1011
Anal. Chem. 1981, 5 3 , 1011-1016 Delahay, P. J. Am. Chem. SOC.1953, 75, 1430. Randles, J. E. 5. Can. J . Chem. 1959, 37, 238. Gerischer. H. Z. E/ektrochem. 1953, 57,604. Timmer, B.; Sluyters-Rehbach, M.; Sluyters, L. H. J. flectroanal. Chem. 1968. 19, 85. Sluyters-Rehbach, M.; Breukd, J. S. M. C.; Sluyters, L. H. J . Nectroanal. Chem. 1968, 19, 73. Macdonald, D. D. “Transient Techniques in Electrochemistry”; Plenum: New York, 1977; pp 132.
(16) Barker, G. C.; Faircloth, R. L.; Gardner, A. W. U.K., At. Energy Res. €stab/., CIR 1958, 1786. (17) Feng, Q. S. HuaXue XuePao 1966, 32, 7.
RECEIVED for review October 26, 1980. Accepted February 23, lg81* This paper was presented at the 180th Meeting Of the American Chemical Society, San Francisco, CA, Aug 1980.
Explicit Finite Difference Method in Simulating Electrode Processes Renato Seeber *
Istituto di Chimlca Generale deli’Universit5 di Siena, Piano dei Mantellini, 44,
53 100 Siena, Italy
Stefan0 Stefani
Istituto di Matematica de8’UniversitS di Siena,
Via del Capltano, 15, 53100 Siena, Italy
The expllclt flnite dlfference method is widely used In slmulating electroanalytical experiments; however, In many cases the required computation amount becomes too large. Two means of overcomlng Nhls dlfflculty are descrlbed In this paper. In the first, a nonunlform space-tlme grld ls built up. If the slre of space elements Increases at Increasing dlstance from the electrode according to a geometrlcal progression, It Is posslble to achieve also a nonunlform tlme dlscretlzatlon, such that the species concentratlons are modlfled wlth a varlable frequency decreaslng at Increasing distance from the electrode. I n the second, the transient behavior of the electrode boundary Is improved leading to accurate values in the computed response even at the early polnts after a sudden change In equilibrium conditions. Efficlency and accuracy of the model are tesUed in simulating different electrode mechanlsms wlth different electroanalytical techniques.
In the last 15 years the digital simulation technique has become a powerful tool in elucidating electrode mechanisms. Its usefulness has been proved in the prediction of the theoretical trend of the significant parameters relative to responses obtained with different electroanalytical techniques and in establishing the nature of an electrode process by comparison between experimental and computed response. The simulation techniques are very useful in solving problems which invollve equations with time-dependent boundary conditions or nonlinear differential equations and, in general, in the cases in which a complete analytical solution becomes either difficult or unfeasible. Among the different possible methods which can be followed in the digital simulation, the explicit finite difference method still retains full validity, as the implicit finite difference method (1-4)seems to be less suitable to solve the boundary value problem if the involved partial differential equations are nonlinear. Although the model involving uniform space and time discretization (5-7) is still satisfactory, also because of its simplicity, perhaps too little attention has been devoted to search possible improvements on it. In particular, some problems arise in the case of the occurrence of homogeneous chemical reactions which are so fast that diffusion layer 0003-2700/81/0353-1011$01.25/0
thickness results much greater than reaction layer. Together with some improving analyses of the classical model (8)we find in literature few examples of valuable suggestions to reduce the amount of computation, introducing a nonuniform space discretization (3) and treating the homogeneous chemical reactions coupled to the charge transfer in an unconventional way (9). In view of these facts the present paper reexamines the classical “Feldberg’s model”, reconsidering both space and time discretization, and the expression of the boundary conditions for the electrode kinetics.
SPACE-TIME DISCRETIZATION In an electrochemical experiment the diffusing perturbation arises at the electrode surface; the concentrationsof the species involved in the electrode process may show abrupt changes near the electrode, while the concentration profiles appear always smoother, increasing the distance from it. This suggests that by considering a nonuniform space discretization the concentrationprofies are still suitably described (3). A similar approach allows the use of a very close-mesh space grid near the electrode, as is for instance required by the occurrence of fast homogeneous chemical reactions coupled to the charge transfer, without a prohibitive increase of the computation amount. One finite-difference approximation of the differential equation for the planar diffusion
following the explicit method ( 1 0 , l l )is, in the case of uniform space discretization, a set of linear equations
+
= P(Ai-lj - 2Aij A i t l j ) (2) where Aij is the concentration value at the ith space element and at the j t h time increment and /3 is the so-called “dimensionless diffusion coefficient”. However, if the space discretization is nonuniform, for every space element two coefficients have to be considered, so that eq 2 becomes A i j t l - Aij = Pr,i(Ai+lj- Aij) + Pl,i(Ai-lj - Ai;) (3) where P,i and are the “right diffusion coefficient” and “left diffusion coefficient”, respectively. A possible way to evaluate Aijtl
- Ai,
0 1981 American Chemical Society
1012
ANALYTICAL CHEMISTRY, VOL. 53, NO. 7, JUNE 1981
SPACE
Flgure 1. Nonuniform space-time grid. Space elements are equal to each other up to X,,; then they form a geometrical progression. X,, X,, and X, are the boundaries of subsequent sets of space elements, each set being characterlzed by a given computation frequency, as shown by the time dimensions of the meshes. The points a, b, c, d, e, f , g, h, and m are mentioned In the text.
these coefficiepts is suggested in ref 3; however, by following this approach a concentration profile of constant slope is not always a steady-state condition. This fact leads one to look for an alternative way 6 determine the quantities and &i. A first relationship, stating the conservation of the diffusing species at the boundary between two space elements, must hold Pr,i-l(Ai,j- Ai-lj)(Xi-1/2- Xi-3/2)
+
Pl,i(Ai-lj- Aij)(Xi+1/2- Xi-,/J
= 0 (4)
i.e.
PI,^ = Pr,i-1(xi-1/2- Xi-3/2)/(Xi+1/2- Xi-1/2)
(5)
where and Xi+llzare the coordinates of left and right boundary of the ith space element, respectively. A further relationship can be drawn out. Since a linear concentration profile of the diffusing species has to be a steady-state condition, the following equation must hold: Pr,i(Xj+l- Xi) + Pl,i(Xi-l- Xi) = 0 (6) i.e. P1,i = Pl,i(Xi- Xi-l)/(Xi+I- Xi) (7) where Xi is the coordinate of the “mean point” of the ith space element. Equations 5 and 7 allow the easy computation of every Pi and 0,value from a given one. An alternative way (12) to evaluate 0,and /31 is to directly write the relationship Aij+i - Aij = ~ [ ( A i +-i jAij)/(Xi+i - Xi) + (Ai-ij Aij)/(Xi - Xi-l)I/(Xi+1/2- Xi-1/2)
(8)
and to compare it with eq 3. y is a “diffusion coefficient” including the length of the time increment. As a geometrical progression appears to be the best criterion in dimensioning an electrical ladder network which simulates the diffusion process (13),it seems a suitable choice for the function X X, = k ( q - 0.5) if q I N + 0.5
X, = k [ N + (ec(P--N-a.5) - l)/c]
if q 1. N
+ 0.5
(9)
both for the integer and half-integer indexes; for integer values it gives the coordinates of the “mean points” of the space elements and for half-integer values those of their boundaries. In eq 9 k is the “space length unit” in the model, N is the number of uniform space elements near the electrode, and c is a parameter determining the “rate” at which the dimension of the space elements changes; it is useful to consider a uniform space discretization near the electrode to state the boundary conditions in a simpler way. For the sake of simplicity, we will indicate A. and Boas the concentrations at the electrode and Xo as the coordinate of the electrode surface (rather than AO.S,BO.S, and X0.J. The approach described above leads to Pr,i and Pl,i values decreasing at increasing i; therefore it becomes possible to use time increments which in their turn increase at increasing i, without injuring the stability of the explicit iteration steps. The simplest way to perform the nonuniform time discretization could be doubling its size whenever it is possible. Figure 1 shows a space-time grid obtained following this procedure. The concentration values of the diffusing species are evaluated at each time increment only up to the XKth space element, every two iterations up to the XLth one, every four iterations up to the XMth one, and so on. Accordingly, from the above defined & i and @l,i parameters new “diffusion coefficients” 9,i and 91,ihave to be defined multiplying Pr,i and Pli by the size, in basic time unities, of the time dimension of the mesh at the Xith space element. The space boundaries at which the computation frequency is halved require a particular handling. First of all, referring to Figure 1, it is evident that Ad, the concentration of the diffusing species in the grid knot “d”, arises from the relationship Ad
= Ab
+ 8,j(AC- Ab) + ~ I , L & - Ab)
(10)
A8 A, has not been explicitly evaluated in the previous time increment, it is replaced by the value obtained by the firstorder extrapolation
Moreover, A, has to be calculated as follows:
ANALYTICAL CHEMISTRY, VOL. 53, NO. 7, JUNE 1981
A, =
1013
length; hence, the following boundary conditions can be written:
(12) We note that a similar evaluation of the concentrations at the boundaries between two subsequent arrays of space elements in which the concentrationsare computed with different frequency ensures the conservation of the diffusing species as well as the steady-state condition which foresees a constant value of the flux through the diffusion layer. Let us reconsider eq 9; setting c = -In 2 (13) 2P where p is a suitable positive integer, it is possible to double the size of the time dimension of the mesh at every p space elements. In fact, from eq 5, 7, 9, and 13 it is evident that for every i > N Pr,i+p
= Pr,i/2
Pl,i+p
= Pl,i/2
(14)
In such a case, for every i > N ar,i+p
= 8r,i
Ql,i+p
=
(15)
&,i
It can be noted that if the parameter c in eq 9 tends to zero, the above described nonuniform space-time discretization becomes the uniform one. The coupled homogeneous chemical reactions can be treated, in the described space-time grid, in a similar way as in the case of a uniform space-time discretization. It is only necessary to take into isccount the size of the time increment a t a given space element. For instance, in the case of an irreversible first-order chemical reaction, if T* is the size of the time mesh at a given space element, the change in concentration, AA, in that space element due to the chemical reaction occurring in the time length T*, is given by
A,j = A* B,j B* (20) Finally, the boundary conditions at the electrode surface imposed to the partial differential equations describing diffusion of species A and B are expressed in the general form as Da a i
- - = fgo - bb, k , ax
} at x = 0, i.e., at the electrode surface Db a6
-- = bF, - fi, k , ax
where k, is the “standard heterogeneousrate constant”, f and b are the potential-dependent part of the kinetic constants for forward and backward charge transfer, respectively, and do and bo are the concentrations of species A and B at the electrode surface. In the following we will assume the time dependence of the potential to be known; i.e., we will deal with potential-controlled techniques. We note that the boundary conditions expressed by eq 21 are, in general, both time dependent and of mixed type; i.e., they involve both function values and their derivatives. Indeed, in the particular case of a perfectly reversible charge transfer, the left-hand sides of eq 21 vanish, obtaining Nernst’s equation. Literature reports some different ways to translate eq 21 into forms suitable for the explicit finite difference method (6-8).
A simple formulation of the electrode boundary in terms of finite differences, which does not imply particular assumptions, is given by eq 22, where h is l / k 8 in the case of hzj = fAoj - bB,j
Da ( A1 j - Aoj) zi=
AA = -A(1 - &*) (16) where A is the “old” concentration and k, is the rate kinetic constant. Similarly, for an irreversible chemical reaction of second order in respect to tho same diffusing species AA = -A[1 - 1/(AkC7* + l)] (17) The computation time required by a uniform space-time discretization is of the same order as !PI2, where Tis the total number of time elements (iteration loops) (6); on the other hand, the size of computation required by a nonuniform space discretization as defined by eq 9 is that of T log T. In the case of the proposed nonuniform space-time grid the order is that of T.
BOUNDARY CONDITIONS To discuss the formulation of the boundary conditions at the electrode surface, let us consider an uncomplicated onestep electron transfer:
ne-&^ kb
,
(18)
The initial conditions for a solution homogeneous in respect to A and B species, in absence of adsorption phenomena, become: A;,o = A* Bi,o = B* (19) where A* and B* are the bulk concentrations. If the number, w ,of the considered space elements is high enough, the diffusion layer is included in the considered space
Db7
BlJ+l= Bl, +
xl-xo
(B,J - BlJ) +
(X2 - XJ2
zj7
x 2-x 1
a not perfectly reversible charge transfer and zero otherwise, 7 the time unit, zj the flux of the diffusing species through a unit electrode area, X1 - X o the half of X 2- X1. Nevertheless, this approximation entails wide spurious oscillations to arise in the response in consequence of transient excitations; some difficulties in dealing with the high-frequency portion of the signals seem inherent in the finite difference method itself (14). Hence, a useful expedient is to filter off this high-frequencyportion of the excitation (input filtering) and, if necessary, taking into account the nonlinearity of the relation between input and output, of the response (output filtering). A simple first move in this direction consists in splitting a potential step in two subsequent ones. If the system can be assumed to be roughly linear in the potential step range, the two halves of the step can be chosen equal to each other. A similar expedient allows the removal from the input signal of the componentwith a frequency equal to 1/27 (15). In such a context we note that the time at which the potential step occurs has to be identified with that of the middle potential
1014
ANALYTICAL CHEMISTRY, VOL. 53, NO. 7, JUNE 1981
point; consequently, the time flowing during a potentiostatic impulse has to be computed from this midpoint. On the contrary, if a similar procedure cannot be followed, the time origin of the step has to be shifted by -712 (8). In order to achieve a more effective suppression of the high frequencies, one must use a low pass filter in addition to the device described. A linear phase filter must be employed to avoid any dispersion, Le., any time distortion of the signal. For this purpose, it seems natural to choose a finite impulse response (FIR) filter. In particular, satisfactory results have been obtained by using 11 signal points, a cut frequency of 1/47, and the Keiser weighing function for an overshot amplitude of (16). It is evident that filtering can lead to an excessive smoothing of excitation and response. This drawback can be overcome by not directly taking as output response the flux computed by using a filtered input but by only using it “to damp” the behavior of the boundary in a further simulation subjected to an unfiltered excitation. The boundary conditions can then be rewritten in the form shown in eq 23, where zj* is the flux hzj = fAoj - bBoj zj
=
Da(Aij - Aoj) x1-
xo
AlJ+l =
BlJ+l=
computed by using the filtered input and 6 is a parameter governing the damping action (0 < 6 < 1). The quantity 6zj* + (1 - S)zj is taken as the output response. In this way a first approximate solution of the differential problem has been refined by means of a second interation. Indeed, it is unnecessary to compute once more the whole concentration profiles: the performing of further diffusion computations can be restricted to few space elements near the electrode.
RESULTS AND DISCUSSION In order to check the validity of the outlined algorithms we have computed the responses for different electrode mechanisms and different potential-controlled techniques. As benchmark techniques we have chosen chronoamperometry, linear sweep and cyclic voltammetry, staircase voltammetry, and cyclic staircase voltammetry; the electrode mechanisms were both uncomplicated charge transfers with different reversibility degree and electrochemical, chemical, electrochemical (ECE) type mechanisms, with first- or second-order irreversible chemical reactions. It is well-known that, in order to ensure the stability of the diffusion simulation, the quantity P = Dm(7/h2) (24) (where D, is the largest diffusion coefficient of the diffusing species) has to be less than 0.5. In the simulations whose results have been reported in the following, we have always assumed = 0.45 for every diffusing species and 6 = 0.1 in eq 23; the temperature was fixed at 298.15 K. All computer programs were written in FORTRAN and executed on an IBM 4331 computer. Simulations of chronoamperometric tests relative to a one-electron uncomplicated electrode process prove the va-
Table I. Simulation of Cyclic Voltammograms for an Uncomplicated Reversible Process, Computer Times and Degrees of Accuracy in Evaluating the Peak Current Function, with Different Nonuniform Space-Time Grids N p computer timea errorYboleo 6.78 X lo-’ -1.112 3 2 3 3 7.54 x -0.627 -0.287 3 6 9.55 x -0.0706 3 10 14.07 X lo-* Computer time = 1 for a simulation performed by the Referred to a simulation standard Feldberg procedure. carried out by the standard Feldberg procedure (nl’Zx(at)max = 0.446 321 2). lidity of the proposed devices in damping spurious oscillations in the response after sudden potential change. So much so that the deviation of the nappvalues from unity is already less than 1%and 0.5% after three iterations (with N = p = 3 in eq 9 and 13); with the standard procedure (6) such accuracy degrees are reached after 22 and 24 iterations, respectively. A similar result allows one to employ a basic time length of a relatively large size, leading to a corresponding saving in computation extent. As regards the behavior of the model at rather long times (e.g., loo00 iteration loops), it can be noticed that nappdeviates from the theoretical value less than 1.5%0and 0.85%0putting p = 2 and p = 3, respectively ( N always equal to 3); correspondingly, the error in the flux integral after 10OOO iterations is of -1.2760 and of -0.65%. Simulations of cyclic voltammetric tests relative to an uncomplicated reversible charge transfer have been performed by using N = 3, p = 2,3,5, and 10, and a potential increment in the time unit of 0.2569 mV, by sweeping a potential range of 308.295 mV centered at Eo of the redox couple involved. The location on the potential axis of the whole response is in any case completely coincident with the theoretical one. The peak current function appears to be the most critical parameter; its deviations from the theoretical value are in any case very small, as Table I shows. A very good agreement with the theoretical data (17,18) has been found for cyclic voltammetric responses relative to an uncomplicated charge transfer with different reversibility degree, as well as for chronoamperometric and cyclic voltammetric tests (19) relative to ECE type mechanisms. Let us consider the computation amount required by the described space-time grid in respect to that of a uniform space-time discretization, with regard to both the number of knots in the grids and the real computation times. First, we draw out the relationships between the number of iteration loops, T, the value of the parameters N and p defining the nonuniform space-time discretization, and the number of knots, 9,in this space-time grid. The thickness of solution layer, L, perturbed by the diffusion, can be evaluated as (6)
L
6~112piz
(25)
where L is expressed in length units, k . On the basis of eq 9 and 13, we determine the number of sets of space elements, each set characterized by a different time dimension of the mesh, as the smallest integer, M , for which the following inequality holds:
M is hence the smallest integer such that
ANALYTICAL CHEMISTRY, VOL.
53, NO. 7, JUNE 1981
1015
Table 11. Cornparisom between the Total Number of Space Elements Considered, by Using Different Space-Time Grids N= 3;p=2 N = 3;p = 3 N = 3;p= 5 N = 3;p = 10 12 915 a1; 6986 22 370 8970 17.58 X l o w 2 5.49 x lo-, 7.05 X lod2 10.15 X lo-’ R, 20 984 68 510 26 970 38 925 a2 RZ 3.17 X lo-* 4.08 X 10” 5.89 X lo-’ 10.36 X lo-’ a3 69 988 89 976 129 895 229 210 R3 1.74 X 2.24 X lo-’ 3.23 X lo-’ 5.69 X 10” a, 699 986 899 970 1 2 9 9 860 2 299 000 R4 0.550 X lo-, 0.707 X lo-’ 1.02 x 10” 1.81 x 10-2 are evaluated for T = 1000, 3000, 10 000, and 100 000, respectively. R, = R , = a2/QZ; a a 1, G?2 , a and R , = a 3 / @R, ,; = Table 111. Comparison between Real Computer Times for Simulations of Chronoamperometric Tests Relative to an Uncomplicated One-Step Charge Transfer, by Using Different Space-Time Grids N=3;p=2 N=.3;p=3N=3;p=5 N=3;p=10 4.27 X l o m 25.24 X lo-’ 7.57 X l o - , 11.07 X lo-, 2.39 X lo-’ 3.12 X 10” 3.95 X 6.47 X lo-* a T = 3000. T = 10 000. Computer time = 1 for a simulation performed by the standard Feldberg procedure. 0 ,a
OZb
If the total number of iteration loops, T, is an integer multiple of 2M-1,the evaluation of s1 is straightforward.
TP
Q = TN+ ~
(
2 1)~
-
(28)
On the other hand, if CI, indicates the number of knots in a uniform space-time grid of the same extension, one has 9 = 6pw973/2
(29)
Table I1 reports the values of Q and s1/@ for different T; it can be noted that, at a fixed p value, the ratio Q / Tis almost constant for a large-enough T. The real computation times for simulations of chronoamperometric and voltammetric tests are reported in Tables I11 and I, respectively. The times reported include the “build up” of the space-time grid, which resulted indeed very short. By extrapolating the computation time at p = 0, we obtain a time value which, in the case of chronoamperometric tests, is almost equal to that spent to solve the electrode boundary equations; at p equal to 2 or 3 it is a large part of the overall computation time. In voltammetric tests it includes also the time used to evaluate the potential-dependent part of the kinetic constants at every iteration: it is the major part of the overall computation time. As a matter of fact, the data in Table I1 show that the average number of space elements considered in a single iteration is less than 7 and than 9 for p = 2 and p = 3, respectively. These facts suggest the uselessness to look for expedients which reduce further the computation time for the simulation of diffusion. We have previously put in evidence that the use of a nonuniform space-time grid and the damping of the spurious oscillations allow a noticeable saving in amount of computation. The importance of these facts is dramatically evident in simulating electrode responses obtained by the staircase and cyclic staircase techniques (20,21). An interesting aspect of these techniques is connected to the sampling of the current at short times after each potential change; it is evident that’ in simulating a response a sample can be taken onIy if the spurious oscillations are damped enough. Consequently, the number of time elements constituting a single step is constrained by the number of iterations required so that the response “becomes stable”. Moreover, it is often useful to
employ staircase waveforms with potential steps of reduced sizes, leading to a great number of stairs. In order to obtain the time length of the overall response the number of time elements constituting a single step should be hence multiplied by a rather great number, implying a very wide space-time grid; in this case, the use of a nonuniform space-time discretization is particularly useful. On the basis of the model described in the present paper we have drawn out cyclic staircase responses relative to an uncomplicated reversible electrode process; they result in very good agreement with those obtained by using the analytical solution reported in ref 21. We have also evaluated with a reasonable amount of computation the staircase responses for ECE type mechanisms with irreversible first-order chemical reactions, verifying that the response does not change a t constant value of the product between the chemical kinetic constant and the time length of each step of the stair. Moreover, responses for ECE type mechanisms with secondorder chemical reactions have been computed. A different approach to the simulation of electroanalytical responses is the “orthogonal collocation technique”, in which, expressing the concentration profiles of the species through a linear combination of orthogonal polynomials, each partial differential equation of the boundary problem is translated into a system of ordinary differential equations (22-24). Although this method appears to be a promising one, its degree of accuracy and efficiency in simulating electroanalytical experiments have not yet been thoroughly investigated in a wide number of electrode mechanisms studied with different electroanalytical techniques. The improvements suggested in the present paper greatly enhance the efficiency of the explicit finite difference method in simulating electroanalytical experiments,mainly in the case of long step times or wide potential sweeps, as also in the presence of sudden potential changes. It follows, for instance, that the simulation time of complex electrode processes by using techniques involving rather complicated waveforms, as the cyclic staircase, becomes reasonable, without loss in accuracy. Supplementary Material Available: Listings of the most significant computer programs (11 pages) will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementarymaterial from this paper or microfiche (105 X 148 mm, 24X reduction, negatives) may be obtained from Business Operations, Books and Journals Division, American Chemical Society, 1155 16th St., N.W., Washington, DC 20036. Full bibliographic citation (journal, title of article, author) and prepayment, check or money order for $11.00 for photocopy ($12.50 foreign) or $4.00 for microfiche ($5.00 foreign), are required. LITERATURE CITED Booman, G. L.; Pence, D. T. Anal. Chem. 1865, 37, 1386-1373. Winograd, N. J . Electroanal. Chem. 1975, 43, 1-8. Josiin, T.; Pletcher, D. J. Electroanal. Chem. 1874, 49, 171-186. Brumleve, T. R.; Buck, R. P. J . Electroanal. Chem. 1878, 90,1-31. (5) Feldberg, S. W.; Auerbach, C. Anal. Chem. 1064, 36, 505-509. (1) (2) (3) (4)
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(6) Feidberg, S. W. In "Electroanalytical Chemistry"; Bard, A. J., Ed.; Marcel Dekker: New York, 1969;Vol. 3. (7) Feldberg, S. W. In "EiectrochemiStry. Calculatlons, Simulations and Instrumentatlons";Mattson, J. S., Mark, H. B., Jr., Mac Donald, H. C., Jr., Eds.; Marcel Dekker: New York, 1972;Vol. 2,Chapter 7. (8) Sandifer, J. R.; Buck, R. P. J . Electroanal. Chem. 1974, 49, 161-170. (9) RuiiE, I.; Feldberg, S. W. J . Nectroanal. Chem. 1974,50, 153-162. (IO)Crank, J. "The Mathematics of Diffusion", 2nd ed.; Ciarendon Press: Oxford, 1975;Chapter 8. (11) Smith, G.D. "Numerical Solution of Partlal Differential Equations: FinIte Difference Methods", 2nd ed.; Clarendon Press: Oxford, 1978. (12) Feidberg, S.W., private communication. (13) Oldham, K. B. Anal. Chem. 1973, 45, 39-47. (14) Parker, I. B.; Crank, J. Comput. J. 1964, 7 , 163-167. (15) Oppenheim, A. V.; Schafer, R. W. "Digital Signal Processing"; Prentice-Hall: Englewood Cliffs, NJ, 1975;Chapter 3.
(16) Walt, J. V. In "Active Filters: Lumped, Distributed, Integrated, Digital and Parametric";Hueisman, L. P.; Ed.; McGraw-HID: New York, 1970; Chapter 5. (17) Nicholson, R. S.; Shain, I. Anal. Chem. 1964,38,706-723. (18) Nlcholson, R. S.; Shain, I. Anal. Chem. 1965,37, 178-190. (19) Nlcholson, R. S. Anal. Chern. 1965, 37, 1351-1355. (20) Ryan, M. D. J. Electroanal. Chem. 1977, 79, 105-119. (21) Miaw Lee-Hua, L.; Boudreau, P. A,; Pichler, M. A,; Perone, S. P. Anal. Chem. 1978, 50, 1988-1996. (22) Whiting, L. F.; Carr, P. W. J. flecfroanal. Chem. 1977, 81, 1-20. (23) Speiser, B.; Rieker, A. J. Nectroanal. Chem. 1979, 102, 1-20. (24) Speiser, B. J . Electroanal. Chem. 1980, 110, 69-77.
RECEIVED for review June 13,1980. Accepted February 17, 1981.
Differential Pulse Polarography at the Static Mercury Drop Electrode J.
E. Anderson, A.
M..Bond," and R. D. Jones
Division of Chemical and Physical Sciences, Deakin University, Waurn Ponds 32 17, Victoria, Australia
The static mercury drop electrode, SMDE, is an Important new electrode for use In polarography. In this paper an approximafe theoretlcal treatment of a reverslble electrode process Is presented for the technlque of dlfferentlal pulse polarography. The considerably dlfferent nature of the DC response at the SMDE compared to that of the conventlonal dropplng mercury electrode (DME) leads to new factors whlch need to be carefully considered in analytlcal appllcations of dlfferential pulse polarography at the SMDE. For example, the DC time dependence of f-"*, SMDE, vs. f'/*, DME, for reverslble electrode processes can lead to a large negative dlstoitii term for reversible electrode processes at the SMDE which is of opposlte sign and far larger In magnltude than at the DME (reduction process assumed). Interestlngly, for irreversible electrode processes the DC dlstortlon term in dlfferential pulse polarography at the SMDE reverts to belng posltive. For a quasi-reversible electrode process the DC dlstortlon term In dlfferential pulse polarography at the SMDE can be negatlve, zero, or positive dependlng on the parameters chosen for the experlment. Differences in DC terms are also shown to be Important In comparing dlfferentlal pulse polarograms obtained at the SMDE and DME for electrode processes exhlbltlng phenomena related to adsorption or film formatlon.
The static mercury drop electrode (ShfDE) is a new and important development in polarography (1). Data (2) indicate that at the SMDE, where area growth terms are absent, use of a constant potential technique (DC polarogrdphy) provides comparable detection limits to using time-dependent waveforms associated with normal and differential pulse polarography. In this paper a simple theoretical interpretation of differential pulse polarography at the SMDE is presented for a reversible electrode process. Results c o n f i i that differential pulse polarography when undertaken at a SMDE suffers from a distortion term ( I ) which for reversible electrode processes manifests itself as a negative offset of the base line at po-
tentids negative of the peak potential (for a reduction process). A related but considerably less severe positive offset is observed with conventional dropping mercury electrodes (DME) ( 3 , 4 ) . However, the absence of the atea growth terms with the SMDE causes the offset to be of opposite sign and substantially larger in magnitude than with a DME. Experimental results and implications of this work to reversible as well as other classes af electrode processes are considered. Interestingly, for irreversible electrode processes, the offset reverts to being negative as is the case at the DME (reduction process assumed).
EXPERIMENTAL SECTION Unless otherwise stated the polarographic experiments described here were performed using an EG&G Princeton Applied Research Corp. (PARC) Model 174 polarographic analyzer modified as described previously (5) to provide the output of the pulse and DC components while in the differential pulse mode. An EG&G PARC 374 microprocessor-controlledinstrument was used for some experiments. Such experiments are specifically designated as being performed with this instrumentation. The EG&G PAkC 174 instrumentwas equipped with an EG&G PARC Model 172 drop timer for experiments at the DME. A platinum auxiliary electrode and a Ag/AgC1(3 M KC1) reference electrode were used in conjunction with a conventionalDME or the EG&G PARC 303 Model SMDE. REAGENTS AND PROCEDURES Analytical reagent grade chemicals were used throughout the experiments. Polarograms were recorded at ambient temperatures of (20 & 1)"C. A11 solutions were degassed with nitrogen for at least 5 min prior to undertakiig the experiments. Electrode areas were calculated by weighing 100 drops and assuming a spherical shape. The drops were grown in the supporting electrolyte at a potential of -0.4 V vs. AgIAgC1. RESULTS AND DISCUSSION The SMDE, as does the DME, consists of a capillary through which mercury flows from a reservoir. However, a t the top of the capillary a solenoid activated plunger mediates the flow of mercury. The plunger normally prevents the flow of mercury into the capillary. When the solenoid is activated, the plunger rises from the end of the capillary allowing
0003-2700/8l /0353-1016$01.2510 0 1981 American Chemical Society
