Influence of hydrogen absorption on the potential dependence of the Faradaic impedance parameters of hydrogen evolution reaction

Electrochimica Acta 201 (2016) 233–239

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Influence of hydrogen absorption on the potential dependence of the Faradaic impedance parameters of hydrogen evolution reaction V.I. Kichigin* , A.B. Shein Perm State University, Bukirev str. 15, Perm, Russia

A R T I C L E I N F O

Article history: Received 4 February 2016 Received in revised form 14 March 2016 Accepted 31 March 2016 Available online 1 April 2016 Keywords: hydrogen evolution hydrogen absorption impedance

A B S T R A C T

The influence of hydrogen absorption reaction (HAR) proceeding through Hads with kinetic or with diffusion control on the dependence of the elements R1, R2, C2 of the equivalent electrical circuit for hydrogen evolution reaction (HER) on the electrode potential E has been considered. It has been shown that if HER proceeds through the Volmer–Heyrovsky mechanism, then the presence of HAR reduces the slope dlogR2/dE at all E and the slope dlogE2/dE at sufficiently high cathodic polarisations. In the case of the Volmer–Tafel mechanism the presence of HAR mainly affects the resistance R2. The character of the dependences of R1, R2, C2 on E is the same for both kinetic and diffusion control of HAR. An interpretation of the experimental impedance data for HER on CoSi2 in 0.5 M H2SO4 has been presented. ã 2016 Elsevier Ltd. All rights reserved.

1. Introduction Previously [1] we suggested the diagnostic criteria for hydrogen evolution reaction (HER) mechanisms that are applicable within the framework of electrochemical impedance spectroscopy. These criteria are the derivatives of the values of the Faradaic impedance elements in equivalent electrical circuit with respect to the electrode potential (or overvoltage) and with respect to the concentration of hydrogen ions (in acidic solutions) or hydroxyl ions (in alkaline solutions). Theoretical development of the criteria is based on a number of assumptions: HER proceeds according to the Volmer–Tafel or Volmer–Heyrovsky mechanism; hydrogen atom adsorption obeys the Langmuir or Temkin isotherm; there is no influence of hydrogen ion diffusion towards the electrode and H2 from the electrode and hydrogen absorption by the electrode material; transfer coefficients of the Volmer and Heyrovsky steps do not depend on the overvoltage; electrode surface is smooth and clean (there are no oxides, organic adsorbates, etc.). It is assumed implicitly that the current and potential distributions over the working electrode surface are uniform. Thus, an idealised situation was analysed in [1]. In practice, we can expect different kinds of deviations from simple relations corresponding to this idealised situation. One can mention some reasons that will lead to more or less significant deviations of this kind: adsorption phenomena (water chemisorption, the presence of different forms of adsorbed hydrogen, the difference of the

* Corresponding author. Tel.: + 7 342 239 6452. E-mail addresses: [email protected] (V.I. Kichigin), [email protected] (A.B. Shein). http://dx.doi.org/10.1016/j.electacta.2016.03.194 0013-4686/ ã 2016 Elsevier Ltd. All rights reserved.

adsorption isotherm from the Langmuir or Temkin isotherm); hydrogen entry into the electrode; the presence of oxide films or chemisorbed oxygen on the electrode surface; different types of surface heterogeneities on solid electrodes; dependence of the transfer coefficients of HER steps on the overvoltage, etc. Some of these effects can change the structure of equivalent electrical circuit. One of the most common factors that can affect the dependences of equivalent circuit elements on the electrode potential E is the absorption of atomic hydrogen by electrode material that can occur simultaneously with HER. The entry of hydrogen into the solid phase is typical for transition metals, alloys, and intermetallic compounds [2–4]. The influence of hydrogen absorption reaction (HAR) on the impedance of the hydrogen electrode was discussed in many studies [5–23]. Many theoretical and experimental results on the HAR impedance are summarised in [24–26]. The main attention in [5–23] was focused on the analysis of frequency dependence of the impedance, the derivation of expressions for the Faradaic admittance, the determination of the equivalent circuit structure, the experimental tests of obtained relationships (using, in most cases, palladium or its alloys [5–9,16,18–22]). In these works two mechanisms of hydrogen absorption
direct (one-step) [7,8,11,12,15,16,23] and indirect (two-step) [5,7–9,13,17,19,23] absorption
were considered, and two types of boundary conditions
permeable [7,8,10,12,19] and impermeable [9,10,12–14,18,19] boundary conditions
were used. In some cases the evolution of molecular hydrogen was not taken into account at the analysis of impedance spectra [5,9,12,13,15]. In the mentioned

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V.I. Kichigin, A.B. Shein / Electrochimica Acta 201 (2016) 233–239

works [5–23] the influence of HAR on the potential dependence of Faradaic impedance elements associated with HER was studied insufficiently. In some papers [6,8,9,17,18,20,22] one gives only the estimated or experimental dependences of the charge transfer resistance Rct on E. Gabrielli et al. [18] presented the dependences of all equivalent circuit elements (including 3 parameters of HER impedance) on E. However, in all these cases there was no analysis of the relationship between HER mechanism (in the presence of HAR) and the character of the dependence of HER impedance parameters on the electrode potential. The purpose of this work is to estimate the changes in the values of diagnostic criteria [1] for HER mechanisms due to the occurrence of HAR in parallel with HER.

Assume that HER in acid solution may include the following steps: Volmer reaction: H+ + e
+ M ! MHads

(1)

Heyrovsky reaction: H+ + MHads + e
! M + H2

(2)

Tafel reaction: MHads + MHads ! 2 M + H2

(3)

The expressions for the rates vi of these steps can be represented respectively by [25]: v1 ¼ k1 ð1
u Þexpð
a1 F h=RTÞ
k
1 uexpðð1
a1 ÞF h=RTÞ 0 0 ¼ k1 ð1
uÞ
k
1 u

ð4Þ

v2 ¼ k2 uexpð
a2 F h=RTÞ
k
2 ð1
uÞexpðð1
a2 ÞF h=RTÞ 0 0 ¼ k2 u
k
2 ð1
uÞ

ð5Þ

v3 ¼ k3 u
k
3 ð1
uÞ2

ð6Þ

where h is the overvoltage, ki and k-i are the rate constants of steps at h = 0 in forward and backward direction respectively, ki0 and k-i0 are the rate constants at given overvoltage, ai are the transfer coefficients, u is the electrode surface coverage by adsorbed hydrogen. The concentration of hydrogen ions is included into the rate constants. Thus, it is assumed that the Langmuir isotherm is fulfilled, the rate constants for the Volmer and Heyrovsky steps are exponential functions of the overvoltage, and the rate constant for the Tafel step does not depend on h. In this work we discuss the Volmer–Heyrovsky and Volmer–Tafel mechanisms, both in combination with the HAR. Although the direct absorption of hydrogen is discussed in many works on impedance of HER + HAR [7,8,11,12,15,16,23], a more likely mechanism is the entry of H into the metal through a common with HER intermediate Hads [2,3,27]. In [28] it is assumed that hydrogen passes consecutively through two adsorbed states (weakly bonded Hads in the on-top sites and strongly bonded Hads in the hollow surface sites) during the absorption by transition metals at cathodic overvoltages. In the present work we assume that HAR proceeds through Hads and this intermediate is identical to the intermediate in HER. In this case we can express the reaction of transition of adsorbed hydrogen atoms across the interface as [17] MHads + S $ M + SHabs

va ¼ ka uð1
X 0 Þ
kd ð1
uÞX 0

(7)

ð8Þ

where ka and kd are the rate constants for the steps of absorption and desorption respectively, X = CH/CH,max, E= is the hydrogen concentration in metal, CH,max is the maximal value of E=; index 0 in (8) indicates the concentration of hydrogen at the metal surface (x = 0). In assumption of low u and X 0, expression (8) is simplified to va ¼ ka u
kd X 0

ð9Þ

In some works [6,27] equation (9) is expressed as va ¼ ka u
kd C s

2. Background

2

where ; is a surface metal atom, S is a free subsurface site. The HAR rate can be expressed as [17,24,29]

ð10Þ

where Cs is the hydrogen concentration at x = 0. There are few data on the values of ka and kd. For Fe electrode the values ka from 3.5 s
1 in alkaline solutions [30] up to 14.75 s
1 in acidic chloride solutions [31] were obtained. The rate constant ka in units of s
1 is used when the amount of adsorbed hydrogen is expressed in mol cm
2 and not in fractions of the surface covered by Hads. In order to compare the values of ka with the values of rate constants for the steps of HER let us express ka in mol cm
2 s
1 by multiplying by the maximal adsorption of atomic hydrogen Gm = q1/F where q1 is the charge required for the formation of adsorbed hydrogen monolayer. The typical value of Gm is equal to 2  10
9 mol cm
2, so the above values of ka will be from 7  10
9 up to 3  10
8 mol cm
2 s
1. The value ka = 2  10
10 mol cm
2 s
1 is reported for Fe in sulphuric acid solution [32]. Thus, the probable range of variation of ka can be 10
10–10
7 mol cm
2 s
1, though it was estimated by a limited amount of experimental data. The step of absorption of hydrogen atoms is followed by hydrogen diffusion into the bulk of electrode. Generally it is assumed that Reaction (7) is in quasi-equilibrium and the rate of hydrogen uptake in metal is limited by the bulk diffusion of H [2,28]. But in some cases the hydrogen absorption process may proceed with the kinetic control [3,21,27]. For example, the hydrogen diffusivity in pure iron is rather high (nearly 10
4 cm2 s
1 at ambient temperature [33,34]) and in film Fe electrode the diffusion process is very rapid as compared with the surface processes [23]. The transition to the kinetic control of HAR is facilitated, e.g. by decreasing foil thickness when using membrane electrodes [27]. In this paper we consider both cases
kinetic as well as diffusion control of HAR. In general, the properties of the surface change as a result of absorption of hydrogen by the metal [35], the energy of hydrogen adsorption E;-= on the metal surface decreases [36] which, in turn, will affect the kinetics of HER. This factor acts in addition to the change of u as a result of H absorption. Decrease in E;-= is probably due to the indirect electronic interaction mediated by the substrate [37]. It may be noted that in many cases, including transition metals that tend to absorb hydrogen, the regularities of HER that are based on the Langmuir adsorption isotherm for Hads are observed [38–42]. The Langmuir isotherm is fulfilled on a homogeneous surface with no interaction between adsorbate atoms and in this case E;-= = const. The interactions between adsorbed hydrogen atoms and the interactions between Hads and subsurface hydrogen atoms Habs may have the same nature being indirect interactions through the electronic system of the substrate. The realisation of the Langmuir isotherm means that we can neglect the interaction between the surface hydrogen atoms, and between surface and subsurface hydrogen. Therefore, we do not take into account the effect of reducing E;-= during hydrogen absorption but, generally speaking, it should be kept in mind.

V.I. Kichigin, A.B. Shein / Electrochimica Acta 201 (2016) 233–239

3. Results and discussion

235

0.5

3.1. Volmer–Heyrovsky mechanism with hydrogen absorption reaction 0.4

-η / V

For simplicity we assume that the transfer coefficients of the Volmer and Heyrovsky reactions are equal to each other: a1 = a2 = 0.5. 3.1.1. Kinetically controlled hydrogen absorption Because of fast diffusion of absorbed hydrogen atoms from the interface into the metal, the absorption Reaction (7) in this case can be considered to be irreversible: va = kau

0.2

0.1

0.0

(11)

-8

Hence, there are grounds for believing that the assumption of low X0 in Eq. (8) is valid. It is assumed that ka does not depend on the overvoltage. 3.1.1.1. Steady-state polarisation curve. The change in the surface coverage by adsorbed hydrogen when taking into account HAR is expressed as follows [29] du F ¼ ðv1
v2
va Þ dt q1

ð12Þ

Hence, taking into account (11), in the steady state (du /dt = 0) and if the Langmuir isotherm is fulfilled

u¼

0.3

0

0

k1 þ k
2 0 0 0 k1 þ k
1 þ k2 þ k
2 þ ka

ð13Þ

0

Then the steady-state current density is equal to 0

0

0

0


j=F ¼ k1 ð1
uÞ
k
1 u þ k2 u
k
2 ð1
uÞ 0 0 0 0 0 0 2k1 k2
2k
1 k
2 þ k1 ka
k
2 ka ¼ 0 0 0 0 k1 þ k
1 þ k2 þ k
2 þ ka

ð14Þ

The calculated polarisation curves for different ratios of k1 and k2 are shown in Figs. 1 and 2. Calculations by equation (14) show that in the case of the Volmer–Heyrovsky mechanism at quasiequilibrium Volmer reaction the presence of HAR leads to increased current densities and Tafel slope b (from 40 mV to 60 mV). The length of straight-line portion with b = 60 mV decreases with increasing ka (especially in the case of k1 < k2) and for sufficiently high ka it is not pronounced (Fig. 2). In the region of irreversible steps of HER the Tafel slope is approximately 120 mV, the current densities in the presence of HAR are higher

-7

-6

-5

-4

-3

-2

-1

0

log (j / A cm-2 ) Fig. 2. Calculated polarisation curves for k1 = 10
10, k
1 = 10
6, k2 = 10
9 mol cm
2 s
1 and different ka: 0 (solid line); 10
7 mol cm
2 s
1 (dashed line); 10
6 mol cm
2 s
1 (dash-dotted line).

than those in the absence of hydrogen absorption if k1 > k2, and they are a few less if k1 < k2. The curves for HER and HER + HAR merge at high overvoltages, so the contribution of HAR to the kinetics of cathodic process becomes very small. The relationships that show the influence of hydrogen absorption on polarisation curves can be obtained also on the basis of Frumkin’s work [43], devoted to the effect of supplying of additional quantities of atomic hydrogen on HER kinetics. In the case of HER + HAR mechanism, some additional amount of atomic hydrogen is removed from the electrode surface, therefore in the expressions obtained in [43] it is necessary to change the sign of the flux associated with the transition of atomic hydrogen across the metal surface. It can be written according to [43]


 @j @j @u ¼ ð15Þ @ja h @u h @ja h where ja is the rate of =ads removal from the surface due to HAR (in the units of current density), ja =
Fva. It follows from Eq. (14):
 @j 0 0 0 0 ¼ Fðk1 þ k
1
k2
k
2 Þ ð16Þ

@u

h

From the steady-state solution of Eq. (12) we obtain  @ja 0 0 0 0 ¼ Fðk1 þ k
1 þ k2 þ k
2 Þ




@u

0.5

Thus, for the Volmer–Heyrovsky mechanism  0 0 0 0 @j k1 þ k
1
k2
k
2 ¼ 0 0 0 @ja h k1 þ k
1 þ k2 þ k
2 0

ð17Þ




0.4

-η / V

h

0.3

ð18Þ

At k-10 >> k10, k20 , k-20 (quasi-equilibrium Volmer reaction) we obtain (@j/@ja)h > 0 regardless of the ratio k10 and k20 . At irreversible steps (@j/@ja)h = (k10
k20 )/(k10 + k20 ); therefore at k10 > k20 the current density in presence of HAR is higher than that in the absence of HAR, but at k10 < k20 it is on the contrary less. The calculations of polarisation curves (Figs. 1 and 2) are consistent with these conclusions.

0.2

0.1

0.0 -8

-7

-6

-5

-4

-3

-2

-1

0

log (j / A cm-2 ) Fig. 1. Calculated polarisation curves for k1 = 10
9,k
1 = 10
6, k2 = 10
10 mol cm
2 s
1 and different ka: 0 (solid line); 10
8 mol cm
2 s
1 (dashed line); 10
7 mol cm
2 s
1 (dash-dotted line).

3.1.1.2. Impedance. As it is known [25] the impedance of an electrode, at which HER occurs, corresponds to the equivalent electrical circuit shown in Fig. 3(a). The physical meaning of the parameters of Faradaic impedance in this case can be expressed by

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V.I. Kichigin, A.B. Shein / Electrochimica Acta 201 (2016) 233–239

C =
(F/q1)(@r1/@u)E,

(24)

r0 = j/F is the net rate of charge transfer in HER; r1 = (q1/F)(du/dt) is the net rate of adsorbed hydrogen formation. In the equivalent circuits in Fig. 3: Rs is the solution resistance, Cdl is the double-layer capacitance. Equivalent electrical circuit for HER + HAR with the kinetic control of HAR is the same as for HER (Fig. 3(a)) [7,44] which makes it impossible to reveal the presence of HAR using the frequency dependence of impedance. However, as will be shown below, determining the dependence of the Faradaic impedance parameters on the electrode potential allows recognising the presence of the HAR that proceeds with kinetic control. The partial derivatives for determination of the values A, % and E in (22)–(24) in the presence of HAR (kinetic control) and at a1 = a2 = 0.5 have the form:
 @r0 F 0 0 0 0 ½k ð1
uÞ þ k
1 u þ k2 u þ k
2 ð1
u Þ ¼ ð25Þ @h u 2RT 1 Fig. 3. Equivalent electrical circuits. (a) For HER or for HER + HAR (kinetic control), (b) for HER + HAR (diffusion control).










the relationships [25] 1 R1 ¼ A

R2 ¼


ð19Þ

B AðAC þ BÞ

A2 C2 ¼
B

@r1 @h @r0 @u @r1 @u

B = (F /q1)(@r0/@u)E(@r1/@E)u, 2

u

 h

 h

¼

F 0 0 0 0 ½k1 ð1
u Þ þ k
1 u
k2 u
k
2 ð1
u Þ 2RT

ð26Þ

0

0

0

0

ð27Þ

0

0

0

0

ð28Þ

¼ k1 þ k
1
k2
k
2

¼ k1 þ k
1 þ k2 þ k
2 þ ka

Substituting (25)–(28) into (22)–(24) and then into (19)–(21) one obtains expressions for R1, R2, C2 which are simplified at certain ratios of the rate constants. At quasi-equilibrium Volmer step: ð20Þ

ð21Þ

R1 ¼

RT 1 F h=2RT e F 2 k1

ð29Þ

R2 ¼

RT k
1 F h=RT e F 2 k1 ka

ð30Þ

C2 ¼

q1 F k1
F h=RT e RT k
1

ð31Þ

where A = F(@r0/@E)u,



(22)

(23)

Fig. 4. Calculated logP–h curves (P = R1, R2, C2) for k1 = 10
9, k
1 = 10
6, k2 = 10
10 mol cm
2 s
1 and different ka: 0 (solid line); 10
8 mol cm
2 s
1 (dashed line); 10
7 mol cm
2 s
1 (dash-dotted line).

V.I. Kichigin, A.B. Shein / Electrochimica Acta 201 (2016) 233–239

In the region of irreversible steps of HER: at k1 >> k2 R1 ¼

RT 1 F h=2RT e F 2 k2

ð32Þ

In general, the assumption of low X0 seems to be poorly fulfilled for diffusion-controlled HAR. However, the values of X0 can be low in this case too, if the values of kd/ka are sufficiently high. Instead of (13) we can now write

u¼ R2 ¼

C2 ¼

RT ka 2F 2 ðk2 Þ2

eF h=RT

ð33Þ

2q1 F ðk2 Þ2
F h=2RT e RT k1 ka

ð34Þ

and at k1 << k2 R1 ¼

ð35Þ

RT ka F h=RT e R2 ¼
2 2F k1 k2

ð36Þ

u¼

2q1 F k1
F h=2RT e RT ka

ulim ¼

k1 k1 0 0 ¼ k1 þ k2 k1 þ k2

0

ð39Þ

where K= = (D/L)(ka/kd); D is the diffusion coefficient for hydrogen in metal, L is the thickness of diffusion layer (in the particular case
the thickness of membrane electrode). Eq. (39) coincides with (13) in form, and the nature of the dependences of R1, R2, C2 on h will be the same as with the kinetic control of HAR (provided that K= does not depend on h).

It is a process in which Hads is removed from the surface by two heterogeneous chemical reactions; one of them has the first order with respect to Hads, and the other has the second order. The steady-state surface coverage by adsorbed hydrogen, which is determined by the equation of mass balance, in this case is equal to

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 0 0 0 ðk1 þ k
1 þ ka þ 4k
3 Þ2 þ 8ðk1 þ 2k
3 Þðk3
k
3 Þ
ðk1 þ k
1 þ ka þ 4k
3 Þ 4ðk3
k
3 Þ

ð37Þ

In the latter case the impedance has an inductive component. A comparison with the expressions for the impedance parameters for HER without HAR [1] shows that the presence of HAR (kinetic control) significantly reduces the slope dlogR2/dh at all h and the slope dlogC2/dh at sufficiently high cathodic polarisations. The calculated dependences of logP on h (P = R1, R2, C2) for one set of rate constants for HER steps and at several values of ka are presented in Fig. 4. Considerable influence of HAR on R2 and E2 is observed even at significant overvoltages (Fig. 4) when the influence of HAR on polarisation curve becomes negligible (Figs. 1 and 2). This, apparently, is because at these h the presence of HAR has a significant impact on the relative changes of the quantity (ulim–u) with h, where ulim is the limiting value of the surface coverage at high cathodic polarisations, which at a1 = a2 = 0.5 is equal to 0

0

k1 þ k
2 0 0 0 k1 þ k
1 þ k2 þ k
2 þ K H 0

3.2. Volmer–Tafel mechanism with hydrogen absorption reaction

RT 1 F h=2RT e F 2 k1

C2 ¼


237

ð40Þ

In the case of the Volmer–Tafel mechanism with quasiequilibrium Volmer step the presence of HAR leads to an increase in current densities and Tafel slope (from 30 to 60 mV). If the Volmer reaction is the rate-limiting step in the Volmer–Tafel mechanism, then the presence of HAR also leads to an increase in current density (at a given h), but the changes in j are significantly less than those for quasi-equilibrium Volmer step. The partial derivatives that determine the values A, % and E in (22)–(24) in presence of HAR (kinetic control) and at a1 = 0.5 are equal to
 @r0 F 0 0 ½k ð1
uÞ þ k
1 u ¼ ð41Þ @h u 2RT 1


@r1 @h




@r0 @u

 u

 h

¼

F 0 0 ½k1 ð1
uÞ þ k
1 u 2RT

ð42Þ

0

0

ð43Þ

0

0

ð44Þ

¼ k1 þ k
1

ð38Þ

The expression (38) has the same form for both HER and HER + HAR, because ka << k10, k20 at high cathodic polarisations. 3.1.2. Diffusion-controlled hydrogen absorption If HAR proceeds with diffusion control then the equivalent circuit is changed [25]
the impedance of hydrogen solid-phase diffusion Zd is connected in parallel to R2 and E2 (Fig. 3(b)). The physical meaning of parameters R1, R2, C2 is still expressed by Eqs. (19)–(24) [25], but the dependence of these parameters on h is changed as the presence of HAR alters the relation u(h). In accordance with the purpose of this work, we will consider only the dependence of parameters R1, R2 and C2 on h, and the element Zd will not be discussed.




@r1 @u

 h

¼ k1 þ k
1 þ 4k3 u þ 4k
3 ð1
u Þ þ ka

Substituting (41)–(44) into (22)–(24) and then into (19)–(21) one obtains expressions for R1, R2, C2, which are simplified at certain ratios of the rate constants. At quasi-equilibrium Volmer step the expressions coincide with (29)–(31). As compared with the case without HAR [1], hydrogen absorption causes the slope dlogR2/dh to decrease by a factor of 2 but the slopes dlogR1/dh and dlogC2/dh are not changed. If the Volmer reaction is the rate-determining step in the sequence of Reactions (1) and (3), then at ka << k10 the dependences R1, R2, C2 on the overvoltage are the same as in the absence of HAR. At ka >> k10 and ka2 >> 8k10 k3, R2  (2RT/F2)(1/ka), i.e.

238

V.I. Kichigin, A.B. Shein / Electrochimica Acta 201 (2016) 233–239

R2 does not depend on the overvoltage. At the rate-limiting Volmer step the presence of HAR has almost no effect on the logR1–h and logE2–h dependences but it leads to a decrease in the values of R2 at all cathodic h. 3.3. Comparison with experimental data For a CoSi2 electrode in 0.5 ; H2SO4 at potentials of hydrogen evolution, we obtained dlogR1/dE, dlogR2/dE, dlogC2/dE that were equal to 9, 14.0 and
12.5 V
1 respectively [1]. The last two values are less than the theoretical values for the Volmer–Heyrovsky mechanism, but they are higher than the theoretical values for the Volmer–Tafel mechanism (with the Volmer reaction as the ratelimiting step). In this regard we supposed [1] that HER on CoSi2 in acidic solution proceeds through the two-route Volmer– Heyrovsky–Tafel reaction mechanism. The calculations for the two-route mechanism of HER (at a1 = a2 = 0.5) show that at quasi-equilibrium of the Volmer reaction the increase in the rate constant of the Heyrovsky step, as compared with the rate constant of the Tafel step, leads to reducing the slope dlogR2/dE, but these changes are limited to the interval from 2F/2.3RT = 34.4 V
1 (at 20oE) to (1 + a2)F/2.3RT = 25.8 V
1 (at a2 = 0.5 and 20oE), i.e. the slope dlogR2/dE is higher than experimental one at any ratio of k2 and k3. At more negative E, when the Volmer reaction is not in quasi-equilibrium, a not wide potential interval exists at k3 > k1 > k2 where dlogR2/dE  4.3 V
1 but the capacitance E2 is practically constant. At even more negative E, when the Tafel reaction can no longer compete with the Heyrovsky reaction in the removal of Hads from the electrode surface, we have for the two-route mechanism dlogR2/dE = F/2.3RT = 17.2 V
1, dlogC2/dE = –F/(22.3RT) =
8.6 V
1. Thus, in all intervals of the electrode potential the calculated data for the two-route mechanism are not consistent simultaneously with the experimental dependences of R2 and E2 on E for CoSi2 in acidic solution. The above results (Section 3.1) show that the decrease of dlogR2/dE and dlogC2/dE as compared with the theoretical values for the Volmer–Heyrovsky mechanism may be caused by the presence of HAR (in this case HAR apparently proceeds with kinetic control since the equivalent circuit shown in Fig. 3(a) fits impedance spectra for CoSi2 well [1]). One may note that the experimental values 14.0 V
1 and
12.5 V
1 for dlogR2/dE and dlogC2/dE respectively are relatively close to the theoretical slopes 17.2 and
17.2 V
1, which result from (30) and (31). The equations (30), (31) are realised in the region of quasi-equilibrium of Volmer reaction where the Tafel slope must be close to 60 mV in presence of HAR. Experimental value of b for CoSi2 in 0.5 ; H2SO4 is equal to 95–100 mV [1]. However, there is no contradiction because in presence of HAR the plot with b = 60 mV can be not pronounced (Fig. 2). Additional information may be obtained from the time constant t = R2C2. As it follows from the experimental slopes dlogR2/dE and dlogC2/dE, the value dlogt /dE for CoSi2 is equal to 1.5 V
1. For the Volmer–Heyrovsky mechanism without HAR the calculated value of dlogt /dE is approximately equal to 8.6 V
1, i.e. it is considerably higher than the experimental value. At the same time the value of t does not depend on E for the Volmer–Heyrovsky mechanism with simultaneously proceeding HAR in some range of electrode potentials (Eqs. (30) and (31)). For the Volmer–Tafel mechanism without HAR dlogt /dE  4.3 V
1, i.e. it is about 3 times higher than the experimental value. In the case of the Volmer–Tafel mechanism with HAR one can obtain dlogt /dE  0, but this equality holds at high values of ka (wherein at sufficiently high cathodic polarisations dlogR2/dE  0 (Section 3.2), which does not agree with the experiment). For the two-route Volmer–Heyrovsky–Tafel

mechanism, the calculated dlogt /dE take the values 17.2, 4.3 and 8.6 V
1 (at 20oE) as the electrode potential decreases. Thus, the experimental value dlogt /dE = 1.5 V
1 is also closer to that for the Volmer–Heyrovsky mechanism with in parallel proceeding HAR. Slight slope of logt –E dependence shows, probably, that the examined range of electrode potentials corresponds to the transition from the quasi-equilibrium Volmer reaction to irreversible steps of HER. If, in spite of the weak dependence of t on E, we use Eqs. (30) and (31) then we may write ka ¼

q1 Ft

ð45Þ

At E =
0.24 V vs. SHE, the highest electrode potential in the studied range of E, t  0.15 s [1]. If we take q1 = 200 mC cm
2 for CoSi2, then ka  1.4  10
8 mol cm
2 s
1 from (45). This value of the absorption rate constant is approximate and it is more likely that it is overestimated. Here two factors act: (i) the observed t were not strictly constant; (ii) the value of q1 may be less than expected (e.g. [45]). Although in the proposed interpretation of the data for CoSi2 electrode (HER, Volmer-Heyrovsky mechanism + HAR, kinetic control) some discrepancy between the calculated and experimental data remains, it is much smaller than in the case of other mechanisms of HER discussed. The remaining discrepancy is apparently due to some unaccounted factors which include: (i) difference of the transfer coefficients a1 and a2 from 0.5; (ii) the presence of weak interaction between adsorbed hydrogen atoms (according to [1], the interaction between Hads that results in deviations from the Langmuir adsorption isotherm affects more markedly the slope dlogC2/dE than dlogR2/dE). 4. Conclusions In this paper we discussed the effect of a frequently encountered process
the penetration of atomic hydrogen into the electrode material (with kinetic or diffusion control)
on the dependence of the elements of equivalent circuit on the electrode potential at various mechanisms of HER, which can provide further insight into the kinetics of such an important process as the electrochemical hydrogen evolution by using impedance spectroscopy. In addition to the impedance characteristics we considered steady-state current-potential curves at different values of hydrogen absorption rate constants. If HAR proceeds with the kinetic control, then the character of the frequency dependence of impedance is the same for both HER and HER + HAR [44]. However these two cases are distinguishable by the nature of potential dependence of the Faradaic impedance elements. It has been shown that if HER proceeds through the Volmer–Heyrovsky mechanism, then the presence of HAR decreases the slope dlogR2/dE at all E and the slope dlogE2/dE at rather high cathodic polarisations. In the case of the Volmer– Tafel mechanism the presence of HAR mainly affects the resistance R2. The character of dependences of R1, R2, C2 on E is the same for both kinetic and diffusion control of HAR. Previously obtained experimental results on the kinetics of the cathodic process on cobalt disilicide CoSi2 in 0.5 M H2SO4 [1] were compared with theoretical results for several mechanisms involved. The experimental results as a whole (the nature of impedance spectra, the values of dlogR1/dE, dlogR2/dE, dlogC2/dE, dlogt /dE) are in a better agreement with the Volmer–Heyrovsky mechanism with parallel proceeding HAR (kinetic control). Acknowledgements The support by Russian Foundation for Basic Research (project No.14-03-96000-ural) is acknowledged.

V.I. Kichigin, A.B. Shein / Electrochimica Acta 201 (2016) 233–239

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