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Solid State Ionics 175

A simple model for solid polymer electrolyte (SPE) water electrolysis

Pyoungho Choia, Dmitri G. Bessarabovb, Ravindra Dattaa,*

aFuel Cell Center, Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, USAbKvaerner Chemetics, 1818 Cornwall Avenue, Vancouver, BC, Canada

Accepted 13 January 2004

Abstract

Solid polymer electrolyte (SPE) water electrolysis is analyzed by a simple model based on Butler–Volmer kinetics for electrodes and

transport resistance in the polymer electrolyte. An equivalent electrical circuit analogy is provided for the sequential kinetic and transport

resistances. The model provides a relation between applied terminal voltage of the electrolysis cell and current density in terms of Nernst

potential, exchange current densities, and conductivity of polymer electrolyte. The overpotentials and resistances at the anode, cathode, and

overpotential due to ohmic resistances are individually analyzed and are in good agreement with experimental results. The reduction kinetics

at the cathode is relatively fast while the anodic overpotential is mainly responsible for the voltage drop.

D 2004 Elsevier B.V. All rights reserved.

Keywords: Water electrolysis; Solid polymer electrolytes; Electrochemical analysis; Polarization; Hydrogen generation

1. Introduction

The solid polymer electrolyte (SPE) has been utilized

in many energy-related fields such as fuel cell [1],

hydrogen compressor [2], and solar cell systems [3].

Electrolysis of water using the SPE [4–6], which serves

as a solid electrolyte that conducts protons and as a

separator of gases, is considered as a promising method-

ology for producing hydrogen as an alternative to the

conventional alkaline water electrolysis. SPE electrolyzer

has certain advantages over the classical alkaline process

in terms of its simplicity, high energy efficiency, and

specific production capacity. It is also creating new

options for the fuel cell system, e.g., a regenerative fuel

cell which operates both as a fuel cell and as an

electrolyzer [7–9].

In principle, SPE water electrolyzer and fuel cells are

basically the same device working in the opposite direction

[10]. Although there are many studies on the theoretical

analysis of fuel cells [11–15], not much has been reported

0167-2738/$ - see front matter D 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.ssi.2004.01.076

* Corresponding author. Tel.: +1 508 831 5250; fax: +1 508 831 5853.

E-mail address: rdatta@wpi.edu (R. Datta).

on the kinetics and polarization characteristics of the SPE

electrolyzer. In order to design and use the SPE electro-

lyzer effectively, analytical models for the device are

necessary so that the system may be optimized. Recently,

Onda et al. [16] have provided a voltage–current relation

wherein the cell voltage is described as the sum of Nernst

voltage, resistive overpotential, and anode and cathode

overpotentials. However, empirical equations were utilized

for the anode and cathode overpotentials as a function of

temperature of the electrolytes and current density of the

cell.

The objective of this study is to propose a simple

but useful first-generation theoretical model to explain

the current-potential characteristics of SPE electrolysis

cell based on the involved charge and mass balances as

well as Butler–Volmer kinetics on the electrode

surfaces.

2. Principle of operation

Electrolysis of water is the dissociation of water

molecules into hydrogen and oxygen gas. A schematic of

SPE water electrolysis is shown in Fig. 1. A potential is

(2004) 535–539

Fig. 1. Cross-section of SPE water electrolyzer.

P. Choi et al. / Solid State Ionics 175 (2004) 535–539536

applied across the electrochemical cell to induce electro-

chemical reactions at both electrodes. Water is introduced at

the anode and dissociated into oxygen, protons and

electrons via the following reaction:

At anode : H2OYp 2Hþþ2e� þ 0:5O2 E8 ð258CÞ ¼ 1:23V

ð1Þ

The protons are driven through the SPE to the cathode under

an electric field where they combine with the electrons

arriving from the external circuit to form hydrogen gas:

At cathode: 2Hþ þ 2e� YpH2 E8 ð258CÞ ¼ 0:00V ð2Þ

Therefore, the net reaction in the electrolysis cell is

Net reaction: H2OYpH2 þ 0:5O2 E8 ð258CÞ¼1:23V ð3Þ

The heart of the SPE water electrolyzer is of course the

membrane-electrode-assembly (MEA). For the solid elec-

trolyte, typically a perfluorosulfonic acid (PFSA) polymer

such as NafionR, has widely been used for water

electrolysis [6–9,16–20]. For the anode, platinum shows

a significant overpotential and thus platinum/ruthenium

[18], iridium [19] and platinum/iridium [16,18,20] have

been investigated. The addition of Ru decreases the anode

overpotential, but Pt–Ru anode is not stable and corrodes

under oxygen evolution [18]. The Pt–IrO2 based alloy

catalysts are relatively stable and preferred as anode

electrocatalyst [20]. For the cathode, platinum provides

the best performance and commonly used for water

electrolysis [16–20].

3. The Model

A simplified mathematical model is developed below

based on appropriate mass balances, transport, and electro-

chemical kinetics applied to the SPE electrolysis cell.

3.1. Steady state conservation equations

3.1.1. Anode and cathode chambers

For the sake of simplicity, the anode chamber is treated

as being well-mixed. The mass balances of water and

oxygen at the anode, and that of hydrogen at the cathode can

then be written as

NNH2O;in � NNH2O;out ¼iA

2Fð4Þ

NNH2;in � NNH2;out ¼ � iA

2Fð5Þ

NNO2;in � NNO2;out ¼ � iA

4Fð6Þ

where N, i, A and F represent the molar flow rates [mol/s],

current density [A/cm2], MEA area [cm2] and Faraday’s

constant, respectively.

3.1.2. Anode and cathode layers

Assuming no transport limitations the Butler–Volmer

expression is utilized for the overall electrochemical

reaction at the anode

i ¼ iA0 expaAve�FgA

RT

� �� exp � 1� aAð Þve�FgA

RT

� �� �ð7Þ

where iA0 is the anode exchange current density [A/cm2],

ve� is the stoichiometric coefficient of electrons in the anode

reaction, aA is the transfer coefficient, and gA is the anode

overpotential. Alternatively, the anode overpotential may be

written in the following form, assuming the effective

transfer coefficient aA=0.5 and ve�=2 [11]

gA ¼ RT

Fsinh�1 i

2iA0

� �ð8Þ

For the cathode, if Butler–Volmer equation is assumed as

well along with aC=0.5 and ve�=�2, the cathode over-

potential is obtained similarly as

gC ¼ � RT

Fsinh�1 i

2iC0

� �ð9Þ

where iC0 is the cathode exchange current density. Here, it

should be noted that the solutions are assumed to be well-

mixed in the chambers and thus the surface concentrations

do not differ appreciably from the bulk phase. If there is a

limitation for mass transfer, e.g., oxygen diffusion from

catalyst site to gas bubble across a diffusion film near

electrode, limiting current density may be incorporated in

Eqs. (8) and (9) [11,14].

Fig. 2. Equivalent circuit for the electrolysis process: V0=internal power

supply, RA=anode resistance, RSPE=membrane resistance, RC=cathode

resistance and RI=interface resistance.

Fig. 3. Comparison of the model with experiments at 80 8C (1: equilibrium

voltage V0, 2: ohmic drop, 3: cathode overpotential, 4: anode overpotential

on Pt–IrO2, 5: anode overpotential on Pt). Experimental data is given by

symbols [18,20].

P. Choi et al. / Solid State Ionics 175 (2004) 535–539 537

3.1.3. Solid polymer electrolyte (SPE)

At steady state, the divergence of current density in the

solid polymer electrolyte is zero, i.e.,

di

dz¼ 0 and i ¼ � r

d/dz

ð10Þ

where r is the conductivity of the electrolyte [S/cm] and /is the potential [V] in the membrane.

3.2. Electrochemical potential of electrolysis cell

Fig. 2 shows as equivalent circuit for electrolysis process

represented by a series of resistances. The overall applied

cell potential is thus composed of the cell Nernst potential

(V0), anode and cathode overpotentials, overpotential due to

membrane, and interfacial resistance as [11,14]

V ¼ V0 þ gA � gC þ gSPE þ g1 ð11Þ

where the Nernst potential V0 is empirically given as [21]

V0 ¼ 1:23� 0:9� 10�3 T � 298ð Þ þ 2:3RT

4Flog P2

H2PO2

� �ð12Þ

The anode and cathode overpotentials in Eq. (11) are

provided by Eqs. (8) and (9). Integration of Eq. (10) gives

overpotential due to the membrane resistance

gSPE ¼ LB

rB

� �i ð13Þ

where LB is the thickness of SPE, rB is conductivity of the

electrolyte. The interfacial overpotential gI may be written

in terms of interfacial resistance RI and current density as

gI ¼ RIi ð14Þ

Therefore, the overall cell voltage–current relation can be

obtained by combining Eqs. (8), (9), (12)–(14) with (11).

V ¼ V0 þRT

Fsinh�1 1

2

i

iA0

� �� �þ RT

Fsinh�1 1

2

i

iC0

� �� �

þ LB

rB

� �iþ RIi ð15Þ

Correspondingly, the required power density is obtained

by P=Vi as

P ¼ V0iþRT

Fsinh�1 1

2

i

iA0

� �� �iþ RT

Fsinh�1 1

2

i

iC0

� �� �i

þ LB

rB

� �i2 þ RIi

2 ð16Þ

4. Simulation

Fig. 3 shows the simulation results obtained by Eq. (15)

based on the parameters provided in Table 1 along with

experimental data [18,20] to validate the adequacy of this

simple model. For Pt based electrodes, the exchange current

density for the oxygen reduction and hydrogen oxidation

reactions is reported as 10�9–10�12 [22–24] and 10�4–10�3

A/cm2 [25], respectively. The exchange current density

depends on the temperature at the electrode surface and also

the roughness factor [14], which is defined as the electro-

chemically determined electrode area divided by the geo-

metric area.

i0 ¼ cMexp � E

R

1

T� 1

Tref

� �� �iref0 ð17Þ

where i0, cM, E and i0ref represent exchange current density,

roughness factor, activation energy and exchange current

density at reference state. The roughness factor can be

determined experimentally [4,22–24] or estimated by

catalyst loading, catalyst particle density and size [14]. Of

course, the microstructure of electrodes affects the rough-

ness factor and reported for oxygen reduction reaction as 2.7

[24], 9.2 [26], 200 [4] for Pt microdisk, Pt wire and Pt

powder electrodes, respectively. The roughness factor for

Table 1

Model parameters for water electrolysis for Pt based anode and cathode

electrodes on NafionR electrolyte at 80 8C

Parameters Values Dimensions Comments and references

iA0,Pt 10�12 A/cm2 anode exchange current

densitya for Pt [22–24]

iA0,Pt–Ir 10�7 A/cm2 anode exchange current

densitya for Pt–Ir [22]

iC0 1�10�3 A/cm2 cathode exchange current

densitya for Pt [25]

LB 178 Am thickness of NafionR 117

electrolyte

rB 0.14 S/cm conductivity of NafionR117 electrolyte [29]

cM 150 dimensionless roughness factor [27,28]

a The exchange densities here are based on the electrochemically active

surface area.

Fig. 5. Power density vs. current density: star=energy supply due to Pt–IrO2

anode overpotential, circle=power supply due to cathode overpotential,

square=power supply due to ohmic drop, and triangle=minimum power

input.

P. Choi et al. / Solid State Ionics 175 (2004) 535–539538

typical electrolysis cell would be in the range of 100–300

[27,28] and here 150 is adopted for both the electrodes.

The conductivity of NafionR depends upon its water

content and is taken to be 0.14 S/cm at 80 8C for water

immersed membrane [29]. The interfacial resistance RI is

assumed to be negligible and hence set equal to zero in

this model.

Fig. 3 shows that the ohmic overpotential increases

steadily with current density and the cathode overpotential

is relatively small because of the fast kinetics at the

electrode surface. The anode reaction is sluggish and the

overall process is limited by the oxygen evolution

reaction. The anode overpotential increases rather sharply

at low current density and slowly thereafter with the

current density. Since the cathode reaction is relatively

fast compared with the anode reaction, the potential

increase of the electrolysis cell with current density is

mainly attributable to the slow kinetics of water dissoci-

ation at the anode. Thus, a current density of 1 A/cm2 is

Fig. 4. Differential resistances for water electrolysis: star=total differential

resistance, circle=anode differential resistance, square=membrane differ-

ential resistance and triangle=cathode differential resistance.

achieved for the applied voltage of 2.1 V at 80 8C for the

Pt anode [18,20]. In order to reduce anode polarization,

iridium, which exists in oxide form under reaction

conditions, has usually been added to Pt for SPE water

electrolysis. Ioroi et al. [20] reported that the mixture of

high surface area IrO2 and Pt black (50:50 mol ratio)

improved the efficiency of water electrolysis from 77% to

95% at 300 mA/cm2. When IrO2 is added to Pt, the

exchange current density is increased and thus the oxygen

evolution reaction at the anode occurs at lower over-

potential [22]. The model predicts overpotentials quite

satisfactorily over the current range of the experiment for

the Pt and Pt–IrO2 anodes.

The electrolysis process has been represented by an

equivalent electrical circuit consisting of a series of

resistances representing each individual steps. In analogy

to the linear Ohm’s law, a differential resistance Rd may be

defined for an electrolysis cell as [11]

Rd ¼d V � V0ð Þ

dið18Þ

where V0 is the Nernst potential and may be thought of as an

internal power supply for the cell. Combining Eq. (11) with

(18) provides each individual resistance associated with the

different steps of the process.

Rd ¼dgAdi

� dgCdi

þ dgSPEdi

þ dgIdi

ð19Þ

Differentiation of the corresponding overpotentials gives

resistance separately for anode, cathode, and the solid

polymer electrolyte.

RA ¼ RT

2FiA0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1

4

i

iA0

� �2s ð20Þ

P. Choi et al. / Solid State Ionics 175 (2004) 535–539 539

RC ¼ RT

2FiC0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1

4

i

iC0

� �2s ð21Þ

RSPE ¼ LB

rB

ð22Þ

and of course, RI=RI. The overall resistance for membrane/

electrode unit calculated for the parameters given in Table 1

is shown in Fig. 4. At low current densities, i.e., at a current

density less than 200 mA/cm2, the anode resistance RA

dominates and thereafter RSPE=0.13 V cm2 becomes a

significant fraction of the total resistance. If there is a

limitation for mass transfer in the cell, diffusional limitation

resistance would dominate at higher current densities, which

has been neglected in this simple model.

Fig. 5 represents the power density input to the

electrolysis cell using a solid polymer electrolyte with

Pt–IrO2 for oxygen electrode and Pt for hydrogen

electrode. The power supply to the cell is proportional to

the current density, and thus the rate of reaction. The

lowest energy supply for water electrolysis is about 1.2 W/

cm2 for 1 A/cm2 at 80 8C. It is impossible to avoid this

minimum power input for water electrolysis because it

comes from the Gibbs free energy change of the reaction

at the experimental condition. An additional 0.3W/cm2 at

1A/cm2 is required due to the high anode overpotential

that may be reduced by developing new electrocatalysts.

5. Conclusions

The performance of SPE water electrolysis is analyzed

by means of a simple analytical model incorporating the

kinetics at the electrodes surfaces and transport in the

membrane. The model analyzes each individual resistance

associated with the different steps of the electrolysis process

in the membrane/electrode unit and predicts overpotentials

over a range of current densities for Pt and Pt–IrO2

electrocatalysts. It clearly shows that the high anode

overpotential is the limiting factor for the whole process

and mainly responsible for the energy supply needed in the

electrolysis cell in addition to the inherent thermodynamic

work. The model represents the experimental data satisfac-

torily and provides useful insights for water electrolysis in a

solid polymer electrolyte cell.

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