J. Phys. Chem. C 2009, 113, 8901–8910
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Electrochemical Determination of Diffusion Anisotropy in Molecularly-Structured Materials Louise A. Evans, Matthew J. Thomasson, Stephen M. Kelly, and Jay Wadhawan* Department of Physical Sciences (CHEMISTRY), The UniVersity of Hull, Cottingham Road, Kingston-upon-Hull HU6 7RX, United Kingdom ReceiVed: July 17, 2008; ReVised Manuscript ReceiVed: March 12, 2009
Theory is presented for the case of two-dimensional diffusion anisotropy in axiosymmetric systems, which, advantageously and indirectly, affords a unified theory of diffusive mass transport at planar, microdisk (or nanodisk) and cylindrical electrodes in isotropic media. A strategy is proposed to determine the extent of diffusion anisotropy in experimental data; proof-of-concept is considered via a lamellar lyotropic liquid crystalline system. 1. Introduction Developments in materials science have enabled the easyfabrication and synthesis of a plethora of substances which have a degree of molecular ordering, such as chlolesterolic, smectic, and nematic liquid crystals,1 mesoporous solids,2 ionic liquids,3 surfaces such as electrodes modified with elaborate architectures (such as those afforded by self-assembled monolayers),4 etc. The synthesis of these “functionally-adapted” materials enables their tailoring to be exploited by their use; indeed, such “designer” media are directly amenable for exploitation in, inter alia, gas sensors,5 membranes,6 highly porous solids to be used as templates for synthesis,7 and technology display devices.8 In all these cases, the function of the ordering is to confer the system with a degree of selectivity, via introduction of anisotropy, to enable the enhanced operation of the particular structure for its purpose. Indeed, the essence of this is exploited by naturally selected structured systems:9 orthotropic solids such as wood,10 plant phloems,11 bilipidic bilayers in cellular membranes,12 etc. all capitalize on the anisotropic geometry for their biological functioning. The transport properties of materials within these systems will also be anisotropic, as has been demonstrated using electrical conductivity.1 However, such measurements generally employ alternating currents so as to avoid electrolysis of the materials. Nevertheless, results presented in this manuscript demonstrate that, for the case of axiosymmetric systems, dc electrolysis at ultramicroelectrodes is able to provide information on this asymmetric transport, since these tools uniquely afford the quantitative monitoring of diffusive chemical flux in two dimensions (normal and tangential to the electrode surface, q.V. Figure 1 for disk geometries).13-15 Electrochemical routes for measuring anisotropic diffusion16 have been pioneered by Abrun˜a,17 by Murray,18 and subsequently by White.19 The work by Murray followed earlier considerations on transport in polymer electrolytes20 and employed dual microband electrodes in a collector-generator mode to determine anisotropic diffusion coefficients via “timeof-flight” voltammetric measurements, or two mutually perpendicular macroelectrodes, so as to measure diffusion in each direction;18 White was interested in determining, via voltammetry compared with finite element simulations, the tortuosity * Author to whom all correspondence should be addressed. E-mail:
[email protected]. Telephone: +44 (0) 14 82 46 63 54. Fax:+44 (0) 14 82 46 64 16. Web: http://www.hull.ac.uk/chemistry/wadhawan.
Figure 1. Schematic illustration of the coordinate system employed for microdisk electrodes. Note the medium into which the electrode is immersed is shown as being heterogeneous, resulting in a diffusion anisotropy (Dr > Dz).
in the diffusion flux in face-centered cubic opals.19 The work presented herein explores the use of a single microdisk electrode to quantify anisotropic diffusion coefficients within a model system. The work employs finite difference methods coupled with a quasi-conformal map developed following principles recently expounded by Amatore,21 to simulate chronoamperometric transients. The choice of calculations presented lies in the ubiquity of finite-difference methods in applications-driven numerical simulations in the literature, coupled with their ease in construction (compared with alternatives such as orthogonal collocation22a or adaptive finite-element methods22b) and reduced computational expenditure. Indeed, the results presented give analogous results when compared with more computationally intensive simulations: there is good agreement (even under the most optimal computational conditions, Viz. for short time electrolysis) between the results obtained via the quasiconformal mapping technique and those calculated using Gavaghan’s geometrically expanding finite difference mesh approach.23 2. Theory Consider the following simple heterogeneous electron transfer reaction occurring at a microdisk electrode (radius r0) immersed in an axially symmetric anisotropic medium, initially containing species Ox (of charge z, and present in solution at a concentration c0).
10.1021/jp806322c CCC: $40.75 2009 American Chemical Society Published on Web 04/27/2009
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Oxz + ne- a Redz-n The reaction formal redox potential, E ′, takes a certain value depending on the electron affinity and the solvation of the species Ox and Red.24 If the working electrode in an amperometric or a voltammetric experiment is potentiostated far beyond the formal electrode potential (Viz. |E| . |E0′|), then the observed current is a function of the mass transfer regime, since this is slower; cf. the rate of heterogeneous electron transfer. In the following, we shall assume that the standard chemical potential does not change with distance from the electrode surface25 and that the Ohmic losses (which occur differentially) across the electrode surface26 are negligible. In an isotropic system, mass transfer (hereafter assumed to be merely due to diffusion27) is identical in all directions. This is not the case for an anisotropic system.28 Considering the particular problem of a material transfer in cylindrical space, such as that offered to disk electrodes, assuming that the principle diffusion axes coincide to the axial (z), radial (r), and angular (φ) coordinates of the cylindrical coordinate system, Fick’s first law describing the diffusion flux (ji) becomes (under the approximation of solution ideality)10 0
jz ) -Dz
∂c ∂z
jr ) -Dr
∂c ∂r
jφ ) -Dφ
1 ∂c r ∂φ
ψ)
i ) 4nFDisor0c0
[
]
Dφ ∂2c ∂c ∂2c ∂2c 1 ∂c + 2 + D ) ∇ · ∇ c ) Dr 2 + z ∂t r ∂r ∂r r ∂φ2 ∂z2
(2) The above equation reduces to the following for an axially symmetric (φ-independent) system, allowing the mass-transport to the solid-liquid interface to be governed by flux in only two mutually perpendicular directions.
[
]
∂c ∂2c 1 ∂c ∂2c + Dz 2 ) Dr 2 + ∂t r ∂r ∂r ∂z
(3)
This latter result reduces to the more familiar nonuniform result obtained for isotropic media when Dr ) Dz ) Diso:28,29
[
∂c ∂2c 1 ∂c ∂2c ) Diso 2 + + ∂t r ∂r ∂z2 ∂r
]
(4)
In this case, the current (i) flowing under potential-step conditions is given by the analytical expressions deduced by Aoki and Osteryoung,30a using the Wiener-Hopf technique,
ψ)
i ) 4nFDisor0c0
{
1+
0.71835 0.05626 0.0064 ... τ g 0.82 + √τ √τ3 √τ5
4τπ + π4 + 0.094√τ...
τ e 1.44
where τ ) (4tDiso)/(r02). Note that the expressions given above overlap in the given time domain. For isotropic systems, Shoup and Szabo30b empirically deduced the following convenient expression to an accuracy better than 0.6%.
∀ξ πξ + 0.2146 exp[- 0.3912 √ξ ]
where
ξ)
tDiso 2
τ 4
)
r0
It is pertinent to note two further limiting cases of eq 3. First, if Dz . Dr, the problem effectively reverts to the case of planar diffusion to the disk and is a problem that has been very well considered and solved previously:31 the spatial and temporal variation in the concentration of species Ox is
(√ )
c ) c0 erf
z
2 Dzt
with the Cottrell equation, Viz. i ) nFSc0[Dz/(πt)]1/2 (or, equivalently, in adimensional form, ψ ) 1/2(π/τ)1/2), describing the temporal variation of the current flowing through the electrode. Conversely, if Dr . Dz, radial diffusion dominates over normal diffusion. This is the case of the cylinder electrode, for which the time-dependent current is32
(1) in which c denotes the concentration of the electroactive reactant. Thus, Fick’s second law, pertaining to the temporal (t)dependence of the concentration, becomes
1 4
0.7854 +
i)
2nFDrc0 2
ξ r2
-ξx ∫0∞ x Jexp 2 (x) + Y 2(x)
(
{
0
2)
0
}
dx
where J0(x) and Y0(x) are, respectively, zeroth order Bessel functions of the first and second kinds, with x being the auxiliary variable. Asymptotic expansion furnishes the following results (γ ) 0.57722, Euler’s constant). Note that, at long-time electrolyses, only a quasi-steady-state is observed: ir0 ) nFDrc0
{
√ξ 1 ξ 1 + + - 0.147ξ3⁄2 + 0.203ξ2 + ... if ξ < 0.18 √πξ 2 4√π 8 2⁄3 2 2γ π - 2γ 2 + ... if ξ > 1200 [ln(4ξ) - 2γ] [ln(4ξ) - 2γ]2 [ln(4ξ) - 2γ]3
}
This expression has been generalized by Aoki32a for 0 e ξ e 106 to within 1% error and by Szabo and co-workers32c for all times to within 1.3%:
( )
√πξ exp ir0 10 ) + nFDrc0 √πξ
1
[
( 35 )]
ln √4 exp(-γξ) + exp
Since the diffusion regime is a function of the diffusion space at an electrode, it follows that the three limiting cases of eq 3 considered above have special inferences for the case of electrodes immersed into isotropic media. The equality of the diffusion coefficients in the perpendicular and tangential directions to the electrode surface is a characteristic of a nonmodified electrode surface. The dominance of planar diffusion over radial diffusion occurs when the edge circumference of the microdisk electrode is “blocked off” by, for example, the electrode being recessed compared with its surrounding insulations. The scenario when radial diffusion dominates over planar diffusion may physically correspond to the case of a large spherical object centrally located on the disk electrode blocking off any normal material transfer to the disk electrode. In other words, the quantitative deconvolution of the diffusion rates in planar and
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radial directions in isotropic media enables the voltammetric sizing of immobilized species. Such concepts have been exploited recently to describe quantitatively, using voltammetry, the structure of “designer interfaces”.33 In a similar manner to that described for solution of eq 4 in isotropic media under conditions of steady-state (Viz. ∂c/∂t ) 0), we first describe the analytical solution of eq 3, thereby anticipating efficient conformal mapping transforms that allow for the full numerical simulation of the problem for any temporal case. 2.1. Two-Dimensional Diffusion Anisotropy at Disk Electrodes under Steady-State. Under steady-state conditions, and assuming Dr is finite, eq 3 transforms to
()
Dz ∂2c ∂2c 1 ∂c + )0 + Dr ∂z2 ∂r2 r ∂r
(5)
The pertinent boundary conditions are as follows. Initial conditions: t e 0 c ) c0
∀ r,z
(Bulk solution)
At steady-state: t f ∞ c)0
r e r 0, z ) 0
∂c / ∂z ) 0 c ) c0
r > r 0, z ) 0 r f ∞, z f ∞
(Transport-limited electrolysis) (Insulation boundary) (Bulk solution)
Suppose eq 5 is separable, and with a solution of the form c(r,z) ) F(r) G(z), where G(z) takes the form G(z) ) exp [-(Dr/ Dz)1/2qz], in which q is a constant. Then, eq 5 transforms into the following Bessel form.
∂2F(r) 1 ∂F(r) + + q2F(r) ) 0 2 r ∂r ∂r
(6)
A solution of eq 6 is F(r) ) AJ0(qr) + BY0(qr), where J0(qr), variable s, is the zeroth order Bessel function of the first kind, Viz. ∞
J0(qr) )
qr ∑ s ! Γ-1 (s + 1) ( 2 ) )s
(
2s
s)0
in which Γ(s + 1) is the gamma function, Y0(qr) is the Bessel function of the second kind, order zero, and A and B are integration constants. Since the problem is for axial symmetry, for F(r) to be finite everywhere, B ) 0. Thus, a general solution of eq 5 is the superposition of solutions of the form ∞ An J0(qnr) c0 - c ) J0(qr) exp[-(Dr/Dz)1/2qz], Viz. c0 - c ) ∑n)1 exp[-(Dr/Dz)1/2qnz]. Exploiting the orthogonality of the Bessel functions under the boundary conditions of z ) 0 enables the deduction of the coefficients An,29,34 so that the general solution of eq 5 becomes
c0 - c )
2c0 π
[ ]
∫0∞ sin(qr0) exp -
Dr dq qz J0(qr) (7) Dz q
Considering the result deduced by Saito for the isotropic case,29a it is seen that eq 7 differs only by the presence of the square root of the diffusion coefficient ratio in the exponential term. While eq 7 satisfies eq 5, it is further demonstrated below, using standard results34 that eq 7 is of the correct form when z ) 0; Viz., it reduces to a real constant (equal to c0 for r e r0, and so that ∂c/∂z ) 0 for r > r0).
(c0 - c)z)0 )
2co π
∫0∞ sin(r0q)J0(qr) dqq )
{
c0 2c0 r0 arcsin π r
()
if 0 < r < r0 if r0 < r < ∞
}
(8a)
( ∂c∂z )
) z)0
2c0 π
Dr Dz
∫0∞ sin(qr0) J0(qr) dq )
{
if r2>r02
0
2c0 π
Dr Dz
1
√r0
2
-r
2
if r02>r2
}
(8b)
Satisfyingly, the expressions in eq 8a are analogous to those derived by Saito29a and Crank and Furzeland;29b the effects of the singularity at the electrode|insulation boundary (Viz. r ) r0) are manifested by, upon implication of this condition, a concentration gradient that tends to infinity (an effect that is physically compensated by finite heterogeneous electron transfer kinetics). Thus, the flux of Ox to the electrode surface under reductive conditions (or equally under oxidative conditions of Red) is jz, which is defined in eq 1. Given that the steady-state current (ilim) flowing between the working and the counter electrodes is given as ilim ) nF∫jz dS, where n is the number of electrons transferred heterogeneously, F is the Faraday constant, and S is the electrode area,
ilim ) 2πnF
∫0r rjz dr ) 4nFc0√DzDr∫0r 0
0
r dr
√r20 - r2
)
4nF√DzDrr0c0
(9)
This equation resembles the familiar steady-state expression in isotropic media and reduces to that form when Dz ) Dr ) Diso. Moreover, eq 9 demonstrates that diffusion coefficients in isotropic media can be expressed by the geometric mean of the individual diffusion coefficients in the axial and radial directions. This important result suggests that any analytical result obtained in isotropic media can be explicitly employed for the special form of two-dimensional anisotropy considered here, using the geometric mean of the direction-dependent diffusion coefficients as the “aVerage” Value; this view will be ratified for the case of potential-step chronoamperometry in subsequent subsections. Furthermore, in returning to the concept presented earlier for electrode surfaces modified with objects and immersed into isotropic media, by measuring diffusion coefficients under steady-state conditions versus planar conditions, it is possible to use differential flux calculations, based on distortions in the diffusion regime, to deduce the size and shape of species (such as nano-objects, microparticles, or microdroplets)33 that may be located on the electrode surface, Viz. Dobs ) Diso(γzγr)1/2, in which Dobs is the measured diffusion coefficient at the partially blocked electrode, Diso is the true diffusion coefficient in the isotropic medium, and γi represents the fraction of the flux in the i-director (Viz. γi ) Di/Diso). A further experimentally important result of eq 9 is that, compared with conditions of ultrashort electrolysis (or, indeed, large electrode size, for which the “edge effects” of the insulation singularity are negligible), which create diffusion layers smaller than the electrode size and thus enable determination of the planar diffusion coefficient (since the measured current property, such as the peak current, contains (Dz)1/2), the limiting current encompasses the diffusion coefficients in both directions. This comparative benefit is capitalized in sections 2.3 and 4 (Vide infra). Last, in order to derive appropriate mapping tools for the numerical solution of the particular problem at hand, eq 7 is solved, ex cathedra, using standard integral35 results ∀ r,z:
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c 2 ) 1 - arcsin c0 π
{( )
2
Dr 2 Z + (1 + R)2 + Dz
( )
Dr 2 Z + (1 - R)2 Dz
Evans et al.
}
c 2β ) arctan c0 π
Z>0
(10a)
where Z and R are dimensionless variables characterizing mass transfer in the normal and radial directions, respectively, Viz. Z ) z/r0 and R ) r/r0. Using this notation, the solutions provided in eq 8a for the concentrations at the electrode surface (z ) 0) become
{
0 c ) 1 2 1 - arcsin c0 π R
()
if 0 e R e 1 if 1 < R < ∞
}
Z)0
(10b)
The effects of the differential diffusion (pertinent only for the case Z > 0, Viz. eq 10a) are illustrated via the admensional concentration profiles depicted in Figure 2, which emphasize the change in the diffusion geometry. 2.2. Simulation of the Temporal Dependence of the Current Using Conformal Mapping Techniques. As previously mentioned, the employment of potential-step chronoamperometry enables the characterization of the temporal dependence of the anisotropic diffusion regime, since the electrode can be held at a driving force, potential (E), sufficiently greater than the formal redox potential, allowing the observed variable, the current (i), to not reflect the effects of the heterogeneous electron transfer process. The axiosymmetric nature of the problem allows for simplification via simulation of only half the disk surface, thereby introducing a new boundary condition due to the presence of the symmetry axis, Viz., (∂c/∂r)r)0 ) 0 ∀ z. Accordingly, first, an appropriate coordinate transform is sought. 2.2.1. DeriWation of a Quasi-Conformal Map for the Simulation of Axially Symmetric Diffusion Anisotropy. A recent work by Amatore21 advocates the employment of “analytical solutions of the concentration at steady-state for the generation of conformal maps”,21b thereby allowing for highly efficient computation, since the effects of the singularity are fully encompassed by the analytical solution. To date, with the exception of that employed by Saito29a for the simulation of microband electrodes and that used for microdisk electrodes by Amatore,36 all conformal maps produced for the case of isotropic diffusion to microdisk electrodes are really quasi-conformal, since they do not follow the Cauchy-Reimann conditions.37 Accuracy and efficiency aside, the most attractive of all the quasi-conformal maps are those developed by Amatore and Fosset,21a based on Newman’s use of rotational cylindrical coordinates,38 and Amatore et al.,21b since these reduce semiinfinite space into a closed box, of dimension [0,1]. Of these two possibilities, the former is adapted for solution herein. One useful aspect of using conformal mapping techniques is that one of the conformal variables, Γ, takes the solution of eq 5, Viz. eq 10a. For ease, eq 10a is converted into the following form.39
{ [
( Z
Dr 1 1 - Z2 - R2 + 2 Dz
Dr Dz
R2 +
)
2 Dr 2 Dr Z - 1 + 4 Z2 Dz Dz
]
}
(11)
In the above equation, β is a constant (Vide infra). Equations 10a and 11 are compared in Figure 2 (under the condition of β ) 1, which corresponds to steady-state, q.V. section 2.2.2), where it is seen that the approximation is reasonably adequate. Thus, the conversion of eq 11 into a closed-grid form, Viz. [0,1], is achieved by splitting eq 11 into the following form,
Θ)
[
Dr 1 1 - Z2 - R2 + 2 Dz
(
Γ)
R2 +
) ] { }
2 Dr 2 Dr Z - 1 + 4 Z2 Dz Dz
2β arctan Θ-1 π
Dr Z Dz
(12)
Hence, rearranging the expressions in eq 12 enables the formation of the following coordinate transform.
( 2βπ Γ) D Z)Θ D tan(2βπ Γ)
R ) (1 - Θ2)1⁄2 sec z
(13)
r
This is an adapted form of the familiar conformal mapping transform pioneered by Amatore and Fosset,21a and it allows the curved isoconcentration lines observed at steady-state to become straight, parallel, and equidistant lines (β ) 1) in the [Γ,Θ] space. It is constructive to note that application of Θ ) cos[(π/2β)Ξ], where Θ is a normalized coordinate that replaces Θ, to the relations in eq 13 (and those proceeding eq 13) results in the formulation of the currently most-efficient quasi-conformal transform and merely improves the angular resolution of the mapped space, thereby leading to a more efficient simulation.21b Attention is next focused on the meaning of β. 2.2.2. The Constant Parameter β in Conformal Space. Since the mappings exploit the steady-state concentration profile, the accuracy of the transform under non-steady-state conditions for isotropic systems tends to diminish, in particular at short electrolytic time scales or when the diffusion coefficients of Ox and Red are sufficiently different, so that it often becomes more efficient to undertake computations based on the real space coordinates (q.V. section 2.3), since that space enables the development of geometrically expanding, adaptive simulation grids which contain a very large number of nodes around the singularities and an increasingly fewer number of nodes toward the semi-infinite limits, allowing for the rapid and accurate simulation of the problem compared with very fine meshes required for the conformal problem. For the anisotropic problem, this also becomes a problem whenever diffusion in onedimension becomes dominant (Viz. Dz/Dr * 1, where Cottrellian effects at short time electrolyses are implicit). One method, proposed by Amatore,40 to bypass this is to introduce an adjustable parameter, β, defined as follows, into the conformal map.
Diffusion Anisotropy in Molecularly-Structured Materials
β)1+
r0
√
R (θ√DzDr)
)1+
2 R√τmax √Dz ⁄ Dr 4
(14)
In the above expression, θ is the maximal duration of the simulated experiment (in real time, units seconds, expressed in adimensional form, τmax), so that β f 1 as θ f ∞ (Viz. steady-state conditions). Note that τmax ) 4θDr/r02. Thus, for short-time electrolyses, θ f 0, the fractional part of eq 14 plays a more dominant role, so that π/2β f 0. Likewise, as
J. Phys. Chem. C, Vol. 113, No. 20, 2009 8905 Dz f ∞, β f 1. Hence, the introduction of this variable accounts for the smaller diffusion layer in the Z-direction that is obtained when small time scales are employed, and this acts in an adaptive sense, in that it adjusts the distance between the nodes in the Γ-direction. Similar arguments apply for variation in Dr. The final term in eq 14, R, an integer, merely signifies that further accuracy is achievable by allowing it to take an arbitrary value (such as 10-20);40 in this work, the value of R is taken to be 20.
Figure 2. Adimensional concentration profiles at the microdisk electrode; keys for the shading are provided in each individual case. In all cases, only one-half of the electrode is shown on the abscissa (in the ranges 0 e R e 1 or 0 e %Θ e 100); (i) is deduced from eq 10a, with the calculation from eq 11 overlayerd onto itsthere is nil difference between the two plots; (ii) is the quasi-conformal representation obtained from calculations using eq 16. (a) Dr ) Dz ) 10-5 cm2 s-1, so that Dr/Dz ) 1; (b) Dr ) 10-4 cm2 s-1, Dz ) 10-6 cm2 s-1, so that Dr/Dz ) 100; (c) Dr ) 10-6 cm2 s-1, Dz ) 10-4 cm2 s-1, so that Dr/Dz ) 0.01.
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The use of the above conformal mapping techniques in simulating chronoamperometry at microdisk electrodes immersed in a two-dimensionally anisotropic medium is next considered. 2.2.3. Form of Fick’S Second Law for the Simulation of Chronoamperometry under Quasi-Conformal Map Conditions. Equation 3 is first rewritten in the following reduced form.
4
∂a ∂2a 1 ∂a Dz ∂2a + ) + ∂τ ∂R2 R ∂R Dr ∂Z2
(15)
where a ) c/c0 is the normalized concentration of species Ox and τ ) 4Drt/r02 is the reduced time variable and exploits the radial diffusion coefficient. Changing the variables as indicated in eq 13 transforms eq 15 to the following form (see section 1 of the Supporting Information).
∂a ) ∂τ
1
[
( 2βπ Γ)]
4 Θ2 + tan2
{
4β2 ∂2a 2 π cos + Γ 2β ∂Γ2 π2
( )
(1 - Θ2)
∂2a ∂a - 2Θ ∂Θ ∂Θ2
}
(16)
The subtle change of the dimensionless variables as indicated transposes the anisotropic problem to the identical form observed for isotropic diffusion. The transformed boundary conditions for this axially symmetric case for chronoamperometry are also identical to those of the isotropic system, Viz., Initial conditions: τ e 0 a)1
∀ Γ,Θ
(Bulk solution)
During electrolysis: τ > 0 a)0 0 e Θ e 1, Γ ) 0 (Transport-limited electrolysis) ∂a / ∂Θ ) 0 Θ ) 0, 0 e Γ e 1 (Insulation plane) ∂a / ∂Θ ) 0 Θ ) 1, 0 e Γ e 1 (Symmetry axis) a)1 b e Θ e 1, Γ ) 1 (Bulk solution)
It is evident that a ) Γ satisfies eq 16 and all other boundary conditions.21 The expression for the observed current flowing through the disk electrode is as eq 9:
i ) 2πnFDzc0r0
∂a ∫01 ( ∂Z )Z)0R dR ) 4βnF√DzDrr0c0
Θ)1 ∂a ∫Θ)0 ( ∂Γ )Γ)0 dΘ
(17)
Hence, the steady-state normalized form of the current, ψ, is deduced as
ψ)β
Θ)1 ∂a ∫Θ)0 ( ∂Γ )Γ)0 dΘ
(18)
This succinct representation of the current demonstrates that ψ f 1 as the experiment approaches steady-state. Simulation of the problem using the ADI method is detailed in sections 2, 2.2, and 2.3 of the Supporting Information. In order to demonstrate that the conformal map employed (Vide supra) is sufficient for the numerical solution of the chronoamperometric problem, and further to verify the general conclusion of section 2.1, calculations were additionally undertaken directly in cylindrical space with a high density grid near the electrode edge. The formulation of this problem, and its simulation, is briefly summarized in sections 2, 2.2, and 2.3 of the Supporting Information.
Figure 3. Reduced variable transients (over four million points are plotted) simulated using the quasi-conformal map (9, see section 2 of the Supporting Information) and via the Shoup-Szabo microdisk approximation (O). All simulations were undertaken using adimensional variables using c0 ) 2.0 mM, r0 ) 10.0 µm, n ) 1, -10 e lg(Dr/(m2 s-1)) e -8, and -10 e lg(Dz/(m2 s-1)) e -8. Note the deviation between the two plots at τ < 4 × 10-4 stems from convergence errors in the simulations (smaller normalized time steps remove these deviations) and is shown here so as to indicate that two types of data are present.
2.3. Simulated Results and Discussion. Figure 3 illustrates adimensional logarithmic chronoamperometric traces obtained for the quasi-conformal map simulations with the directiondependent diffusion coefficient ratio being varied in the range -2 e lg(Dz/Dr) e 2. Pleasingly, in reduced coordinates, all traces lie on the same line (within the convergence error observed), demonstrating equivalence in computation (the trends in the reduced form of the transient are independent of Dr and Dz), as expected from eq 16. These traces match the normalized form of the Shoup-Szabo expressions for microdisk transients, to a very good agreement (after convergence issues when τ < 4 × 10-4), as evidenced by the Cottrellian part of the transient at short times. Likewise, Table 1 compares the simulated data via the conformal technique with the geometrically expanding grid. The close self-consistency of the data is reassuring. Simulated concentration profiles (in conformal space) are also shown in Figure 2. These form straight lines as envisaged, with their position on the ordinate axis being affected by the magnitudes of Dr and Dz. This ratifies the outcomes of the steady-state analysis (q.V. section 2.1). In turning attention to exploiting these results for experimental systems, it is noteworthy that the reduced time variable is a function of the radial diffusion coefficient, while the dimensionless current depends on the geometric mean of the diffusion coefficients in the axial and radial directions. Thus, the following strategy is proposed. (i) Obtain steady-state data, and using eq 9, estimate (DrDz)1/2. (ii) Use the results from (i) to calculate ψexpt from the experimental transient data, using eqs 17 and 18. (iii) Use Figure 3 (or equivalently the reduced form of the Shoup-Szabo microdisk transient expression) to convert ψexpt - t data into ψexpt - τ. (iv) The mapping t f τ enables the determination of Dr, and thence Dz, using the results from (i). In the case when instrumentation limitations, uncompensated resistance, or capactitative charging precludes the measurement of high-quality short-time Faradaic data, the isotropic analytical results or approximations may be used to simulate the experimental current|time data, iterating through a range of Dr and Dz, noting that DrDz replaces the isotropic diffusion coeffiicent. These theoretical data can be compared with the experimental results, and the best fit, via a minimum mean-scaled absolute
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TABLE 1: Comparison of the Short-Time Transient Calculations Undertaken Using the Quasi-Conformal Map and the Geometrically-Expanding Finite Difference Grid Dr ) Dz ) 10-5 cm2 s-1
Dr ) 10-4 cm2 s-1, Dz ) 10-6 cm2 s-1
Dr ) 10-6 cm2 s-1, Dz ) 10-4 cm2 s-1
τ
ψa
ψb
ψa
ψb
ψa
ψb
ψc
0.01 0.04
9.63758 5.23579
9.33598 5.00875
9.63505 5.23548
9.29411 4.98951
9.63532 5.23552
9.59207 5.13611
9.648607 5.218413
a Obtained using the quasi-conformal map (see section 2 of the Supporting Information). b Obtained using the geometrically expanding finite difference grid (see section 2 of the Supporting Information). c Obtained using the short time Aoki-Osteryoung expression for microdisk chronoamperometric transients.
deviation (MSAD ) (1/∑s)∑s[(ithy - iexpt)/iexpt]), can be found, enabling the deduction of Dr and Dz from the experimental data. 3. Experimental Section All chemical reagents were purchased from Fluka in the purest commercially available grade and used as received. Water, with a resistivity of not less than 18 MΩ cm, was taken from an Elgastat system (Vivendi, Bucks, U.K.). Following the methods of Bowden et al.41 and Murray et al.,18a cesium pentadecafluorooctanoate was prepared via neutralization of cesium pentadecafluorooctanoic acid with aqueous cesium hydroxide to a pH of 7.0. The salt solution was evaporated to dryness and recrystallized from a 50:50 (v/v) solution of n-hexane and n-butanol, to afford glossy white crystals. Heavy water was heated to approximately 60 °C, and cesium pentadecafluorooctanoate was added to it such that the composition of this mixture was 43:57 wt % cesium pentadecafluorooctanoate/D2O. The mixture was degassed with oxygen-free nitrogen (BOC Gases, U.K.) and maintained under a nitrogen blanket. At this temperature, the system affords an isotropic micellar solution.41 The mixture was cooled to ambient temperature (23 ( 1 °C), so that it would pass through the discotic nematic phase and into the lamellar lyotropic liquid crystalline phase. Although the intermediate nematic phase exhibits positive diamagnetic susceptibility anisotropy,18 monodomain lamellar liquid crystal samples were not prepared. Potassium ferricyanide was employed as a redox probe. Murray,18a using 2H NMR spectroscopy, has demonstrated that the reduced form of this probe (potassium ferrocyanide) and other electroactive probes, such as (ferrocenylmethyl)trimethylammonium perchlorate, does not perturb the lamellar phase prepared (when they are present at c0 ) 2.5 mM). Electrochemical measurements were undertaken using a commercially available potentiostat (Autolab, PGSTAT30, Eco Chemie, The Netherlands) controlled by a Pentium IV computer. The working electrode was a 25.0 µm diameter platinum disk (fabricated in house), cleaned using increasingly finer grades of carborundum paper (Presi, France), followed by polishing on a napped polishing pad using a 0.3 µm aqueous alumina slurry. A nickel wire spiral formed the counter electrode, and a silver wire was employed as the reference, so as to avoid the introduction of 1H2O into the system. 4. Results and Discussion The electrochemical reduction of ferricyanide (c0 ) 12.5 mM) in the cesium pentadecafluorooctanoate/D2O mixture under steadystate conditions is shown in Figure 4a. It is clear that a single, one-electron wave with half-wave potential E1/2 ) 0.17 V vs Ag (E1/4 - E3/4 ) 78 ( 3 mV, only slightly larger than the Tomesˇ criterion for reversibility, with a hysteresis of ca. 5.0 mV), is observed, as anticipated.42 The limiting current observed enables a value of (DrDz)1/2 ) 2.1 ( 0.2 × 10-6 cm2 s-1 to be deduced
Figure 4. (a) Steady-state voltammogram (solid line, V ) 5.0 mV s-1) for the reduction of c0 ) 12.5 mM potassium ferricyanide in 43:57 cesium pentadecafluorooctanoate/D2O at a platinum disk microelectrode (r0 ) 12.5 µm) at 23 °C. The arrow indicates the direction of the initial potential sweep. The open circles (O) plot the steady-state voltammogram expected for an electrochemically quasi-reversible reduction with (DrDz)1/2 ) 2.1 × 10-6 cm2 s-1 and E0′ ) 0.22 V vs Ag (see text). (b) Chronoamperometric transients in dimensionless variables corresponding to the experimental system in part assee text. The experimental points are plotted as open circles (O); the data close to τ ) 0.01 are due to instrumental artifacts. Also plotted (solid line) is the Shoup-Szabo expression in normalized coordinates. Normalization parameters for the experimental data were obtained from the iterative procedure detailed in the text and were Dr ) 2.16 × 10-6 cm2 s-1, Dz ) 1.34 × 10-6 cm2 s-1, r0 ) 12.5 µm, and c0 ) 12.5 mM. The solid squares (9) represent the experimental data when only a single diffusion coefficient is employed, and the dotted line is the associated theoretical resultssee the text for full details.
from eq 9. This value is, given the viscosity of this system, reassuringly smaller than the diffusion coefficient of ferricyanide
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in (infinitely dilute) aqueous solution43 (8.96 × 10-6 cm2 s-1 at 298 K). Analysis of the voltammetric wave shape using this value of the diffusion coefficient via the method proposed by Bard and Mirkin44 suggests the following Butler-Volmer parameters: ksapparent ) 2.0 ( 0.1 × 10-4 cm s-1, R ) 0.67 ( 0.1, and E0′ ) 0.22 V vs Ag. The deduced value of the transfer coefficient is in agreement with that obtained from mass-transport-corrected Tafel plots. A simulated wave shape45 is also shown in Figure 4a; it is clear there is a reasonable match between experiment and theory. In turning to data obtained via potential-step chronoamperometry (q.V. Figure 4b), holding the working electrode at a potential corresponding to the transport-limited current (and therefore not affected by electrode kinetics), the very short time data are dominated by instrumental artifacts. Assuming that, at the electrode singularity, there is no solution heterogeneity, analysis via the iterative method suggested in section 2.3 affords best fit transients with Dr ) 2.2 ( 0.2 × 10-6 cm2 s-1 and Dz ) 1.3 ( 0.2 × 10-6 cm2 s-1. Thus, (DrDz)1/2 ) 1.7 ( 0.2 × 10-6 cm2 s-1, which is within experimental error from that deduced from the steady-state datum. These deconvoluted diffusion coefficient values suggest an anisotropic ratio of 1.6 ( 0.3. This value is reasonably close to the ratio of g2.0 reported by Murray and co-workers18a,46 for ferrocyanide, albeit as a monodomain and at 30 °C, suggesting that, in the system studied, the radial direction of the electrode is more closely aligned parallel to the lamellae (Viz. perpendicular to the macroscopic director) rather than perpendicular to them. Also plotted in Figure 4b are the experimental data and associated fitting using the method suggested in section 2.3 for the isotropic case. This method furnishes a best-fit Diso ) 1.8 ( 0.1 × 10-6 cm2 s-1, which is in agreement with the two-diffusion coefficient approach above. However, it is important to note that there is a worse fitting of the experimental data when there are fewer “degrees of freedom” for the fitting. The reason this occurs is because, in the case of the two-diffusion coefficient fitting, the dimensionless time parameter depends only on Dr and the normalized current on both Dr and Dz. In contrast, in the isotropic case, both adimensional parameters depend on the geometric mean of Dr and Dz. It is insightful to enquire as to what is the true physicochemical representation of the observed data. The lamellar phase of cesium pentadecafluorooctanoate has been considered to be either as infinite lipid bilayers (in which pore-type defects are randomly dispersed) separated by sheets of water47a,b or as planes of disks separated by layers of water (a smectic A phase of disk micelles).47b There is much experimental evidence (neutron diffraction, NMR, electrical conductivity) to suggest that it is the former model which is more realistic.47 In this system, the bilayers are ca. 2.3 nm thick, are separated by an ca. 4.5 nm 2 H2O phase, and consist of continuous lamellae pierced by 2H2Ofilled defects which are of width ca. 6.0 nm. This is consistent with the steady-state voltammetry, since this structure causes a “partial-blocking” of the electrode surface, noting that the hydrophilic and highly charged ferricyanide anion is thought to exist in the aqueous phase.18a Accordingly, considering transport to partially blocked electrode surfaces,48 these effects are manifested in the sluggish electrode kinetics observed: for aqueous electrolyte solutions of equivalent ionic strength to that of the macroionic electrolyte considered herein (Vide infra) and at similar temperatures, ksapparent,true ∼ 0.1 cm s-1. This corresponds to a fractional coverage of the electrode (θ′) of θ′ ) 1 - ksapparent/ksapparent,true ≈ 0.998. It follows that the value determined for Dz must refer to this restricted “pore-diffusion”.49 The alternative is that electron transfer occurs via electron-
Evans et al. hopping through the lipid bilayer. However, the thickness47a of this layer (ca. 2.3 nm) coupled with the associated electrical double layers on either side of it (estimated from the Debye length, xDL, pertinent for this concentration of macroionic electrolyte as ca. 0.3 nm) precludes the feasibility of this pseudodiffusion pathway,50 as electrons would have to tunnel distances of approximately 3 nm to acceptor species, which would then physically diffuse to the electrode surface.51 Compared with a steady-state diffusion layer thickness of δ ≈ r0π/4 ≈ 10 µm, this would involve transport across 1350 lipid bilayers. Although the anisotropic diffusion model appears to fit well with the steady-state voltammetry, the shape fitting is less good for the case of the potential step transients. On a physical level, this can be understood by the fact that before any potential difference perturbation commences, an electrical double layer exists only between the lipid bilayers and the 2H2O phase (and between the latter and the walls of the glass cellsthis will not be considered further, since the cell walls are assumed to be at infinity with reference to the electrode surfaceswith the electrode being assumed to be at the potential of zero charge), due to the excess negative charge at the anionic lipidic bilayer surface. Accordingly, since the salt is a strong electrolyte,47b the Cs+ ions will “shield” this surfacial potential of the macrocounterions, forming a diffuse layer of Debye length xDL ) [(2F2I)/(RTεε0)]1/2, where I is the ionic strength (ca. 1.3 M), R is the molar gas constant, T is the absolute temperature, ε0 is the permittivity of free space, and ε is the permittivity of the 2H2O phase. Assuming the latter is unchanged via the presence of ionic species,43 ε ) 78.06 at 298 K, so that xDL ) 0.3 nm. This is in agreement with other calculations.47 It has been noted via Brownian dynamics simulations that, in media such as this,52 the Cs+ ions will retain a high self-diffusion rate. As the ferricyanide analyte is hydrophilic, it will likely stay in the 2H2O phase, and, assuming ligand lability does not occur, the potassium counterions will also participate in the double layer.53 When the experiment commences, as a negative potential is applied to the electrode, it will start to take on an electrical double layer structure, and this may cause an osmotic stress in the 2H2O phase.52 Note that the charge on the electrode may be greater or smaller than that on the lipid bilayer surface to which it is opposite. This in turn may lead to transient changes in the number of “edge defects” (or pores) in the bilayer structure during the course of the experiment,12 and ultimately, given the transformation of a species of -3 charge into -4 charge, the ionic strength will change as a function of electrolysis extent. Such effects may lead to an increase in the bilayer separations and/or cause the diffusion coefficients to change with electrolyte concentration.54 These effects would be more pronounced when the perturbation is via a potential step excursion rather than voltammetry, since the former is a transient technique which causes a “shock” to the system, while the slower perturbation rate for voltammetry would allow for sufficient time for the system to adjust to the most stable state. In light of the experimental data, it could be postulated that osmotic repulsions are dominant at short times (since then the current decays faster than expected), while, at longer perturbation times, electroporation effects may become more important (since the observed current is larger than anticipated). The testing of this hypothesis requires a rigorous quantification of the fractal dimension and associated crystal surface of the microelectrode, in addition to the use of a fast risetime potentiostat, should the edge defects be very transient; this is beyond the current scope of this work. Note that the electrical migration of the ferricyanide may not
Diffusion Anisotropy in Molecularly-Structured Materials be important given the small size of the electrical double layers that can be formed in this system. 5. Conclusions Axiosymmetric anisotropic diffusion to disk shaped microelectrodes has been examined. It is found that, for these cases, theoretical expressions for the isotropic case are amenable for determining diffusion anisotropy, provided the temporal domain is characterized by using radial diffusion only, and the isotropic diffusion coefficient is replaced by the geometric mean of diffusion coefficients in each of the two mutually perpendicular directions. The theory has been examined in the light of an experimental system, and anisotropy ratios deduced electrochemically are analogous with those reported in the literature. Acknowledgment. The authors thank The University of Hull for financial support and EPSRC for a DTA studentship for L.A.E. J.W. gratefully acknowledges Professor D. J. Gavaghan (Oxford University) for supplying code for the expanding grid simulations in the isotropic case. The authors additionally thank I. Cutress and R. Bourne (both Hull University) for undertaking preliminary experiments. Supporting Information Available: Derivation of eq 16 from eq 15, and simulation methods and procedures. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) See, for example: de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals, 2nd ed.; Oxford University Press: Oxford, 1993. (2) (a) Bartlett, P. N.; Gollas, B.; Guein, S.; Harwan, J. Phys. Chem. Chem. Phys. 2002, 4, 3835. (b) Walcarius, A.; Sibottier, E.; Etienne, M.; Ghanbaja, J. Nat. Mater. 2007, 6, 602. (3) Schro¨der, U.; Wadhawan, J. D.; Compton, R. G.; Marken, F.; Suarez, P. A. Z.; Consorti, C. S.; de Souza, R. F.; Dupont, J. New J. Chem. 2000, 24, 1009. (4) (a) Amatore, C.; Maisonhaute, E.; Scho¨llhorn, B.; Wadhawan, J. ChemPhysChem 2007, 8, 1321. (b) Amatore, C.; Genovese, D.; Maisonhaute, E.; Raouafi, N.; Scho¨llhorn, B. Angew. Chem. Int. Ed. 2008, 47, 5211. (c) Amatore, C.; Maisonhaute, E.; Nierengarten, J.-F.; Scho¨llhorn, B. Isr. J. Chem. 2008, 48, 203. (d) Amatore, C.; Maisonhaute, E.; Scho¨llhorn, B. Actual. Chim. 2008, 69. (5) Buzzeo, M. C.; Hardacre, C.; Compton, R. G. Anal. Chem. 2004, 76, 4583. (6) Gin, D. L.; Bara, J. E.; Noble, R. D.; Elliott, B. J. Macromol. Commun. 2008, 29, 367. (7) Senthilkumar, S.; Adisa, A.; Saraswathi, R.; Dryfe, R. A. W. Electrochem. Commun. 2008, 10, 141. (8) Tsoi, W. C.; O’Neill, M.; Aldred, M. P.; Kitney, S. P.; Vlachos, P.; Kelly, S. M. Chem. Mater. 2007, 19, 5475. (9) (a) Amatore, C. Chem.sEur. J. 2008, 14, 5449. (b) Amatore, C.; Olenick, A.; Klymenko, O. V.; Delacote, C.; Walcarius, A.; Svir, I. B. Anal. Chem. 2008, 80, 3229. (10) (a) Griffiths, E.; Kaye, G. W. C. Proc. R. Soc. A 1923, 104, 71. (b) Eckert, E. G., Jr. Analysis of Heat and Mass Transfer; MacGraw-Hill: New York, 1972. (11) Ksenzhek, O. S.; Volkov, A. G. Plant Energetics; Academic Press: San Diego, 1998. (12) Jiang, F. Y.; Bouret, Y.; Kindt, J. T. Biophys. J. 2004, 87, 182. (13) Kanoufi, F. Actual. Chim. 2007, 311. (14) (a) Amatore, C.; Maisonhaute, E. Anal. Chem. 2005, 77, 303A. (b) Amatore, C.; Bouret, Y.; Maisonhaute, E.; Abrun˜a, H. D.; Goldsmith, J. I. C. R. Chim. 2003, 6, 99. (15) See, for example: (a) Szunerits, S.; Thouin, L. In Handbook of Electrochemistry; Zoski, C., Ed.; Elseveir: Amsterdam, 2006; p 391. (b) Amatore, C.; Arbault, S.; Maisonhaute, E.; Szunerits, S.; Thouin, L. In New Trends in Molecular Electrochemistry; Amatore, C., Pomberio, A. J., Eds.; FontisMedia: Lausanne, 2004; p 385. (c) Basha, C. A.; Rajendran, L. Int. J. Electrochem. Sci. 2006, 1, 268. (16) The term diffusion is employed here in the sense of a physical movement of a species along a gradient in chemical potential or in a Dahms-Ruff-type electron hopping sense.
J. Phys. Chem. C, Vol. 113, No. 20, 2009 8909 (17) Serra, A. M.; Mariani, R. D.; Abrun˜a, H. D. J. Electrochem. Soc. 1986, 133, 2226. (18) (a) Postlethwaite, T. A.; Samulski, E. T.; Murray, R. W. Langmuir 1994, 10, 2064. (b) Chen, C.; Postlethwaite, T. A.; Hutchinson, J.E.; Samulski, E. T.; Murray, R. W J. Phys. Chem. 1995, 99, 8804. (c) Amatore, C.; Sella, C.; Thouin, L. J. Phys. Chem. B 2002, 106, 11565. (19) Newton, M. R.; Morey, K. A.; Zhang, Y.; Snow, R. J.; Diwekar, M.; Shi, J.; White, H. S. Nano Lett. 2004, 4, 875. (20) Terrill, R. H.; Murray, R. W. In Molecular Electronics; Jortner, J., Ratner, M. A., Eds.; Blackwell Science; Oxford, 1997; p 215. (21) (a) Amatore, C.; Fosset, B. J. Electroanal. Chem. 1992, 328, 21. (b) Olenick, A.; Amatore, C.; Svir, I. B. Electrochem. Commun. 2004, 6, 588. (22) (a) Britz, D. Digital Simulation in Electrochemistry, 2nd ed.; Springer: 2005. (b) Gavaghan, D. J.; Gillow, K.; Suli, E. Langmuir 2006, 22, 10666. (23) (a) Gavaghan, D. J. J. Electroanal. Chem. 1998, 456, 1. (b) Gavaghan, D. J. J. Electroanal. Chem. 1998, 456, 13. (c) Gavaghan, D. J. J. Electroanal. Chem. 1998, 456, 25. (d) See also: Welford, P. J.; Brookes, B. A.; Climent, V.; Compton, R. G. J. Electroanal. Chem. 2001, 513, 8. (24) (a) Butler, J. A. V.; Orr, W. J. C.; Laurence, D. J. R. Chemical Thermodynamics, 4th ed.; Macmillan: London, 1946. (b) Howell, J. O.; Goncalves, J. M.; Amatore, C.; Klasinc, L.; Wightman, R. M.; Kochi, J. K. J. Am. Chem. Soc. 1984, 106, 3968. (25) Kocherginsky, N.; Zhang, Y. K. J. Phys. Chem. B 2003, 107, 7830. (26) (a) Amatore, C.; Olenick, A.; Svir, I. Anal. Chem. 2008, 80, 7947. (b) Feldberg, S. W. J. Electroanal. Chem. 2008, 624, 45. (27) In actuality, diffusion is not only responsible for mass transfer; effects due to migration have been largely ignored heretofore. It is highly possible that this additional transport pathway can occur. These effects have not been incorporated into the current work, since their inclusion requires the solution of Poisson’s equation in the form (∂φ/∂R)(∂ε/∂R) + (∂ε/∂Z)(∂φ/∂Z) + ε∇2φ )-F∑i zici, where φ is the (local) electrostatic potential, F is the Faraday constant, zi and ci are the charge and concentration of species i, respectively, and ε is the permittivity of the medium. This form of Poisson’s equation recognizes the heterogeneity of the medium; its solution is not trivial and will be reported in a future work from this laboratory. Nevertheless, it suffices to note that normally under classical voltammetric conditions, where the inert background electrolyte is in a sufficiently high concentration, the size of the electrical double layer is considered to be vanishingly small compared with the thickness of the diffusion layer, so that effects due to the latter can be considered merely via a simple Frumkin correction to the electrode boundary condition (q.V. Frumkin, A. N. Z. Phys. Chem. 1933, 164, 121 and Parsons, R. Trans. Faraday Soc. 1951, 47, 1332). This approximation ceases to be valid when the diffusion layers are considered to be thin and when the electrolyte concentration is small. For the experimental case considered in this work, the electrolyte concentration is largesthe surfactant is ionic, and the double layer is approximately one order of magnitude smaller than the lipid bilayer. Accordingly, electrophoretic effects, primarily observed under conditions of very short time scales (when the currents are large), are not realized herein, owing to the bandwidth and rise time of the commercial equipment employed. Nevertheless, other electrokinetic phenomena may be observed; q.V. section 4. (28) See, for example: (a) Crank, J. The Mathematics of Diffusion, 2nd ed.; Oxford University Press: Oxford, 1975. (b) Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids, 2nd ed.; Oxford University Press: Oxford, 1959. (29) (a) Saito, Y. ReV. Polarogr. 1968, 15, 177. (b) Crank, J.; Furzeland, R. M. J. Inst. Math. Its Appl. 1977, 20, 355. (30) (a) Aoki, K.; Osteryoung, J. G. J. Electroanal. Chem. 1981, 122, 19. Correction: 1984, 160, 335. (b) Shoup, D.; Szabo, A. Z. J. Electroanal. Chem. 1982, 140, 237. (31) Compton, R. G.; Banks, C. E. Understanding Voltammetry; World Scientific: London, 2007. (32) (a) Aoki, K. Electroanalysis 1993, 5, 627. (b) Wightman, R. M.; Wipf, D. O. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1989; Vol. 15, p 267. (c) Szabo, A. Z.; Cope, D. K.; Tallman, D. E.; Kovach, P. M.; Wightman, R. M. J. Electroanal. Chem. 1987, 217, 417. (33) (a) Amatore, C.; Bouret, Y.; Maisonhaute, E.; Goldsmith, J. I.; Abrun˜a, H. D. Chem.sEur. J. 2001, 7, 2206. See, for example:(b) Streeter, I.; Compton, R. G. J. Phys. Chem. C 2007, 111, 18049. (c) Amatore, C.; Olenick, A.; Svir, I. B. J. Electroanal. Chem. 2005, 575, 103. et hoc genus omne. (34) (a) Weber, H. J. Reine Angewan. Math. 1873, 75, 75. (b) Weber, H. J. Reine Angewan. Math. 1873, 76, 1. (c) Gray, A.; Matthews, G. B.; MacRobert, T. M. A Treatise on Bessel Functions and their Applications to Physics, 2nd ed.; Macmillan: London, 1922. (35) Gradshtein, I. S.; Ryzhik, I. M. Tables of Integrals, Series and Products, 2nd ed.; Academic Press: New York, 1980.
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(36) Michael, A. C.; Wightman, R. M.; Amatore, C. A. J. Electroanal. Chem. 1989, 267, 33. (37) A comprehensive comparison of all conformal maps for disk ultramicroelectrodes is quoted in truth-table format in ref 21b. (38) Newman, J. J. Electrochem. Soc. 1966, 113, 501. (39) An alternative representation is
2 c ) arctan c0 π
[ {
1 Dr 2 Z + R2 - 1 + 2 Dz
(
)
}]
2 Dr Dr 2 Z + R 2 - 1 + 4 Z2 Dz Dz
(40) Deakin, M. R.; Wightman, R. M.; Amatore, C. A. J. Electroanal. Chem. 1986, 215, 49. (41) Boden, N.; Corne, S. A.; Jolley, K. W. J. Phys. Chem. 1987, 91, 4092. (42) (a) Bard and co-workers42b have suggested that changing the viscosity of a solution can affect the heterogeneous electron transfer kinetics, since it affects the dynamic properties of the solvent. (b) Zhang, X.; Yang, H.; Bard, A. J. J. Am. Chem. Soc. 1987, 109, 1916; see also ref 42c. (c) Clegg, A. D.; Rees, N. V.; Klymenko, O. V.; Coles, B. A.; Compton, R. G. J. Am. Chem. Soc. 2004, 126, 6185. (43) (a) Lide, D. R., Ed. CRC Handbook of Chemistry and Physics, 76th ed.; CRC Press: Boca Raton, FL, 1995. (b) Note that the value quoted is for solutions made using 1H2O; 2H2O is less viscous,43c so diffusion coefficients are expected to be larger. (c) Bell, R. P. The Proton in Chemistry, 2nd ed.; Chapman & Hall: London, 1973. (44) Mirkin, M. V.; Bard, A. J. Anal. Chem. 1992, 64, 2293. (45) (a) Oldham et al.45b report the current for a quasi-reversible electron transfer process at microdisk electrodes within the framework of the Butler-Volmer electrode kinetics formulation as being i/ilim ) (1/θ){1 + (π/κθ)[(2κθ + 3π)/(4κθ + 3π2)]}-1, where the current i is normalized by the limiting current ilim, with θ being a dimensionless potential parameter, θ ) 1 + (DOx/DRed) exp[nF/RT(E- E0′)], in which Di is the diffusion coefficient of species i, and in this case, is assumed identical, and the adimensional kinetics parameter κ ) κ0 exp[-R(nF)/ (RT)(E- E0′)], where the normalized standard heterogeneous rate constant is given by the expression κ0 ) πksapparentr0/4DOx. (b) Oldham, K. B.; Myland, J. C.; Zoski, C. G.; Bond, A. M. J. Electroanal. Chem. 1989, 270, 79. (46) Indeed, the observed anisotropy ratio for Cs+ measured using ionic conductivity is in the region 1.4-1.6. This range is that observed herein, within experimental error.
(47) (a) Leaver, M. S.; Holmes, M. C. J. Phys. II 1993, 3, 105. (b) Boden, N.; Corne, S. A.; Jolley, K. W. Chem. Phys. Lett. 1984, 105, 99. (c) Holmes, M. C.; Reynolds, D. J.; Boden, N. J. Phys. Chem. 1987, 91, 5257. (d) Boden, N.; Parker, D.; Jolley, K. W. Mol. Cryst. Liq. Cryst. 1987, 152, 121. (48) (a) Brookes, B. A.; Davies, T. J.; Fisher, A. C.; Evans, R. G.; Wilkins, S. J.; Yunus, K.; Wadhawan, J.; Compton, R. G. J. Phys. Chem. B 2003, 107, 1616. (b) Davies, T. J.; Brookes, B. A.; Fisher, A. C.; Yunus, K.; Wilkins, S. J.; Greene, P. R.; Wadhawan, J.; Compton, R. G. J. Phys. Chem. B. 2003, 107, 6431. (c) Amatore, C.; Save´ant, J.-M.; Tessier, D. J. Electroanal. Chem. 1983, 147, 39. (49) Amatore, C. Chem.sEur. J. 2008, 14, 5449. (50) (a) Blauch, D. N.; Save´ant, J.-M. J. Am. Chem. Soc. 1992, 114, 3323. (b) Blauch, D. N.; Save´ant, J.-M. J. Phys. Chem. 1993, 97, 6444. (c) Andrieux, C. P.; Save´ant, J.-M. J. Electroanal. Chem. 1980, 111, 377. (d) Laviron, E. J. Electroanal. Chem. 1980, 112, 1. (51) (a) The rate of electron hopping will depend on the thickness and chemical structure of the lamellae, since the probability of tunneling through a Gamow barrier decreases exponentially with the barrier height.51b In modern theories of electron transfer, this probability is manifested as the square of the overlap matrix, which is proportional to e-β(z-ζ), where β is the decay constant for tunneling and ζ is the distance of closest approach (Viz. the bilayer thickness). For proteins and organic species, β is in the range 1.0-1.1 Å-1. A unified theory for electron transfer and electronic conduction through an intervening medium is given in ref 51c. (b) Gamow, G. Z. Phys. 1928, 51, 204. (c) Edwards, P. P.; Gray, H. B.; Lodge, M. T. J.; Williams, R. J. P. Angew. Chem., Int. Ed. 2008, 47, 6758. See also, ref 51d, Figure 1.5. (d) Moser, C. C.; Dutton, P. L. In Protein Electron Transfer; Bendall, D. S., Ed.; BIOS Scientific: Oxford, 1996; p 1. (52) (a) Marry, V.; Gru¨n, F.; Simon, C.; Jardat, M.; Turq, P.; Amatore, C. J. Phys.: Condens. Matter 2002, 14, 9207. (b) Sogami, I.; Ise, N. J. Chem. Phys. 1984, 81, 6320. (c) Durand-Vidal, S.; Simonin, J.-P.; Turq, P. Electrolytes at Interfaces; Kluwer: Dordrecht, 2000. (53) (a) Cs+ is comparable to K+ssee refs 53b and 53c. (b) Peter, L. M.; Durr, W.; Bindra, P.; Gerischer, H. J. Electroanal. Chem. 1976, 71, 31. (c) Bindra, P.; Gerischer, H.; Peter, L. M. J. Electroanal. Chem. 1974, 57, 435. (54) Amatore, C.; Paulson, S. C.; White, H. S. J. Electroanal. Chem. 1997, 439, 173.
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