On stability of model electrocatalytic process with frumkin adsorption isotherm occurring on spherical electrode

ISSN 1023�1935, Russian Journal of Electrochemistry, 2012, Vol. 48, No. 2, pp. 154–162. © Pleiades Publishing, Ltd., 2012.Original Russian Text © V.V. Pototskaya, O.I. Gichan, 2012, published in Elektrokhimiya, 2012, Vol. 48, No. 2, pp. 171–180.

154

INTRODUCTION

Two universal experimental criteria for predictionof dynamic instabilities in electrochemical systems atpresent are the method of cyclic voltammetry [1, 2]and the impedance technique [3–10]. The negativereal component in the impedance spectrum and theintersecting cycle in cyclic voltammograms in a givenrange of potentials are typical characteristics of exist�ence of instabilities.

The impedance spectrum yields linear informationabout the system in a rather wide frequency range[11–13] and is related to bifurcation analysis of itsinstability [3–10]. The intersecting cycle in cyclic vol�tammograms reflects the phenomenological kineticsof the system including the processes with positive andnegative feedback between two bistable states [1].

The impedance spectroscopy technique was usedin this work to study the stability of the model electro�chemical system towards the Hopf bifurcation causingspontaneous oscillations in the system. As well known,appearance of the Hopf bifurcation in nonequilibriumsystems is due to existence of fast and slow nonlinearprocesses with negative and positive feedback [14]. Inelectrochemical systems, the components affectingthe system stability are the nonlinear dependence ofthe charge transfer rate on the applied potential, masstransport to the electrode surface, chemical reactionsand external cell components [15].

The electrochemical reaction in the chosen modelis related to potential–dependent adsorption/desorp�tion on the spherical electrode and the precedingchemical reaction in the Nernst diffusion layer underpotentiostatic experimental conditions. The controlsystem parameters are the spherical electrode size andeffective rate of the preceding chemical reaction.

THEORETICAL SECTION

The studied model process of electrocatalytic oxi�dation can be schematically presented as follows:

(1)

where k1, k2 are the rate constants of the direct andreverse chemical reactions and ka, kd, Ke are the rateconstants of adsorption, desorption, and electrontransfer, accordingly.

The effect of ohmic losses and the double layereffect being neglected, the adsorption kinetics equa�tion is:

(2)

Here, c(r0, t) the concentration of electroactive parti�cles on the electrode surface, θ(t) is the electrode sur�face coverage by the adsorbate, Γ is the maximum sur�face concentration at θ = 1, r0 is the spherical elec�trode radius, γ is the attraction constant in the

a e

d

bulk bulk

diffsurf ads

B A

A A P e

1

2

,

k

k

k K

k

⎯⎯⎯→←⎯⎯⎯

⎯⎯⎯→⎯⎯⎯→ ⎯⎯⎯→ +←⎯⎯⎯

a

d

1 0 0( ( ), ( , )) exp[ ( ) 2] ( , )

[1 ( )] exp[ ( ) 2] ( ).

t c r t k t c r t

t k t t

θ = Γ γθ

× − θ − Γ −γθ θ

v

On Stability of Model Electrocatalytic Process with Frumkin Adsorption Isotherm Occurring on Spherical Electrode

V. V. Pototskayaa, z and O. I. Gichanb, z

aVernadskii Institute of General and Inorganic Chemistry, National Academy of Sciences of Ukraine, Kiev, UkrainebChuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine, Kiev

Received December 8, 2010

Abstract—The method of impedance spectroscopy was used for theoretical studies of the conditions ofappearance of Hopf instability in a model electrochemical system with a preceding homogeneous chemicalreaction in the Nernst diffusion layer and electrocatalytic reaction on the spherical electrode surface underpotentiostatic conditions. It is shown within the suggested electrochemical instability model based on thepotential–dependent adsorption/desorption that the effective rate of the preceding homogeneous chemicalreaction may affect the system stability. The effect diminishes at a decrease in the electrode radius. The insta�bility region grows at an increase in the thickness of the Nernst diffusion layer.

Keywords: N�NDR systems, electrocatalytic mechanism, oscillations, Hopf bifurcation, impedance, spher�ical electrode, Nernst diffusion layer

DOI: 10.1134/S1023193512020140

z Corresponding authors: [email protected](V.V. Pototskaya); [email protected] (O.I. Gichan).

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 48 No. 2 2012

ON STABILITY OF MODEL ELECTROCATALYTIC PROCESS 155

Frumkin adsorption isotherm that we assume to beconstant. In general, γ depends on the potential[16, 17].

As follows from equation (2), adsorption/desorp�tion of particles A on the surface under the steady–state conditions occurs in accordance with theFrumkin isotherm [18].

Let us write the electron transfer rate as

(3)

where ke is the electron rate transfer constant at E(t) = 0,β = 1 – α, α is the electron transfer symmetry factor;E is the electrode potential; b = F/RT, F is the Faradayconstant, R is the gas constant, T is the absolute tem�perature.

Changes in the electrode surface coverage θ by theadsorbate and concentration c(r, t) = c0 + u (u is theconcentration deviation from the equilibrium value,c0 is the equilibrium concentration coinciding with thebulk concentration) satisfy the following equations:

(4)

(5)

with the following boundary conditions:

(6)

(7)

where k is the effective rate constant of the precedinghomogeneous chemical reaction, Jc is the diffusionflux of adsorbing particles to the electrode surface, D isthe diffusion coefficient, δ = r0 + d, d is the thicknessof the Nernst diffusion layer. The origin of coordinatescoincides with the spherical electrode center.

In diffusion equation (5) in spherical coordinates,we omitted in the rightmost part the angular compo�

nent of Laplacian assuming

that derivatives by the sphere surface are low as com�pared to derivatives by the radius vector.

As follows from equations (3) and (4), electrodesurface coverage θ depends on the potential, soadsorption/desorption preceding charge transfer anddescribed by equation (2) is potential–dependent.

The faradaic current density was determinedaccording to the following equation:

(8)

The solution of equations (4), (5) for steady–stateconditions yields the following expressions for steady–state concentration on the electrode surface,

e e2( ) ( ) ( ) exp[ ( )] ( ),t K t t k bE t t= Γ θ = Γ β θv

1 2( ) ( ),d t tdtθ

Γ = −v v

( )2

2

( , ) ( , )1c r t c r tD r kc

t r rr

∂ ∂∂= −

∂ ∂ ∂

0( , ) ,c t cδ =

c 00 1( , )

( , ) ( ),r rc r t

J r t D tr

=

∂= − = −

∂v

( )21 sin ,

sinc

r∂ ∂

ϑ∂ϑ ∂ϑϑ

f e 2( ) exp[ ( )] ( ) ( ).i t F k bE t t F t= Γ β θ = v

st 0( )c r r=

steady–state potential Est, and steady–state faradaiccurrent density:

(9)

(10)

(11)

where

(τd is the diffusion relaxation

time).

The potential is presented vs. the zero–chargepotential of the electrode free of adsorbed particles inthe supporting solution.

Under potentiostatic experimental conditions, thestudies of linear stability of the electrochemical systemnear the steady state are based on the analysis of varia�tion in the impedance zero values under variation inthe electrode potential [3, 4]. The Hopf bifurcationmay occur in the system, when its impedance is equalto zero at a frequency differing from zero.

In calculations of faradaic impedance of the sys�tem, its behavior was analyzed under low periodic sig�nal ΔE(t) = ΔE0e

jωt applied in the given point of thesteady–state voltammetric curve

(12)

where ω is the angular frequency (ω = 2πf,f is the frequency), ΔE0 is the amplitude of a low peri�odic signal.

In response to this perturbation, electrode surfacecoverage θ(t) oscillates near the steady state:

(13)

Faradaic current if(t) and concentration c(r0, t) oscil�late according to (12), (13) as follows:

The expression for faradaic impedance in the Laplace

image space as a function of com�

plex frequency s takes the form of

stc d st

stst

c st a

e

e

20

0 2

*( ) ,

* (1 )

m c kc r

m k

−γθ

γθ

+ Γ θ=

+ − θ Γ

[ ]c stst

e st

1 0 0* ( )

( ) ln ,m c c r

E bk

−

⎧ ⎫−⎪ ⎪= β ⎨ ⎬

Γ θ⎪ ⎪⎩ ⎭

fst c st0 0*[ ( )],i Fm c c r= −

c c0

0

1* ,G

m mG

+ ε= c ,Dm

d=

0

,dr

ε =

d

d

tanh0 ,

kG

k

τ=

τ

d

2dD

τ =

st e0( ) ,j tE t E E ω

= + Δ

1,j = −

st e0( ) .j tt ω

θ = θ + Δθ

f fst f( ) ( , ),i t i i E= + Δ θ

st0 0 0( , ) ( ) ( , ).c r t c r c r= + Δ θ

ste0

( ) ( )F s f t dt∞

−

= ∫f

( )( ) .

( )E s

Z si s

Δ=Δ

156


POTOTSKAYA, GICHAN

Omitting the calculations similar to those per�formed earlier [8, 9], let us write the final expressionfor faradaic impedance in the Laplace image space

(14)

where partial derivatives are designated as

and the following notations are also introduced:

(charge transfer resistance),

(finite Gerischer impedance),

(15)

To determine the Hopf bifurcation points, imped�ance zero points were studied under electrode poten�tial variation. The impedance zero points were foundaccording to the following equation:

(16)

where function Ψ(s, k, θ) represents the numerator ofequation (14).

In order to satisfy condition (16), it is necessarythat

(17)

One can solve the system of equations (17) onlynumerically.

f

cct

c

2 1

1 1

( )

[1 (1 )]1 ,

[1 (1 )] (1 )

Z s

GR

s G Gθ

θ

⎧ ⎫∂ + ε + μ∂= +⎨ ⎬Γ + ε + μ∂ − ∂ + ε⎩ ⎭

v v

v v

xu∂ =ux

∂

∂

( )fct e st st

st

1 / 1 [ exp( ) ]i

R F bk bEE

∂= = Γβ β θ

∂

d

d

tanh ( )

( )

k sG

k s

τ +=

τ +

d st st

a st st st

1 { exp( 2)( 2 1)

exp( 2) (0)[ (1 ) 2 1]},

k

k cθ∂ = Γ −γθ γθ −

+ γθ γ − θ −

v

c a st st1 (1 )exp( 2),k∂ = Γ − θ γθv

e st2 exp( ),k bEθ∂ = Γ βv

0 .r Dµ =

c2 1

1

( , , ) ( )[1 (1 )]

(1 ) 0,

k s s G

Gθ

θ

Ψ θ = Γ + ∂ + ε + μ∂

− ∂ + ε =

v v

v

Re[ ( , , )] 0

Im[ ( , , )] 0.

s k

s k

Ψ θ =⎧⎨

Ψ θ =⎩

Transition from the Laplace space into the Fourierspace was carried out through substitution of s = jω.

In the case of and the

expression for impedance takes the form of

(18)

where . The impedance zero points (18) aredetermined according to the equation of

(19)

In the calculations, we neglected the resistance ofelectrolyte and did not take into account the doublelayer impedance. In this case, the impedance of themodel system under consideration is equal to thefaradaic impedance. In the model calculation, thefollowing values of system parameters were assumed:Γ = 10–9 mol/cm2, γ = 8, Γka = 0.1 cm/s, Γkd =10⎯5 mol/cm2 s, ke = 10 s–1, D = 10–5 cm2/s, d =10⎯3 cm, α = 0.5, c0 = 10–5 mol/cm3, F = 96484 C/mol,

R = 8.314 J/mol K, T = 300 K, = 38.7 V–1.

All numeric calculations were carried out using theMathematicaTM software [19].

RESULTS AND DISCUSSION

The steady–state polarization �curves ofmodel process (1) have the N�type shape with the neg�ative differential resistance region (NDR). Anincrease in rate k of the preceding chemical reactionleads to an increase in the current density (Fig. 1) anddecrease in the charge transfer resistance (Fig. 2).These effects are weaker at a decrease in sphericalelectrode radius r0.

d → ∞

0

* ,cDm kDr

= +

f ct

c

c

2 1

1 1

( ) 1

[ (1 )],

[ (1 )] ( )

Z s R

k s

s k s k sθ

θ

⎧= ⎨⎩

⎫∂ + + χ + μ∂+ ⎬Γ + + χ + μ∂ − ∂ + + χ ⎭

v v

v v

0D rχ =

c2 1

1

( , , ) ( )[ (1 )]

( ) 0.

s k s k s

k s

θ

θ

Ψ θ = Γ + ∂ + + χ + μ∂

− ∂ + + χ =

v v

v

( )Fb

RT=

fst st,i E

0.025

0.015

0.005

–0.3 0.3 0.6Est(θ), V

ist(θ), A cm–2

1

20.03

0.02

0.01

–0.2 0.2 0.6Est(θ), V

ist(θ), A cm–2

1

2 0.08

0.04

0.02

–0.2 0.2 0.6Est(θ), V

ist(θ), A cm–2

1

20.06

(а) (b) (c)

Fig. 1. Steady–state polarization curves of a model process for two values of effective rate k of the preceding chemical reaction, s–1:(1) 0.5; (2) 85 for spherical electrode radii r0, cm: (a) 10–2, (b) 10–3, (c) 10–4.



Figure 3 shows the surfaces of the real and imaginarycomponents of function Ψ(s, k, θ) in equation (16) forthe three spherical electrode radii r0 going to zero.Intersection of surfaces and

yields the Hopf bifurcation points ofsystem (1), namely, the bifurcation values of rate kH ofthe preceding chemical reaction, frequency ωH, andelectrode surface coverage by the adsorbate θH. Thereis a certain threshold value of the rate of the precedingchemical reaction k = kth, above which surfaces

and do not inter�sect, i.e., system (1) becomes stable towards the Hopfbifurcation point and no oscillations occur in it at thevalues of parameter k > kth. The value of kth diminishesat a decrease in spherical electrode radius r0.

Re[ ( , , )] 0s kΨ θ =

Im[ ( , , )] 0s kΨ θ =

Re[ ( , , )] 0s kΨ θ = Im[ ( , , )] 0s kΨ θ =

Threshold values kth for the chosen spherical elec�trode radii r0 can be observed in Fig. 4, where cross–sections of surfaces and

are shown at the given values ofparameter k. Hopf bifurcation points are at the inter�section of these lines and have the coordinates of (θH,ωH). At an increase in parameter k, they draw togetherand vanish at the rate of the preceding chemical reac�tion being k = kth.

The table contains bifurcation values of electro�chemical system parameters for the two chosen ratesof the preceding chemical reaction and two values ofthe diffusion layer thickness: d = 10–3 cm and d ∞.The obtained data show that the range of potentials inwhich instability of system (1) is observed, diminishesboth at a decrease in spherical electrode radius r0 and

Re[ ( , , )] 0s kΨ θ =

Im[ ( , , )] 0s kΨ θ =

→

(а) (b)

2

4

10

8

0.40.30.2 0.5Est(θ), V

Rct(θ), Ohm cm2

1

2

2

6

4

00.40.30.2 0.5

Est(θ), V

Rct(θ), Ohm cm2

1

2

2.0

1.5

0.40.3 0.5Est(θ), V

Rct(θ), Ohm cm2

1

2

(c)

Fig. 2. Dependence of charge transfer rate Rst(θ) on steady–state electrode potential Est(θ). See curve designations in the captionto Fig. 1.

0.5

0

80

160

300600

1.0θ

k, s–1

ω, Hz

0.5

0

80

160

300600

1.0θ

k, s–1

ω, Hz

0.5

0

10

20

300

600

1.0θ

k, s–1

ω, Hz(а) (b) (c)

Fig. 3. Zero surfaces of the imaginary (dark) and real (light) components of the system impedance as functions of frequency ω,surface coverage by the adsorbate θ, and rate k of the preceding reaction for chosen spherical electrode radii r0, cm: (a) 10–2,(b) 10–3, (c) 10–4.

158


POTOTSKAYA, GICHAN

at an increase in rate k of the preceding chemical reac�tion in the bulk. An increase in the diffusion layerthickness results in extension of the system instabilityregion, which is most pronounced at low values ofparameter k. Therefore, in the case with a low elec�trode radius, the limiting value kth increases. The val�ues of kth for the other two chosen radii remainunchanged. For the case of k = 0, bifurcation points ofelectrochemical parameters of system (1) with aspherical electrode coincide with those obtained ear�lier in [8]. The bifurcation values of the electrochemi�cal system parameters under semiinfinite diffusion,d ∞, approach those for the case of d = 10–2 cm. At thelow values of this parameter, namely, at d = 10–4 cm, sys�→

tem (16) has no solution, surfaces and do not intersect; the conditions forimplementation of the Hopf bifurcation are not ful�filled.

The key cause for instabilities in electrochemicalsystems is the negative impedance in the faradaic pro�cess on the electrode. The effect of the effective rate ofthe preceding chemical reaction, electrode radius, anddiffusion layer thickness values can be traced inFigs. 5, 6, where the impedance Nyquist diagrams areshown for Hopf bifurcation points 1, 3, 5 (table). Adecrease in parameter r0 leads to a decrease in theinductive loop with a negative real component of fara�daic impedance. At an increase in parameter k, this

Re[ ( , , )] 0s kΨ θ =

Im[ ( , , )]s kΨ θ

700

350

1.00.50θ

ω, Hz700

350

1.00.50θ

ω, Hz700

350

1.00.50θ

ω, Hz

700

350

1.00.50θ

ω, Hz700

350

1.00.50θ

ω, Hz700

350

1.00.50θ

ω, Hz

700

350

1.00.50θ

ω, Hz700

350

1.00.50θ

ω, Hz700

350

1.00.50θ

ω, Hz

2 22

2 2

222

2

11

1

111

1 1 1

k = 0.5 s–1

k = 125 s–1

k = 133 s–1

k = 0.5 s–1k = 85 s–1 k = 110 s–1

k = 18 s–1k = 15 s–1k = 0.5 s–1

(а)

(b)

(c)

Fig. 4. Surface sections: (1) (2) at fixed values of parameter k given in each figure and chosenspherical electrode radii r0 for the case of limited diffusion layer thickness d = 10–3 cm: (a) 10–2, (b) 10–3, (c) 10–4.

Im[ ( , , )] 0,s kΨ θ = Re[ ( , , )] 0s kΨ θ =



Values of electrochemical system parameters in the Hopf bifurcation points

Values of parameters d, cm, and k, s–1 r0, cm Bifurcation

points ωH, Hz θH ifH, A/cm2 EH, V

10–3 0.5 10–2 1 143.10 0.562 0.00899 0.14512

2 190.55 0.192 0.00495 0.16972

10–3 3 199.64 0.512 0.01525 0.17724

4 243.31 0.209 0.00883 0.19527

10–4 5 253.91 0.372 0.05314 0.25827

6 240.61 0.297 0.04370 0.25979

10–3 15 10–2 1 186.39 0.517 0.01196 0.16418

2 232.34 0.205 0.00691 0.18364

10–3 3 234.91 0.480 0.01777 0.18848

4 272.90 0.220 0.01079 0.20302

10–4 5 260.52 0.364 0.05359 0.25987

6 240.87 0.325 0.04856 0.26058

∞ 0.5 10–2 1 36.71 0.696 0.00283 0.07435

2 68.56 0.168 0.00145 0.11319

10–3 3 110.90 0.583 0.00997 0.14855

4 161.35 0.190 0.00537 0.1745^

10–4 5 239.95 0.402 0.05362 0.25479

6 222.62 0.277 0.03916 0.25772

∞ 15 10–2 1 155.62 0.538 0.0104244 0.15502

2 206.02 0.200 0.00586981 0.17653

10–3 3 209.74 0.495 0.0164563 0.18292

4 252.80 0.214 0.00974596 0.19919

10–4 5 252.86 0.370 0.0536103 0.25900

6 238.44 0.300 0.0446436 0.26038

160


POTOTSKAYA, GICHAN

–0.4

–0.2

0.4

0.2

1.00.5–0.5–1.0Re [Z(ω)/Rct]

–Im [Z(ω)/Rct]

1.00.5–0.5–1.0Re [Z(ω)/Rct]

–Im [Z(ω)/Rct]

–1.5

–0.6

–0.3

0.3

1

2

3

2

1

3

(а)

(b)

Fig. 5. Nyquist diagrams of faradaic impedance in the complex plane in the Hopf bifurcation points given in the text for the caseof limited diffusion layer thickness d = 10–3 cm at r0, cm: (1) 10–2, (2) 10–3, (3) 10–4. The k values, s–1: (a) 15, (b) 0.5.

–0.4

0.4

1.00.5–0.5–1.0Re [Z(ω)/Rct]

–Im [Z(ω)/Rct]

1

2

3

–1.0

1–1–2Re [Z(ω)/Rct]

–Im [Z(ω)/Rct]

1

2

3

–3

–0.5

(а)

(b)

Fig. 6. Nyquist diagrams of faradaic impedance in the complex plane in the Hopf bifurcation points given in the text for the caseof semiinfinite diffusion layer thickness d ∞. See curve designations in the caption to Fig. 5.→



loop decreases further. An increase in parameter dleads to an increase in the inductive loop with a nega�tive real component of faradaic impedance. However,the effect of parameters k and d on the system stabilitybecomes weaker at a decrease in parameter r0.

Figures 7, 8 show dependences of the absolutevalue of faradaic impedance and itsphase angle on logω for the Hopfbifurcation points shown in Figs. 5, 6. The absolutevalue of faradaic impedance turns to zero in the Hopfbifurcation point at ω = ωH. An increase in parameterk leads to a shift in bifurcation frequency ωH to therange of higher frequencies and also to a decrease inthe faradaic impedance absolute value. By contrast tothis, an increase in parameter d leads to a shift in bifur�cation frequency ωH to the range of lower frequenciesand also to an increase in the faradaic impedanceabsolute value. The functional dependence of the fara�

f ctAbs[ ( ) ]Z Rω

f ctArg[ ( ) ]Z Rω

daic impedance phase angle on frequency ω changesin the Hopf bifurcation point.

In the system under consideration, oscillationsoccur when faradaic impedance tends to zero on thenegative side of the Re(Zf(ω)) values at ω → ∞.

Polarization resistance Zf(ω → 0) of the system inthe Hopf bifurcation point is negative.

CONCLUSIONS

Thus, the obtained results show that the instabilityregion of the model electrochemical system with anelectrocatalytic reaction on a spherical electrode pre�ceded by a homogeneous chemical reaction in the dif�fusion layer may be regulated by the values of sphericalelectrode radius r0, rate k of the preceding chemicalreaction and depends on diffusion layer thickness d.The system becomes more stable towards the Hopfbifurcation at a decrease in parameters r0 and d, andalso at an increase in parameter k. However, the effect

1.0

0.5

–1–3 31logω [Hz]

Abs[Z(ω)/Rct]

3

1

–1–3 31

logω [Hz]

Arg[Z(ω)/Rct], rad

–1

–3

3

1

–1–3 31

logω [Hz]

Arg[Z(ω)/Rct], rad

–1

–3

1.0

0.5

–1–3 31logω [Hz]

Abs[Z(ω)/Rct]

1.5

12

3

3

3

2

1

1

1 2

2

3

(а)

(b)

Fig. 7. Dependence of the absolute value of faradaic impedance and its phase angle on logω in the

Hopf bifurcation points given in the text for the case of limited diffusion layer thickness d = 10–3 cm.ctAbs[ ( ) ]Z Rω ctArg[ ( ) ]Z Rω

162


POTOTSKAYA, GICHAN

of parameters k and d on the system stability becomesweaker at a decrease in parameter r0.

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Wesley, 1988.

1.2

0.8

–1–3 31logω [Hz]

Abs[Z(ω)/Rct]

3

1

–1–3 31

logω [Hz]

Arg[Z(ω)/Rct], rad

–1

–3

3

1

–1–3 31

logω [Hz]

Arg[Z(ω)/Rct], rad

–1

2

1

–1–3 31logω [Hz]

Abs[Z(ω)/Rct]

3

12

3

3

3

2

1

1

1

2

2

3

(а)

(b)

0.4

Fig. 8. Dependence of the absolute value of faradaic impedance and its phase angle , rad, on logωin the Hopf bifurcation points given in the text for the case of semiinfinite diffusion layer thickness d ∞. See curve designationsin the caption to Fig. 5.

ctAbs[ ( ) ]Z Rω ctArg[ ( ) ]Z Rω

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