A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 47
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The critical dimension is, of course, the radius, TQ, which must besmaller than —25 /mi. Electrodes made of Pt wire with a radius of 5 fxm are commercially available. Disks with r 0 as small as 0.1 /im have been made from wire, and disklike exposed areas with dimensions in the range of a few nm have been inferred forelectrodes made by other means. The geometric area of a disk scales with the square ofthe critical radius and can be tiny. For r 0 = 1 /im, the area, A, is only 3 X 10~ 8 cm 2 , sixorders of magnitude smaller than the geometric area of a 1-mm diameter microelectrode.
The very small scale of the electrode is the key to its special utility, but it alsoimplies that the current flowing there is quite low, often in the range of nanoamperes orpicoamperes, sometimes even in the range of femtoamperes. As we will see in Section5.9 and Chapter 15, the small currents at UMEs offer experimental opportunities as wellas difficulties.Spherical UMEs can be made for gold (16), but are difficult to realize for other materials. Hemispherical UMEs can be achieved by plating mercury onto a microelectrodedisk. In these two cases the critical dimension is the radius of curvature, normally symbolized by r 0 . The geometry of these two types is simpler to treat than that of the disk, but inmany respects behavior at a disk is similar to that at a spherical or hemispherical UMEwith the same r 0 .Quite different is the band UME, which has as its critical dimension a width, w, in therange below 25 /mi.
The length, /, can be much larger, even in the centimeter range. BandUMEs can be fabricated by sealing metallic foil or an evaporated film between glassplates or in a plastic resin, then exposing and polishing an edge. A band can also be produced as a microfabricated metallic line on an insulating substrate using normal methodsof microelectronic manufacture. By these means, electrodes with widths ranging from 25fim to about 0.1 /гт can be obtained. The band differs from the disk in that the geometricarea scales linearly with the critical dimension, rather than with the square. Thus electrodes with quite small values of w can possess appreciable geometric areas and can produce sizable currents.
For example, a band of l-/mi width and 1-cm length has ageometric area of 10~4 cm 2 , almost four orders of magnitude larger than that of a l-/midisk.A cylindrical UME can be fabricated simply by exposing a length / of fine wire withradius r 0 . As in the case of the band, the length can be macroscopic, typically millimeters.The critical dimension is r 0 .
In general, the mass-transfer problem to a cylindrical UME issimpler than that to a band, but operationally, there are many similarities between a cylinder and a band.5.3 Diffusion-Controlled Currents at Ultramicroelectrodes171Responses to a Large-Amplitude Potential StepLet us consider an ultramicroelectrode in a solution of species O, but initially at a potentialwhere О is not reduced. A step is applied at / = 0, so that О becomes reduced to R at thediffusion-controlled rate.
What current flows under these Cottrell-like conditions?(a) Spherical or Hemispherical UMEWe already know the current-time relationship for the simplest case, the sphere, whichwas treated fully in Section 5.2.2. The result was given in (5.2.18) asHFAD^CQnFADoC$l/2>l/2' *Fnwhere the first term dominates at short times, when the diffusion layer is thin compared tor 0 , and the second dominates at long times, when the diffusion layer grows much largerthan r 0 . The first term is identically the Cottrell current that would be observed at a planarelectrode of the same area, and the second describes the steady-state current flowachieved late in the experiment.
The steady-state condition is readily realized at a UME,where the diffusion field need only grow to a thickness of 100 ^ m (or perhaps even muchless). Many applications of UMEs are based on steady-state currents.At the sphere, the steady-state current /ss is,LrvHFADQCQlss(5.3.2a)r0'Uoriss = 47rnFDoCor0(5.3.2b)A hemispherical UME bounded by a planar mantle has exactly half of the diffusionfield of a spherical UME of the same rg, so it has half of the current of the correspondingsphere. Equation 5.3.2a compensates for the difference through the proportionality witharea, so it is accurate for the hemisphere as well as the sphere. Equation 5.3.2b appliesonly to the sphere.(b) Disk UMEThe disk is by far the most important practical case, but it is complicated theoretically bythe fact that diffusion occurs in two dimensions (17, 18): radially with respect to the axisof symmetry and normal to the plane of the electrode (Figure 5.3.1).
An important consequence of this geometry is that the current density is not uniform across the face of thedisk, but is greater at the edge, which offers the nearest point of arrival to electroreactantdrawn from a large surrounding volume. We can set up this problem in a manner similarto our approach to the one-dimensional cases of Section 5.2.
The diffusion equation forspecies О is written as follows for this geometry (see Table 4.4.2):dCo(r,z,t)_dtUlp 2 C o (r,z,Q2дг!+Г'dC0(r,z,t)дг+<?2Co(r,z,Ql2dzJ&•*•»where r describes radial position normal to the axis of symmetry at r = 0, and z describeslinear displacement normal to the plane of the electrode at z = 0.Five boundary conditions are needed for a solution. Three come from the initial condition and two semi-infinite conditionsC0(r, z, 0) = Cglim Co(r, z, t) = Cnlim Co(r, z, t) = С%(5.3.4)(5.3.5)172Chapter 5. Basic Potential Step Methodsz axis (r = 0)r axis (z = 0)Inlaid UME diskMantle in z = 0 plane (extendsbeyond diffusion layer boundary)Figure 5.3.1 Geometry ofdiffusion at an ultramicroelectrodediskA fourth condition comes from the recognition that there can be no flux of О into or out ofthe mantle, since О does not react there:dC0{r, z, г)dzz=0= 0(5.3.6)(r > r 0 )The conditions defined to this point apply for any situation in which the solution is uniform before the experiment begins and in which the electrolyte extends spatially beyondthe limit of any diffusion layer.
The final condition defines the experimental perturbation.In the present case, we are considering a large-amplitude potential step, which drives thesurface concentration of О to zero at the electrode surface after t = 0.Co(r, 0, t) = 0(r < r 0 , t > 0)(5.3.7)This problem can be simulated in the form given here (19, 20), but an analytical approach is best made by restating it in terms of other coordinates. In no form is it a simpleproblem. Aoki and Osteryoung (21) addressed it in terms of a dimensionless parameter,т = 4Dot/rQ29 representing the squared ratio of the diffusion length to the radius of thedisk.
Given any particular experimental system, т becomes an index of t. The current-timecurve is(5.3.8)гоwhere the function/(т) was determined as two series applicable in different domains of т(21-23). At short times, when т < 1,f(r) =rl/277'„(5.3.9a)4or, with the constants evaluatedf{r) = 0.88623т" 1/2 + 0.78540 + 0.094т1/2At long times, when т > I,/(T) = 1 + 0.71835т~2(5.3.9b)21/2+ 0.05626т~3/2- 0.00646т"5/2• • •(5.3.9c)Aoki and Osteryoung (23) show that the two versions of/(r) overlap for 0.82 < т < 1.44. The dividing pointgiven in the text is convenient and appropriate.5.3 Diffusion-Controlled Currents at Ultramicroelectrodes173Shoup and Szabo provided a single empirical relationship covering the entire range of r,with an accuracy better than 0.6% at all points (22):/(r) = 0.7854 + 0.8862r" 1/2 + 0.2146^" a 7 8 2 3 r ~ 1 / 2(5.3.10)The current-time relationship for a UME disk spans three regimes, as shown in Figure 5.3.2.
If the experiment remains on a short time scale (Figure 5.3.2a), so that thediffusion layer remains thin compared to TQ, the radial diffusion does not manifest itselfappreciably, and the diffusion has a semi-infinite linear character. The early currentflowing in response to a large amplitude potential step is therefore the Cottrell current,(5.2.11). This intuitive conclusion is illustrated graphically in Figure 5.3.2a, where thetwo sets of symbols are superimposed.
One can also see it mathematically as the limitof (5.3.8) and (5.3.9a) when т approaches zero. For an electrode with r 0 = 5 fxm andDo = 10~5 cm2/s, the short-time region covered in Figure 5.3.2a is 60 ns to 60 /JUS. Inthis period, the diffusion layer thickness [taken as 2(Dot)l/2] grows from 0.016 ^ m to0.5 jam.As the experiment continues into an intermediate regime where the diffusion layerthickness is comparable to r 0 , radial diffusion becomes important. The current is largerthan for a continuation of pure linear diffusion, that is, where this "edge effect" (17) could1.0E-51.0Е-41 .OE-31.0Е-2{a) Short time regime0.010.1010.001.00(b) Intermediate time regime1.4,1.21.0• • ••Q _ XL _Q _НАААП_ XL _• _ XL _J0.80.60.40.21-o.o I—10100Н1000(c) Long time regimeП10000Figure 5.3.2 Current-timerelationships at a disk UME.Current is expressed as ///ss andtime is expressed as т, which isproportional to t.
Triangles, Cottrellcurrent. Filled squares, (5.3.8) and(5.3.9b). Open squares, (5.3.8) and(5.3.9c). Dashed line at ///ss = 1 issteady-state.174 • Chapters. Basic Potential Step Methodsbe prevented. Figure 532b illustrates the result. For r 0 and Do values of 5 ^ m and 10~5cm2/s, this frame corresponds to an experimental time between 60 ^s and 60 ms. The diffusion layer thickness is in the range from 0.5 д т to 16 /xm.At still longer times, when the diffusion field grows to a size much larger than r 0 , itresembles the hemispherical case and the current approaches a steady state (Figure5.3.2c).
For the specific values of r 0 and Do used as examples above, Figure 5.3.2c describes the time period from 60 ms to 60 s, when the diffusion layer thickness enlargesfrom 16 jam to 500 /xm.3The experimental time ranges discussed here relate to practical values of electrode radius and diffusion coefficient and are all readily accessible with standard commercialelectrochemical instrumentation. A distinguishing feature of a UME is the ability to operate in different mass-transfer regimes. Indeed, we used, in essence, the ability to approachor to achieve the steady-state as the basis for our operational definition of a UME in theopening paragraphs of this section.The steady state for the disk can be seen easily as the limit of (5.3.8) and (5.3.9c)when T becomes very large,(5.3.11)It has the same functional form as for the sphere or hemisphere, however the /ss at a diskis smaller (by a factor of 2/тг) than at a hemisphere with the same radius.
This differencemanifests the different shapes of the concentration profiles near the electrode surface.4In the intermediate and late time regimes, the current density at a UME disk is intrinsically nonuniform because the edges of the electrode are more accessible geometricallyto the diffusing electroreactant (17). This non-uniformity affects the interpretation of phenomena that depend on local current density, such as heterogeneous electron-transfer kinetics or the kinetics of second-order reactions involving electroactive species in thediffusion layer.(c) Cylindrical UMEWe return to a simpler geometry by considering a cylindrical electrode, which involvesonly a single dimension of diffusion.
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