Scherson, Daniel A.

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Lawrence Berkeley National LaboratoryLawrence Berkeley National LaboratoryTitleTHEORY OF TRANSITION TIMES: CATALYSIS AT ROTATING DISK ELECTRODESPermalinkhttps://escholarship.org/uc/item/6wf2m604AuthorScherson, Daniel A.Publication Date1980-03-01eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaSubmitted to the Journal of Electroanalytical and Interfacial Electrochemistry THEORY OF TRANSITION TIMES: CATALYSIS AT ROTATING DISK ELECTRODES Daniel A. Scherson and Philip N. Ross LBL-10700~ · Prepri nt lAWRENCE 8f::RKF1 f.Y '"'·· · LABORATORY March 1980 APR 2 5 1980 TWO-WEEK LOAN COP This is a Library Circulating Copy which may be borrowed for two For a personal retention copy~ call Tech. Info. Division, Ext. 6782. Prepared for the U.S. Department of Energy under Contract W-7405-ENG-48 DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflec~ those of the United States Government or any agency thereof or the Regents of the University of California. "Theory of transition times: Catalysis at rotating disk electrodes" Daniel A. Scherson and Philip N. Ross Materials and Molecular Research Division, Lawrence Berkeley Laboratories Berkeley, California 94720 (U.S.A.) LBL-10700 -2-Introduction The Sand equation1 has been applied to the study of anodic passiva-tion of metals and mass-transfer processes of electrochemically active species in oxidation-reduction reactions at electrode surfaces. It has been observed experimentally2- 5 , however, that in some cases a constant must be subtracted from the classical current density-transition time rela-tionship for the lines to intercept the origin. Recently, Littauer and Tsai 4 have reported a model that successfully accounts for this anomalous behavior. Their treatment involves solving Fick's law with initial condi-tions representing an already developed concentration profile brought about by a heterogeneous process at the surface. Based on this simplified model, and assuming an exponential form for the initial profile, these authors 6 were able to determine rates of catalytic decomposition of hydrogen peroxide using rotating disk electrodes of different materials. This particular forced convection system has been widely utilized in the field of electro-chemistry, because the characteristics of its hydrodynamic flow are well understood. This communication analyzes in detail the problem of convective dif-fusion to a rotating disk with specific application given for an electr·o-active species undergoing a catalytic reaction at the electrode surface. As a result, rate constants for this catalytic step can be determined by chronopotentiometric techniques, without making any assumptions as to the form of the rate law or the relative magnitude of the kinetics. A rigorous mathematical derivation will establish the range of validity of the model proposed by Littauer and Tsai 4 , and allows for a precise interpretation of transition times. -3-Mathematical Formalism Within the framework of approximations specified elsewhere7 , the equation of convective diffusion to a rotating disk electrode is given by, a2c ac ac -+3z;;2--= ( 1 ) az;; 2 az;; ae where c is the concentration of the electrochemically active species. The dimensionless variables z;; and e , are related to y , the axial distance from the disk, and the time t, by (2) Initial and boundary conditions corresponding to a current step at the surface of the electrode can be expressed as, e > o (3) lim c( z;;, 8) = c co where c 0 is any prescribed initial profile, c the bulk concentra-co tion, and (4) the reduced flux at the surface. The solution to this problem can be formally written as a sum of two functions which independently satisfy the differential equation + (5) -4-where c1 and c2 are subject to initial and boundary conditions specified in EQ(6) and EQ(7) respectively. lim c 1 (~,6) = C00 I;; + 00 ::: c (l;;) 0 lim c2 ( z;;, e) ::: 0 z;; -+ 00 = ::: e > o (6) 0 e > o (7) If the electroactive species is undergoing a catalytic {chemical) reaction at the surface of the electrode, a steady state concentration pro-file will be established prior to the current step. assumes an explicit functional form8 { 1 - c /c =-c oo CX) r (4/3l Consequently, c (rl 0 (8) where c0 corresponds to the unknown concentration of the species at the surface at time zero. Nisancioglu and Newman9 have reported a solution for c1 ( z;; , e ), valid for long times, in terms of numerically evaluated coefficients, eigenfunctions and eigenvalues of the resulting Sturm-Liouville system, that is c 00 <P { I z;; 3 e-x dx B Z { z;;l n n ->- e } e n (9) n=O -5-It is interesting to note that due to the unique form of c 0 1: ) a solu-tion for c2 ( 1: , e ) can be obtained from the same eigenfunction expan-sion, yielding 00 c ... _(_c...;;..../_c ___ , ) '""" 00 0 00 L...J r (4/3) n::O B Z ( 1:) n n ->- e e n ( 10) Based on E0(9), EO(lO), and E0(4), the solution for the time dependent concentration at the electrode surface, c(O, e ), can be shown to be given by :: f(4/3) - { 1 - ( 11 ) c - c(0,6) 00 Given that measurements of transition times will require applying currents in excess of the limiting current it is convenient to express ¢ in those terms. For.the particular dimensionless variables chosen the reduced flux at the surface at the limiting current, is ac c 00 ( 12) Therefore, it is natural to define :: ]JC /f(4/3) 00 , where ]J > 1 and EO(ll) can then be rearranged to yield, ( 13) where c - c(0,8) c - c e 00 11.0 00 0 II. = :: and e = ( 14) 0 0 0 2 c c (f{4/3}) 00 00 -6-it Although EQ(l3) represents the solution for any arbitrary time e0 would be necessary to include an increasing number of eigenfunctions and eigenvalues in ordec to generate acceptable values for short times, i.e. 00< .1. However, a rapidly convergent series valid for short times can b bt · d b · s· ' · t· 10 e o a1ne y us1ng 1ver s approx1ma 10n . As it has been shown by ll Hale , this approximation yields excellent results in this regime and is only up to 4% different from the exact numerical solution in the whole domain. Introducing Siver's variable, c; w =I 0 3 e-x dx ( 15) into EQ(l), and assuming a value of unity for the slowly varying function, -21:;3 e , the governing differential equation is transformed into the heat equation ( 16) Defining initial and boundary conditions corresponding to those specified in E0(5) and EQ(6), and making use of Rosebrugh and Lash-Miller12 solution in terms of Fourier series, an expression analogous to EQ(l3) is obtained. 1\ = 0 The predicted values of 1\ arising from the two diffet'ent approximations 0 i.e., EQ(l3 and EQ(l7), arc compared with Hale's numerical solution in Table I, for the speci fie case lJ = 1 and -7-Based on this analysis, a dimensionless transition time, T0 , corres-ponding to 11.0 = l can be defined, and a generalized expression relating the current density to the transition time is obtained. where Discussion B n f(4/3) \.1 1 = 1 - A T e n o --ro 11.0 ~,1 X 1, 2 (To) 0 ( 18) ~ xl, 2 (To) n=O, 1 and, 8 X (T) = exp(--t(2n-1) 2 n 2 T) 2 o TI 2 ( 2n_ 1 ) 2 o ( 19) The exact treatment presented here enables a proper interpretation of diagrams in a regime where transition times are more accessible to measure-ment. As a means of illustration, selected values of 11.0 have been chosen 0 1 to generate T- 2 vs ]J curves through the combined use of Xl (TO) and 0 X (T ) 2 0 in EQ(l8) (see Figure 1). A more detailed graph of the region in which departures from the linear behavior are observed is given in Figure 2. The curves for 11.0 = l correspond to the simpler case of 0 chronopotentiometry in the absence of catalysis9 . By plotting experimental results 0 in the format of Figures 1 and 2 a unique value of 11.0 is prescribed and, thus, the catalytic rate. The functional relationship between cataly-tic rate and surface concentration can be easily established either by changing the rotation speed of the disk, or by varying the bulk concentra-tion. Standard analysis of this relationship determines the rate law and I -8-relavant kinetic parameters. This model is valid for any transition time or current sU:p above the limiting current giving rise to considerable experimental flexibility and minimizing the difficulties associated with high currents (small transition times). In the limit of very small times, i.e. 60 << .1 00 1 """' 8 - t....J -- exp{ n2 n=1 (2n-1 ) 2 and EQ(l7l reduces to In particular, for . nF \r:rri5 i =----2 c 0 I 1 - 2VS:0 I) TI c + 00 c ) nFD 0 where 6 , the thickness of the diffusion layer, is equal to 6 = (v~~ /3D )1/3 f(~)- L_ 3 Q \av (20) (21) (22) (23) Consequently, EQ(22) can be rearranged to yield the same expression re-ported by Littauer and Tsai 4 , namely (24) -9-'where in this case, the critical current density, i , and k* are given 1 by i c c :: --00----0- nFD nF\!riD and, k* = -----c 0 (25) 2 Therefore, the introduction of adjustable parameters in order to derive EQ(24), as proposed by these a~thors is unnecessary. If the initial concentration is constant throughout the solution, i.e. no catalysis, the second term in EQ(22) vanishes, yielding as expec-ted the Sand equation1 • Finally, it is important to note that EQ(22) could have also been obtained by solving Fick's law in a semi-infinite medium and expanding c2 ( s1 0) as defined in EQ(9), in terms of a Taylor series around s =0 . Summary An exact solution to the problem of convective diffusion to a rota-ting disk electrode with a prescribed initial profile and current step conditions at the surface is presented. Based on this solution a current density-transition time relationship is established which in the limit reduces to a previously proposed expression that accounts for experimen-tally observed deviations from the Sand equation. Applications of this theory in connection with the determination of rate parameters for electroactive species undergoing a catalytic reaction at the electrode surface are discussed. Acknowledgement This work was supported by the Division of Solar, Geothermal, Elec-tric and Storage Systems, Office of the Assistant Secretary of Energy Technology, U.S. Department of Energy under contract no. 7405-ENG-48. -10-TABLE I Values of the Dimensionless Surface Concentration 1\0 as a Function of the Dimensionless Time eo 0 = 1 ) • [\ =0, J1 0 * * eo 1\0 1\0 Exact EO. (11) EQ. (13) Solution .1 0.355 0.357 0.352 .2 0.494 0.504 0.492 .3 0.594 0.613 0.593 .4 0.672 0.697 0.671 .5 0.734 0.764 0.733 . 6 0.783 0.815 0.783 .7 0.824 0.855 0.824 .8 0.856 0.887 0.857 .9 0.883 0.912 0.884 1.0 0.905 0.931 0.905 *These values have been calculated using only the first three terms of the sums. -11-References l. H. J. s. Sand, Phil. Mag., 1' 45 (1901). 2. c. Iwakura, Y. Matsuda and H. Tamura, E1ectrochim. Acta, 16, 471 (1971). 3. E. D. Farmer and A. H. Webb, J. Appl. Electrochem., 2, 123 (1972). 4. 5. 6. E. E. E. L. Littauer L. Littauer L. Littauer and K. c. Tsai, and K. c. Tsai, and K. c. Tsai, J. E1ectrochem. Soc., 123, 964 (1976). Electrochim. Acta, 24, 953 (1979). J. E1ectrochern. Soc., 126, 1925 (1979). 7. R. Hornsy and J. Newman, J. E1ectrochern. Soc., 121, 521 (1974). 8. J. Newman, Electrochemical Systems, Prentice Hall, Englewood Cliffs, 1973, pg. 308. g. K. Nisancioglu and J. Newman, J. Electroanal. Chern., 50, 23 (1974). 10. Yu. G. Siver, Zh. Fiz. Khirn., 34, 577 (1960). 11. J. M. Hale, J. Electroanal. Chern., 6, 187 (1963). 12. T. R. Rosebrugh and W. Lash-Miller, J. Phys. Chern., 14, 816 (1910). -12-Nomenclature a B n D F i k* n t y z n A A n \) T e X ]J 6 0.51023 Coefficients of Sturm-Liouville system Concentration/mol cm-3 D"ff . ff" . t! 2 -l 1 us1on coe 1c1en em s -1 Faraday's constant/cb mol -2 -1 Current density/amp em s nFVTiiS 2 c , constant defined in EQ.(l9) 0 Number of electrons transferred Time/s Axial distance from disk/em Eigenfunction of Sturm-Liouville system Dimensionless concentration Eigenvalues of Stur~-Liouville system K. t. . . . t I 2 -l 1nema 1c v1scos1 y em s -l Angular rotation speed/radian s Transition time/s Dimensionless time, see EQ.(2) Siver's variable, see EQ.(l2) Terms in sum, see EO.(l4) i/ilim Diffusion layer thicknesc/cm, see EQ.(l8) Dimensionless distance normal to the disk, see E0.(2) -3 Reduced flux at the surface/mol em -13-FIGURE CAPTIONS: Figure 1: Figure 2: Square root of inverse transition time vs. number of limiting currents plotted for selected values of A0 as 0 calculated through the combined use of x1 (T0 ) and x2 (T0 ) in EO. (18). Expanded form of Figure 1 in the region closer to the limiting current as evaluated from x1 (T0 l in EO. (18). 0 0 f'() 0 L() 0 1"-. 0 (j) 0 II oo < N N 0 C\J ro -14-lQ ~ 2/1-0~ N Q ro 0 0 N 01 ~ 1"--(.() 0 w (j) I -0 co -' m X 10 9 8 7 C\.1 ' 6 I 0 t--> 4 3 2 0 A 0 =0.9 2 3 0.7 4 -15-0.5 5 f-L 6 0.3 0.1 0 7 8 9 XBL 801-97 

THEORY OF TRANSITION TIMES: CATALYSIS AT ROTATING DISK ELECTRODES

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