Electrochemical Impedance of Ion-Exchange Membranes with

Article pubs.acs.org/JPCC

Electrochemical Impedance of Ion-Exchange Membranes with Interfacial Charge Transfer Resistances A. A. Moya* Departamento de Física, Universidad de Jaén, Edificio A-3, Campus Universitario de Las Lagunillas, 23071 Jaén, Spain ABSTRACT: The characteristics of the electrochemical impedance and the effect of nonlinearity on the validity of impedance measurements of ion-exchange membrane systems constituted by the membrane and two diffusion boundary layers adjacent to the membrane, including charge transfer resistances at the solution−membrane interfaces, have been investigated. Ionic transport processes are described by the Nernst−Planck and Poisson equations, while the interfacial charge transfer processes are described by means of the Chang−Jaffé equations. The electrochemical impedance and total harmonic distortion associated with the decomposition into Fourier components of the potential−time response of the system to a sine electric current have been obtained by using the network simulation method. The influence of the kinetic rate constants of the interfacial charge transfer processes on the electrochemical impedance and the linearity of the current− voltage relationship of ion-exchange membrane systems is analyzed and discussed.

1. INTRODUCTION Studies on the electrochemical properties of ion-exchange membranes are receiving considerable attention because of the interesting environmental applications of these systems in different fields of science and technology such as electrodialysis for brackish water or seawater desalination1 or renewable energy harvesting such as pressure-retarded osmosis or reverse electrodialysis.2 In addition, these membranes are often used as separators in fuel cells3 and redox flux batteries4 or as elements joined to the porous electrodes in the systems for energy production from salinity differences5 or capacitive deionization.6 Electrochemical impedance spectroscopy is a powerful method of characterizing many of the electrical properties of a great variety of electrochemical systems, such as electrochemical cells and membrane systems.7 The standard measurement technique is to perturb the system with a single-frequency sine electric current and to measure in a frecuency response analyzer the resulting voltage at that frequency. Although the usual electrochemical systems are strongly nonlinear, the impedance measurements are useful and meaningful in general only for input signals of magnitude such that the system response is electrically linear. Theoretically, the perturbation amplitude should tend to zero, but this condition is not practical because the signal-to-noise ratio of the measurement will become unacceptably small. The choice of the amplitude of the perturbation signal is then both a theoretical and an experimental problem, being useful to study the effects of nonlinearities in electrochemical systems on the validity of impedance measurements.8 Since the work of Sistat et al.,9 a number of papers dealing with the electrochemical impedance of systems constituted by © XXXX American Chemical Society

an ion-exchange membrane and two diffusion boundary layers (DBLs) adjacent to the membrane have appeared in the literature.10−20 These works include detailed numerical studies based on the Nernst−Planck and Poisson equations10−14 and specific topics in the field of ion-exchange such as diffusion coefficients dependent on the ionic concentrations,15 degree of inhomogeneity of the membrane fixed charge,16 degree of asymmetry of the bathing concentrations,17 rectangular structure of fluidic nanochannels,18 competitive transport of counterions,19 and water splitting.20 Impedance plots are usually interpreted on the basis of the Randles equivalent circuit where the relationship between the circuit elements and the physical and chemical parameters of the system is determined by the Nernst−Planck−Donnan equations under electroneutrality on the basis of the Teorell−Mayer−Sievers model. In addition, the electric double-layer differential capacitance at the membrane−solution interfaces as given by Gouy−Chapman theory can be included according to the Randles electric equivalent circuit. However, these previous studies have ignored any kind of interfacial charge transfer resistance at the membrane−solution interfaces, although they are common in systems under membrane control,21 particularly in the field of ion-selective electrodes,22,23 which ignore the DBLs adjacent to the membrane. A number of experimental studies on electrochemical impedance of ion-exchange membranes24−30 have evidenced the presence of three arcs in the full range of frequencies or two arcs in the low-frequency region of the Nyquist plot. The Received: December 10, 2015 Revised: March 1, 2016

A

DOI: 10.1021/acs.jpcc.5b12087 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

is represented by ϕ(x,t), the electric permittivity by ε, the electric displacement by D(x,t), and the electric field by E(x,t). The constants F, R, and T have their usual meanings: Faraday constant, ideal gas constant, and absolute temperature, respectively. On the other hand, the faradaic electric current, If, is

intermediate-frequency arc is usually interpreted as associated with interfacial charge transfer resistances due to the presence of active films at the surface of the membrane, where ionic transport processes are different than those inside of the membrane. However, these behaviors are not investigated in detail from the Nernst−Planck and Poisson equations and the corresponding interfacial charge transfer kinetics. The main purpose of this paper is to contribute to this area by doing a novelty theoretical study of the characteristics of the electrochemical impedance and the effect of nonlinearity on the validity of impedance measurements of ion-exchange membrane systems constituted by the membrane and two DBLs adjacent to the membrane, including charge transfer resistances at the solution−membrane interfaces. The system under study is constituted by a cation-exchange membrane with negative fixed charge and two DBLs on both sides of the membrane, ionic transport being described by the Nernst−Planck and Poisson equations. Interfacial charge transfer resistances are allowed at the membrane−solution interfaces, and they are described by means of the Chang−Jaffé equations. The electrochemical impedance and the total harmonic distortion associated with the decomposition into Fourier components of the potential−time response of the system to a sine electric current have been obtained by using the network simulation method. The influence of the kinetic rate constants of the interfacial processes on the electrochemical impedance and the linearity of the current−voltage relationship of ion-exchange membrane systems are analyzed and discussed.

2

I f (x , t ) =

∂x

∂c (x , t ) =− i , i = 1, 2 ∂t

and the displacement electric current, Id, is given by Id(x , t ) =

∂I(x , t ) =0 ∂x

(6b)

i.e., I is not a function of x. It is a well-known fact that eqs 6a and 6b are equivalent to eqs 3 and 4. Therefore, according to the previous ideas of Brumleve and Buck,21 the Nernst−Planck and Poisson equations systems can be solved from eqs 1, 2, 6a, and 6b. Since the electric current will be the externally controlled variable, the boundary conditions can be expressed as ci( −δ , t ) = ci(d + δ , t ) = c0 , i = 1, 2

(7a)

⎡ X⎤ J1(0, t ) = k1L⎢c1(0, t ) − ⎥ ⎣ z⎦

(7b)

⎡ X⎤ J1(d , t ) = −k1R ⎢c1(d , t ) − ⎥ ⎣ z⎦

(7c)

J2 (0, t ) = J2 (d , t ) = 0

(7d)

I ( − δ , t ) = I (t )

(8a)

ϕ(d + δ , t ) = 0

(8b)

I(0, t ) = I(d , t ) = (1)

ϕM RM

+ CgM

dϕM dt

(8c)

where ϕM(t ) = ϕ(0, t ) − ϕ(d , t )

(8d)

is the electric potential drop through the membrane, RM is the membrane resistance, and CgM is the geometric capacitance of the membrane, which is given by ε CgM = (8e) d

(2)

(3)

The above equations specify all the conditions to be imposed on the solution of the Nernst−Planck and Poisson equations in the two DBLs. Equations 7a−7c are referred to the Nernst− Plank flux equations, while eqs 8a−8c correspond to the Poisson equation. In particular, eq 7a indicates that the system is electrically neutral at the outer boundaries of the DBLs, because c1 and c2 have the constant concentration c0 there. Equations 7b and 7c represent the Chang−Jaffé boundary

where ∂ϕ(x , t ) ∂x

(6a)

However, from the above equations one obtains

2

D(x , t ) = εE(x , t ) = −ε

(5b)

I(x , t ) = If (x , t ) + Id(x , t )

and the Poisson equation ∂D(x , t ) = F ∑ zici(x , t ) ∂x i=1

∂D(x , t ) ∂E(x , t ) =ε ∂t ∂t

Now, the total electric current, I, is the sum of the faradaic and displacement currents

the Nernst−Planck flux equations written for dilute solutions ⎡ ∂c (x , t ) F ∂ϕ(x , t ) ⎤ Ji (x , t ) = −Di⎢ i + zici(x , t ) ⎥ ⎣ ∂x ∂x ⎦ RT

(5a)

i=1

2. THEORETICAL BASIS Let us consider a membrane that extends from x = 0 to x = d and two identical DBLs adjacent to the membrane lying from x = −δ to x = 0 and from x = d to x = d + δ. The membrane is bathed by two bulk solutions of a z:z symmetric binary electrolyte, and it will be assumed to have a negative fixed charge. We denote by c0 the salt concentration in the bulk solutions and by X the fixed charge concentration inside the membrane. If the ionic transport is one dimensional and perpendicular to the membrane−solution interface, with x being the direction of transport, the equations determining the behavior of the system in the two DBLs at time t are the two laws of mass conservation or continuity equations ∂Ji (x , t )

∑ ziFJi (x , t )

(4)

Here Ji(x,t), ci(x,t), Di, and zi denote the flux, the molar concentration, the diffusion coefficient, and the charge number of ion i (z1 = z and z2 = −z), respectively. The electric potential B


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If we ignore the displacement electric current, denoted by c, the ionic concentrations and define the transport number of ion i, ti, as follows

conditions modeling the charge transfer resistances at the membrane−solution interfaces,21 respectively, with k1L and k1R being the kinetic rate constants at the left and right interfaces, while eq 7d indicates that the co-ion flux is zero at the boundaries of the membrane. The charge transfer resistances can be due to the existence of active layers or chemical reactions at the membrane surface. It must be noted that in this work the ions are considered charged points and the usual Butler−Volmer equations modeling the electrochemical charge transfer lead to Chang-Jaffé equaions.21,31 On the other hand, in eq 8a the externally applied electric current at x = −δ is represented by I(t), while eq 8b defines the reference level for the electric potential at the outer boundary of the right DBL. Finally, eq 8c corresponds to the equation for the electric current through the membrane, which is assumed to be a membrane fully impermeable to the co-ions and which is only described by their resistance and geometric capacitance. Under small-signal ac conditions the system variables may be separated into steady-state and time-dependent parts by using the algebra of complex numbers. In particular, the ionic concentration in an electroneutral region, c, and the electric potential can be written as c(x , t ) = c DC(x) + Δc(x)e jωt

(9a)

ϕ(x , t ) = ϕDC(x) + Δϕ(x)e jωt

(9b)

ti =

E=

dc I + t1 dx zF

(15a)

J2 = −DS

dc I + t2 dx zF

(15b)

2D1D2 D1 + D2

(16)

In order to study the response of the two DBLs to an externally applied electric current perturbation, the boundary conditions can be expressed as c( −δ , t ) = c(d + δ , t ) = c0

(17a)

⎡ X⎤ I J1(0, t ) = k1L⎢c L(t ) − ⎥ = ⎣ z ⎦ zF

(17b)

⎡ X⎤ I J1(d , t ) = −k1R ⎢c R (t ) − ⎥ = ⎣ z ⎦ zF

(17c)

J2 (0, t ) = J2 (d , t ) = 0

(17d)

where cL and cR, respectively, are the ionic concentrations at the left and right surfaces of the membrane, as schematically shown in Figure 1. If we define the reference level for the electric potential at the right bathing solution, the electric potential of the membrane system, ϕ(−δ,t) = ϕT(t), is

(11a)

(11b)

ϕT(t ) = ϕL(t ) + ϕdL(t ) + RMI + ϕR (t ) + ϕdL(t )

Now, the electrochemical impedance, Z, is a complex quantity given by the following equation7 Z(jω) =

J1 = −DS

DS =

and the electric potential at the outer boundary of the DBL, ϕT(t) = ϕ(−d,t), can be written as ϕT(t ) = ϕTDC + ΔϕTe jωt

(14)

where DS is the salt diffusion coefficient

In order to study the electrochemical impedance, the externally controlled electric current through the system is expressed as I(t ) = IDC + ΔIe

D1 − D2 dc t t1 + 21 I zD1 c dx z cD1

and the ionic fluxes can be written as

(10)

jω t

(13)

the electric field in each DBL, E, is given by

where j = −1 is the imaginary unit and ω is the circular or angular frequency, which can be written as a function of the conventional frequency, f ω = 2πf

D1 D1 + D2

(18)

ΔϕT ΔI

(12a)

and it can be expressed in the Euler or polar and rectangular forms as follows Z(jω) = |Z|e jφ = Zr(ω) + jZ i(ω)

(12b)

where |Z|, φ, Zr, and Zi are modulus, phase, real, and imaginary parts of the impedance, respectively.

3. ELECTRONEUTRAL ION-EXCHANGE MEMBRANE SYSTEM 3.1. General Governing Equations. Single analytical expressions for a great number of variables can be easily obtained in an ideal membrane system by assuming the general following hypothesis:31 • Electrical neutrality inside the two DBLs. • Electric potential obeying the Donnan equilibrium relations at the two membrane−solution interfaces.

Figure 1. Scheme of an ion-exchange membrane system by assuming electrical neutrality inside the two DBLs and by ignoring the electric double layers by means of the Donnan equilibrium relations at the interfaces. When an electric current passes through the system, the interfacial resistances lead to different values of the concentrations at the surface membrane. C


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The Journal of Physical Chemistry C where ϕL(t) and ϕR(t), respectively, are the potential differences through the electroneutral zones of the left and right DBL ϕ L (t ) =

∫0

ϕ R (t ) =

dL ϕDC

δ

∫d

E(x , t )dx

(19a) dR ϕDC

d+δ

E(x , t )dx

(19b)

RT ⎡ c L(t ) ⎤ ln⎢ ⎥ zF ⎣ c(0, t ) ⎦

RT ⎡ c(d , t ) ⎤ ln⎢ ϕdR (t ) = ⎥ zF ⎣ c R (t ) ⎦

(20b)

RDC =

RTL =

RTR =

RT 1 2 F zk X + 1L

IDC Fk1L

)

(26a)

RT 1 2 F zk X − 1R

IDC Fk1R

)

(26b)

(

(

3.3. Alternative Current. According to the modified Randles model, which ignores the displacement electric current at low frequencies under diffusion control and following the previous studies by Nikonenko and Kozmai,11 the impedance of the ion-exchange membrane system can be written as the sum of the geometric, interfacial, and diffusion impedances

(20c)

Z = Zg + Z T + Zd

(27)

In the high-frequency range dominates the geometric impedance given by

(20d)

Zg =

R acL R acR RM + + 1 + jωR acLCgL 1 + jωRMCgM 1 + jωR acR CgR (28a)

where the ohmic resistances of the left and right DBLs, respectively, are

(21a)

while that in the right DBL is

R acL = − (21b)

Then by using eq 14 the electric potential differences in the electroneutral zones of the DBLs are given by RT =− ln(1 − βL ) zF

(22a)

RT ln(1 + βL ) zF

(22b)

R ϕDC =

2RT ⎛ 1 1 ⎞ ⎜⎜ ⎟⎟ + RTL + RTR + zFIL ⎝ 1 + βL 1 − βL ⎠

where the charge transfer resistances are given by

from eq 15b with J2 = 0, one finds that the ionic concentration in the left DBL is

L ϕDC

dIDC

(25)

If we define βL as

⎛ x − d − δ⎞ ⎟ c DC(x) = c0⎜1 − βL ⎝ ⎠ δ

⎞ ⎟ ⎟ ⎠

dϕMDC

= RM +

while from eq 15b with J2 = 0 one obtains that the limiting current in a DBL is

⎛ x + δ ⎞⎟ c DC(x) = c0⎜1 − βL ⎝ δ ⎠

IDC Fk1L IDC Fk1R

(24)

(19d)

IKR = Fk1R X

IDC IL

⎛ 2RT ⎛ 1 + βL ⎞ RT ⎜ X + ⎟⎟ + + ln⎜⎜ ln zF ⎝ 1 − βL ⎠ zF ⎜ X − ⎝

and the dc resistance of the system, RDC, is

(20a)

βL =

(23b)

(19c)

IKL = Fk1LX

2zD1c0 δ

⎡ ⎤ RT ⎢ c0(1 + βL ) ⎥ ln = zF ⎢ X − IDC ⎥ ⎣ z zFk1R ⎦

ϕTDC = RMIDC

The analytical solution of the transport equations for the steady-state electric potential and the impedance under smallsignal ac conditions can be easily obtained following the work of Sistat et al.9 For the sake of clarity we expose here the most important expressions in order to provide a better understanding of our numerical results. 3.2. Steady State. The limiting electric current through the system is obtained when the concentration gradient reaches its maximum value into a DBL or at one of the surfaces of the membrane. This can be due to the electrodiffusion process or to the interfacial process. By taking into account eqs 17b and 17c, the absolute values of the interfacial limiting currents are

IL =

(23a)

In this way, the steady-state electric potential of the system is

and ϕdL(t) and ϕdL(t), respectively, are the Donnan potential differences at the left and right solution−membrane interfaces ϕdL(t ) =

IDC ⎤ ⎡ X RT ⎢ z + zFk1L ⎥ ln = zF ⎢ c0(1 − βL ) ⎥ ⎦ ⎣

R acR =

ln(1 − βL ) δ βL z (D1 + D2)c0 2

ln(1 + βL ) δ βL z (D1 + D2)c0 2

and the geometric capacitances are ε CgL = CgR = δ

(28b)

(28c)

(28d)

The intermediate frequency range is controlled by the interfacial processes, and the interfacial charge transfer impedance is

and the Donnan electric potential differences at the interfaces are D


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The Journal of Physical Chemistry C ZT =

RTL RTR + 1 + jωRTLC EL 1 + jωRTR C ER

(29a)

where the electric double-layer capacitances at the interfaces are C EL =

C ER =

⎛ zFϕ dL ⎞ ε DC ⎟ cosh⎜⎜ ⎟ LD ⎝ 2RT ⎠

(29b)

⎛ zFϕ dR ⎞ ε DC ⎟ cosh⎜⎜ ⎟ LD 2 RT ⎝ ⎠

(29c)

where LD given by LD =

εRT 2z 2F 2c0

(29d)

is the Debye length in the solution. Finally, in the low-frequency range the diffusion impedance is given by Zd = (R dL + R dR )

tanh α α

Figure 2. Complex-plane impedance plot for a system with k1 = k1L = k1R, X/c0 = 10, IDC = 0, for k1/k0 = 0.001, 0.002, and 0.005. The arrow indicates increasing values of k1.

(30a)

frequencies in the left of the plot, an interfacial semicircle at intermediate frequencies in the middle of the plot, and a diffusion arc at low frequencies in the right of the plot, which is a Warburg-type impedance since it presents a 45° slope straight line at high frequencies and a semicircle at low frequencies. The geometric arc is like a distorted semicircle, and it arises from the superposition of the three semicircles corresponding to the geometric impedances of the membrane and of the two DBLs. In Figure 2 it is clearly appreciated that this arc is not a function of the kinetic rate constants. It is worth noting that the usual experimental Nyquist plots of the impedance of ion-exchange membranes show two arcs in addition to the uncompensated resistance of the system,26 because the geometric arc is only experimentally observed when the measurement instruments operate in the range of very high frequency. Particularly interesting is the fact that the three arcs appear when operating in the overlimiting current range due to the water splitting process.20 Although from a given Nyquist-type plot like that in Figure 2 one can to easily obtain and tabulate different parameters, we will focus on the following. • The dc resistance of the system, RDC = Zr( f = 0), i.e., the real part of the impedance at the limit of zero frequency. • The characteristic frequency of the diffusion arc, f1, which is obtained from the relative maximum value of the imaginary part of the impedance with minus sign at low frequencies. • The characteristic frequency of the interfacial arc, f 2, and the imaginary part of the impedance at this frequency, Zi2 = Zi(f 2). These values are obtained from the relative maximum value of the imaginary part of the impedance with a minus sign at intermediate frequencies. • The frequency at the intersection point between the interfacial and diffusion arcs, f 3, and the imaginary part of the impedance at this frequency, Zi3 = Zi(f 3). These values are obtained from the relative minimum value of the imaginary part of the impedance with a minus sign at intermediate frequencies between f1 and f 2. In Table 1 the numerical results for the cited parameters are compared with the solutions obtained from the approximated analytical expressions derived in section 3 for electroneutral

where R dL =

ln(1 − βL ) δ 2RT ⎛ 1 ⎞ ⎜⎜ ⎟⎟ + 2 βL zFIL ⎝ 1 − βL ⎠ z (D1 + D2)c0 (30b)

R dR =

ln(1 + βL ) δ 2RT ⎛ 1 ⎞ ⎜⎜ ⎟⎟ − 2 βL zFIL ⎝ 1 + βL ⎠ z (D1 + D2)c0 (30c)

and α=δ

jω DS

(30d)

4. RESULTS AND DISCUSSION In this paper, the electrochemical impedance and the total harmonic distortion of an ion-exchange membrane system with interfacial charge transfer resistances are numerically obtained by using the network simulation method.19 Simulations have been performed for systems with z = 1, ε = 80ε0, c0 = 50 mM, D1 = 1.32 × 10−9 m2/s, and D2 = 1.96 × 10−9 m2/s. Then the Debye length in solution is LD = 1.373 nm at T = 298 K. The thicknesses of the DBLs and the membrane are chosen as d = δ = 105LDS. The chosen values for the parameters could correspond to typical membranes used in power harvesting such as reverse electrodialysis when they are immersed in NaCl diluted solutions. The following other scaling factors have been used: ϕ0 = 25.7 mV for potential, I0 = 0.92 A/cm2 for current, R0 = 27.7 mΩ·cm2 for resistance, f 0 = 0.7 MHz for frequency, k0 = 0.096 cm/s for kinetic rate constant, and CS0 = 51.6 μF/ cm2 for capacitance. 4.1. Electrochemical Impedance. Figure 2 shows the complex-plane impedance plots in a system with identical values of the kinetic rate constant at both interfaces k1 = k1L = k1R, X/c0 = 10, IDC = 0, for different values of k1, namely, k1/k0 = 0.001, 0.002, and 0.005. In this figure, −Zi(ω) is plotted against Zr(ω) with the angular frequency, ω, as a parameter increasing from the right to the left of the plot (Nyquist-type plot). Figure 2 shows three regions: a geometric arc at high E


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interfacial arc in the Nyquist plot, which increases as the rate constant decreases. This can be clearly appreciated in Figure 2 because the width of the interfacial arc decreases as the rate constant increases. As the kinetic rate constant increases, the transfer resistance decreases and the characteristic frequency of the interfacial process increases. Therefore, the difference between the characteristic frequencies of the diffusion and interfacial processes increases and the imaginary part of the impedance at the intersection between the interfacial and the diffusion arc, −Zi3, decreases. This behavior can be roughly estimated by considering the basic ideas of the Randles circuit.32 The straight line at the highest frequencies in the Warburg diffusion impedance can be written as σ Z WH(jω) = W (1 − j) (33a) ω

Table 1. Numerical and Analytical Results for the Characteristic Parameters Used in the Interpretation of the Impedance Plots of Ion-Exchange Membrane Systems with X/c0 = 10, IDC = 0, and Different Values of k1 k1/k0 = 0.001 RDC/R0 ( f1/f 0) × 108 ( f 2/f 0) × 104 −Zi2/R0 ( f 3/f 0) × 106 −Zi3/R0

k1/k0 = 0.002

k1/k0 = 0.005

num.

anal.

num.

anal.

num.

anal.

820 4.85 4.57 201.1 5.62 14.8

820 4.85 4.57 200 8.96 7.84

620 4.85 9.12 100.8 14.1 9.35

620 4.85 9.15 100 22.57 4.94

500 4.85 22.6 40.5 47.9 5.08

500 4.85 22.9 40 76.6 2.68

ion-exchange membrane systems with IDC = 0 and different values of k1, namely, k1/k0 = 0.001, 0.002, and 0.005. It must be noted that IL = 0.01, and the values for the rate constant have been chosen in such a way that k1/k0 ≥ 0.001, and the interfacial limiting currents given by eqs 20a and 20b take higher values than IL. Moreover, the values for the rate constant have been chosen in order to obtain comparable values for the interfacial, geometric, and diffusional resistances in the systems, i.e., to make observable three arcs in the Nyquist plot. In Figure 2 and Table 1 it can be clearly appreciated that the dc resistance of the system decreases as the rate constant increases, and the numerical results are in excellent agreement with the theoretical expression given by eq 25. In Table 1 it can be seen that the characteristic frequency of the diffusion arc, f1, is not a function of the kinetic rate constant. This behavior is expected because the two DBLs are identical under open circuit dc conditions, IDC = 0, and the diffusion arc obeys the classical expression for the Warburg impedance. In particular, the characteristic frequency, fd, is given by D 2πfd = 2.54 2S (31) δ −8 and the numerical result, f1/f 0 = 4.85 × 10 , is in excellent agreement with the theoretical expectations. Since the steady state of the system is an equilibrium one and the two rate constants are identical, the impedance corresponding to the interfacial processes is a semicircle in the Nyquist plot, where the radius is the interfacial charge transfer resistance RT = RTL = RTR given by RT =

RT 1 F 2 zk1X

where the Warburg coefficient, σW, is given by σ W = (R dL + R dR )

Z ig = −2RT2C Eω

ε ⎛ X ⎜⎜ + 2L D ⎝ zc0

zc0 X

⎞ ⎟⎟ ⎠

(33b)

(33c)

By equaling eq 33c to the imaginary part in eq 33a, the frequency at the intersection point between the geometric semicircle and the Warburg arc, f C, can be estimated from the following relation ⎞2/3 ⎛ σ W 2πfC = ⎜ 2 ⎟ ⎝ 2RTC E ⎠

(33d)

with this result being in agreement with the results previously published by Rubinstein and co-workers.10 In this way, in Table 1 it is observed that frequency f 3 increases as the rate constant increases, according to the approximated theoretical expectations. Figure 3 partially shows the complex-plane impedance plots in systems with X/c0 = 10, IDC = 0, and k1L/k0 = 0.001 for different values of k1R, namely, k1R/k0 = 0.005, 0.01, and 0.05. These values are arbitrary, but they illustrate asymmetric situations where the interfacial resistance is higher at the left interface than in the right. The reader must note that the geometric arcs have not been represented in this figure. The main difference between these plots and those in Figure 2 is that the interfacial arc is like a distorted semicircle for the closest values of k1L and k1R. As the value of k1R increases with respect to that of k1L, the interfacial resistance at the right interface becomes very small and the interfacial arc is a semicircle corresponding to the charge transfer process at the left interface. The dc resistance of the system decreases as the rate constants increase, but this effect in less significant in Figure 3 than in Figure 2 because k1L is kept constant and the interfacial arc corresponding to the left interface does not vanish when the right rate constant increases. In Table 2 the numerical results for the characteristic parameters used in the interpretation of the Nyquist plots are compared with estimations or approximated analytical solutions. In Figure 3 and Table 2 it can be clearly appreciated that the dc resistance of the system decreases as the rate constant at the right interface increases, and the numerical

(32b)

where the differential capacitance of the electric double layer obeys an identical expression at each interface and can be expressed as32 CE =

2δ 2

and from a Taylor series the imaginary part of the impedance of the geometric semicircle at the lowest frequencies can be expressed as

(32a)

and the characteristic frequency, f T, is 1 fT = 2πRTC E

DS

(32c)

In this way, in the chosen system one obtains CE/C0 = 1.74 and the characteristic parameters of the geometric arc −Zi2 and f 2, respectively, are in excellent agreement with the theoretically expected values for RT and f T. The rate constant can be determined from the measurement of the radius of the F


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The Journal of Physical Chemistry C ⎛ σ ⎞2/3 2πfC = ⎜ 2 W ⎟ ⎝ RTLC E ⎠

(34b)

In Table 2 it can be observed that frequency f 3 is not a function of the kinetic rate constant at the right interface, according to the approximated theoretically results. On the other hand, experimental Nyquist plots of electrochemical systems are often obtained under different dc bias conditions. First, Figure 4 shows the steady-state current−

Figure 3. Partial complex-plane impedance plot for a system with X/c0 = 10, IDC = 0, and k1L/k0 = 0.001 for k1R/k0 = 0.005, 0.01, and 0.05. The arrow indicates increasing values of k1R. Geometric arcs have not been represented.

Table 2. Numerical and Analytical Results for the Characteristic Parameters Used in the Interpretation of the Impedance Plots of Ion-Exchange Membrane Systems with X/c0 = 10, IDC = 0, k1L/k0 = 0.001, and Different Values of k1R k1R/k0 = 0.005 RDC/R0 ( f1/f 0) × 108 ( f 2/f 0) × 104 −Zi2/R0 ( f 3/f 0) × 106 −Zi3/R0

k1R/k0 = 0.01

Figure 4. Steady-state current−voltage characteristic of a system with X/c0 = 10, k1L = 0.001, and k1R/k0 = 0.002 with IL/I0 = 0.01 being the limiting current. Theoretical and numerical curves appear overlapped. The curve corresponding to a system without interfacial resistance is represented by means of a dashed line.

k1R/k0 = 0.1

num.

anal.

num.

anal.

num.

anal.

660 4.85 4.90 109 8.71 11.9

660 4.85 4.93 108 18.8 5.40

640 4.85 4.62 103.8 8.91 11.8

640 4.85 4.67 102 18.8 5.40

622 4.85 4.57 101.1 8.91 11.8

622 4.85 4.57 100 18.8 5.40

voltage characteristic of a system with X/c0 = 10, k1L/k0 = 0.001, and k1R/k0 = 0.002, these values leading to the limiting current IL/I0 = 0.01. This curve is typical in ion-exchange membrane systems showing two regimes: an underlimiting current regime at the lowest potential and a limiting current regime at the highest potential. The obtained numerical curve is in excellent agreement with that theoretically given by eq 24 because the DBL thickness is large enough. In fact, theoretical and numerical curves appear overlapped in Figure 4. The steadystate current−voltage characteristic corresponding to a system without interfacial resistances has been also represented in Figure 4 by means of a dashed line. Obviously, the slope of this curve in the underlimiting current regime is higher than that of the system including interfacial resistances. In order to study the impedance under different bias conditions, we marked on the current−voltage characteristic the chosen points, which are IDC/I0 = 0.0025 for ϕTDC/ϕ0 = 1.765, IDC/I0 = 0.005 for ϕTDC/ ϕ0 = 3.683, and IDC/I0 = 0.0075 for ϕTDC/ϕ0 = 6.343. Figure 5 shows the complex-plane impedance plots in a system with X/c0 = 10, k1L/k0 = 0.001, and k1R/k0 = 0.002 for different values of IDC, namely, IDC/IL = 0.25, 0.5, and 0.75. In this figure it can be appreciated that the width of the three arcs (geometric, interfacial, and diffusion) increases as the dc component of the electric current IDC increases. In Table 3 the numerical results for the characteristic parameters used in the interpretation of the Nyquist plots are compared with estimations or approximated analytical solutions. In Figure 5 and Table 3 it can be clearly appreciated that the dc resistance of the system increases as the dc

results are in excellent agreement with the theoretical expression given by eq 25. In Table 2 it can be seen that the characteristic frequency of the diffusion arc, f1, is not a function of the kinetic rate constant, and it is in excellent agreement with that given by eq 31. Again, the electric double-layer capacitance is the same at both interfaces, CE/C0 = 1.74, and the characteristic parameters of the geometric arc, −Zi2 and f 2, are in excellent agreement with those theoretically obtained from eq 29a. However, it is worth noting that the numerical results corresponding to the characteristic frequency of the interfacial arcs, f 2, are nearly independent of the rate constant k1R. Then the interfacial process is controlled by the charge transfer resistance at the left interface because its interfacial resistance is higher than that in the right interface, according to the chosen values for the parameters of the system, and the characteristic parameters of the interfacial arc −Zi2 and f 2, respectively, are in good agreement with the theoretically expected values for RTL/2 and the frequency f TL given by 1 fTL = 2πRTLC E (34a) Now the frequency at the intersection point between the geometric semicircle and the Warburg arc, f C, can be estimated from the following relation G


Article

The Journal of Physical Chemistry C ∞

ϕT(t ) = d0 +

∑ dn sin(2πnft + φn) n=1

where d0 is usually called the faradaic rectification and dn and φn are the amplitude and phase of the nth harmonic, respectively. Equation 35b constitutes the harmonic response of the system for the electric potential, and the first harmonic (n = 1) is known as the fundamental one. The evaluation of the different coefficients dn and φn allows for a simple and precise determination of the degree of nonlinear distortion which is present in the response of the system. Since the output of a linear system to a sinusoidal input is another sinusoid with the same frequency, the only existence of higher order harmonic terms (n > 1) in eq 35b would indicate the presence of nonlinear effects in the process of ionic transport through the considered ion-exchange membrane system.35,36 The nonlinear effects in the waveforms of the electric potential can be also quantified by means of the total harmonic distortion (THD), which is defined as37,38

Figure 5. Complex-plane impedance plot for a system with X/c0 = 10, k1L/k0 = 0.001, and k1R/k0 = 0.002 for IDC/IL = 0.25, 0.5, and 0.75. The arrow indicates increasing values of IDC.

THD =

Table 3. Numerical and Analytical Results for the Characteristic Parameters Used in the Interpretation of the Impedance Plots of Ion-Exchange Membrane Systems with X/c0 = 10, k1L/k0 = 0.001, k1R/k0 = 0.002, and Different Values of IDC IDC/I0 = 0.0025 RDC/R0 ( f1/f 0) × 108 ( f 2/f 0) × 104 −Zi2/R0

IDC/I0 = 0.005

100 d1

∞

∑ dn2 n=2

(36)

Figure 6 shows the THD as a function of the amplitude of the perturbing signal in systems with different value of the

IDC/I0 = 0.0075

num.

anal.

num.

anal.

num.

anal.

713.3 4.85 6.61 133.3

713.3 4.85 6.52 129.3

853.3 4.85 5.89 150.3

853.3 4.85 6.05 148.4

1412 4.85 3.24 230.2

1414 4.85 3.92 240

component of the electric current increases, the numerical results being in excellent agreement with the theoretical expression given by eq 25. In Table 3 it can be seen that the characteristic frequency of the diffusion arc, f1, is not a function of the dc current, and it is in excellent agreement with that given by eq 31. Now the electric double-layer capacitances are different at both interfaces, but the characteristic parameters of the interfacial arc, −Zi2 and f 2, are in excellent agreement with those theoretically obtained from eq 29a. 4.2. Total Harmonic Distortion. In this section we will analyzed the effect of nonlinearity on the validity of impedance measurements in the intermediate frequency range of an ionexchange membrane system with identical values of the kinetic rate constant k1 = k1L = k1R, X/c0 = 10, IDC = 0, for different values of k1, namely, k1/k0 = 0.001, 0.002, and 0.005. Now the system is perturbed around the equilibrium state with a large amplitude sine electric current, and the perturbing signal can be expressed as I(t ) = IA sin(2πft )

(35b)

Figure 6. Evolution of the THD with the perturbing current amplitude in a system with k1/k0 = 0.001, 0.002, and 0.005 by using the fundamental frequencies (f/f 0) × 106 = 5, 10, and 50, respectively. The arrow indicates increasing values of k1.

kinetic rate constant, namely, k1/k0 = 0.001, 0.002, and 0.005, by using the fundamental frequency approximately corresponding to frequency f 3 in Table 1, i.e., ( f/f 0) × 106 = 5, 10, and 50, respectively. In previous papers14,39 we found that in the absence of interfacial resistances the linear behavior is kept at the highest frequencies, even for frequencies lying on the 45° straight line of the diffusion arc. Now the results obtained show the great influence that the interfacial resistances have on the linear behavior of an ion-exchange membrane system. In Figure 6 it can be seen that the linear behavior is obeyed for the highest values of the kinetic rate constant because THD is always smaller than 1%. However, the THD increases as the rate constant increases for the highest values of the current amplitude. In Figure 6 it can be seen that the THD has a value

(35a)

It is well known that such system response contains two terms: transient and periodic. The transient term will be over in a time after the diffusion characteristic time of the system. As soon as the transient is over, the electric potential of the membrane system ϕT(t) can be represented by Fourier series in the following way33,34 H


Article


the system decreases as the rate constant increases, but the characteristics of the geometric and diffusion arcs are unaffected. This semicircle becomes distorted when the kinetic rate constant takes a different value at each interface. Moreover, increasing values of the dc component of the electric current strongly increase the width of the interfacial arc and the dc resistance of the system. On the other hand, it can be established that interfacial charge transfer resistances increase the nonlinear distortion, which is present in the response of an ion-exchange system for high values of the perturbing current amplitude, although the linear behavior is obeyed for frequencies higher than that characteristic of the interfacial arc.

superior to 1% for amplitudes of the input sine signal greater than 0.42IL for k1/k0 = 0.002 and 0.2IL for k1/k0 = 0.001. It can be established that, in general, the nonlinear distortion greatly increases as the current amplitude, IA, increases for the frequency at the intersection point between the interfacial semicircle and the Warburg arc in the impedance plot of an ionexchange membrane system. The above values of the current amplitude can be chosen as the threshold current values beyond which the membrane system response is nonlinear. In this way, the results obtained show that the above threshold value of the current amplitude decreases as the kinetic rate constant decreases. Figure 7 shows the THD as a function of frequency in a system with k1/k0 = 0.001 for different amplitudes of the

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The author acknowledges the financial support from the Ministry of Economy and Competitiveness of Spain, Project FIS-47666-C3-2-R, the University of Jaén, and Caja Rural de Jaén, Project UJA2015/06/23.

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REFERENCES

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Figure 7. Evolution of the THD with frequency in a system with k1/k0 = 0.001 for IA/IL = 0.2, 0.3, and 0.4. The arrow indicates increasing values of IA.

perturbing current, namely, IA/IL = 0.2, 0.3, and 0.4. In this plot, the characteristic frequency of the interfacial arc has been pointed out by means of a dotted vertical line, while the frequency at the intersection point between the interfacial and the diffusion arcs has been marked by circles. The results show the great effect that the current amplitude IA has on the nonlinearity of the system. However, in Figure 7 it can be seen that the linear behavior is maintained at the system, regardless of the input sine signal amplitude, for the highest frequencies. This behavior is due to the linear nature of the Nernst−Planck and Poisson equations at the highest frequencies,8 and it is obeyed for frequencies higher than the characteristic frequencies of the interfacial arc.

5. CONCLUSIONS The electrochemical impedance and the linearity of the current−voltage relationship in ion-exchange membrane systems, including interfacial charge transfer resistances, have been studied by using the network simulation method on the basis of the Nernst−Planck and Poisson equations and the Chang−Jaffé boundary conditions. Interfacial charge transfer resistances lead to the presence of a third arc at intermediate frequencies in the Nyquist plot of an ion-exchange membrane system. The width of this interfacial arc decreases and their characteristic frequency increases as the kinetic rate constant increases. As a result, the dc resistance of I


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Electrochemical Impedance of Ion-Exchange Membranes with

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