A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 45
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We will consider our instantaneous step to terminate in this region.(a) Solution of the Diffusion EquationThe calculation of the diffusion-limited current, i^ and the concentration profile, CQ(X, t),involves the solution of the linear diffusion equation:) _d2Co{x, t)nunder the boundary conditions:Co(x, 0) = C%C o (0,f) = 0(5.2.2)(forr>0)(5.2.4)The initial condition, (5.2.2), merely expresses the homogeneity of the solution before the experiment starts at t = 0, and the semi-infinite condition, (5.2.3), is an assertionthat regions distant from the electrode are unperturbed by the experiment. The third condition, (5.2.4), expresses the condition at the electrode surface after the potential transition, and it embodies the particular experiment we have at hand.Section A.
1.6 demonstrates that after Laplace transformation of (5.2.1), the application of conditions (5.2.2) and (5.2.3) yieldsCo(x, s) = ^+ A(s) e-^^o*(5.2.5)By applying the third condition, (5.2.4), the function A(s) can be evaluated, and thenCQ(X, S) can be inverted to obtain the concentration profile for species O. Transforming(5.2.4) givesC o (0, s) = 0(5.2.6)which implies that^^VJ(5.2.7)^In Chapter 4, we saw that the flux at the electrode surface is proportional to the current;specifically,]х=0which is transformed toThe derivative in (5.2.9) can be evaluated from (5.2.7).
Substitution yieldsl(s) =:£—^(5.2.10)5.2 Potential Step Under Diffusion Control163and inversion produces the current-time response(5.2.11)which is known as the Cottrell equation (3). Its validity was verified in detail by the classic experiments of Kolthoff and Laitinen, who measured or controlled all parameters (1,2). Note that the effect of depleting the electroactive species near the surface leads to aninverse t^2 function.
We will encounter this kind of time dependence frequently in otherkinds of experiments. It is a mark of diffusive control over the rate of electrolysis.In practical measurements of the i-t behavior under "Cottrell conditions" one must beaware of instrumental and experimental limitations:1. Potentiostatic limitations. Equation 5.2.11 predicts very high currents at shorttimes, but the actual maximum current may depend on the current and voltageoutput characteristics of the potentiostat (Chapter 15).2. Limitations in the recording device. During the initial part of the current transient, the oscilloscope, transient recorder, or other recording device may be overdriven, and some time may be required for recovery, after which accuratereadings can be displayed.3. Limitations imposed by Ru and Cd.
As shown in Section 1.2.4, a nonfaradaic current must also flow during a potential step. This current decays exponentially witha cell time constant, RuCd (where Rn is the uncompensated resistance and Q isthe double-layer capacitance). For a period of about five time constants, an appreciable contribution of charging current to the total measured current exists, andthis superimposed signal can make it difficult to identify the faradaic current precisely.
Actually, the charging of the double layer is the mechanism that establishes a change in potential; hence the cell time constant also defines the shortesttime scale for carrying out a step experiment. The time during which data are collected after a step is applied must be much greater than RuCd if an experiment isto fulfill the assumption of a practically instantaneous change in surface concentration at t = 0 (see Sections 1.2.4 and 5.9.1).4. Limitations due to convection.
At longer times the buildup of density gradientsand stray vibrations will cause convective disruption of the diffusion layer, andusually result in currents larger than those predicted by the Cottrell equation.The time for the onset of convective interference depends on the orientation ofthe electrode, the existence of a protective mantle around the electrode, and otherfactors (1, 2). In water and other fluid solvents, diffusion-based measurementsfor times longer than 300 s are difficult, and even measurements longer than 20 smay show some convective effects.(b) Concentration ProfileInversion of (5.2.7) yields(5 .2.,2)or(5.2.13)164Chapter 5. Basic Potential Step Methods0.40.81.21.6x, cm x 10 32.02.42.8Figure 5.2.1 Concentration profiles for several times after the start of a Cottrell experiment.
Do= 1 X 1(Г 5 cm2/s.Figure 5.2.1 comprises several plots from (5.2.13) for various values of time. The depletion of О near the electrode is easily seen, as is the time-dependent falloff in the concentration gradient at the electrode surface, which leads to the monotonically decreasing idfunction of (5.2.11).One can also see from Figure 5.2.1 that the diffusion layer, that is the zone nearthe electrode where concentrations differ from those of the bulk, has no definite thickness. The concentration profiles asymptotically approach their bulk values. Still, it isuseful to think about the thickness in terms of (D o 0 1 / 2 > which has units of length andcharacterizes the distance that species О can diffuse in time t.
Note that the argumentof the error function in (5.2.13) is the distance from the electrode expressed in units of2(Dot) ' . The error function rises very rapidly toward its asymptote of 1 (see SectionA.3). When its arguments are 1, 2, and 3 (i.e., when x is 2, 4, and 6 times (£>o01/2)> ithas values, respectively, of 0.84, 0.995, and 0.99998; thus the diffusion layer is completely contained within a distance of 6(Dot)1^2 from the electrode. For most purposes,one can think of it as being somewhat thinner.
People often talk of a diffusion layerthickness, because there is a need to describe the reach of the electrode process into thesolution. At distances much greater than the diffusion layer thickness, the electrodecan have no appreciable effect on concentrations, and the reactant molecules therehave no access to the electrode.
At distances much smaller, the electrode process ispowerfully dominant. Even though no consistent or accepted definition exists, peopleoften define the thickness as 1, 2 1/2 , тг1/2, or 2 times (£>o01/2- Any °f these ideas suffices. We have already seen diffusion lengths defined in different ways in Sections1.4.3 and 4.4.1.Of course the thickness of the diffusion layer depends significantly on the time scaleof the experiment, as one can see in Figure 5.2.1. For a species with a diffusion coefficientof 1 X 10~5 cm2 s" 1 , (Dot)1/2 is about 30 fxm for an experimental time of 1 s, but only 1ixm at 1 ms, and just 30 nm at 1 ^ts.5.2 Potential Step Under Diffusion Control5.2.2165Semi-Infinite Spherical DiffusionIf the electrode in the step experiment is spherical rather than planar (e.g., a hanging mercurydrop), one must consider a spherical diffusion field, and Fick's second law becomes2дСо(г, t)^(д Со(г,D°[r2 <?Q)( 't)2~dr+T<^~where r is the radial distance from the electrode center.
The boundary conditions are thenC0(r, 0) = CQlim C0(r,f) = C$Co(r0,t) = 0(r > r 0 )(5.2.15)(5.2.16)(f>0)(5.2.17)where r 0 is the radius of the electrode.(a) Solution of the Diffusion EquationThe substitution, v(r9 t) = rCo(r, t), converts (5.2.14) into an equation having the sameform as the linear problem. The details are left to the reader (Problem 5.1). The resultingdiffusion current isid(0 = nFADoC% \—±—+1(5.2.18)which can be written/d(spherical) = /d(linear) HnFADoC%(5.2.19)Thus the diffusion current for the spherical case is just that for the linear situation plus aconstant term.
For a planar electrode,lim id =0(5.2.20)but in the spherical case,_ nFADoC%The reason for this curious nonzero limit is that one converges on a situation in whichthe growth of the depletion region fails to affect the concentration gradients at the surfacebecause the diffusion field is able to draw material from a continually larger area at itsouter limit. In actual experiments with working electrodes of millimeter diameters orlarger, convection caused by density gradients or vibration becomes important at longertimes and enhances the mass transfer, so that the diffusive steady state is rarely reached.On the other hand, it is easy to reach this condition with UMEs (radius of 25 fim orsmaller), and the ability to exploit the steady state is one of their principal advantages (seeSection 5.3).(b) Concentration ProfileThe distribution of the electroactive species near the electrode also can be obtained fromthe solution to the diffusion equation, and it turns out to beГ"(5.2.22)166Chapter 5.
Basic Potential Step MethodsBecause r - r 0 is the distance from the electrode surface, this profile strongly resemblesthat for the linear case (equation 5.2.12). The difference is the factor ro/r and, if the diffusion layer is thin compared to the electrode's radius, the linear and spherical cases are indistinguishable. The situation is directly analogous to our experience in living on aspherical planet. The zone of our activities above the earth's surface is small compared toits radius of curvature; hence we usually cannot distinguish the surface from a roughplane.At the other extreme, when the diffusion layer grows much larger than r0 (as at aUME), the concentration profile near the surface becomes independent of time and linearwith 1/r. One can see this effect in (5.2.22), where error function complement approachesunity for (r - r0) < < 2(D o 0 1/2 . In that case,C0(r, t) = Cg(l - ro/r)(5.2.23)The slope at the surface is CgAo, which gives the steady-state current, (5.2.21), from thecurrent-flux relationship for the spherical case,dr(5.2.24)(c) Applicability of the Linear ApproximationThese ideas indicate that linear diffusion adequately describes mass transport to a sphere,provided the sphere's radius is large enough and the time domain of interest is smallenough.
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