71 a
James 6.Flanagan and Lynn Marcoux
Digital Simulation of Tubular Electrode Response in Stationary and Flowing Solution James B. Flanagan and Lynn Marcoux* Department of Chemistry, Texas Tech University, Lubbock, Texas 79409 (Received March 26, 1973; Revised Manuscript Received November 27, 1973)
Publication costs assisted by The Robert A . Welch Foundation
The various approximations made by Levich in order to calculate electrochemical responses at tubular electrodes have been tested using digital simulation. Specifically Levich assumed steady-state conditions, linear rather than cylindrical diffusion, no axial diffusion, and a linear rather than parabolic velocity profile. Using digital simulation a model for tubular electrodes was developed which makes possible the calculation of electrochemical response without these assumptions. These calculations quantitatively define the limits of the approximations as well as predict response beyond those limits. Transient response both under potentiostatic and galvanostatic conditions is obtained via this model. In addition to this the effect of potential scan rate upon voltammograms produced at flowing tubes is presented.
Hydrodynamic voltammetry is an extremely popular electrochemical experiment because it offers to the student of electrode reactions controlled and reproducible mass transfer conditions under which steady-state measurements may be accomplished. To the analytical chemist the convective component of mass transfer offers increased analytical sensitivity. I t is an unfortunate fact of electrochemical life that this same very useful convective component also greatly complicates the derivation of those equations which describe electrode response. Convective diffusion systems are described mathematically by combining the diffusion and convection terms in an equation of the form
earlier experiments involving planar electrodes fixed in flowing solution^,^^^ and are represented by the work of Bain and Arvia839 and Ross and Wragg.lo Electrochemically oriented experiments have been carried out almost entirely by Blaedel and his past and present coworkers. This work began with the construction of a simple gravity flow tubular platinum electrode which was used to verify Levich’s theoretical analysis of convective diffusion to the surface of a tube.11 This was followed by a more complete derivation of the expressions for steady-state current-voltage curves for reversible12 as well as quasireversible and irreversible electrode r e a ~ t i 0 n s . lThe ~ only homogeneous reaction studied at a tubular electrode has been the classical catalytic ~ a s e . 1 ~ O + n e C L R
where C is the concentration of the electroactive species; D, the diffusion coefficient; and ux, uy, uz are the components of the flow velocity. Clearly it is necessary to have available hydrodynamic expressions for the various flow velocities and it is this complication which is largely responsible for the limited development of hydrodynamic voltammetry. Although plate1,2 and conical3 electrodes situated in flowing solutions have been given some attention, most hydrodynamic voltammetry has been carried out at rotating disk electrodes4J simply because of the availability of solutions for the hydrodynamic equations. The rotating disk electrode and its variant, the rotating ring-disk electrode, have been fully developed as tools by which to investigate both charge transfer at the electrode and chemical reactions following charge transfer. The extent of their usefulness is best demonstrated by the extensive special bibliography devoted to rotating electrodes which appeared in a recent monograph.6 Another hydrodynamic electrode system, the tubular electrode, has received some attention, but in view of its great potentialities, its usefulness is largely unexplored. The few existing papers which describe tubular electrode experiments may be divided into two categories, namely, those which are principally hydrodynamic in nature and those which are more directly concerned with electrochemical results. Studies in the first category use the electrode reaction principally as an indicator of the flow conditions. Studies of this sort were stimulated by The Journal of Physical Chemistry, Vol. 78, No. 7, 1974
k
R + Z - 0
(2) (3 1
With the exception of the theoretical treatment of current-time curves at tubular electrodes in quiet solutionl4 all other studies have been involved with either the development of new electrode m a t e r i a l ~ l or ~ . with ~ ~ the application of tubular electrodes to actual analyses.18-21 Thus far all theoretical analyses of tubular electrode response have relied upon Levich’s treatment of the mass transfer problem.22 By necessity this treatment contained several approximations and limitations. The finite difference method as developed and extensively used by Feldberg23924 has proved extremely helpful in the past for the solution of complicated electrochemical problems. This has been especially true in the case of convective diff~sion~5-28 and for difficult geornetrie~.~9-3~ In order to fully explore current-potential-time relationships at tubular electrodes we have utilized this technique and developed a finite difference model for tubular electrode calculations. The equation describing mass transfer to the inner surface of a tube through which a solution is flowing in a laminar regime is at
where C = C ( r , Z , t ) is the concentration a t any point within the tube, D is the diffusion coefficient, r is the ra-
71 9
Digital Simulation of Tubular Electrode Response dial coordinate, Z is the axial coordinate, R is the inside radius of the tube, and VOis the axial flow rate. The use of the Poiseuille velocity profile (5) in the above expression requires that a sufficiently long inlet region exist in order for this parabolic profile to establish itself. It is also to be carefully noted that eq 4 is valid only in the case of laminar flow and that this restriction will be imposed throughout this discussion. It is obvious from eq 4 that the analytical solution of the mass transfer problem with appropriate electrochemical boundary and initial conditions is a difficult task. In order to simplify this problem several approximations have been made. The solutions obtained under these approximations were of course limited to a very restricted set of conditions. The best known set of approximations are probably those used by L e ~ i c h . ~ ~ Levich solved the equation
using the boundary conditions appropriate to constant applied potential in the region of convective-diffusion control. These are
C(Z,r,O) = Cb C(2,r.t) = 0 r
5
0
0
(8)
~ ( z , r , t =) (9) Here X is the electrode length, t is the time after the m
boundary condition is imposed, and Cb is the concentration of the electroactive species in the solution bulk. Comparison of eq 6 with eq 4 reveals that the following approximations were made in order to facilitate solution. (1) aC/at = 0. This is of course the steady-state assumption which although it permits the useful calculation of steady-state currents, precludes the possibility of studying transient phenomena. ( 2 ) ( l / r ) ( d C / d r )= 0. This assumes that the diffusion process may be approximated by linear diffusion which is tantamount to saying that the diffusion layer is much smaller than the tube radius. This places a physical limit on the tube geometry and flow rate. (3) a2C/dZ2 = 0. The neglect of axial diffusion places a restriction on the lower end of flow rates which may be utilized, because this assumption requires axial mass transport to be due predominantly to flow. (4) Vo[l - ( r 2 / R 2 ) ]N 2Vo[(R - r ) / r ] . The approximation of the Poiseuille velocity profile by a linear one also limits the tube size to large tubes since it again requires that the diffusion layer be small in comparison to the tube radius. Using these assumptions and the above boundary conditions (7-9) an expression (10) was obtained for the steady-state current a t a tubular electrode.
i
=
2.01nFirCbD23R23X2/3V013
(10) More recently a similar problem has been treated and extensively discussed using a different set of assumptions and boundary condition^.^^ In the present notation the equation which was solved was
C ( Z , R ) = C1 C ( Z , R ) = C,
z5
X 0
ac/az = o
( 12)
(13)
(14)
The flow field, V, is assumed to be constant. The inclusion of axial diffusion as well as the use of the cylindrical Laplacian certainly increases the rigor of this solution; however, the assumption of a uniform flow field which is independent of radius is questionable. This assumption is probably not as good as the linear gradient employed by Levich. I t must also be noted that the boundary condition described by eq 13 is not strictly speaking the case for a tubular electrode surrounded by insulating surfaces since the surface concentration on the upstream side should not be the same as that on the downstream side.
Method The computational method used was the digital simulation technique which has been applied to a variety of electrochemical problems. This technique is mathematically equivalent to the explicit difference method which has been used to solve the boundary value problems of diffusion. The mathematical considerations are fully discussed in several standard numerical analysis t e ~ t s . The ~ ~ , ~ ~ methodology for the application of this procedure has been thoroughly described by Feldberg.24 The electrode model consists of a net of points in a coordinate system of proper symmetry. For tubular electrodes two-dimensional circular cylindrical coordinates constitute the appropriate system. This same geometry was recently used in a theoretical discussion of finite planar disk electrode^.^^ A model of the simulation grid is shown in Figure 1. The model electrode is L units in length and P units in radius. There are P + 1 radial divisions. The surface of the electrode is taken to be at the points where p = 1, 0 < { 5 L where p and ( are the radial and axial coordinates, respectively. The tube surface corresponding to net points { 5 0 or { > L was taken to be nonelectroactive. In this model solution resistive drops have been ignored. This assumption was made in order to remain consistent with previous treatments so as to facilitate comparison. These effects will depend markedly on geometry and will certainly be important in the case of nonaqueous electrolytes. The finite difference formalism for cylindrical diffusion has been discussed in detail.29 Very briefly the finite difference equations for diffusion used in this work are given below. Linear diffusion was necessary to model the assumptions made by Levich, and the difference equation for one-dimensional diffusion is given by where K is the iteration counter, 6 is the dimensionless diffusion coefficient 6 = DAt/(Ar), (16) and Fp,c, is the dimensionless concentration a t the point ( p , [ ) and a t time K FPJ,K = CP,,E/Cb (17) For one-dimensional circular radial diffusion the diffusion equation becomes for 1 < p < P 1
+
and the boundary conditions used were The Journal of Physical Chemistry, Vol. 78, No. 7, 1974
720
James B. Flanagan and Lynn Marcoux
Real Coordinates
X
0
o1
Simulation Coordinates
C mleetroactlve.l
4.0. Cylindrical Diffusion. Figure 3 makes it quite apparent that the equivalent time, t' defined by eq 29, very nearly describes a condition of the diffusion layer at a flowing tube which corresponds to that for a stationary electrode at time t'. This suggests that it might be profitable to substitute t' into eq 34 and thereby estimate the change in steady-state current due to the cylindrical geometry. Figure 5 shows the results of such an experiment. The lower curve represents a simulation including cylindrical diffusion and the upper curve is the result of the above approximation. This makes clear the region in which cylindrical geometry must be taken into account and provides a good approximation for much of this region. Additionally the difference between the two curves provides an estimate of the departures expected when Poiseuille flow is neglected. Clearly linear diffusion may be safely used whenever t'D/Rn < In terms of experimental variables this requires that X D / V f < 10-6 and if we take the conventional value of 1 x 10-5 cm2/sec for D then one would expect Levich theory to be perfectly adequate when X / V , is less than 0.1 sec/cm2. Axial Diffusion. Axial diffusion was seen to be important in the zero flow limit; however, its significance in the case of flowing solution will depend upon flow rate and geometry. Figure 6 shows the simulation for an electrode with an X / R ratio of 1/1. The lower curve in this figure is for an X / R ratio of infinity, i.e., no possible axial diffusion. The departure of the lower curve is due entirely to cylindrical diffusion and parabolic flow. The geometry chosen is an extreme one and even here the influence of axial diffusion is slight. In fact it is even somewhat compensatory. For electrode and flow rates for which the other Levich assumptions held it was impossible to produce an axial diffusion effect. Not surprisingly the neglect of axial diffusion is seen to be the best of the Levich assumptions. Scan Rate. The derivation of the equation of the current-voltage curve for a tubular electrode with flowing solution is relatively easy if one makes the steady-state assumption.l2J3 From an experimental point of view it would be useful to know at what potential scan rate the The Journal of Physical Chemistry, Vol. 78, No. 7, 1974
Figure 6. Departures from Levich equation d u e to axial diffusion. Curve a is for an electrode with X / R = 1/1 and curve b represents the limiting case X / R = a.
I IE-EO!
Figure 7. Voltammograms of a reversible system at a flowing tube electrode as a function of scan rate. steady-state assumption becomes valid. Figure 7 shows the results of the simulation of this problem. The assumptions used were identical with those used for the treatment of galvanostatic transients. Various voltammograms shown as a function of nVt' where V is the voltage scan rate in mV/sec. Both the current and potential axes have been normalized. This calculation demonstrates that the steady-state assumption is valid for cases in which nVt' < 5 . Perhaps more important is the fact that one might
Ion Pairing of Naphtho[b]cyclobutene
723
naively assume that the absence of a current maximum would experimentally confirm steady-state behavior when this, in fact, is not the case. The voltammogram obtained in the case nVt' = 25 has no maximum but the value of Ellm/2 obtained from this curve would be in error by 20/n mV. With respect to this parameter it should be recalled that for a tubular electrode E112 differs from the polarographic E112 by (RTIGnF) In (Dox/Drerj)'l2. Acknowledgment. The support of the Robert A. Welch Foundation (Grant No. D-511) is gratefully acknowledged.
Appendix I. Explanation of Notation r = actual radial coordinate p = radial coordinate of model R = inside tube radius Z = actual axial coordinate S = axial coordinate of model X = electrode length D = actual diffusion coefficient 6 = dimensionless diffusion coefficient Vo = axial flow rate V, = volume flow rate Wo = maximum axial velocity in units of 1:per time step L = number of length units of model P = number of radial units of model K = iteration counter R = dimensionless time F,,J,~= dimensionless concentration Z = dimensionless current parameter t' = equivalent time (see eq 27 and 29) 8' = dimensionless equivalent time References and Notes (1) V. G. Levich, Discuss. FaradaySoc., 1, 37 (1947). (2) G. Wranglen and 0. Nilsson, Electrochim. Acta, 7, 121 (1962). (3) J. Jordan, R. A. Javick, and W. E. Ranz, J. Amer. Chem. Soc., 80, 3846 (1958). (4) R. N. Adarns. "Electrochemistry at Solid Electrodes," Marcel Dekker. New York, N. Y., 1969. (5) V. G. Levich, "Physicochemical Hydrodynamics." Prentice-Hall. Englewood Cliffs, N. J., 1962. (6) Reference 4, pp 110-1 14.
(7) C. S. Lin, E. B. Denton, H. S. Gaskill, and G. L. Putnarn, ind. Eng. Chem., 43,2136 (1951). (8) J. C. Bazan and A. J. ArJia, Electrochim. Acta, 9, 17 (1964). (9) J. C. Bazan and A. J. ArJia, Electrochim. Acta, 9, 667 (1964). (IO) T. K. Ross and A. A. Wragg, Electrochim. Acta, 10, 1093 (1965). (11) W. J. Blaedel, C. L. Olson, and L. R. Sharma, Anal. Chem., 35, 2101 (1963). (12) W. J. Blaedel and L. N. Klatt, Anal. Chem., 38, 879 (1966). (13) L. N. Klatt and W. J. Blaede1,Anal. Chem., 39, 1065 (1967). (14) L. N. Klatt and W. J. Blaedel, Anal. Chem., 40, 512 (1968). (15) T. 0. Oesterling and C. L. Olson, Anal. Chem., 39, 1546 (1967). (16) T.0. Oesterling and C. L. Olson, Anal. Chem., 39, 1543 (1967). (17) W. D. Mason and C. L. Olson, Anal. Chem., 42,548 (1970). (18) W. J. Blaedel and C. L. Olson, Anal. Chem., 36, 343 (1964). (19) W. D. Mason and C. L. Olson, Anal. Chem., 42,488 (1970). (20) D. B. Easty, W. J. Blaedel, and C. Anderson, Anal. Chem., 43, 509 (1971). (21) W. J. Blaedel and S.C. Boyer, Anal. Chem., 43,1538 (1971). (22) Reference 5.p 112. (23) S. W. Feldberg and C. Auerbach.AnaL Chem., 36, 505 (1964). (24) S. W. Feldberg in "Electroanalytical Chemistry-A Series of Advances, Vol. I I i , A. J. Bard, Ed., Marcel Dekker, New York, N. Y., 1969. (25) L. S. Marcoux, R. N. Adarns, and S. W. Feldberg, J. Phys. Chem., 73,2611 (1969). (26) K. B. Prater and A. J. Bard, J. Electrochem. SOC., 117, 207 (1970). (27) K. B. Prater and A. J. Bard, J. Electrochem. SOC., 117, 335 (1970). (28) K. 6. Prater and A. J. Bard, J. Electrochem. SOC., 117, 1517 (1970). (29) J. B. Flanagan and L. Marcoux, J. Phys. Chem., 77, 1051 (1973). (30) I. B. Goldberg and A. J. Bard, J. Electroanal. Chem., 38, 313 (1972). (31) I. B. Goldberg and A. J. Bard, J. Phys. Chem., 75, 3281 (1971). (32) I. 6.Goldberg, A. J. Bard, and S. W. Feldberg, J. Phys. Chem., 76, 2550 (1972). (33) Reference 5, pp 112-113. (34) H. E. Wilhelm, Z.Angew. Phys., 30, 376 (1971). (35) B. Carnahan, H. A. Luther, and J. 0.Wilkes, "Applied Numerical Methods," Wiley, New York. N. Y., 1969, p 429. (36) C.-E. Froberg, "Introduction to Numerical Analysis," 2nd ed, Addison-Wesley, Reading, Mass., 1965, p 294. (37) K. B. Prater in "Applications of Computers to Chemical Instrurnentation," J. S. Mattson, H. B. Mark, Jr., and J. C. MacDonald, Jr.. Ed., Marcel Dekker, New York. N. Y., 1972. (38) R. M. Barrer, "Diffusion I n and Through Solids," Macrnillan, New York, N. Y.. 1941, p34. (39) W. Jost, "Diffusion in Solids, Liquids, and Gases." Academic Press, New York, N. Y., 1960, p 52. (40) P.J. Lingane, Anal. Chem., 36, 1723 (1964). (41) 2.G. Soos and P. J. Lingane, J. Phys. Chem., 68,3821 (1964). (42) M. von Stackelberg, M. Pilgram, and V. Toorne. Z. Elektrochem.. 57, 342 (1953). (43) J. M. Hale, J. Electroanal. Chem., 6, 187 (1963). (44) J. M. Hale, J. Electroanal. Chem., 8, 332 (1964). (45) R. P. Buck and H. E. Keller,Anal. Chem., 35, 900 (1963)
Ion Association between Naphtho[b]cyclobutene Radical Anion and Alkali Metal Ions' Reuben D. Rieke*
and Stephen E. Bales
William Rand Kenan, Jr. laboratories of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27574 (ReceivedAugust 77, 7972)
Alkali metal reduction of naphtho[b]cyclobutene has been carried out in a variety of ethereal solvents in order to study ion-pairing effects. The methylene protons of naphtho[b]cyclobutene were found t o be equivalent under all conditions involving tight ion pairs. The origin of this equivalency is discussed.
Interest in the detailed structure of ion pairs has been ion p a i r ~ . ~ Esr c - ~studies of pyracene4 and acenaphthene5 c ~ n s i d e s a b l e .Of ~ particular interest has- besn. the strtdy ~ f . radical. snion~-haxve prov.ide%.exmqdes- of. !ine+idth. slterthe kinetics and equilibria between structurally different nation and nonequivalency of methylene protons as a reThe Journal of Physical Chemistry, Vol. 78, No. 7, 1974
