International Journal of Hydrogen Energy 33 (2008) 248–251www.elsevier.com/locate/ijhydene

Technical communication

A new method for determining kinetic parameters by simultaneouslyconsidering all the independent conditions at an overpotential

in case of hydrogen evolution reaction followingVolmer–Heyrovsky–Tafel mechanism

Mukesh Bhardwaj, R. Balasubramaniam∗

Department of Materials and Metallurgical Engineering, Indian Institute of Technology, Kanpur 208016, India

Received 20 February 2007; received in revised form 7 August 2007; accepted 18 September 2007Available online 24 October 2007

Abstract

Hydrogen evolution reaction following Volmer–Heyrovsky–Tafel mechanism and not under diffusion control can be characterized usingTafel polarization and AC admittance data at various frequencies and at various overpotentials. Such reaction has four independent kineticparameters. One empirical constant related to charge required for complete surface coverage is also involved. A new approach to determinekinetic parameters utilizing these data and neglecting Heyrovsky and Tafel backward reaction rates has been proposed. This involves determiningall the four kinetic constants using experimental data at a single overpotential. The empirical constant, i.e., charge required for complete surfacecoverage is determined by validating obtained kinetic constants at other overpotentials. The Levenberg–Marquardt algorithm to solve couplednonlinear equations has been utilized for such purpose. The approach has been validated using the literature data.� 2007 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved.

Keywords: Hydrogen evolution reaction; Kinetics; Adsorbed intermediate; Electrochemical impedance spectroscopy; Tafel polarization; Levenberg–Marquardtalgorithm

1. Introduction

The kinetic rate constants and mechanism of electrochemi-cal reactions involving adsorbed intermediate can be character-ized using steady state Tafel polarization and electrochemicalimpedance spectroscopy (EIS). Generally, electrical equivalentcircuits are fitted to EIS impedance data using a complex non-linear least square algorithm. The circuit parameters thus ob-tained are used for determining the transfer function. The trans-fer function is then used to determine kinetic constants for thereaction using constitutive equations.

Hydrogen evolution reaction (HER) involves three reactionsteps, Volmer (electrosorption), Heyrovsky (electrodesorption)and Tafel (recombination) [1]. In case of HER, Harrington and

∗ Corresponding author.E-mail address: bala@iitk.ac.in (R. Balasubramaniam).

0360-3199/$ - see front matter � 2007 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.ijhydene.2007.09.017

Conway [1] represented faradaic admittance as Yf =A+B(j�+C)−1, where A, B and C are constants derived using frequencyresponse data and are defined below assuming cathodic currentas negative

A = F [�r0/��]�, (1)

B = (F 2/q1)[�r0/��]�[�r1/��]�, (2)

C = −(F/q1)[�r1/��]�, (3)

where r0 is charge transfer rate and r1 is mass transfer rate [1].� is the fractional surface coverage of adsorbed species and � isthe net applied overpotential obtained after adding steady stateoverpotential and a small amplitude sinusoidal modulation ofpotential [1,2]. In order to determine r0 and r1, constitutiveequations are required. In case of HER, constitutive equationsare given as Eqs. (4)–(6), respectively, assuming Langmuir

M. Bhardwaj, R. Balasubramaniam / International Journal of Hydrogen Energy 33 (2008) 248–251 249

adsorption isotherm

vv = kv(1 − �) exp

(−�vF�

RT

)− k−v� exp

((1 − �v)F�

RT

)

= k′v(1 − �) − k′−v�, (4)

vh = kh� exp

(−�hF�

RT

)− k−h(1 − �) exp

((1 − �h)F�

RT

)

= k′h� − k′−h(1 − �), (5)

vt = kt�2 − k−t (1 − �)2. (6)

r0 and r1 can be defined as follows. r0 = −(vv + vh), r1 =vv − vh − 2vt . At steady state, r1 = 0. This gives � = {−b +√

b2 − 4ac}/(2a), where a = 2(kt − k−t ), b = k′v + k′−v + k′

h +k′−h +4k−t and c=−(k′

v +k′−h +2k−t ). However, k′−h and k−t

are usually found negligible and can be neglected [2]. Negativeroot is rejected as � can only be positive.

At �=0, vv=vh=vt=0. This implies that k−h=khkv/k−v andk−t = ktk

2v/k2−v . Therefore, out of six rate constants, only four

are independent. Therefore, the independent kinetic constants,i.e., kv , k−v , kh and kt are determined using variation of A, B, Cand iss (obtained from Tafel polarization) as the function of �.

The charge required for complete surface coverage, q1 is de-termined through empirical means as the value obtained throughindependent sources like experimental measurement or theo-retical calculations generally does not result in best fit [3].

The method to determine individual rate constants is notwell established in case of hydrogen evolution reaction withoutdiffusion control. Atleast four approaches have been tried, asmentioned below:

(1) Simultaneous fitting of iss and (A + B/C) [3,4].(2) Simultaneous fitting of iss and A [5].(3) Factorial fitting and minimizing residuals of iss , Rinf and

Rp [6].(4) Manual fitting by adjusting rate constants [2].

Four independent conditions are required to obtain four ki-netic constants. However, in all four approaches, experimentaldata were simulated using less than four conditions. It indicatesthat other kinetic parameters might have been adjusted man-ually through trial and error approach. Also the procedure tochoose q1 was not explicitly mentioned.

The estimation of kinetic parameters for the case of HER willbe addressed in this communication through a novel approach.The approach will be validated using the literature data [5].

2. Theory

Heyrovsky and Tafel backward reaction rate can be neglectedand �v ≈ �h ≈ 0.5 [2]. For convenience, k′

v , k′−v , k′h and kt

will be represented as x1, x2, x3 and x4, respectively. Therefore,

constitutive equations (4)–(6) can be re-written as Eqs. (7)–(9)

vv = x1(1 − �) − x2�, (7)

vh = x3�, (8)

vt = x4�2. (9)

Using Eqs. (1)–(3) and (7)–(9), Eqs. (10)–(13) are obtained

x1(1 − �) − x2� + x3� = −iss/F = m1, (10)

x1(1 − �) + x2� + x3� = 2ART /F 2 = m2, (11)

[x1 + x2 − x3][x1(1 − �) + x2� − x3�]= −2Bq1RT /F 3 = m3, (12)

x1 + x2 + x3 + 4x4� = Cq1/F = m4. (13)

In the above equations, m1, m2, m3 and m4 are constantsrelated to iss , A, B, C and q1 and are defined for convenience. Ata more cathodic overpotential, x2 → 0. This results in m1=m2.Solving Eqs. (10)–(13), following equations are obtained:

x1 = [(m2 + m1)/2 − x3�]/(1 − �), (14)

x2 = (m2 − m1)/(2�), (15)

4�2x23 − 2�

[2m2 − m1(1 − 2�)

]x3

+ 2m3�2 + 2(m1m2 − m3)� + m2(m2 − m1) = 0, (16)

x4 = (m4 − x1 − x2 − x3)/(4�). (17)

Let the positive root of Eq. (16) be named as n3. To solvethese equations, the following issues need to be addressed:

• Eqs. (14)–(17) are coupled and nonlinear as � is dependenton x1, x2, x3 and x4.

• Constants m3 and m4 are not available at all �. For example,this has been noticed in case of Pt in 0.5 M NaOH electrolyte.The values of B and C could be obtained only at less cathodicoverpotentials [5].

• Charge required for full surface coverage, q1 is unknown.• Nonlinear equations may possess multiple solutions and need

to be eliminated based on physical grounds.

The approach to solve the problem is to first solve four equa-tions (Eqs. (14)–(17)) to determine four unknowns (x1, x2, x3and x4) at a less cathodic overpotential (higher �) assuminga guess value for q1 and then to adjust q1 for fitting iss or Aat lower �. Several different approaches were tried and out ofmany approaches, the one which worked is explained below.

The Levenberg–Marquardt algorithm to solve these equa-tions was used. The algorithm source code [7] (in MATLAB)and the booklet on algorithm [8] can be downloaded from theinternet. The algorithm requires error functions fi(x) (where iis equation number), and Jacobian of the error function, fi(x).Jacobian J is defined by Eq. (18), where j is variable numberto be solved

(J (x))i,j = �fi

�xj

(x). (18)

250 M. Bhardwaj, R. Balasubramaniam / International Journal of Hydrogen Energy 33 (2008) 248–251

Fig. 1. Plot of iss , A, B and C for HER at (1 1 1) plane of Pt. The scattered dots represent experimental data of Barber and Conway [5]. The dotted lineshows fitting using kinetic constants obtained by Barber and Conway. The solid line shows fitting using kinetic constants obtained by novel approach for dataat overpotential of −26 mV.

Let n1 = (m2 + m1)/2, n2 = (m2 − m1)/(2�) and n4 = m4 · n3was defined earlier as positive root of x3. Error function f andJacobian J of f for Eqs. (14)–(17) are defined as Eqs. (19) and(20), respectively. In order to obtain good conditioning of thematrices, f and J were obtained by normalizing.

f =⎡⎢⎣

{x1(1 − �) + x3�}/n1 − 1x2/n2 − 1x3/n3 − 1

{x1 + x2 + x3 + 4x4�}/n4 − 1

⎤⎥⎦ , (19)

J =⎡⎢⎣

(1 − �)/n1 0 �/n1 00 1/n2 0 00 0 1/n3 0

1/n4 1/n4 1/n4 4�/n4

⎤⎥⎦ . (20)

3. Results and discussion

Any nonlinear algorithm, if it converges, converges to localminimum of error near to initial guess value. It is thereforenecessary to try many combinations of initial guess values.One should test the correctness of algorithm as f and J forEqs. (14)–(17) can be represented in many forms. If there aremany zeros in J, the converged kinetic data are dependent oninitial guess values even in small range and algorithm cannotbe used. Good fitting of iss , A, B and C in the entire potentialrange should be obtained by manipulating q1.

For validation of this novel approach by numerical simula-tion using nonlinear Levenberg–Marquardt algorithm for HER,Tafel polarization and EIS data of (1 1 1) plane of single crys-tal Pt in 0.5 M NaOH solution at 296 K [5] was utilized. Thedata were extracted from the available graphs with the helpof software OriginPro 7.5 (OriginLab Corporation) using itsscreen reader utility. It was assumed that Heyrovsky and Tafelbackward reaction rates are negligible while calculating kineticparameters. This assumption was dropped for obtaining calcu-lated iss , A, B and C using the obtained kinetic parameters. Thebest fit was obtained at � = −26 mV using initial guess valueskv = 10−7, k−v = 10−10, kh = 10−14 and kt = 1.31 × 10−8. Thevalues of kinetic parameters obtained are kv = 829 × 10−10,k−v = 284 × 10−10, kh = 1.9 × 10−10, kt = 78 × 10−10 andq1 = 84 × 10−6. Calculated values of iss , A, B and C usingthese kinetic parameters were compared with respect to the ex-perimental data and also to the simulated data using kineticconstants and q1 obtained by Barber and Conway [5] and areshown in Fig. 1. It can be noticed that curves obtained usingnovel approach pass quite closely to the experimental data at� = −26 mV (data used for fitting). This validates accuracy ofthe novel approach. Major contribution in fitting is due to sim-plification by reducing degree of quadratic equation. Nonlinearmethod (Levenberg–Marquardt algorithm) assured minimiza-tion of error. The speedy fitting technique helped in choosingappropriate initial guess values and q1.

M. Bhardwaj, R. Balasubramaniam / International Journal of Hydrogen Energy 33 (2008) 248–251 251

References

[1] Harrington DA, Conway BE. AC impedance of faradaic reactionsinvolving electrosorbed intermediates—I. Kinetic theory. ElectrochimActa 1987;32(12):1703–12.

[2] Bai L, Harrington DA, Conway BE. Behavior of overpotential depositedspecies in faradaic reactions—II. AC impedance measurements on H2evolution kinetics at activated and unactivated Pt cathodes. ElectrochimActa 1987;32(12):1713–31.

[3] Choquette Y, Brossard L, Lasia A, Menard H. Investigation of hydrogenevolution on raney nickel composite coated electrodes. Electrochim Acta1990;35(8):1251–6.

[4] Dumont H, Los P, Brossard L, Lasia A, Menard H. Lanthanumphosphate bonded composite nickel rhodium electrodes for alkaline waterelectrolysis. J Electrochem Soc 1992;139(8):2143–8.

[5] Barber JH, Conway BE. Structural specificity of the kinetics of thehydrogen evolution reaction on the low-index surfaces of Pt single-crystal electrodes. J Electroanal Chem 1999;461:80–9.

[6] Krstajic N, Popovic M, Grgur B, Vojnovic M, Sepa D. On the kineticsof the hydrogen evolution reaction on nickel in alkaline solution.Part I. The mechanism. J Electroanal Chem 2001;512:16–26.

[7] Nielsen HB. Marquardt’s method for least squares: Marquardt.m,〈http://www2.imm.dtu.dk/hbn/Software/〉.

[8] Madsen K, Nielsen HB, Tingleff O. Methods for non linear least squaresproblems, 〈http://www2.imm.dtu.dk/〉; 2004.