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Limit cycle in electrochemical oscillation—potential oscillation during anodic oxidation of H2
LIMIT CYCLE IN ELECTROCHEMICAL OSCILLATIONPOTENTIAL OSCILLATION DURING ANODIC OXIDATION OF H,* TAKURO KODERA+, TADAYOSHI
YAMAZAKI,
Research Institute for Catalysis, (Received
29 July
MAKIHIKO
Hokkaido
1987; in reuised
MASUDA and RYUICHIRO
University,
Sapporo,
form6 October
OHNISHI
060, Japan.
1987)
A limit cycle was obtained by simulating kinetic equations describing the potential oscillation due to coupling of the anodic oxidation of Hz with Ag deposition and dissolution. Attractive interaction between Ag adatoms on Pt test electrodes was found to he essential for producing the limit cycle.
Abstract
INTRODUCTION
depositiondissolution results in potential oscillation. The present paper aims to give a kinetic model representing the potential oscillation and to report a limit cycle obtained from the simulation.
Periodic phenomena during the anodic oxidation of H, at Pt test electrode have been reported[l, 23. The coupling of H, oxidation with adsorption (deposition) and desorption (dissolution) of an inhibitor for the H, oxidation was supposed to result in potential oscillation. Inhibitors reported were metals such as Ag and Cu, metal oxides, organic compounds. In a previous report[3], the deposition and dissolution processes of Ag metal were directly observed by measuring the rest potential of a Ag monitor electrode during potential oscillation as shown in Fig. 1. From the above result, it was clarified that the coupling of H2 oxidation with Ag
KINETIC
ANALYSIS
The coupling reactions are analyzed on the basis of the following four conditions. (i) The constant anodic current, I, is the sum of the oxidation current of H,, I u, the cathodic current of Ag deposition, I,, the anodic current of deposited-silver dissolution, I,, and the charging or discharging current of the double layer, 1,,,[2]: I = In-le+la+I*,_
-460
In the above first condition, the current of oxide formation-reduction on Pt was assumed to be negligible compared with I,, I, and I,, because the waves of oxide formation-reduction do not appear in the voltammogram at a H, bubbling rate of 200 cm3 min - I even at high potential up to 1.6 V, and moreover, in the solution containing Cu2+ instead of Ag’, the shape of potential oscillation is similar to that shown in Fig. 1, although the maximum potential in oscillation is about 600 mV[2, 71, at which oxide formation does not visibly occur. (ii) The deposited-silver, Ag(a), covered less than a monolayer of the Pt test electrode surface under the experimental conditions[3]. (iii) The oxidation current of H, on Ag(a) is negligible because the activity of Ag for the hydrogen electrode reaction is two or three orders of magnitude less than that of Pt[d], and hydrogen does not adsorb on Ag[S]. (iv) The attractive interaction is introduced between silver adatoms on Pt, based on the results that the group IB metal (Cu, Ag, and Au) adatoms on Pt( 111) single crystal surface show attractive interactions and coalesce into 2d islands[6]. The I,, is given by
- +u/mV
400-
(1)
- 420
t /set Fig. 1. Sustained potential oscillation of Pt test electrode (+ USrhe) and of Ag monitor electrode (+,,, DSrhe). Solution: 5.2 x 10m6 M AgClO, in 0.70 M HCIO, aqueous solution, H, bubbling rate: 200 cm3 mitt- ‘, constant anodic current: 6.0 mA.
*Part of a Dissertation. ‘Author to whom correspondence should be addressed Muroran Institute of Technology, Muroran, 050, Japan.
I,, = C, W/dr), 537
(2)
TAKURO KODERA et al.
538
where Cd is the double layer capacity, and 4 is the electrode potential. Combination of equations (1) and (2) leads to d4/dt = (I-lIH+le--1,)/C.,. The time dependence as
(3)
#/mV
of Ag(a) coverage, 0, is expressed
doldt
= (I, -
I,YQ.
(4)
where Q is the quantity of electricity which is required to complete the monolayer of Ag(a). The values of C, and Q were estimated to be 18 x S JIF and 210 x S PC, respectively, where S, true surface area of the Pt test The simultaneous differential electrode, is 45 cm’. equations (3) and (4) can be solved by giving the rate equations for I,, I, and I,. Figure 2 shows the polarization curve for H, oxidation in 0.7 M HClO., aqueous solution without Ag l. From Fig. 2, it is suggested that the overall rate of the reaction is controlled by the rate of surface reaction at low potential region and by the rate of diffusion at high potential region, respectively. Thus, the following reaction scheme is considered: H,(b)
+ H,(s)
(5.1)
H,(s)=2H(a) H(a)-+
(5.2)
H+ +e-
2oc
IOC
0 I /mA
Fig. 2. Polarization curve for H2 oxidation in 0.70 M HClO*. H2 bubbling rate: 200 cm”min- .‘ -: observed, A: calculated.
(5.3)
where (b) and (s) stand for solution bulk and for the vicinity of the electrode surface, respectively. From the scheme and the conditions (ii) and (iii), the rate equation of I, is expressed as I
30(
H=
determining step, (5.3), and equal to 2, and (&$/al), = 0 is the polarization resistance at equilibrium potential. The unknown value of K was determined to be 0.1 so that the calculated curve might fit closely to the observed one, as shown in Fig. 2. The following scheme is adopted for Ag deposition and dissolution at Pt test electrode: Ag+ (b) -
(6.1) where
I =ka+K 0 __l+K
k,-
(6.2)
l+K’
K = Ckz+CH,(Wllk,-I”“,
(6.3)
I3 = Cl -Z,/{I,(l
(6.4)
- B)}]“2,
Ag+ (s),
Ag* (s) + e
(8.1)
+ Ag(a).
(8.2)
The exact rate equation is still not completely elucidated especially in the presence of Hz oxidation but the rate of deposition, I,, and that of dissolution, I,,,may be given from the above scheme and the conditions (ii) and (iv):
I ~..&CAg+(bIl(l--8)exp(G)exp(
-(liG)oF)
I, =
(9) ZLA,+k,CAg+(b)]exp($)exp(
-(liG)dF)
’
I,, = k,,,exp(w)exp(g)
and kj+ or kj_ is the constant of step 5.3. The tion in Ag+ free solution the H, bubbling rate of current of H2 diffusion,
forward or backward rate rate equation of Hz oxidais given by (6.1) at 8 = 0. At 200 cm3 min _ ,‘ the limiting I,, was determined to be
67 mA from the steep rising of current in Fig. 2. The exchange current, I,, was derived as 13.8 mA by using Horiuti’s equation [8]. I,, = (vRT/2F)/(hb/aI where
Y is the stoichiometric
(7)
),+=,, number
of
the rate
where k, or k, is the rate constant of the cathodic or anodic process, and u is the interaction energy between Ag adatoms. The limiting current of Ag+ diffusion, >was roughly estimated to be 0.23 mA by Fick’s 1 fi%aw under the experimental conditions depicted in Fig. 1. The symmetry factor, 6, and the anodic Tafel constant, y, were taken to be the same as 0.5. Now, prior to the analysis of the simultaneous differential equations (3) and (4) by using three rate equations (6), (9) and (IO), it is necessary to examine whether a computed solution shows a sustained oscil-
Limit cycle in electrochemical oscillation lation or not, in other words, to check necessary condition(s) that limit cycle(s) will be obtained. Two inequalities (11) should be satisfied to give a limit cycle on the basis of the linearized normal mode analysis[9]: all
+a22
’
(11.1)
0
(all -+#+4a12+, aI1 = ~12 =
(11.2)
< 0
(l/~)(ar,iae-az,iae),~.,,(12.1) (12.2) (lIQ)(a~,l~~l,-~Z,la~),~,,
uzl = (l/C,)(--Z,/as+aZ,/ae-aZ,/ae),~.,
(12.3)
az2 = (l/C,)(
(12.4)
- aZ,ia+
+ aZ,dlp - aZ,/aa),ij
where 4 and 6 are C#J and 0 at steady state, respectively, which satisfy d+ldt = d0ldt = 0 of equations (3) and (4). It should be emphasized here that the inequalities (11) were not fulfilled except for the case where the condition (iv) is adopted: u < 0. Predetermined constants are summarized as follows: C, = 0.81 mF, Q = 9.45 mC, 6 = y = 0.5, I,
539
= 13.8mA, K =O.l, I,=67mA, I,.,,=0.23mA, [Ag+] = 5.2 x 10 6 mol I- ‘, Z = 6.0 mA. Three parameters, k,, k, and u were selected, so as to satisfy the necessary conditions (1 l), and the simultaneous differential equations (3) and (4) were calculated by using Runge-Kutta-Gill method. The cycle of selection of parameters and the calculation was repeated until the q’-t curve observed experimentally as shown in Fig. 1[3] was reproduced. The final values of k,, k, and II were 8.57 x 10’ mA obtained 1 mol-‘, 4.31 x 10m4 mA and - 17.7 Kcal mol-‘, respectively. Figure 3 shows the phase curves calculated by the use of the three parameters and the predetermined constants. Every phase curve starting from points A, B and C winds into a cycle with a passage of time. Therefore, the cycle is a limit cycle, which corresponds to the stable periodic oscillations of 4 and 0. Integral curves of + and 0 are illustrated in Fig. 4, where phase shift between two curves can be seen clearly. Compared with the curve in Fig. 1, oscillation frequency is different in some measure, but the slow
::,: 1;/
Fig. 3. Phase curve of d OS8. Initial point (4 mV, 0): A (950, 0.95X B (400, 0.40), C (100, 0.01). values of parameters: see in text.
. :
___-*--. *.-.,: ;_.”
80(
4ot 201
,
:
____-----., _.--
_-_._-----..*
,’
**
.’
: :
IOOC +/mV
601
,’ a’
/
J
:
L
: : : : : : : :
::/
600
400
t/see
Fig. 4. Integral curves of rp and 8. Initial point: A in Fig. 3.
E
540
TAKURO
KODERA
ups and the sudden drops and amplitude on 4-t curve reappear in the same manner. For passive film formation, there have been several reports dealing with model calculations[lO]. Some of them described a limit cycle, but the limit cycle is not a true one in the sense that the ups and the downs of 4 or I-t curve were explained by two separate equations, respectively. On the other hand, recently, Talbot and Oriani[l l] have reported the models provided steadystate multiplicity and oscillatory solutions. The minimum requirement for the models was to have a multiple-valued isotherm arising from the desorption functionality exp(B8) with attractive interaction of p c 0. It was also concluded, as was already described in the present report, that no limit cycle was obtained without condition (iv) in more complex systems consisting of reactions of H2 oxidation and Ag deposition and dissolution. The conclusion was drawn from the strong connection of the model with the experimental results. Further proof of attractive interaction between metal adatoms may be obtained from the experiments on Pt( 111) single crystals with group of IB or VB metal adatoms because it is known that group of the former adatoms interact attractively and the latter repulsively[6]. Including the above subjects, further investigation is going on and will be reported in a separate paper.
et al.
REFERENCES
1. M. Thalinger and M. Volmer, Z. phys. Chem. 150, 401 (1930); G. Armstrong and J. A. V. Butler, Disc. Faraday
Sot. I, 122 (1947); D. T. Sawyer and E. T. Sea, J.
2. 3. 4. 5. 6. 7. 8.
9.
10.
11.
electroanal. Chem. 5, 23 (1963); B. E. Conway and D. M. Novak, J. phys. Chem. 81, 1459 (1977). G. Horanyi and Cs. Visy, J. electroanal. Chem. 103, 3.53 (1979). T. Kodera, T. Yamazaki and N. Kubota, Electrochim. Acta, 31, 1477 (1986). H. Kita, J. Rex Inst. Catal. Hokkoido Univ. 13,151(1965). G. C. Bond, Catalysis by Metals, Academic Press, (1962). M. T. PatTett, C. T. Campbell and T. N. Taylor, J. Vat. Sci. Technol. 3, 8 12 (1985). T. Yamazaki and T. Kodera to be oublished. J. Horiuti, J. Res. Inst. Catnl. Hokkdido Univ. I,8 (1948); T. Kodera, H. Kita and M. Honda, Electrachim. Acta, 17, 1361 (1972). G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems-From Dissipative Structures for to Order through Fluctuations, John Wiley (1977). V. U. F. Frank and R. Fitzhugh, Z. Elektrochemie, 65,156 (1961); .I. Keizer and D. Scherson, J. ghys. Chem. 84,2025 (1980); P. Russell and Newman, J. electrochem. Sac. 134, 1051 (1987). J. B. Talbot and A. Oriani, Electrochim. Acta, 30, 1277 (1985).
YAMAZAKI,
Research Institute for Catalysis, (Received
29 July
MAKIHIKO
Hokkaido
1987; in reuised
MASUDA and RYUICHIRO
University,
Sapporo,
form6 October
OHNISHI
060, Japan.
1987)
A limit cycle was obtained by simulating kinetic equations describing the potential oscillation due to coupling of the anodic oxidation of Hz with Ag deposition and dissolution. Attractive interaction between Ag adatoms on Pt test electrodes was found to he essential for producing the limit cycle.
Abstract
INTRODUCTION
depositiondissolution results in potential oscillation. The present paper aims to give a kinetic model representing the potential oscillation and to report a limit cycle obtained from the simulation.
Periodic phenomena during the anodic oxidation of H, at Pt test electrode have been reported[l, 23. The coupling of H, oxidation with adsorption (deposition) and desorption (dissolution) of an inhibitor for the H, oxidation was supposed to result in potential oscillation. Inhibitors reported were metals such as Ag and Cu, metal oxides, organic compounds. In a previous report[3], the deposition and dissolution processes of Ag metal were directly observed by measuring the rest potential of a Ag monitor electrode during potential oscillation as shown in Fig. 1. From the above result, it was clarified that the coupling of H2 oxidation with Ag
KINETIC
ANALYSIS
The coupling reactions are analyzed on the basis of the following four conditions. (i) The constant anodic current, I, is the sum of the oxidation current of H,, I u, the cathodic current of Ag deposition, I,, the anodic current of deposited-silver dissolution, I,, and the charging or discharging current of the double layer, 1,,,[2]: I = In-le+la+I*,_
-460
In the above first condition, the current of oxide formation-reduction on Pt was assumed to be negligible compared with I,, I, and I,, because the waves of oxide formation-reduction do not appear in the voltammogram at a H, bubbling rate of 200 cm3 min - I even at high potential up to 1.6 V, and moreover, in the solution containing Cu2+ instead of Ag’, the shape of potential oscillation is similar to that shown in Fig. 1, although the maximum potential in oscillation is about 600 mV[2, 71, at which oxide formation does not visibly occur. (ii) The deposited-silver, Ag(a), covered less than a monolayer of the Pt test electrode surface under the experimental conditions[3]. (iii) The oxidation current of H, on Ag(a) is negligible because the activity of Ag for the hydrogen electrode reaction is two or three orders of magnitude less than that of Pt[d], and hydrogen does not adsorb on Ag[S]. (iv) The attractive interaction is introduced between silver adatoms on Pt, based on the results that the group IB metal (Cu, Ag, and Au) adatoms on Pt( 111) single crystal surface show attractive interactions and coalesce into 2d islands[6]. The I,, is given by
- +u/mV
400-
(1)
- 420
t /set Fig. 1. Sustained potential oscillation of Pt test electrode (+ USrhe) and of Ag monitor electrode (+,,, DSrhe). Solution: 5.2 x 10m6 M AgClO, in 0.70 M HCIO, aqueous solution, H, bubbling rate: 200 cm3 mitt- ‘, constant anodic current: 6.0 mA.
*Part of a Dissertation. ‘Author to whom correspondence should be addressed Muroran Institute of Technology, Muroran, 050, Japan.
I,, = C, W/dr), 537
(2)
TAKURO KODERA et al.
538
where Cd is the double layer capacity, and 4 is the electrode potential. Combination of equations (1) and (2) leads to d4/dt = (I-lIH+le--1,)/C.,. The time dependence as
(3)
#/mV
of Ag(a) coverage, 0, is expressed
doldt
= (I, -
I,YQ.
(4)
where Q is the quantity of electricity which is required to complete the monolayer of Ag(a). The values of C, and Q were estimated to be 18 x S JIF and 210 x S PC, respectively, where S, true surface area of the Pt test The simultaneous differential electrode, is 45 cm’. equations (3) and (4) can be solved by giving the rate equations for I,, I, and I,. Figure 2 shows the polarization curve for H, oxidation in 0.7 M HClO., aqueous solution without Ag l. From Fig. 2, it is suggested that the overall rate of the reaction is controlled by the rate of surface reaction at low potential region and by the rate of diffusion at high potential region, respectively. Thus, the following reaction scheme is considered: H,(b)
+ H,(s)
(5.1)
H,(s)=2H(a) H(a)-+
(5.2)
H+ +e-
2oc
IOC
0 I /mA
Fig. 2. Polarization curve for H2 oxidation in 0.70 M HClO*. H2 bubbling rate: 200 cm”min- .‘ -: observed, A: calculated.
(5.3)
where (b) and (s) stand for solution bulk and for the vicinity of the electrode surface, respectively. From the scheme and the conditions (ii) and (iii), the rate equation of I, is expressed as I
30(
H=
determining step, (5.3), and equal to 2, and (&$/al), = 0 is the polarization resistance at equilibrium potential. The unknown value of K was determined to be 0.1 so that the calculated curve might fit closely to the observed one, as shown in Fig. 2. The following scheme is adopted for Ag deposition and dissolution at Pt test electrode: Ag+ (b) -
(6.1) where
I =ka+K 0 __l+K
k,-
(6.2)
l+K’
K = Ckz+CH,(Wllk,-I”“,
(6.3)
I3 = Cl -Z,/{I,(l
(6.4)
- B)}]“2,
Ag+ (s),
Ag* (s) + e
(8.1)
+ Ag(a).
(8.2)
The exact rate equation is still not completely elucidated especially in the presence of Hz oxidation but the rate of deposition, I,, and that of dissolution, I,,,may be given from the above scheme and the conditions (ii) and (iv):
I ~..&CAg+(bIl(l--8)exp(G)exp(
-(liG)oF)
I, =
(9) ZLA,+k,CAg+(b)]exp($)exp(
-(liG)dF)
’
I,, = k,,,exp(w)exp(g)
and kj+ or kj_ is the constant of step 5.3. The tion in Ag+ free solution the H, bubbling rate of current of H2 diffusion,
forward or backward rate rate equation of Hz oxidais given by (6.1) at 8 = 0. At 200 cm3 min _ ,‘ the limiting I,, was determined to be
67 mA from the steep rising of current in Fig. 2. The exchange current, I,, was derived as 13.8 mA by using Horiuti’s equation [8]. I,, = (vRT/2F)/(hb/aI where
Y is the stoichiometric
(7)
),+=,, number
of
the rate
where k, or k, is the rate constant of the cathodic or anodic process, and u is the interaction energy between Ag adatoms. The limiting current of Ag+ diffusion, >was roughly estimated to be 0.23 mA by Fick’s 1 fi%aw under the experimental conditions depicted in Fig. 1. The symmetry factor, 6, and the anodic Tafel constant, y, were taken to be the same as 0.5. Now, prior to the analysis of the simultaneous differential equations (3) and (4) by using three rate equations (6), (9) and (IO), it is necessary to examine whether a computed solution shows a sustained oscil-
Limit cycle in electrochemical oscillation lation or not, in other words, to check necessary condition(s) that limit cycle(s) will be obtained. Two inequalities (11) should be satisfied to give a limit cycle on the basis of the linearized normal mode analysis[9]: all
+a22
’
(11.1)
0
(all -+#+4a12+, aI1 = ~12 =
(11.2)
< 0
(l/~)(ar,iae-az,iae),~.,,(12.1) (12.2) (lIQ)(a~,l~~l,-~Z,la~),~,,
uzl = (l/C,)(--Z,/as+aZ,/ae-aZ,/ae),~.,
(12.3)
az2 = (l/C,)(
(12.4)
- aZ,ia+
+ aZ,dlp - aZ,/aa),ij
where 4 and 6 are C#J and 0 at steady state, respectively, which satisfy d+ldt = d0ldt = 0 of equations (3) and (4). It should be emphasized here that the inequalities (11) were not fulfilled except for the case where the condition (iv) is adopted: u < 0. Predetermined constants are summarized as follows: C, = 0.81 mF, Q = 9.45 mC, 6 = y = 0.5, I,
539
= 13.8mA, K =O.l, I,=67mA, I,.,,=0.23mA, [Ag+] = 5.2 x 10 6 mol I- ‘, Z = 6.0 mA. Three parameters, k,, k, and u were selected, so as to satisfy the necessary conditions (1 l), and the simultaneous differential equations (3) and (4) were calculated by using Runge-Kutta-Gill method. The cycle of selection of parameters and the calculation was repeated until the q’-t curve observed experimentally as shown in Fig. 1[3] was reproduced. The final values of k,, k, and II were 8.57 x 10’ mA obtained 1 mol-‘, 4.31 x 10m4 mA and - 17.7 Kcal mol-‘, respectively. Figure 3 shows the phase curves calculated by the use of the three parameters and the predetermined constants. Every phase curve starting from points A, B and C winds into a cycle with a passage of time. Therefore, the cycle is a limit cycle, which corresponds to the stable periodic oscillations of 4 and 0. Integral curves of + and 0 are illustrated in Fig. 4, where phase shift between two curves can be seen clearly. Compared with the curve in Fig. 1, oscillation frequency is different in some measure, but the slow
::,: 1;/
Fig. 3. Phase curve of d OS8. Initial point (4 mV, 0): A (950, 0.95X B (400, 0.40), C (100, 0.01). values of parameters: see in text.
. :
___-*--. *.-.,: ;_.”
80(
4ot 201
,
:
____-----., _.--
_-_._-----..*
,’
**
.’
: :
IOOC +/mV
601
,’ a’
/
J
:
L
: : : : : : : :
::/
600
400
t/see
Fig. 4. Integral curves of rp and 8. Initial point: A in Fig. 3.
E
540
TAKURO
KODERA
ups and the sudden drops and amplitude on 4-t curve reappear in the same manner. For passive film formation, there have been several reports dealing with model calculations[lO]. Some of them described a limit cycle, but the limit cycle is not a true one in the sense that the ups and the downs of 4 or I-t curve were explained by two separate equations, respectively. On the other hand, recently, Talbot and Oriani[l l] have reported the models provided steadystate multiplicity and oscillatory solutions. The minimum requirement for the models was to have a multiple-valued isotherm arising from the desorption functionality exp(B8) with attractive interaction of p c 0. It was also concluded, as was already described in the present report, that no limit cycle was obtained without condition (iv) in more complex systems consisting of reactions of H2 oxidation and Ag deposition and dissolution. The conclusion was drawn from the strong connection of the model with the experimental results. Further proof of attractive interaction between metal adatoms may be obtained from the experiments on Pt( 111) single crystals with group of IB or VB metal adatoms because it is known that group of the former adatoms interact attractively and the latter repulsively[6]. Including the above subjects, further investigation is going on and will be reported in a separate paper.
et al.
REFERENCES
1. M. Thalinger and M. Volmer, Z. phys. Chem. 150, 401 (1930); G. Armstrong and J. A. V. Butler, Disc. Faraday
Sot. I, 122 (1947); D. T. Sawyer and E. T. Sea, J.
2. 3. 4. 5. 6. 7. 8.
9.
10.
11.
electroanal. Chem. 5, 23 (1963); B. E. Conway and D. M. Novak, J. phys. Chem. 81, 1459 (1977). G. Horanyi and Cs. Visy, J. electroanal. Chem. 103, 3.53 (1979). T. Kodera, T. Yamazaki and N. Kubota, Electrochim. Acta, 31, 1477 (1986). H. Kita, J. Rex Inst. Catal. Hokkoido Univ. 13,151(1965). G. C. Bond, Catalysis by Metals, Academic Press, (1962). M. T. PatTett, C. T. Campbell and T. N. Taylor, J. Vat. Sci. Technol. 3, 8 12 (1985). T. Yamazaki and T. Kodera to be oublished. J. Horiuti, J. Res. Inst. Catnl. Hokkdido Univ. I,8 (1948); T. Kodera, H. Kita and M. Honda, Electrachim. Acta, 17, 1361 (1972). G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems-From Dissipative Structures for to Order through Fluctuations, John Wiley (1977). V. U. F. Frank and R. Fitzhugh, Z. Elektrochemie, 65,156 (1961); .I. Keizer and D. Scherson, J. ghys. Chem. 84,2025 (1980); P. Russell and Newman, J. electrochem. Sac. 134, 1051 (1987). J. B. Talbot and A. Oriani, Electrochim. Acta, 30, 1277 (1985).