Supplementary Materials for
Effects of species concentration and external resistance on
spatiotemporal dynamics during the electro-oxidation of sulfide ion
on a platinum disk electrode
Jiaping Yang, You Jia, Qingyu Gao,* Changwei Pan, Yuemin Zhao, Wenyan Bi,
Guangyuan Xie*
College of Chemical Engineering, China University of Mining and Technology,
Xuzhou 221008, China
E-mail: gaoqy@cumt.edu.cn, xgywl@cumt.edu.cn
From the electrochemical reaction mechanism, we can construct a simplified
model to explain and represent the dynamic behavior of the sulfide electro-oxidation
system. First, we consider the general equivalent circuit of an electrochemical cell, as
shown in Fig. S1.
Fig. S1 Equivalent circuit of an electrochemical cell. C: double layer capacitance; DL: double
layer potential; ZF: Faradaic impedance; Rs: series resistance (including the uncompensated ohmic
electrolyte resistance and all external resistances); U: externally applied potential
Based on the charge balance of the circuit, the charging current of the electrical
double layer capacitor is obtained from the total current minus the Faradaic current,
i.e.,
AC
dDL U  DL

 nFAcs k (DL )
dt0
Rs
(E1)
Here, A is the active area of the working electrode (WE), n denotes the number of
electrons in the charge transfer process, cs is the concentration of interface sulfide ions,
F is the Faradaic constant, and k(DL) is the rate constant of the electrochemical
reaction. As the reaction occurs, the sulfide ion concentration at the interface will
decrease through (R4)–(R6). Then, the mass transportation process (R3), including
the diffusion and migration, replenishes the interface sulfide. Thus, the net interface
concentration (cs) evolution rate of the sulfide can be described as
dcs 2 D(cbulk  cs ) 2  cs (U  DL )


 cs k (DL )  k1cs
dt0
2
l
(E2)
Here, D is the diffusion constant; cbulk denotes the bulk concentration of sulfide ions;
 denotes the sulfide ion mobility;  represents the diffusion layer thickness; l is the
distance between the working and the reference electrodes; and k1 is the rate constant
of (R6).
Using the following dimensionless variables:

nF
2D
c
DL , c  s , t  2 t0 ,
RT

cbulk
and also with dimensionless rate constant: k ( ) 
2
2D
k (DL ) ,
we obtain the following dimensionless equations:
d u  

 mp ck ( )
dt
r
dc
u 
 1 c   c
 ck ( )  k2c
dt
r
(E3)
(E4)
with
u
nF
n 2 F 2 cbulk
U , mp 
,
RT
CRT
k ( )  k0 ( xp 2 
b
M  (  ed ) 2

2  RTACRs
,
nF  l
r
2DACRs
2
,
k2 
 2 k1
2D
and
) , where u and r denote the dimensionless external
potential and the dimensionless resistance respectively, and k0, xp, b, M and ed are
constants.
In the reaction mechanism, the sulfide ion is oxidized to generate the deposited
sulfur. It has been reported that the sulfur coverage is dependent on the electrode
potential and decreases with increasing electrode potential [1]. Based on this
relationship between the sulfur coverage and the potential, we obtained a fitted
dimensionless formula to describe the sulfur coverage () dependence on the
electrode potential:
d
1
 P[
 ]
(  4.7)/0.2
dt
1 e
(E5)
where P denotes the adsorption rate.
This indicates that reactions can only occur on the unoccupied platinum sites.
When we consider the sulfur coverage, the dimensionless three-variable equations are
composed of:
d u  

 mp c(1   )k ( )
dt
r
dc
u 
 1 c   c
 c(1   )k ( )  k2c
dt
r
d
1
 P[
 ]
(  4.7)/0.2
dt
1 e
with k ( )  k0 ( x p 2 
b
M  (  ed ) 2
(1)
(2)
(3)
).
In this model, the parameter mp contains the initial bulk concentration of sulfide
ions, and the hydroxide ion concentration is related to the parameter b in k(), i.e., the
tendency for  positive feedback. Subsystems of equations (1) + (2) and (1) + (3)
represent the N-NDR and HN-NDR dynamics, respectively. Using this three-variable
model, the oscillatory behaviors of both HN-NDR and N-NDR can be obtained.
References:
1. Fu J, Hou M, Du C, Shao Z, Yi B (2009) Potential dependence of sulfur dioxide poisoning and
oxidation at the cathode of proton exchange membrane fuel cells. J Power Sources 187:3238