, 2nd ed.; Academic Press: New York, 1970. (20) Morse, P.; Feshbach, H. Methods of Theoretical Physics; McGrawHill: New York, 1953; Chapter 10. (21) Alfred, L. C. R.; Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 1990, 280, 1-25. (22) Spanier, J.; Oldham, K. B. An Atlas of Functions; Hemisphere Publishers: Washington; Springer Verlag: Berlin, 1987; (a) Chapter 59; (b) Section 44:13; (c) Section 44:4. (23) Stegun, I. A. In Handbook of Mathematical Functions; Abramowitz, M., Stegun, I. A., Eds.; National Bureau of Standards: Washington, DC, 1964; p 337. (24) Oldham, K. B.; Myland, J. C. Fundamentals of Electrochemical Science; Academic Press: San Diego, 1994; Chapter 8. (25) Saito, Y. ReV. Polarogr. 1968, 15, 177. (26) Bobbert, P. A.; Wind, M. M.; Vlieger, J. Physica 1987, 141A, 58. (27) Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 1990, 288, 1-14. (28) Alfred, L. C. R.; Oldham, K. B. J. Electroanal. Chem. 1995, 396, 257-263. (29) Lebedev, N. N. Special Functions and Their Applications; Dover Publishers: New York, 1972; p 229. (30) Erde´lyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G., Eds. Higher Transcendental Functions (Bateman Manuscript Project) Vol. 1; McGraw-Hill: New York, 1955; p 20. (31) Gradshteyn, I. S.; Ryshik, I. M. Tables of Integrals, Series and Products; Academic Press: New York, 1980.
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