J. Phys. Chem. B 2007, 111, 6761-6765
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Dimensional Analysis of Chronoamperometry: Proof and Verification of Cottrellian Behavior of Systems with Composition-Dependent Diffusion Coefficients† Stephen W. Feldberg* Chemistry Department, BrookhaVen National Laboratory, Upton, New York 11973 ReceiVed: December 22, 2006; In Final Form: February 13, 2007
Chronoamperometry with a Dirichlet boundary condition and semi-infinite linear diffusion exhibits Cottrellian behavior, that is, the product it1/2 is constant as a function of time as long as the system is initially homogeneous, a conclusion that can be reached using only dimensional analysis; no detailed mathematical analysis is required. The generality of this result is known to include purely diffusional systems and systems in which transport also involves migration. In the present work, it is shown that Cottrellian behavior obtains, even when the system diffusion coefficients are a function of system composition, regardless of the exact form of that function. These conclusions are confirmed by simulations of examples for purely diffusional systems as well as for systems with migration. Some experimental examples from the literature are cited.
Introduction Dimensional Analysis and Proof. Dimensional analysis can provide the functional relationship among the physical parameters describing a physical chemical system.1 The electrochemical protocol known as chronoamperometry2 is typically executed on a system comprising a soluble redox moiety, for example, Ox, with a bulk concentration, cox,bulk. The concentration of Ox at the electrode surface, cox,x)0, is instantaneously changed to a new, constant Value by instantaneously changing the electrode potential (in this type of problem, the constant interfacial value is referred to as the Dirichlet boundary condition). The result of the interfacial perturbation is a flow of current associated with the interfacial redox process (Ox + e ) Red). In the present work, the focus is on systems in which cox,x)0 and semi-infinite linear diffusion obtains. When the system is initially homogeneous and the diffusion coefficients are spatially and temporally constant, Cottrellian behavior obtains, that is, the system response can be characterized by the single dimensionless parameter (DP), P1, defined by
P1 )
|fe|t1/2 D1/2 ox cox,bulk
)
|i|t1/2 aFD1/2 ox cox,bulk
(1)
where fe (moles cm-2 s-1) is the flux of electrons across the electrode/medium interface; t (s) is the time after the initial, instantaneous establishment of the Dirichlet boundary condition; Dox,bulk (cm2 s-1) is the diffusion coefficient; F (C mol-1) is Faraday’s constant; a (cm2) is the electrode area; and i (A) is the current. Buckingham’s Π theorem1,3 tells us that given npp physical parameters and npu physical units, only npp - npu DPs can be constructed. For this system, there are four physical parameters (fe, t, Dox,bulk and cox,bulk) and three physical units (cm, s, and moles); thus only one DP can be constructed. This result has been long known for purely diffusional semi-infinite linear transport for which P1 ) 1/xπ. Readers may be less †
Part of the special issue “Norman Sutin Festschrift”. * To whom correspondence should be addressed. E-mail: Feldberg@ BNL.gov.
familiar with the work of Andrieux and Save´ant,4 who pointed out that P1 is also constant for systems with migrational transport, including systems in which electron-hopping contributes to the effective transport of the redox moieties.5 The numerical value of P1 will vary depending upon the charges and diffusion coefficients of the various species as well as upon the kinetics for electron-hopping. There is now a burgeoning interest in electrochemistry in ionic liquid systems, often involving very high concentrations of a redox analyte, and in some cases, a redox moiety that is one of the component ions of the ionic liquid itself (see, e.g., Wang et al.6 who studied the electrochemical behavior of ferrocenated imidazolium ionic liquids). In these types of systems, it seems reasonable to expect that electrochemically induced changes in composition will induce significant changes in the viscosity and, consequently, in the diffusion coefficients of some or all of the species (see, for example, the recent study of Matsumoto et al.7). One might also expect the rate constant for electron-hopping to depend upon the electrochemically induced structure changes in systems in which the redox species are virtually immobile and electron-hopping effects an apparent, diffusive-like, movement of the redox species5 along with the concomitant movement of counter-ions required to sustain electroneutrality.8 In the present work, I examine what happens to the chronoamperometric response when the diffusion coefficient is a function of the electrochemically induced changes in the composition of an initially homogeneous system: transport may be governed by diffusion and migration, as described by the Nernst-Planck equation2 or by Save´ant’s modified expression describing electron-hopping.5 The notion that diffusion coefficients can be a function of the composition of a system is hardly new (or surprising) and has long been a concern of polymer chemists (see, e.g., Crank and Park, “Diffusion in Polymers”).9 Morris, Fischer, and White10 and Stevenson and White11 have examined the effects of composition-dependent diffusion coefficients in electrochemical systems. Although they do some detailed analysis, they do not discuss the generality of the behavior predicted by dimensional analysis, which can prove that Cottrellian behavior must obtain for a chronoamperometric response. This conclusion may seem a bit anti-intuitive, but the
10.1021/jp0688963 CCC: $37.00 © 2007 American Chemical Society Published on Web 05/10/2007
6762 J. Phys. Chem. B, Vol. 111, No. 24, 2007
Feldberg
Figure 1. Plots of xπP1 vs log[N] for γ1 ) 0, 1, 10.0, and 100.0 with Dox,bulk∆t/∆x2 ) 10.
results are confirmed by simulations of systems, with and without migration, assigning different specified functionalities for the dependence of the diffusion coefficients on the system composition. This will be discussed in detail in the next segment. It is instructive to distinguish the behavior of a system that is initially heterogeneous. Two examples come to mind: 1. A microelectrode operating in the brain tissue of a rat, as discussed by Amatore and Wightman,12 and 2. An ultramicroelectrode small enough to probe the nonbulk properties of water within nanometers of the electrode surface, as suggested by and White and co-workers.13,14 Dimensional analysis can also show that Cottrellian behavior may not obtain for the initially heterogeneous system. Consider a simple example analogous to that discussed by Amatore and Wightman12 in which the diffusion coefficient has one value for x > δ and another for x e δ , e.g.: Dox,x>δ ) Dox,bulk and Dox,xeδ * Dox,bulk. Two new physical parameters have been added to the system, δ and Dox,xeδ. The number of physical parameters, npp, is now six while the number of physical units, npu, is still three. Thus, Buckingham’s Π theorem tells us that there must be two additional DPs, P2 and P3, which can be defined by
P2 )
Dox,bulkt δ2
(2)
and
P3 )
Dox,xeδ Dox,bulk
Figure 2. Plots of xπP1 vs log[N] for γ2 ) 0, 1, 10.0, and 100.0; Dox,bulk∆t/∆x2 ) 1000.
Figure 3. Plots of xπP1 vs log[N] for γ3 ) 10 with Dox,bulk∆t/∆x2 ) 10 and γ4 ) 10 with Dox,bulk∆t/∆x2 ) Dred,bulk∆t/∆x2 ) 100.
to use the implicit Euler method (also known as the backward Euler16 or Laasonen17 method), which can accept a wide dynamic range of the dimensionless diffusion parameter, Dox,bulk∆t/∆x2. The backward Euler is known for its stability and ease of programming but it is also known for slow convergence. Programs were written in QuickBasic, and simulations were executed on a PC for systems comprising only a single species, Ox. The boundary condition at t g 0 is that cox,x)0 ) 0, and the flux at any distance from the electrode, fox,x, is given by Fick’s law: fox,x ) -Dox,x dcox,x/dx. The functionality of the diffusion coefficient, Dox,x, was described using the following two equations:
(3)
P1 can now be a function of P2 and P3 and, therefore, is not necessarily constant. The relevant relationships could then be displayed as a plot of P1 vsP2 for different values of P3.15 Simulations. The dimensional analysis just discussed in the previous segment proved that P1 (see eq 1) remains constant during a chronoamperometric experiment, even when a diffusion coefficient depends upon the composition of the system, as long as the system is initially homgeneous. Because that result may be viewed as surprising or at least not intuitively obvious, simulations were executed with the objective of confirming the constancy of P1 for some selected examples. The simplest system to consider is that comprising a single species, Ox with the Dirichlet boundary condition cox,x)0 ) 0. Semi-infinite linear geometry obtains. For a convincing demonstration that P1 remains constant, even when the diffusion coefficient may be dramatically changing within the depletion region, it is necessary to implement an implicit finite difference simulation. I’ve opted
Dox,x ) Dox,bulk
1 + γ1 cox,x 1 + γ1 cox,bulk
(4)
or
Dox,x ) Dox,bulk
1 + γ2
cox,x
cox,bulk 1 + γ2
(5)
The design of these functions is not based on any particular physical chemical model. However, certain properties of the functions were deemed essential: • Dox,x/Dox,bulk must be a continuous function of cox,x/cox,bulk, • Dox,x/Dox,bulk must approach unity when cox,x/cox,bulk approaches unity, and • Dox,x)0/Dox,bulk must be positive and finite when cox,x/cox,bulk ) 0. With eq 4, Dox,x)0/Dox,bulk ) (1 + γ1) when cox,x)0/cox,bulk
Dimensional Analysis of Chronoamperometry
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TABLE 1: Tabulations of Simulation Parameters and the Corresponding Limiting Value of xπP1 no.
cA,bulk
DA,bulk × ∆t/∆x2
ZA
cB,bulk
DB,bulk × ∆t/∆x2
ZB
cC,bulk
DC,bulk × ∆t/∆x2
ZC
γ
limiting value of xπP1
1 2 3 4
1 1 1 1
0.4 0.1 0.2 0.4
2 2 2 2
0 0 0 0
0.4 0.2 0.4 0.4
1 1 1 1
2 2 2 2
0.4 0.4 0.4 0.4
-1 -1 -1 -1
γ5 ) 0 γ5 ) 0 γ5 ) 1 γ6 ) 1
1.328 1.286 1.602 1.113
) 0, the Dirichlet condition at x ) 0; with eq 5, Dox,x)0/Dox,bulk ) 1/(1 + γ2) when cox,x)0/cox,bulk. Plots of xπP1 vs log[N] (N is the number of simulation iterations) for γ1 ) 0, 1.0, 10.0, and 100.0, and Dox,bulk∆t/∆x2 ) 10 are shown in Figure 1. The analogous plots for γ2 ) 0, 1.0, 10.0, and 100.0, and Dox,bulk∆t/ ∆x2 ) 1000 are shown in Figure 2. The value of Dox,bulk∆t/∆x2 is very large to ensure that Dox,x∆t/∆x2 g 10, even at x ) 0. It is apparent that the simulations require a few iterations to converge; this is common for any simulation of a response involving a discontinuous step-perturbation. For additional confirmation, I assumed that the system comprises both redox components with identical diffusion coefficients, both of which exhibit the same dependence on the concentrations (thus, cox,x + cred,x ) cox,bulk; cred,bulk ) 0). The responses were computed using two algorithms:
that can be investigated using implicit methods. Nevertheless, the case for the constancy of P1 can be adequately supported. I’ve selected a three-species system comprising the redox species AzA and BzA-1, and a counterion, CzC , with corresponding charges zA, zB, and zC, and diffusion coefficients DA, DB, and DC. The operative interfacial redox reaction is AzA + e ) BzA-1. In all examples, I will assume that the Dirichlet condition is established by setting cA,x)0 ) 0 and that electroneutrality obtains throughout the system. The previously discussed constraints obtain. The Nernst-Planck expression was used to describe species transport,
Dredox,x ) Dredox,bulk
where the subscript J denotes the species A, B, or C; x is the electric field, F is Faraday’s constant, R is the gas constant, and T is the temperature (K). The electroneutrality constraint C (∑J)A cJ,xzJ ) 0) leads directly to
4γ3
1 cox,xcred,x cox,bulk2
) +1
4γ3
1 cox,x(cox,bulk - cox,x) cox,bulk2
+1 (6)
fJ ) -DJ,x
dcA F + D c z dx RT x J,x J,x J
C
or
RT cox,x(cox,bulk - cox,x) Dredox,x ) 4γ4 +1 2 Dredox,bulk c
(7)
∑D
J,x
(9)
C
F
ox,bulk
fJzJ
J)A
x )
(8)
∑ cJ,xzJ
2
J)A
For both expressions, the diffusion coefficient ratio, Dredox,x/ Dredox,bulk, equals 1 when cox,x ) 0 or cox,x ) cox,bulk; when cox,x ) cox,red, Dredox,x/Dredox,bulk ) 1/(γ3 + 1) (eq 6) or Dredox,x/ Dredox,bulk ) γ3 + 1. The results shown in Figure 3 again demonstrate that xπP1 is constant, despite a considerably more complex dependency of Dredox,x/Dredox,bulk on the system composition. Perhaps the startling result of these simulations is that the change in the value of xπP1 is considerably smaller than might be expected considering the change in the magnitude of the diffusion coefficient at the electrode surface. Andrieux and Save´ant4 have invoked a dimensional-analysis argument to show that P1 is also constant for systems involving migration (including electron-hopping). Even when the diffusion coefficients are composition-dependent, no additional DPs can be constructed for this system, so P1 should remain constant. This, too, can be confirmed by simulation. Migrational transport problems can also be simulated using implicit finite difference methods (see, e.g., the works of Rudolph18 or Bieniasz19). However, I am unaware of any attempts to model the effects of composition-dependent diffusion coefficients. Consequently, for the present purposes, I’ve opted to use the simpler explicit finite difference (EFD) approach. The EFD algorithm requires that the dimensionless diffusion parameter, DJ,x∆t/∆x2, must be less than 0.50.17,20 Values of DJ,x∆t/∆x2 that are too small are also problematic. Thus, for practicality, 0.10 e DJ,x∆t/∆x2 e 0.4. As a result, the dynamic range of diffusion coefficients that can be explored in any given EFD simulation will be severely restricted as compared to the virtually unlimited dynamic range
The operative set of flux equations is then C
fJ ) -DJ,x
dcA
+
dx
fK z K
∑ K)A D
K,x
DJ,xcJ,xzJ
C
(10)
cK,xzK2 ∑ K)A
with the constraint (remembering that fC,x)0 ) 0) C
∑ fJzJ ) fA,x)0zA + fB,x)0zB
(11)
J)A
The composition dependence will be described by
DA,x ) DA,bulk
1 + γ5 cA,x 1 + γ5 cA,bulk
(12)
or
DA,x ) DA,bulk
1 + γ6
cA,x
cA,bulk 1 + γ6
(13)
Parameter values used in the simulations are summarized in Table 1. Simulated results are shown in Figure 4. Imposing the
6764 J. Phys. Chem. B, Vol. 111, No. 24, 2007
Feldberg sion coefficients. The relevant dimensionless parameter, Pume,ss, would be
Pume,ss )
Figure 4. Results of explicit finite difference simulations of migrational systems. The values of input parameters for each curve are summarized in Table 1.
additional constraints that γ5 e 1 and γ6 e1 ensures that DA,x∆t/ ∆x2 will not vary by more than a factor of 2. As predicted, xπP1 is constant for all examples tested. Conclusions The primary conclusion of the present work is that chronoamperometry of an initially homogeneous system with Dirichlet boundary conditions and semi-infinite linear diffusion will always exhibit Cottrellian behavior. The simulations suggest that the value of the Cottrellian constant, P1 (eq 1), is relatively insensitive to the changes in the functional dependence of the diffusion coefficients on composition. This was confirmed by simulation of a number of examples of semi-infinite linear diffusional and migrational systems with compositionally dependent diffusion coefficients. Establishing Dirichlet conditions other than those effecting a zero concentration of one of the redox moieties may be experimentally difficult because of uncompensated resistance. Experimental confirmation of the constancy of the Cottrellian constant has also been obtained for systems in which one might expect significant composition-dependent changes in the diffusion coefficients. Wang et al.6 report Cottrellian behavior at longer times for chronoamperometry of a ferrocenated imidazolium ionic liquids. Longmire et al.8 showed that Cottrellian behavior obtained (also at longer times) in a virtually solid system comprising undiluted 0.43 M CpFeCpC(O)NH-MPEG polymer with ∼0.03 M LiClO4. In this system, physical movement of the polymer is virtually nonexistent, but electron-hopping effects quasi-diffusional/migrational transport of the ferrocene moieties. Electroneutrality is sustained by appropriate relocation of Li+ and ClO4-. The deviation from Cottrellian behavior at shorter times is most likely caused by uncompensated resistance which precludes the instantaneous establishment of the Dirichlet boundary condition. Careful cell design (e.g., proper positioning of the counter and reference electrodes) and electronic compensation (positive feedback)2 could alleviate this problem, as could the use of ultramicroelectrodes for chronoamperometry. In addition, the steady-state behavior at a small disk, hemispherical, or spherical electrode should be linearly proportional to the electrode radius, even with composition-dependent diffu-
iss FDA,bulkcA,bulkr0
(14)
The algorithms used in the present work to describe the dependence of the diffusion coefficients upon composition were arbitrary, so I made no effort to characterize the resultant values of the Cottrell constant, P1 (eq 1), other than to note that P1 is time-independent. It would not be difficult to develop more realistic algorithms if there were experimental data indicating the relationships between composition, visicosity, and conductivity (and by inference, the diffusion coefficients) for a system of interest. Matsumoto et al.7 have tabulated viscosity and conductivity data for several room temperature ionic liquids neat and with 0.3 mol/kg LiClO4 (which increased the viscosity significantly). I cite this as an example of the type of data that would be useful: these systems contained no redox moieties. By extending the data base for systems of interest that do contain redox species, some useful generalizations about the composition dependence of diffusion coefficients might be made. Extension of the implications of the present results for cyclic voltammetry of these types of systems with reversible electron transfer is currently being considered. The experimental challenge is how best to correct for uncompensated resistance. Activity coefficients will likely be changing significantly during the course of an experiment and will modify the relationship between the Dirichlet boundary condition and the applied potential. Consequently, the shape of the reversible voltammogram may not be the classical shape described by Nicholson and Shain ;21 however, the voltammograms plotted as icv/(area FcA,bulkxFDA,bulkV/RT) vs FE/RT should be congruent for a given initial composition. In the present analysis, I have not considered the possible complications associated with the interactions of counter fluxes in concentrated media.22 Onsager’s fundamental assumption for the phenomenological transport equations23 is that the frictional force between any pair of species is linearly proportional to the difference in their velocities, Vj - Vk(*j), where Vj ) fj/cj and fj and cj are the flux and concentration of the jth species. Despite these interactions, Cottrellian behavior should obtain for a chronoamperometric experiment as long as the force-velocity relationships remain linear. One additional conclusion that I hope the reader will draw from this work is that dimensional analysis is a powerful and underutilized analytic tool that is perhaps most familiar to chemical engineers.1 Acknowledgment. Michael Horne and Tony Hollenkamp (CSIRO Minerals, Clayton, Australia), James Wishart (Brookhaven National Laboratories, Upton, LI), Royce Murray and Mark Wightman (University of North Carolina, Chapel Hill, NC), Henry White (University of Utah), and Robert de Levie (Bowdoin College) are thanked for very helpful comments, discussions, and suggestions. References and Notes (1) Zlokarnik, M. Dimensional Analysis and Scale-up in Chemical Engineering; Springer-Verlag: Berlin, Heidelberg, New York, 1991. (2) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications, 2nd ed.; John Wiley and Sons: New York, 2001. (3) Buckingham, E. Physical ReView, 2nd Series 1914, 4, 345-376. (4) Andrieux, C. P.; Save´ant, J. M. J. Phys. Chem. 1989, 92, 67616767. (5) Save´ant, J. M. J. Electroanal. Chem. 1988, 242, 1-28.
Dimensional Analysis of Chronoamperometry (6) Wang, W.; Balasubramanian, R.; Murray, R. W. Manuscript in preparation. (7) Matsumoto, H.; Sakaebe, H.; Tatsumi, K.; Kikuta, M.; Ishiko, E.; Kono, M. J. Power Sources 2006, 160, 1308-1313. (8) Longmire, M. L.; Watanabe, M.; Zhang, H.; Wooster, T. T.; Murray, R. W. Anal. Chem. 1990, 62, 747-752. (9) Crank, J.; Park, G. S. Diffusion in Polymers; Academic Press: London, 1968. (10) Morris, R. B.; Fischer, K. F.; White, H. S. J. Phys. Chem. 1988, 92, 5306-5313. (11) Stevenson, K. J.; White, H. S. J. Phys. Chem. 1996, 100, 1881818822. (12) Amatore, C.; Kelly, R. S.; Kristensen, E. W.; Kuhr, W. G.; Wightman, R. M. J. Electroanal. Chem. 1986, 213, 31-42. (13) Morris, R.; Franta, D.; White, H. S. J. Phys. Chem. 1987, 91, 35593564. (14) Seibold, J. D.; Scott, E. R.; White, H. S. J. Electroanal. Chem. 1989, 264, 281-289.
J. Phys. Chem. B, Vol. 111, No. 24, 2007 6765 (15) A dimensionless parameter (DP) which evolves from combinations of other DPs does not count as an additional DP; however, it is possible that the resultant DP may have a more convenient form. (16) Ferziger, J. H. Numerical Methods for Engineering Application; John Wiley and Sons, Inc.: New York, 1981. (17) Britz, D. Digital Simulation in Electrochemistry: 2nd, Revised and Extended Edition; Springer-Verlag: Berlin, 1988. (18) Rudolph, M. Digital Simulations With a Fast Implicit Finite Difference Algorithm: The Development of a General Simulator for Electrochemical Processes. In Physical Electrochemistry; Rubinstein, I., Ed.; Marcel Dekker: New York City, 1995; pp 81-129. (19) Bieniasz, L. K. J. Electroanal. Chem. 2004, 565, 273-285. (20) Bond, A. M.; Feldberg, S. W. J. Phys. Chem. B 1998, 102, 99669974. (21) Nicholson, R. S.; Shain, I. Anal. Chem. 1964, 36, 706. (22) Newman, J. Electrochemical Systems; Prentice Hall: Englewood Clffs, NJ, 1973. (23) Onsager, L. Ann. N. Y. Acad. Sci. 1945, 46, 241-265.
