Diffusion impedance of electroactive materials, electrolytic solutions and porous electrodes: Warburg impedance and beyond

Electrochimica Acta 281 (2018) 170e188

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Electrochimica Acta journal homepage: www.elsevier.com/locate/electacta

Diffusion impedance of electroactive materials, electrolytic solutions and porous electrodes: Warburg impedance and beyond Jun Huang College of Chemistry and Chemical Engineering, Central South University, Changsha 410083, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 4 January 2018 Received in revised form 20 April 2018 Accepted 22 May 2018 Available online 24 May 2018

The diffusion coefficient is a key property of materials. Electrochemical impedance spectroscopy (EIS) is a routine tool to determine the diffusion coefficient. Albeit being versatile for varied electrochemical systems and powerful in distinguishing multiple processes in a wide frequency spectrum, the EIS method usually needs a physical model in data analysis; misuse of models leads researchers to provide unwarranted interpretation of EIS data. Regarding diffusion, the simple and elegant formula developed by Warburg has been serving as the canonical model for more than a century. The classical Warburg model has very strict assumptions, however, it is used in a wide range of scenarios where assumptions may not be satisfied. It is the main purpose of the present article to define the boundary of applicability of the Warburg model and develop alternative models for cases beyond the boundary. In so doing, the Warburg model is revisited and its limitations and assumptions are scrutinized. Afterwards, new impedance models for more complicated and realistic scenarios are developed. The present article features: (1) generalization of the boundary condition when treating diffusion in bounded space and geometrical variants; (2) diffusion impedance in porous electrodes and fractals; (3) the effect of electrostatic interactions and coupling between diffusion and migration on the diffusion impedance in electrolytic solutions; (4) introduction of homotopy perturbation method to treat the convective diffusion; (5) physical interpretations of diffusion impedance behaviors. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Electrochemical impedance spectroscopy Diffusion impedance Warburg impedance Material characterization Diffusion coefficient Porous electrode

1. Introduction The diffusion coefficient is a key transport property of electroactive materials and electrolytic solutions. Electrochemical impedance spectroscopy (EIS) has been serving as a powerful and routine tool in estimating the diffusion coefficient [1e17], and characterizing electrochemical devices in a wider scope [18e21], due to its advantage of separating the diffusion processes from other physico-chemical processes in a wide frequency spectrum. In so doing, an EIS model is necessary. In most cases, the conventional wisdom, Warburg impedance [22], is adopted. This practice is, on one hand, prompted by the observation of an oblique line with an angle close to 45
in the low frequency range. On the other hand, the simplicity and elegancy of the Warburg impedance also underpin its popularity. While it is nowadays customary to link a 45
line in the EIS plot with diffusion, there are caveats on this. The assumptions of the classical Warburg impedance are very

E-mail address: [email protected]. https://doi.org/10.1016/j.electacta.2018.05.136 0013-4686/© 2018 Elsevier Ltd. All rights reserved.

strict, that is, semi-infinite, planar, purely concentration-gradientdriven, Fickian diffusion of neutral species in the dilute limit. In reality, we are, however, frequently encountered with ‘non-ideal’ diffusion processes, including but not limited to, diffusion in bounded or irregular space [23e37], diffusion of charged species which is coupled with migration [38e43] and/or convection [44e54], and multi-scale and multi-phase diffusion in porous electrodes [55e58]. In consequence, it shall not be surprising that non-idealities of the diffusion impedance, including but not limited to oblique lines deviating from a 45
line and even arced curves, are frequently observed in experiments. Non-idealities are also encountered in other cases, e.g. the constant phase element phenomenon [59]. There is a long history of recognizing the limitations of the classical Warburg impedance model, and many derivative models have been developed in the literature [26,30,40,41,60e68]. This study presents a systematic extension of the classical Warburg impedance model in two dimensions: new structures and new physics, and intends to develop new models for the aforementioned ‘non-ideal’ cases. For the purpose of completeness, existing

J. Huang / Electrochimica Acta 281 (2018) 170e188

efforts in the literature are reproduced, while they are modified to ensure the uniformity of nomenclature, and generalized in a new framework in this work. In addition, this study also presents several original contributions, including a unified expression for the bounded diffusion with different boundary conditions and new resultant features observed on the EIS plot, consideration of the coupling between diffusion and migration of charged species in electrolytic solutions, introduction of the homotopy perturbation method to treat the diffusion-convection impedance, discussion on the double layer effect on the diffusion impedance, models for diffusion in fractals, discussion on diffusion in porous electrodes, and new physical interpretations of diffusion impedance phenomena. As this article stands in between a review paper and an original paper, it is important for us to give enough credits to original sources. The reminder of this article is organized as follows. First, derivation, assumptions and limitations of the classical Warburg impedance are presented and discussed. Second, the classical Warburg impedance is extended to different structures, including the bounded planar space, other geometrical variants (cylinders and spheres), and porous electrodes. Third, the classical Warburg impedance is extended to incorporate new physics, including transport in electrolytic solutions in which diffusion and migration are strongly coupled, the convection effect, the double layer effect, non-Fickian diffusion and diffusion in fractals. This article is intended to help researchers in choosing an appropriate model for interpreting their data. As a result, implications of theoretical analysis for experimentalists are discussed in the penultimate section.

vci ðx; tÞ v2 ci ðx; tÞ ¼ Di ; vt vx2

We begin with the canonical scenario considered by E. Warburg in 1899 [22], as schematically shown in Fig. 1. The species in the solution diffuse away or towards an electrode-electrolyte interface (EEI). At the EEI, x ¼ 0, the following redox reaction takes place,

Ox þ ne4Re:

ci ¼ c0i :

(4)

At the EEI, the reactant/product is consumed/formed in the charge transfer reaction. The consumption/formation rate is related to the reaction rate, which is usually measured by the current density, i. As a result, the diffusion process is linked with the charge transfer reaction at the EEI,

Di

vci i ¼± ; nF vx

E ¼ f ði; cOx ; cRe Þ;

(2)

where i is the current density, E the potential across the EEI, cOx ; cRe the concentration of oxidant and reductant, respectively. In the electrolyte solution, the species flux is exclusively driven by the concentration gradient, which is described by the Fick's law. The mass conservation law gives,

(5)

where ‘  ’ is for the oxidant and ‘ þ ’ is for the reductant and i is positive for oxidation, F is Faraday constant and n is the number of transferred electrons in Eq. (1). There are two widely adopted methods to solve for the impedance response from Eq. (3): a time-domain method and a frequency-domain method. In the time-domain method, as employed in the original study of E. Warburg [22], the voltage response to a sinusoidal perturbation of the current density, ~i ¼ A sinðutÞ (A the amplitude,u the frequency), is obtained, which i

~ ¼ AV sinðut þ jÞ. As a result, the impedance is expressed as, V response is given by, Z ¼ AV =Ai $expðijÞ with AV =Ai being the amplitude and j the phase angle. The frequency-domain method is comparatively more concise; it first transforms Eq. (3) to,

ju~ci ¼ Di

(1)

The current-potential relation of the reaction in Eq. (1) is written in a general form,

(3)

where Di is the diffusion coefficient of oxidant (i ¼ Ox) or reductant (i ¼ Re). We take the bulk electrolyte at infinite distance, x ¼ ∞, where ci has its bulk value,

i

2. Warburg impedance

171

v2 ~ci ; vx2

(6)

in the frequency-domain via Fourier transformation, where ~ci is the perturbation in the concentration of species i. Thorough this paper, variables marked with an over-tilde represent the perturbed quantity. Accordingly, Eq. (4) is transformed to,

~ci ¼ 0:

(7)

because the bulk concentration is invariant. Eq. (5) in the frequency domain is rephrased as,

Fig. 1. Semi-infinite diffusion coupled with charge transfer reaction at the electrode-electrolyte interface and the representative diffusion impedance in the frequency range of 1  103 Hz with.n ¼ 1; cOx ¼ cRe ¼ 1M; DOx ¼ DRe ¼ 1010 m2 s1

172

J. Huang / Electrochimica Acta 281 (2018) 170e188

~i v~c Di i ¼ ± ; nF vx

(8)

with ~i being the perturbation of the current density. The solution to Eq. (6) is given by,


 
  ~ci ðxÞ ¼ f1 exp x ld;i þ f2 exp x ld;i ;

(9)

pffiffiffiffiffiffiffiffiffiffiffiffi with ld;i ¼ Di =ju being the frequency-dependent diffusion length. The magnitude of ld;i decreases at higher frequencies. The two coefficients, f1 and f2 , can be determined by invoking boundary conditions in Eqs. (7) and (8),

ld;i ~ i; f1 ¼ 0; f2 ¼ ± nFDi

(10)

where ‘ þ ’ is for the oxidant and ‘  ’ is for the reductant. By differentiating Eq. (2) with respect to ~i and ~c , the perturba~ is expressed as, tion in the electrode potential, E,

~ ¼ Z ~i þ E int

X

bd;i ~ci ðx ¼ 0Þ;

i

(11)

i

~ ~i is the interfacial impedance comprising a Farawhere Zint ¼ vE=v daic term corresponding to the charge transfer across the EEI and a non-Faradaic term corresponding to the double layer charging (these two processes are usually decoupled as a prior assumption, ~ ~c while ref [69] discussed the coupling between them), bd;i ¼ vE=v i is the coefficient describing how the electrode potential changes as a function of concentration of the oxidant/reductant. ~ ~i, we arrive Substituting Eq. (9) into Eq. (11) and defining Z ¼ E= at,

Z ¼ Zint þ

1 nF



bd;Ox DOx

ld;Ox 

bd;Re DRe

 ld;Re :

(12)

By invoking the Nernst equation (valid when the species concentration is not very low),

E ¼ E0 þ

  RT g c ln ox Ox ; nF gRe cRe

(13)

with E0 being the equilibrium potential under standard state, and gi being the activity, we obtain,

bd;Ox ¼

RT nF



   1 1 vgOx RT 1 1 vgRe ; bd;Re ¼  : þ þ nF cRe gRe vcRe cOx gOx vcOx (14)

In the dilute limit, gOx ¼ gRe ¼ 1. Thereby, Eq. (12) is rephrased as,

Z ¼ Zint þ

RT ðnFÞ2

! 1 1 pffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffi ðjuÞ0:5 : cOx DOx cRe DRe

(15)

with the prefactor being,

.pffiffiffi 2;

ðnFÞ2

! 1 1 pffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffi : cOx DOx cRe DRe

(17)

pffiffiffiffiffi For a specific species, sw is given by, sw ¼ RT=ðnFÞ2 ci Di . The quintessential feature of the Warburg impedance can be readily seen from Eq. (15), viz., a 45
line in the Nyquist plot, as shown in Fig. 1. Moreover, diffusion properties, DOx and DRe , can be obtained from the relation between the real part of Zw , ReðZw Þ, and pffiffiffi u, given by, ReðZw Þ ¼ sw u0:5 = 2. The simplicity of the Warburg impedance renders popularity in materials scientists and chemists. The main assumptions involved in the original Warburg impedance model are five-fold: (1) Semi-infinite diffusion. As a result, the Warburg impedance is applicable when the characteristic diffusion length ld;i is much smaller than the thickness of the diffusion region Ld , namely when the frequency is sufficiently high, u[Di =L2d . Otherwise, we have to treat the finite-length diffusion problem, which is termed bounded diffusion [23,24,27,29,30,70]. (2) Infinite dilute solution. The Warburg impedance model invokes the approximation of ionic activities gi ¼ 1. Churikov et al. demonstrated that such approximation may bring large errors for diffusion in solid materials [14e17]. (3) Planar geometry. The Warburg impedance describes diffusion of a species towards a planar surface. Therefore, it cannot be used to describe diffusion in spherical [26,33,35], cylindrical [26,33,35] geometry and other more complicated ones such as fractals [25,37,67,71e73]. (4) Pure diffusion. The Warburg impedance accounts for the diffusion current only. However, we are usually encountered with multiple transport mechanisms, including diffusion, migration and convection, involving multiple species such as cations and anions, in many cases, e.g. in electrolytic solutions [38e40,42,68,74]. Moreover, the classical Warburg impedance is, in some cases, physically ill-posed, as incisively realized by Barbero and Lelidis [68]. The perturbation of the ionic current density is given by, ~iðxÞ ¼  P zi FDi V~ci ðxÞ. ~iðxÞ is position-dependent, which contradicts i with the conservation law of charge, as readily derived from Eq. (9). As a result, it is essential to account for other transport mechanisms, such as migration, to preserve the charge conservation for transport of charged species. In other words, the classical Warburg impedance is limited to pure diffusion of neutral species, and it fails for the case of ionic species (see detailed discussion in Section 4.1). (5) Fickian diffusion. The Warburg impedance is based on the classical Fick's law which leads to the relation Zw  ðjuÞ0:5 . This cannot explain the frequently observed relation, Zw  ðjuÞa with as0:5, which compels researchers to consider non-Fickian diffusion [62,65,67]. In what follows, the classical Warburg impedance model is extended by releasing the above assumptions.

3. Extension in structure 3.1. Bounded diffusion

The Warburg impedance is expressed as,

Zw ¼ sw ðjuÞ0:5 ¼ sw u0:5 ð1  jÞ

sw ¼

RT

(16)

The Warburg impedance assumes a semi-infinite diffusion region. This assumption is released by considering a finite diffusion region with a thickness of Ld . The Warburg formula is the limiting case of Ld /∞. The diffusion impedance in bounded space has been

J. Huang / Electrochimica Acta 281 (2018) 170e188

considered by many researchers in the literature [23,24,27e30,70]. For example, Bisquert and co-workers have published a series of influential works on bounded diffusion, in which different expressions are derived for different boundary conditions [27,28,30]. Herein, the novelty here is twofold: (1) a unified expression for bounded diffusion impedance considering a general boundary condition which reduces to that considered by Bisquert et al. as limiting cases, and (2) physical interpretations of diffusion impedance phenomena, for example, why the diffusion process with a totally reflecting boundary behaves like a capacitor. At x ¼ Ld , the boundary condition can be generally expressed as,

v~ci þ kbc~ci ¼ 0; vx

(18)

with kbc ½m1  being the coefficient of boundary condition. Eq. (18) is reduced to a totally reflecting boundary, v~ci =vx ¼ 0, when kbc ¼ 0, and to a totally adsorbing boundary, ~ci ¼ 0, when kbc ¼ ∞. The totally-reflecting boundary condition is applicable when the plane at x ¼ Ld is symmetrical or impermeable. The totally adsorbing boundary condition is applicable when the diffusion region is interfaced with an ionic reservoir at x ¼ Ld . There are different names for boundary conditions in the literature. Jacobsen and West termed it the impermeable boundary for kbc ¼ 0 and the Nernstian diffusion layer for kbc ¼ ∞ [26]. Note that Eq. (18) is a homogeneous boundary condition. It may need to use a nonhomogeneous boundary condition in certain scenarios, which is not considered here. Applying boundary conditions in Eqs. (8) and (18) to the general solution in Eq. (9) results in,


 ~il 1  kbc ld;i exp lLd d;i d;i    ; f1 ¼ ± nFDi
1  k l exp  Ld 
1 þ k l exp Ld bc d;i bc d;i l l d;i

d;i

d;i

(20) where ‘  ’ is for the oxidant and ‘ þ ’ is for the reductant. In the similar manner, the impedance of finite diffusion corresponding to species i is given by,

pffiffiffiffiffiðjuÞ0:5 ; ðnFÞ ci Di

(21)



  1 þ kbc ld;i exp lLd þ 1  kbc ld;i exp lLd  d;i
 d;i ; ¼
  1 þ kbc ld;i exp lLd  1  kbc ld;i exp lLd d;i

(22)

d;i

which is reduced to,

lL;i

for kbc ¼ ∞ (a totally adsorbing boundary). It is important to note pffiffiffiffiffiffiffiffiffiffiffiffi that ld;i ¼ Di =ju is frequency-dependent. lL;i in Eqs. (23) and (24) approximates unity and Eq. (21) is reduced to the Warburg impedance in Eq. (16) when Ld /∞.



When u is sufficiently high, ld;i ≪Ld . Therefore, lL;i /1. In other words, the Warburg impedance phenomenon, viz. a 45
tilted line, can be observed in the high-frequency range. In plain English, the diffusion process does not ‘sense’ the boundary condition at x ¼ Ld at such high frequencies, hence, behaves as a semi-infinite diffusion as described by the classical Warburg model. Moreover, the diffusion process does not ‘sense’ the specific geometry at sufficiently high frequencies, as will be seen in section 3.2.



In the low-frequency range, ld;i [Ld , and thereby,

! !1 Ld Ld 1 Ld coth þ ; z 3 ld;i ld;i ld;i

tanh

(25)

! !3 Ld L 1 Ld L z d: z d  ld;i ld;i 3 ld;i ld;i

(26)

As a result, the asymptotic behavior of the diffusion impedance in Eq. (21) in the low-frequency range is given by,

for

kbc ¼ 0

! Ld ¼ coth ; ld;i

for kbc ¼ 0 (a totally reflecting boundary), and to,

(23)

(a

Rd;i ¼ RTLd =ðnFÞ2 ci Di

(27) totally is

the

reflecting effective

boundary), diffusion

where

resistance,

Cd;i ¼ ðnFÞ2 ci Ld =RT is the effective diffusion capacitance. Eq. (27) indicates that the diffusion impedance shows a vertical line, viz. the capacitive phenomena, in the low-frequency range. Combined, there is a transition from a 45
tilted line to a vertical line when the frequency decreases. In terms of kbc ¼ ∞ (a totally adsorbing boundary), the asymptotic behavior of the diffusion impedance in Eq. (21) in the lowfrequency range is described by,

Zd;i ¼ Rd;i ;

lL;i

2

where lL;i is a frequency-dependent coefficient describing the effects of the finite diffusion length and the boundary condition,

lL;i

(24)

d;i


 ~il 1 þ kbc ld;i exp lLd d;i d;i    ; f2 ¼ ± nFDi
1  k l exp  Ld 
1 þ k l exp Ld bc d;i bc d;i l l

RT

! Ld ¼ tanh ; ld;i

1 1 Zd;i ¼ Rd;i þ ; 3 juCd;i (19)

Zd;i ¼

lL;i

173

(28)

Eq. (28) indicates that the diffusion impedance converges to a resistance in the low-frequency range, which is sharply different from the totally reflecting case. In Fig. 2, the effect of kbc on the diffusion impedance is investigated. In line with the above theoretical analysis, Zd;i shows a 45
tilted line in the high-frequency range, regardless of kbc . However, Zd;i in the low-frequency range depends on kbc : it exhibits a vertical line for the cases of kbc /0 and a semicircle for the cases of kbc /∞. Moreover, the real part of the impedance in the low frequency range for the totally reflecting case is one third of that for the totally absorbing case, as readily seen from Eqs. (27) and (28). The above phenomena have been well-documented in the literature. However, physical implications behind the phenomena have not been clearly disclosed yet. Why does the low-frequency impedance behave like a capacitor for kbc ¼ 0? For this case, we have v~ci =vx ¼ 0 at x ¼ Ld . By integrating Eq. (6), we obtain,

174

J. Huang / Electrochimica Acta 281 (2018) 170e188

of the double layer in the frequency domain is in the same form as in Eq. (29). This is an intuitive explanation why capacitive behavior manifests in the low-frequency diffusion impedance for the case of kbc ¼ 0. Integrating Eq. (6) for the case of kbc ¼ ∞, that is, ~ci ðx ¼ Ld Þ ¼ 0, Z Ld gives, ~ið0Þ  ~iðLd Þ ¼ ju nF ~ci dx. At very low frequencies, ~ci is 0

uniform, namely, ~ci ðxÞz0. As a result, ~ið0Þz~iðLd Þ, and the system behaves like a resistor. 3.2. Geometrical variants In this section, the Warburg impedance is extended from the planar symmetry to cylindrical and spherical symmetries, as shown in Fig. 3. This extension is of practical importance, for example, battery electrodes are made up of cylindrical or spherical active particles. In the literature, the geometrical effect on the diffusion impedance with totally reflecting and adsorbing boundary conditions has been considered by Jacobsen and West [26]. The novelty here is application of a general boundary condition and new resultant impedance features under certain circumstances. In cylindrical symmetry, the diffusion equation in Eq. (6) is written as,

Fig. 2. Impedance response of finite-length diffusion with a series of kbc simulated using Eqs. (21) and (22). Other parameters are n ¼ 1; ci ¼ 1M; Di ¼ 108 m2 s1 ; Ld ¼ 1 cm. The frequency range is 1  105 Hz.

~iðx ¼ 0Þ ¼ ju

ZLd

nF ~ci dx;

(29)

ju~ci ¼ Di

! v2 ~ci 1 v~ci : þ r vr vr 2

(30)

By introducing the dimensionless variable, r nd ¼ r=ld;i with ld;i ¼ pffiffiffiffiffiffiffiffiffiffiffiffi Di =ju, Eq. (30) is transformed to a modified Bessel equation of zero order [75],

v2 ~ci 1 v~ci þ nd nd  ~ci ¼ 0; 2 vr nd r vr

(31)

0

of which the general solution is given by, ~ with ~iðxÞ ¼  nFDi vvxci . At very low frequencies, ~ci is nearly uniform and weakly dependent on u. Regarding an electric double layer, the current density is defined as, i ¼ dQ =dt, where Q is the charge PR stored in the double layer carried by ions, Q ¼ zi Fci dx, with zi being the charge number. It is readily seen that the charge density

  ~ci ¼ f1 I0 r nd þ f2 K0 r nd ;

(32)

where I0 and K0 are modified Bessel functions of the first and second kind.

Fig. 3. Diffusion in cylindrical and spherical symmetry. Re is the radius of the cylinder/sphere and Ld locates the other boundary.

J. Huang / Electrochimica Acta 281 (2018) 170e188

As shown in Fig. 3, we designate Ld > Re as the outward diffusion case, and Ld < Re as the inward diffusion case. At r ¼ Re , the boundary condition in Eq. (8) is rewritten as,

Di

~i v~ci ¼ ± $sgnðRe  Ld Þ: nF vr

(33)

where ‘ þ ’ is for oxidant and ‘  ’ is for reductant, sgnðRe  Ld Þ ¼ 1 for Re > Ld and sgnðRe  Ld Þ ¼ 1 for Re < Ld, and ~i is positive for oxidation. At r ¼ Ld , we apply the following general boundary condition,

v~ci þ kbc ~ci ¼ 0: vr nd

(34)

Two coefficients f1 and f2 in Eq. (32) are solved by invoking boundary conditions in Eqs. (33) and (34). The diffusion impedance of species i is written as,

Zd;i ¼

lc;i

RT 2

ðnFÞ ci

pffiffiffiffiffiðjuÞ0:5 Di

(35)

where the coefficient lc;i , describing effects of the cylindrical geometry, the boundary condition and the diffusion direction, is given by,

lc;i

Re ld;i

K1

 Ld ld;i

d;i

þ I1

d;i



 n P ðx2 =4Þk ∞ In ðxÞz 2x k¼0 k!Gðnþkþ1Þ where GðxÞ is the Gamma function, and Zd;i is approximated as,

Zd;i

! 1 2Di j 2 ; 4 uRe

RT

Re ¼ 2 cD ðnFÞ i i

Ld ld;i

K0

d;i

lc;i ¼ K0

Re ld;i

!, K1

! Re ; ld;i

I0

Re

K1

 Ld

l l lc;i ¼ ±  d;i  d;i

I1

Re ld;i

Ld ld;i

K1

þ I1  I1



for both kbc ¼ 0 and kbc ¼ ∞. The independence of diffusion impedance on the boundary condition at r ¼ Ld can be understood in the following way. The diffusion process does not sense the boundary condition at the infinite r ¼ Ld /∞. In the high-frequency limit, ld;i ≪Re ,   qffiffiffiffi p expðxÞ 1 þ 4n2 1 for jxj/∞, hence, l z1. ConseKn ðxÞz 2x c;i 8x quently, the Warburg impedance phenomenon is retained in the high frequency limit, as shown in Fig. 4 (a). In the low frequency limit, ld;i [Re , K0 ðxÞz  lnðxÞ; K1 ðxÞz1=x; for jxj/0, hence, Zd;i is approximated as,

Re ld;i d;i

Ld l

K0

Ld ld;i

K1

 d;i

d;i

d;i

d;i

 Re l

 d;i ; Re ld;i

(37)

for kbc ¼ 0, and,

I0

 Re ld;i

K0

Re ld;i

K0

 Ld ld;i

 I0

Ld ld;i

þ I0

lc;i ¼ ±   I1

 Ld ld;i

K0

Ld ld;i

K1



 Re l

 d;i ; Re ld;i

(38)

for kbc ¼ ∞. We further consider two limiting cases: Ld ¼ 0 and Ld /∞. The case of Ld ¼ 0, so-called the central diffusion, is relevant for battery electrodes that are comprised of cylindrical active materials. In this scenario, Liþ reacts at the cylindrical surface and diffuses into active materials. In this case, kbc ¼ 0, as restricted by the symmetrical condition at the axis. Thereby, Eq. (37) is simplified as,

lc;i ¼ I0

Re ld;i

!, I1

! Re ; ld;i

(41)



where ‘ þ ’ is for inward diffusion (Re > Ld ) and ‘  ’ is for outward diffusion (Re < Ld ). Eq. (36) is reduced to,



(40)

which indicates a capacitive phenomenon, as shown in Fig. 4 (a). For the case of Ld /∞, Zd;i is reduced to,

      kbc I0 lRe K0 lLd  I0 lLd K0 lRe     d;i  d;i  d;i  d;i ; ¼±   I1 lRe K1 lLd  I1 lLd K1 lRe  kbc I1 lRe K0 lLd þ I0 lLd K1 lRe I0



175

(39)

which has the asymptotic value of lc;i /1 in the high-frequency range, namely, ld;i ≪Re . As a result, the Warburg impedance phenomenon is retained in the high frequency limit, as shown in Fig. 4 (a). In the low frequency limit, namely, ld;i [Re ,

(36)

d;i

Zd;i

RT

Re ¼ 2 cD ðnFÞ i i

! pffiffiffiffi ! p Di pffiffiffiffi  j ; ln 4 Re u

(42)

which indicates that the imaginary part is constant, as shown in Fig. 4 (a). Fig. 4 (b)-(d) evaluates the effects of the ratio of Ld =Re and kbc on the diffusion impedance in the cylindrical symmetry. For the cases of kbc ¼ 0 and kbc ¼ ∞, the structure of diffusion impedance in the cylindrical case resembles that in the planar case (Fig. 2); the ratio of Ld =Re only influences the magnitude of the impedance. However, for the case of kbc ¼ 10, a combined boundary condition, the impedance structure shows new features. For the inward diffusion (Ld =Re < 1), three regions can be discerned: (i) a 45
line in the high frequency range, (ii) a semicircle in the intermediate frequency range, and (iii) again a 45
line in the very low frequency range. To the best knowledge of the author, this newly emerging 45
line in the very low frequency range has not been reported in neither experimental nor theoretical studies. In comparison, for the outward diffusion (Ld =Re > 1), the 45
line in the very low frequency range stretches out in an opposite direction. What are the physics behind the low-frequency 45
line? We first rule out the possibility that it is an essential property of the special boundary condition (kbc ¼ 10), as the low-frequency 45
line is absent in Fig. 2 for bounded diffusion with the same boundary condition. Fig. 4 (b) and (d) implies that 0 < kbc < ∞ is essential to the presence of low-frequency 45
line. Moreover, Fig. 4 (a) indicates that 0 < Ld =Re < ∞ is essential to the presence of lowfrequency 45
line. As a result, it is concluded the low-frequency

176

J. Huang / Electrochimica Acta 281 (2018) 170e188

Fig. 4. Diffusion impedance in the cylindrical symmetry: (a) the central diffusion (Ld ≪Re ) and the infinite outward diffusion (Ld [Re ); and the effect of Ld =Re in the range between 0.1 and 2 with (b) kbc ¼ 0 (a totally adsorbing boundary), (c) kbc ¼ 10, and (d) kbc ¼ ∞ (a totally reflecting boundary). Other parameters are n ¼ 1; ci ¼ 1M; Di ¼ 108 m2 s1 ; Re ¼ 1 cm. The frequency range is 1  105 Hz for (a), (b) (d) and 1  106 Hz for (c).

45
line stems from the convoluted interplay between the special boundary condition (0 < kbc < ∞) and the special geometry (cylindrical and 0 < Ld =Re < ∞). However, the mathematical complex in Eq. (36) makes it extremely difficult to obtain a simple interpretation. In the spherical symmetry, the diffusion equation in Eq. (6) is written as,

conditions in Eqs. (33) and (34). The diffusion impedance of species i is written as,

Zd;i ¼

ls;i

RT 2

ðnFÞ ci

pffiffiffiffiffiðjuÞ0:5 ; Di

(45)

where the coefficient ls;i , describing effects of the spherical geometry, the boundary condition and the diffusion direction, is given by,



ls;i

   ð1  kbc Þ lLd þ 1 exp RelLd þ ð1 þ kbc Þ lLd  1 exp RelLd d;i d;i d;i d;i    ;   ¼ ± l l ð1  kbc Þ lLd þ 1 1  Rd;ie exp RelLd  ð1 þ kbc Þ lLd  1 1 þ Rd;ie exp RelLd d;i

ju~ci ¼ Di

! v2 ~ci 2 v~ci : þ r vr vr 2

d;i

d;i

d;i

(43)

where ‘ þ ’ is for inward diffusion (Re > Ld ) and ‘  ’ is for outward diffusion (Re < Ld ). Eq. (46) is simplified to,

By introducing the dimensionless variable, r nd ¼ r=ld;i with ld;i ¼ pffiffiffiffiffiffiffiffiffiffiffiffi Di =ju, Eq. (30) has the solution,

  f f ~ci ¼ 1 exp r nd þ 2 exp r nd : nd nd r r

(46)

(44)

In a similar manner as in the cylindrical symmetry case, the two coefficients f1 and f2 in Eq. (44) are solved by invoking boundary

ls;i ¼ ±

 þ tanh RelLd  d;i  ; l  Rd;ie tanh RelLd  RLde  1 Ld ld;i

Ld ld;i

for kbc ¼ 0, and,

d;i

(47)

J. Huang / Electrochimica Acta 281 (2018) 170e188

Zd;i ¼

RT

Re ; c ðnFÞ i Di 2

177

(53)

for Ld ¼ ∞, indicating that the diffusion impedance exhibits a capacitive phenomenon for Ld ¼ 0 (central diffusion) and a resistive phenomenon for Ld /∞ (infinite diffusion) in the low frequency range, as shown in Fig. 5. By comparing Fig. 4 (a) and Fig. 5, we note a key difference between the cylindrical and spherical symmetry for the case of infinite outward diffusion in the low-frequency range, that is, a plateau with a constant imaginary part is found in the cylindrical case, while a converged semicircle with a zero imaginary part is found in the spherical case. 3.3. Porous electrodes

Fig. 5. Diffusion impedance in the spherical symmetry for the case of central (Ld ≪Re ) and infinite outward diffusion (Ld [Re ). Other parameters are n ¼ 1; ci ¼ 1M; Di ¼ 108 m2 s1 ; Re ¼ 1 cm; the frequency range is 1  105 Hz.

tanh



Re Ld ld;i



;  l 1  Rd;ie tanh RelLd

ls;i ¼ ±

(48)

d;i

for kbc ¼ ∞. As in the cylindrical symmetry, two limiting cases, namely Ld ¼ 0 (central diffusion) and Ld /∞ (infinite diffusion), are discussed below. From Eqs. (47) and (48), we obtain,

ls;i

 tanh lRe d;i  ; ¼ ld;i 1  Re tanh lRe

(49)

d;i

Battery electrodes (100 um) are usually porous and made up of microscopic active particles ( 1 mm), which can be pictured as cylindrical or spherical particles, however, little is known about the effect of the porous structure on the diffusion impedance. In this scenario, we are encountered with a two-scale problem: diffusion inside active particles at the microscopic scale (10 nm  1 mm) and ion transport across porous networks at the macroscopic scale (10 mm  1 mm), as depicted in Fig. 6. In the literature, the Warburg model has been widely used to extract the diffusion coefficient of active particles from the impedance data of the porous electrode [4e7,12]. Such a practice is not well justified. On one hand, diffusion in active particles, viz., finite-length, non-planar diffusion, does not meet the assumptions of Warburg impedance. On the other hand, the diffusion impedance measured at the macroscopic electrode scale is affected by the structural and transport properties of the porous electrode, which shall be, therefore, different from the diffusion impedance of the microscopic active particles. In this section, I treat a rudimentary example illustrating the interference effect caused by the porous structure. In spirit of the De Leive model [76], I make following assumptions: (1) the electrical conductivity of the electrical phase is infinite, therefore, the potential of the electrical phase is constant across the porous electrode; (2) the concentration gradient in the electrolyte phase is neglected, that is, ions transport via migration in the electrolyte phase; (3) the porous electrode is simplified as a one-dimensional

for Ld ¼ 0, and,

ls;i ¼

1 l

1 þ Rd;ie

;

(50)

for Ld ¼ ∞. In the high frequency range, ld;i ≪Re , therefore, Zd;i is approximated to,

Zd;i ¼

RT ðnFÞ2 ci

1 pffiffiffiffiffiðjuÞ0:5 ; Di

(51)

for Ld ¼ 0 and Ld ¼ ∞, indicating that the Warburg impedance phenomenon can be observed in the high frequency range for both the infinite and central diffusion, as shown in Fig. 5. In the low frequency range, ld;i [Re , therefore, Zd;i is approximated as,

Zd;i

RT

Re ¼ 2 cD ðnFÞ i i

for Ld ¼ 0 and

! 1 3Di j 2 ; 5 uRe

(52) Fig. 6. Two-scale diffusion in porous electrodes. Note that microscopic active particles can be in any shape (As a representative case, they are shown in cylinders).

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J. Huang / Electrochimica Acta 281 (2018) 170e188

cylindrical pore. Later, I will discuss in detail the convoluted ‘crosstalk’ between the microscale and the macroscale using a more involved model with all above assumptions released [58]. The ion migration in the electrolyte phase is described as,

  ~ v vf ke e ¼ av jmicro ; vx vx

(54)

where ke is the ionic conductivity in the electrolyte phase, fe the potential in the electrolyte phase, av the electrochemical surface area per volume in the porous electrode, jmicro the current density generated by reactions and transport at the microscopic scale and poured into the macroscopic scale as a source term, which is given by,

~ ~ ~jmicro ¼ fs  fe ; micro Z

(55)

~ s ¼ 0 according to the assumption of an infinite electrical with f conductivity and Z micro being the microscopic impedance. The boundary conditions to close Eq. (54) are: (i) an impene~ e =vx ¼ 0, and (ii) an applied current density trable wall at x ¼ 0, vf ~ e =vx ¼ japp . Solving Eq. (54) gives the distribution at x ¼ lp ,  ke vf ~ e , expressed as, of f

qffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi av micro cosh Z ke Z micro x ~ e ¼ japp : qffiffiffiffiffiffiffiffiffiffiffiffi f av av ke sinh k Z micro lp

(56)

e

The impedance of the porous electrode is defined as,


 ~ e x ¼ lp ~s  f f Zp ¼ ; japp

(57)

which is given by,

0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sffiffiffiffiffiffiffiffiffiffiffiffiffi av l2p Z micro A: Zp ¼ coth@ av ke ke Z micro

(58)

Eq. (58) clearly demonstrates how the structure (av ; lp ) and transport properties (ke ) of the porous electrode affect the relation between the intrinsic material property (Z micro ) at the microscopic scale and the observable electrode property (Zp ) at the macroscopic scale. Eq. (58) is asymptotic to,

sffiffiffiffiffiffiffiffiffiffiffiffiffi Z micro Zp ¼ ; av ke

(59)

when the modulus of Z micro is very small (usually obtained in the high frequency range), and to,

Zp ¼

lp Z micro þ ; 3ke av lp

(60)

when the modulus of Z micro is very large (usually obtained in the low frequency range). As aforementioned, we have known that Zd;i has a small magnitude and shows a 45
line in the high frequency range, regardless of the geometry and boundary conditions. If we take Z micro ¼ Zd;i , Eq. (59) indicates that the 45
line in the high frequency range will be transformed to a 22.5
line in the EIS measured at the porous electrode scale. However, the 22.5
line is

difficult to observe in real battery electrodes. The reason is that Z micro in the high frequency range is usually not dominated by the diffusion impedance, but by the interfacial impedance. Z micro is usually dominated by the diffusion impedance in the low frequency range, that is, Z micro zZd;i . Eq. (60) indicates that Z micro at the microscopic scale should be multiplied by a factor of 1/ av lp to obtain Zp at the macroscopic scale. That is to say, if the impedance expression derived for the microscopic active materials is employed to fit the EIS data of a porous electrode, one must multiply the obtained diffusion coefficient by a factor of av lp to compensate the interference effect of the porous electrode. As discussed in the literature, the situation becomes more complicated when ionic diffusion in the electrolyte is accounted for [57,58]. In this scenario, the co-existence and competition between the solid phase diffusion and the electrolyte phase diffusion generate new features of the diffusion impedance [57,58]. I will give an example in the penultimate section. 4. Extension in physics 4.1. Transport in electrolytic solutions An electrolytic solution contains cations, anions and solvents. Ion transport in electrolytic solutions may involve diffusion, migration and convection. However, the Warburg impedance accounts for pure diffusion only. The limitations of Warburg impedance in describing diffusion in electrolytic solutions have been acknowledged by many researchers, and new models have been presented [38,42,43,58,68,74]. The novelty is that an expression connecting the diffusion coefficient measured with that of constituent ions is derived, Eq. (89). In this section, we are to derive the diffusion impedance of ions in an electrolytic solution. In so doing, we make following assumptions: (i) the electrolyte is binary and symmetric; therefore, the ionic concentration of cations and anions are equal, cþ ¼ c ¼ ce , under the premise of the electroneutrality condition; (ii) the convection effect is neglected here but will be discussed in the next section; (iii) ion transport takes place in an integral dimensional space; (iv) there is no reactions in the bulk electrolyte. We begin with writing the electrochemical potential of species i in the electrolyte phase which is expressed as,

mi ¼ m0i þ kB Tlnðai Þ þ zi efe ;

(61)

where m0i is the standard chemical potential, kB the Boltzmann constant, T the temperature, zi the charge number, e the elementary electron charge, fe the potential in the electrolyte phase, ai the absolute chemical activity,

ai ¼ gi ci ;

(62)

with ci being the concentration and gi the activity coefficient. The transport of species i is driven by the gradient of mi . Assuming that the electrochemical system is close to equilibrium, the flux of species i is written as,

Ni ¼ Mi ci Vmi ;

(63)

where Mi is the mobility. Substituting Eq. (61) into Eq. (63) gives,

Ni ¼ Dchem Vci  i

zi eDi ci Vfe ; kB T

(64)

J. Huang / Electrochimica Acta 281 (2018) 170e188

where

 Dchem i

¼ Di

tþ ¼

 vln gi 1þ ; vln ci

(65)

zþ eDþ ce Vfe ; kB T

(66)

N ¼ Dchem Vce  

z eD ce Vfe ; kB T

(67)

Je ¼ zþ FNþ þ z FN ;

(68)

which is expanded to,

 z2þ FeDþ ce z2 FeD ce þ Vfe  zþ Dchem þ kB T kB T FVce ; þ z Dchem 

Je ¼ 

(69)

by substituting Eqs. (66) and (66) and (67) into Eq. (68). Let us define the observed conductivity of the electrolytic solution as,

z2þ FeDþ ce z2 FeD ce þ : kB T kB T

(70)

Then, we have the general relation,

 FVce ; Je ¼ se Vfe  zþ Dchem þ z Dchem þ 

(71)

(77)

Given the definitions in Eqs. (75)e(77), Eq. (74) is transformed to,

Vce ¼

t Nþ  tþ N : Damb

Je ¼ se Vfe ;

(72)

when zþ Dchem þ z Dchem ¼ 0, namely, Dchem ¼ Dchem because we þ  þ  have zþ ¼ z for a binary z : z electrolyte. Furthermore, we can define a diffusivity difference between cations and anions of the electrolyte,

(73)

Vce can be expressed as a function of Nþ and N by eliminating Vf via algebraic calculations of Eqs. (66) and (67),



z D Dchem þ

tþ Je : z F

vcþ ¼ V$Nþ  Rþ ; vt

  vce tþ Je ¼ V$ðDamb Vce Þ þ V$ ; vt z F

(78)

:

(74)

An ambipolar diffusivity Damb of a binary z : z electrolyte is defined as,

zþ Dþ Dchem  z D Dchem  þ : zþ Dþ  z D

The cation transference number is given as,

(79)

(80)

(75)

(81)

by assuming Rþ ¼ 0. Eq. (81) describes ion transport in an electrolyte solution with coupled diffusion and migration in an integral dimensional space. Compared with Eq. (3) used in the Warburg impedance, Eq. (81) shows two differences: (i) a new term describing migration, the second term on the rhs, comes into play; (ii) the diffusion coefficient is the ambipolar diffusivity Damb, as expressed in Eq. (75), which embodies the combined effects of cations and anions. Usually, tþ depends on the ionic concentration. If we take, however, tþ as a constant and invoke the charge conservation law, V$Je ¼ 0, Eq. (81) is reduced to,

(82)

Eqs. (82) and (3) share the same form; they differs only in the diffusion coefficient. However, the boundary conditions are largely different. As in usual case, we assume that only cations are involved in the following reaction occurring at the electrode-electrolyte interface,

Mzþ þ ne4M;

Dchem ¼ zþ Dchem þ z Dchem : ± þ 

z D Nþ  zþ Dþ N

Nþ ¼ Damb Vce 

vce ¼ V$ðDamb Vce Þ: vt

and the specific relation,

Damb ¼

z D : zþ Dþ  z D

where Rþ is the consumption rate of cations in the homogeneous reactions occurring in the electrolyte, gives,

!

zþ Dþ Dchem 

t ¼

Substituting Eq. (79) into the continuity equation for cations, that is,

for anions, respectively. The current density in the electrolyte phase is defined as,

Vce ¼

(76)

Combining Eqs. (68) and (78) leads to,

for cations and,

se ¼

zþ Dþ ; zþ Dþ  z D

and the anion transference number is given as,

is the chemical diffusivity considering the concentrationdependent activity coefficient in concentrated electrolytes, and Di ¼ Mi kB T is the dilute-solution limit diffusivity. Applying Eq. (64) as well as the relation cþ ¼ c ¼ ce for cations and anions leads to,

Nþ ¼ Dchem Vce  þ

179

(83)

with n ¼ zþ . At the electrode-electrolyte interface, the faradaic current density is carried by cations only,

~i zþ eDþ ce ¼ Dchem Vfe ; Vce  þ nF kB T

(84)

while the contribution of anions is zero, namely the flux of anions is zero,

Dchem Vce  

z eD ce Vfe ¼ 0: kB T

(85)

Some mathematical rearrangements assemble Eqs. (84) and

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J. Huang / Electrochimica Acta 281 (2018) 170e188

Accordingly, Eq. (81) is modified to [56],

(85) into,





~i Dþ ¼ Damb 1 þ Vce : nF D

(86)

In the classical Warburg formalism, the diffusion coefficient in the controlling equation (Eq. (6)) and that in the boundary condition (Eq. (8)) are identical. In terms of an electrolytic solution, however, different diffusion coefficients are used in the controlling equation (Eq. (82)) and the boundary condition (Eq. (86)). As a result, the diffusion impedance should be modified accordingly. As the inter-electrode distance is much larger than the characteristic diffusion length, diffusion in electrolyte cells is well described by the semi-infinite planar diffusion, with the new diffusion impedance given by,

Zd ¼



RT ðnFÞ

2

1 1 v gþ þ ce gþ vcþ



1  pffiffiffiffiffiffiffiffiffiffiffiðjuÞ0:5 ; Dþ 1 þ D Damb

(87)

which is reduced to,

Zd ¼

RT ðnFÞ2 ce

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiðjuÞ0:5 ; Dchem þ

(88)

when D [Dþ and gþ ¼ 1. As a result, the diffusion coefficient extracted from the impedance response of ion transport in an electrolytic solution is given by,

Dmea

  Dþ 2 ¼ Damb 1 þ : D

(89)

Eq. (89) is degenerated to Dmea zDchem only when D [Dþ . In þ other words, if one wants to measure the diffusion coefficient of ‘participating ions’ that are involved in electrochemical reactions, cations in this case, it is desirable that ‘spectating ions’ that are not involved in the electrochemical reactions, anions in this case, have a much higher diffusion coefficient. Otherwise, what is extracted from the impedance data is a composite diffusion coefficient due to the coupling between cations and anions. In addition, the relation between the diffusion coefficient and the ionic conductivity in Eq. (70) shows that they are closely related with each other. Therefore, a word of caution should be added if one makes assumptions on diffusion and migration. It is usually assumed that the migration effects are neglected, that is, the ionic conductivity is infinite or the potential gradient is zero, when treating the diffusion impedance. Eq. (70) indicates that such an assumption can be made when the ionic concentration ce is sufficiently large (e.g. in the presence of supporting electrolyte). Above treatment is limited to a bulk electrolyte. Now we extend our discussion to a porous electrode immersed in a binary symmetric electrolyte. In this case, new factors at play include: (i) electrochemical reactions at the electrode-electrolyte interface; (ii) porous structure parameters such as the volume fraction of the electrolyte phase εe . By using the volume averaging method, Eq. (71) is transformed to [56],

 chem;eff V$ seff Vce ¼ av jmicro ; e Vfe  FD±

(90)

which means that the charge conservation in the electrolyte phase is broken as there is net charge poured into the electrolyte phase due to electrochemical reactions at the electrode-electrolyte interface. Moreover, effective transport properties should be used to account for the porous structure.

a jmicro  vεe ce v ¼ V$ Deff ð1  tþ Þ: Vce þ amb vt zþ F

(91)

As jmicro depends on the reactant concentration ce and the potential fe , Eqs. (90) and (91) are strongly coupled and hence should be solved together. It is readily seen that ionic transport in the electrolyte contained in a porous electrode cannot be described by the simple equation in Eq. (3). Therefore, it is, generally, erroneous to use the Warburg impedance to fit the impedance data of a porous electrode. We will come to this issue in detail in the section of “implications”, based on the theoretical framework for impedance response of porous electrodes developed by the present author together with Jianbo Zhang [58]. 4.2. Convection effect Hydrodynamic methods, such as the rotating disc electrode (RDE), are routine practices to control mass transport in electrochemical experiments. In such devices, the convective diffusion comes into play. Regarding a symmetrical binary electrolyte, the general form of the convection-diffusion equations is given by,

vce ¼ V$ðDamb Vce Þ  v$Vce ; vt

(92)

in which the migration term in Eq. (81) is neglected by invoking V $Je ¼ 0 according to the charge conservation law, v is the velocity vector. A key assumption says that radial convection and diffusion are negligible [44]. In other words, we treat the diffusion-convection impedance under the assumption of uniform accessibility of the RDE. As a result, we only consider the problem in the direction normal to the RDE surface,

vce vce v2 ce þ vz ¼ Damb 2 ; vt vz vz

(93)

where vz can be described by the Cochran series of von Kaman equations,

vz ¼

!  3   pffiffiffiffiffiffi aU 1 U 2 3 b U 2 4 vU  z2 þ z  z þ/ ; 3 v 6 v v

(94)

with a ¼ 0:51023; b ¼ 0:61592, v being the kinematic viscosity, U the rotation speed of the disk. By applying Fourier transformation and introducing the dimensionless distance,

z

znd ¼

dcd

;

(95)

with the characteristic length dcd being,



dcd ¼

3Damb av

1=3 rffiffiffiffi v ;

U

(96)

and the dimensionless frequency,

und ¼



u 9v U a2 Damb

1=3 ;

(97)

nd the dimensionless perturbation in ce , cee ¼ cee =cref e , and the Schmidt number, Sc ¼ v=D, Eq. (93) becomes,

J. Huang / Electrochimica Acta 281 (2018) 170e188

nd v2 cee 2 nd vz

 jund cee

nd

2

þ 3 znd 



1

1 9a4 Sc

3

! 3

znd

nd

vcee ¼ 0: vznd

(98)

The boundary conditions to close Eq. (98) are, nd cee ¼ 0;

(99)

in the bulk solution, znd ¼ ∞, and, nd cee ¼ 1;

(100)

at the electrode/electrolyte interface, znd ¼ 0. By applying Eq. (100), we mean that the system is perturbed with a sinusoidal ionic concentration at the electrode/electrolyte interface. On the contrary, we have assumed that the system is perturbed with a sinusoidal current density, as expressed in Eq. (5), in foregoing cases. However, we emphasize that the impedance response is an intrinsic property of the system and it should be independent on the form how the perturbation is applied as long as it is in the linear regime. Herein, Eq. (100) is used for the sake of simplicity. According to the Nernst equation in Eq. (13), the perturbation in the electrode potential aroused by the perturbation in the surface ~ ¼ RT=nFce nd (note that cee nd ¼ 1 ionic concentration is written as, E d

at znd ¼ 0 and ce nd is the dimensionless ionic concentration at steady state. Here, we focus on the oxidant and assume the activity is unity). The perturbation in the current density corresponding to   þ diffusion is given by, jed ¼ nFDamb 1 þ D D Vce ð0Þ according to Eqs. (84) and (85). As a result, the convective diffusion impedance is defined as,

Zcd

11 0 1 rffiffiffiffi  nd 3 RT 3Damb v Bvcee C  ¼ A ; @ þ av U vznd n2 F 2 ce Damb 1 þ D D nd

where vcee =vznd is calculated at znd ¼ 0. In the case of neutral species (e.g. oxygen) diffusion, Eq. (101) is modified by replacing Damb with Di (the diffusion coefficient of the neutral species) and dropping the term, Dþ =D . Unfortunately, Eq. (98) is not tractable analytically. In consequence, approximations have to be made to obtain analytical results. Instead of solving Eq. (98) directly, the first approximation of the convective diffusion impedance is calculated in the context of bounded diffusion. The basic idea is to transform the semi-infinite convective-diffusion problem to a bounded diffusion problem. The so-called stagnant Nernst diffusion layer has a finite thickness of Leff , d eff

Ld

which is expressed as,

  4 ¼G d ; 3 cd

approximation that is valid for large perturbation frequency, und / ∞ [44]. One year later, Levart and Schuhmann published another improved approximation for und /∞ [45]. The first analytical approximation for a wide range of low and high frequencies was first developed by Scherson and Newman [46]. Their results are,

nd

vcee

 nd

vz

(102)

with G being the Euler gamma function. In this scenario, the convective diffusion impedance can be readily calculated using the expressions derived in the section of ‘Bounded diffusion’ by replacing Ld with Leff given in Eq. (102). d Next, we present several approximations for the solution of Eq. (98), including those in the literature and a newly derived one as well. Firstly, Eq. (98) is simplified by dropping the second term in the bracket, which is based on the assumption, Sc/∞. With this simplification, in 1974, Homsy and Newman proposed an

znd ¼0

0:5  3j 9  nd 2:5 ju ¼ jund  þ ; 4und 32

(103)

for high frequencies, and

nd

vcee

 nd

vz

X 1 ¼
4 þ jund

G

znd ¼0

3

n¼0

Bn ; jund þ ln

(104)

for low frequencies, with Bn and ln being coefficients and eigenvalues of the Sturm-Liouville system, which has been tabulated by Nisancioglu and Newman [46]. Note that Eqs. (103) and (104) are based on the simplification of dropping the second term in the bracket, viz. an infinite Schmidt number. In addition, Eqs. (103) and (104) are valid for high and low frequencies respectively, while the intermediate frequencies are not covered. In what follows, we will present an analytical solution to Eq. (98) in the whole frequency range. Our approach is based on the homotopy perturbation method, which is developed by two Chinese mathematicians Shijun Liaw [77] and Jihuan He [78] and has been introduced by Jansi Rani et al. to treat the convective diffusion problem recently [54]. However, Jansi Rani et al. did not discuss the diffusion-convection impedance [54]. We begin by constructing a homotopy as follows,

0 ð1 

(101)

181

1

2 nd Bv cee pÞ@ nd2 vz

0

 2 2 nd nd C nd Bv cee  jund cee A þ p@ nd2  jund cee þ 3 znd vz

1 nd e 3 vce C  kznd A vznd ¼ 0: (105)  with k ¼

1 1 9a4 Sc

3

and p2½0; 1 is an imbedding parameter. Eq. (105)

is reduced back to Eq. (98) when p ¼ 1. The solution of Eq. (105) is written as, nd cee ¼ c0 þ pc1 þ p2 c2 þ /:

(106)

which is, therefore, the solution of Eq. (98) when p ¼ 1. Substituting Eq. (106) into Eq. (105) leads to,

þ p2

!

vc  2 3 v2 c1 0  jund c1 þ 3 znd  kznd 2 nd vz vznd ! vc  2 v2 c2 1 nd nd nd3  ju c2 þ 3 z  kz þ/ 2 vznd vznd

v2 c0  jund c0 2 vznd

!

þp

¼ 0: (107) Eq. (107) is satisfied over the range of p2½0; 1. Therefore, we have,

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J. Huang / Electrochimica Acta 281 (2018) 170e188

v2 c0  jund c0 ¼ 0: 2 vznd

(108)

vc  2 3 v2 c1 0  jund c1 þ 3 znd  kznd ¼0 2 nd vz vznd

(109)

vc  2 3 v2 c2 1  jund c2 þ 3 znd  kznd ¼0 2 nd vz vznd

(110)

The boundary conditions are,

  c0 znd ¼ 0 ¼ 1; c0 znd ¼ ∞ ¼ 0

(111)

  c1 znd ¼ 0 ¼ 1; c1 znd ¼ ∞ ¼ 0

(112)

  c2 znd ¼ 0 ¼ 1; c2 znd ¼ ∞ ¼ 0

(113)

Solving Eq. (108) closed by Eq. (111) gives,

  qffiffiffiffiffiffiffiffiffi c0 ¼ exp  jund znd ;

(114)

Substituting Eq. (114) into Eq. (109), we obtain,

 qffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffi 2 3 v2 c1 nd nd znd  kznd exp  jund znd ¼ 0  j u c  3 j u 1 2 vznd (115) The solution to Eq. (115) is written as,

 pffiffiffiffiffiffiffiffiffi 3 1  3 exp  jund znd 3k 4 3k nd nd 2 z j znd c1 ¼ þ u  2
8 4 2 jund   qffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffi 3 nd 9k ju  jund znd znd jund þ 1  : 4 8

(116)

This iteration can go on. However, the expression of ci becomes more and more tedious. Hence, the expression of c2 is omitted here. Given the expressions of c0 , c1 and c2 , we obtain,

nd

vcee

 nd

vz

znd ¼0

1  3j 9k nd 32 9  nd 52 2 ju ju ¼ jund    nd 8 32 4u þ

27k nd 3 81k2  nd 72 ju ju  : 32 124 (117)

Then, the convective diffusion impedance can be calculated according to Eq. (101). It is noted that the new expression, Eq. (117), is akin to Eq. (103). However, the new expression includes terms involving k, therefore, the new expression is believed to be applicable for cases with a finite k, viz., finite Schmidt number. Fig. 7 presents the simulated diffusion impedance in presence of convection flow. Three approximations are compared, including the stagnant Nernst diffusion layer approximation in Eq. (102), the piece-wise approximation by Newman et al. in Eqs.(103)e(104), and the new approximation accounting for finite k effect in Eq. (117). In addition, numerical simulation by solving Eq. (98), labeled as the dashed line, is given as the reference. In the high frequency range, three approximations all exhibit a 45
line. Note that the high-frequency wing of the Newman approximation is overlapped with the curve corresponding to Eq. (117). In the low-frequency range, a semicircle, a feature of bounded diffusion with a totally adsorbing boundary condition, manifests. The Newman approximation in Eq. (104) almost overlaps with the stagnant Nernst diffusion layer approximation. The new approximation in Eq. (117) embodies the high-frequency 45
line and the low-frequency semicircle using a single simple expression. However, Eq. (117) gives rise to an inductive loop at extremely low frequencies. This artificial effect is attributed to the truncation of the iteration. We find that the stagnant Nernst diffusion layer approximation in Eq. (102) is usually a good approximation (the error is within 10%) for the diffusion-convection impedance, when the Schmidt number Sc is larger than 100. In the low n range (Sc<100), the new approximation based on the homotopy perturbation method improves over the Nernst diffusion layer approximation in the high frequency range (results not shown). However, the agreement in the low frequency range is poor due to the artificial inductive loop, an issue remaining to be resolved in future studies (e.g, by increasing the iteration number when solving Eq. (107)). The most prominent advantage of the new method is its capability to treat the problem with an arbitrary Schmidt number in the full frequency range. 4.3. Double layer effect

Fig. 7. Diffusion impedance in presence of convection flow. Three approximations are shown, including the stagnant Nernst diffusion layer approximation, the piece-wise approximation by Newman et al., and the new approximation accounting for finite k effect. Numerical simulation is labeled as the dashed line. Note that the high-frequency wing of the Newman approximation shows a 45
line, which is overlapped with the curve corresponding to Eq. (117). Other parameters are n ¼ 1; ci ¼ 0:01 M; Damb ¼ 1010 m2 s1 ; Dþ ≪D ; n ¼ 102 cm2 s1 ; U ¼ 100 rad; the 5 3 frequency range is 10  10 Hz.

In preceding models, the electroneutrality condition ensures that the ionic concentration of cations and ions of a symmetrical binary electrolyte are equal everywhere, that is, cþ ¼ c ¼ ce . Such a treatment is applicable in the bulk phase, while it is violated in the vicinity of an electrified interface. In the interfacial region, the surface charge on the electrode induces a very large potential gradient, up to 109 Vm1 , which breaks down the electroneutrality condition. As a result, counter-ions are accumulated, while co-ions

J. Huang / Electrochimica Acta 281 (2018) 170e188

are repelled, in the interfacial region. The electroneutrality assumption can be released by invoking the Poisson equation,

ε0 εe V$Vfe ¼ zFðcþ  c Þ;

(118)

with ε0 being the vacuum permittivity, εe the relative dielectric constant of the electrolyte phase. A new boundary condition relating the potential distribution in the electrolyte phase with the surface charge density on the electrode surface says,

! ε0 εe ð n $Vfe Þ ¼ ss ;

4.4. Non-Fickian diffusion and fractality The original Warburg impedance predicts a 45
line in the Nyquist plot. The observation of the deviation from the 45
line compels researchers to study Non-Fickian diffusion [36,62,65,67]. One approach is to generalize the continuity equation in the following way,

vg ci vN ¼  i: vt g vx

(121)

As a result, the pure diffusion equation in Eq. (3) is generalized

(119)

! where n is the normal vector pointing towards the bulk solution and ss is the surface charge density on the electrode surface. Based on Eq. (64), the continuity equation of cations (i ¼ þ ) and anions (i ¼  ) can be written as,

  vci z FD c ¼ V$ Dchem Vci þ i i i Vfe : i vt RT

183

(120)

Combining Eqs. (118) and (120) gives the controlling equation set describing ion transport in an electrolytic solution accounting for the double layer effect, namely the Poisson-Nernst-Plank (PNP) equation set. Impedance of electrolytic cells governed by the PNP equation set has been studied by Macdonald in last seventies [79] and there is very recent discussion on it [80].

to,

vg ci v2 ci ¼ D ; i vt g vx2

(122)

which is transformed to the following expression in the frequencydomain,

ðjuÞg ~ci ¼ Di

v2 ~ci : vx2

(123)

It is readily seen from Eq. (123) that the frequency-dependent pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi characteristic diffusion length is now written as, ld;i ¼ Di =ðjuÞg : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi By substituting the original expression of ld;i ¼ Di =ðjuÞ with this new expression into Eq. (12), the generalized Warburg impedance is now written as,

2

Debye length, lD ¼ ε0 εe RT=ð2z2 F 2 ce Þ, is the characteristic length of double layer effects. lD z0:3 nm for a 1 M aqueous solution (εe ¼ 78:5). Electroneutrality is achieved beyond the diffuse layer, which extends ca. 5lD away from the electrode (usually within several nm). Therefore, diffusion impedance models under the promise of electroneutrality are applicable in most cases (exceptions include nanopores). If the diffusion region is so small that it is within the diffuse layer, the diffusion impedance would be then negligibly small in the usual frequency range (for example, see Eq. (27)). In other words, double layer effects on the diffusion impedance are usually trivial, while that on the interfacial impedance such as the charge transfer resistance are non-trivial in the spirit of the Frumkin correction. Employing the supporting electrolyte further justifies the electroneutrality assumption. A characteristic frequency of the double layer effects is the Debye frequency, uD ¼ Di =l2D . With a usual value for Di ¼ 106 m2 s1, we obtain

uD  1014 s1 , which is far beyond the usual frequency range of ½105 ; 103  Hz.

Zw ¼ sw ðjuÞg=2 :

(124)

Accordingly, diffusion impedance expressions for bounded cases and other geometries can be modified in a similar manner. Bisquert published a series of paper on the generalized diffusion impedance, which is termed anomalous diffusion [65,81]. Interested readers are directed to the original works of Bisquert and co-workers for more details and physical origins [82]. Fig. 8 shows the simulated impedance of non-Fickian diffusion. In this case, we concern more about the structure of the impedance than the magnitude of the structure. In consequence, I erase numbers in the labels. From Fig. 8 (a), the angle of the tilted line for the infinite-diffusion is correlated with the coefficient g, q ¼ pg=4. For the totally reflecting bounded diffusion, as shown in Fig. 8 (b), the low-frequency tilted line deviates from a vertical line, and it is readily seen that the angle follows, q ¼ pg=2. For the totally adsorbing bounded diffusion, the low-frequency semicircle is suppressed when g < 1.

Fig. 8. Impedance of non-Fickian diffusion: (a) semi-infinite; (b) bounded (black lines for the totally reflecting case, and dark blue lines for the totally adsorbing case). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

184

J. Huang / Electrochimica Acta 281 (2018) 170e188

here. The schematic illustration of aW , b and g can be found in Ref. [73]. In this formalism, the dimensionless diffusion equation in frequency domain is written as,

  nd V~ cnd juL2d nd V$ dosðn1Þ g; x i ~c ¼
 : Di i dosn dF ; xnd

(128)

We consider a rudimentary case where dF ¼ 3  dd; g ¼ 2  dd;

Fig. 9. Impedance of totally adsorbing bounded diffusion in a fractal with the dimensionality deficiency dd ¼ 0; 0:1; 0:2, respectively. Di ¼ 1010 m2 s1 and Ld ¼ 100 mm. The frequency range is over 100  106 Hz.

The introduction of non-Fickian diffusion brings about an issue of dimensionality. Balance of the dimensionality of two sides of Eqs.(121)e(123) gives Di ¼ ½m2 sg . A question naturally arises, how can Di obtained from the non-Fickian diffusion be transformed to the normal diffusion coefficient Di ¼ ½m2 s1 ? We may learn from the classical problem of transformation of the constantphase-element (CPE) coefficient to the capacitance. A possible 1=g 22=g Ld

transformation formulation is given by, Dtran ¼ Di i

with Ld

Dtran i

being the diffusion length. For this case, is a function of Ld . Electrodes in the real world can be fragmented, irregular and fractal. As a result, there has been continuous interests in studying diffusion in fractal electrodes. Diffusion toward/from a fractal diffusion has been investigated by a number of researchers, as reviewed in two book chapters by Pyun and co-workers [83,84]. In their approach, the generalized diffusion equation in Eq. (122) is used. Moreover, the exponential factor g is related with the fractal dimension of the electrode, dF , as,

g ¼ 3  dF :

(125)

In a different scenario, the present author considered diffusion through a fractal media towards/from a regular and smooth electrode surface [73]. In so doing, the present author generalized the porous electrode theory to non-integral dimensional space. The key point is to generalize the continuity equation to,

 V$ Ni dosðn1Þ ðg; xÞ dosðn1Þ ðb; xÞ vci ¼ a R; dosn ðdF ; xÞ i vt dosn ðdF ; xÞ

(126)

for objects with non-integral dimensions, where, i n Y p 2 ai 1
a xi ;

a

dosn ðaW ; xÞ ¼

i¼1

G

i

nd ¼ 0 and ~ nd ¼ 1. dd is defined as the V~cnd cnd i ¼ 1 at x i ¼ 0 at x dimensionality deficiency of the fractal. This case corresponds to the totally adsorbing bounded diffusion. Fig. 9 shows the diffusion impedance derived from Eq. (128) for different values of dd (only qualitative trend is shown). dd ¼ 0 corresponds to the regular diffusion, in which case the impedance structure consists of a 45
line in the high-frequency range and a semicircle in the lowfrequency range, which has been discussed before in the section of ‘Bounded diffusion’. When the object transforms from regular to fractal, that is dd goes larger than zero, the structure of diffusion impedance changes accordingly. The tilted line in the highfrequency range becomes steeper and the total diffusion impedance becomes larger when dd goes larger. It is interesting to compare the non-Fickian diffusion that is treated in the preceding section and diffusion in fractal considered in this section. In nonFickian diffusion, the angle of the tilted line in the high-frequency range is usually smaller than 45
when g < 1, while it is usually larger than 45
when dd > 0 in this case. This feature can be used to recognize the diffusion scenario.

(127)

2

describes how permitted states of elementary particles constituting P the fractal are packed in the space ℝn [85]. aW ¼ ni¼1 ai and ai ði ¼ 1; /; nÞ is the fractional dimensionality of the fractal in each Cartesian coordinate in ℝn ðn ¼ 1; 2; 3Þ. GðxÞ is the gamma function. b is the dimension of the electrode-electrolyte interface in the fractal. g is the dimension of the electrolyte phase in a 2D section of the fractal. Ri is the reaction rate occurring at the electrodeelectrolyte interface in the fractal electrodes, which is neglected

5. Implications 5.1. Diffusion in electroactive materials As discussed in section 2, the original Warburg impedance model is strictly applicable to semi-infinite, planar, purely concentration-gradient-driven, Fickian diffusion in the dilute limit. However, electroactive materials are usually in spherical shape. Therefore, at a minimal level of approximation, Eq. (49) derived for the inward spherical diffusion should be used in data interpretation. In addition, the concentration-activity relation is very important in concentrated cases, such as solid state particles [14e17]. Structural complexity of the investigated system constitutes another source of difficulties in determining the diffusion coefficient of electroactive materials using the EIS method. In most cases, a porous electrode consisting of electroactive material particles is employed in the EIS measurement. In this scenario, new interference factors come into play. On one hand, the porous electrode is usually impregnated with an electrolytic solution. Hence, diffusion of ions in the electrolytic solution may be coupled with diffusion of ions in the electroactive materials. In consequence, a vexing question arises: how to separate the diffusion impedance of the electroactive material particle from the total diffusion impedance? On the other hand, the porous electrode is composed of many active particles with distributions in the particle size and shape. Therefore, it is not straightforward to determine the diffusion coefficient of microscopic active particles from the impedance data measured on a macroscopic porous electrode; additional mathematical treatments are required. Compared with porous electrodes, the single particle technique is preferable for the determination of the diffusion coefficient of electroactive materials [4e6]. From the data of Dokko et al., the EIS of a single LiCoO2 particle exhibits a 45
degree line followed by a transition to a vertical line in the low-frequency range, which is

J. Huang / Electrochimica Acta 281 (2018) 170e188

well-described by the spherical diffusion impedance in Eq. (49), as shown in Fig. 5 [5]. It is worth noting that the so-called ‘single particle’ is actually an assembly of many primary particles at a smaller scale. Therefore, the ‘single particle’ should be treated as an agglomerate [56]. The charge transfer reaction takes place on both the agglomerate surface and the electrode-electrolyte interfaces inside the agglomerate. As a result, the practice of treating the ‘single particle’ as a solid particle with charge transfer reactions occurring only on its surface is misleading and constitutes a latent source of error. A more rigorous treatment is to develop a new model accounting for intra-agglomerate reactions and mass transfer, as in Ref. [56]. 5.2. Diffusion in electrolytic solutions We distinguish two scenarios of species diffusion in the electrolytic solution: neutral species such as gas molecules and charged species such as cations and anions. The diffusion of dissolved oxygen in lithium-oxygen batteries and RDE test systems for the evaluation of oxygen reduction reaction electrocatalysts belongs to the first type. The diffusion of Li-ions in the electrolyte belongs to the second type. Regarding the scenario of neutral species, the original Warburg impedance expression can be used directly when the diffusion region is sufficiently large. Otherwise, the expressions derived for bounded diffusion must be used, which is the case for the diffusion of dissolved oxygen in the cathode of lithium-oxygen batteries, and the diffusion of oxygen in the catalyst layer of polymer electrolyte fuel cells [86,87]. Special attention should be paid to the boundary condition when choosing the model, as the expression for the diffusion impedance can be very different for different boundary conditions. In RDE tests, convection is introduced to enhance mass transfer. In this case, the diffusion impedance model considering the convection effect should be used, see section 4.2. The scenario of charged species is much more complicated. On one hand, the concentration-gradient-driven diffusion is now coupled with the potential-gradient-driven migration. On the other

Fig. 10. Impedance response of a lithium-ion battery electrode simulated using the model developed in Ref. [54] with parameters therein. The diffusion coefficients of Liions in solid and electrolyte phase are adjusted in three cases: (Ds ;De ), (Ds ;105 De ), and (Ds ; 101 De ). The frequency range is 105  104 Hz.

185

hand, the transport of cations and anions are coupled. As a result, equations that are much more complicated than the simple Fick's law are involved in describing the mass transport in electrolytic solutions, see section 4.1. Regarding the classical semi-infinite planar diffusion case, a new diffusion impedance expression in Eq. (87) has been derived, in which three diffusion coefficients, namely the diffusion coefficient of cations, that of anions, and the ambipolar diffusion coefficient, are included. Hence, it is extremely difficult to obtain the diffusion coefficient of a certain species. The situation becomes more complicated when the double layer effect comes into the play, see section 4.3. 5.3. Diffusion in porous electrodes Porous electrode structure is a commonality shared by many electrochemical energy systems such as batteries, supercapacitors and fuel cells. Acquiring transport properties in porous electrodes is of fundamental importance to the diagnostic and optimization of electrochemical energy systems. However, it is extremely challenging to do that, mainly due to following two reasons. First, porous electrodes usually have a multi-scale structure. Second, porous electrodes usually involve multi-physics processes, such as charge transfer reactions at the electrode-electrolyte interface and species transport in the electrolyte and electrode phases. In terms of the diffusion impedance, a very challenging issue is that multiscale and multi-phase diffusion processes are coupled. It is, therefore, oversimplified to employ the original Warburg impedance model in interpretation of diffusion impedance measured on a porous electrode. In a recent publication, the present author together with J. Zhang have developed a theoretical framework for the impedance response of porous electrodes [58]. In this framework, the multiscale structure and the multi-phase, multi-physics processes are treated using the porous electrode theory. Herein, this framework is employed as a theoretical tool to elucidate the emerging complexity in the diffusion impedance. Fig. 10 shows the impedance response of a lithium-ion battery electrode simulated using the model developed in Ref. [58] with parameters therein. The diffusion coefficients of Li-ions in solid and electrolyte phase are adjusted in three cases: (Ds ; De ), (Ds ; 105 De ), and (Ds ; 101 De ). The impedance response is structured by three regimes, including a 45
line in the high frequency range corresponding to the ionic migration and electron transport in the porous structure, a semicircle in the medium frequency range corresponding to charge transfer reactions and double-layer charging on the electrode-electrolyte interface, and the diffusion regime in the low frequency range. In terms of the diffusion regime, three sub-regimes can be observed; there is a short tilted line, followed by an arced curve, which is further developed into a vertical line. This peculiar structure cannot be described by any elementary diffusion models developed previously. The complexity originates from the coupling of the solid phase diffusion, namely the diffusion of Li atoms in active particles ( 1 mm), and the electrolyte phase diffusion, namely the diffusion of Li-ions in the electrolytic solution ( 100 mm). The solid phase diffusion can be described by the inward spherical diffusion expressed in Eq. (49). Increasing the electrolyte phase diffusion coefficient De by a factor of 105 recovers the diffusion impedance to the simple inward spherical diffusion case, as shown in Fig. 10. Whereas, the electrolyte phase diffusion belongs to the case of bounded diffusion with a totally adsorbing boundary, see Fig. 2. In consequence, the arced curve is attributed to the electrolyte phase diffusion, which is further confirmed by the enlarged arced curve when De is decreased by a factor of 10.

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J. Huang / Electrochimica Acta 281 (2018) 170e188

In the literature, it is a common practice to estimate the diffusion coefficient of active particles based on the diffusion impedance measured on a porous electrode. A tacit assumption in such a practice is that the diffusion impedance of the porous electrode is dominated by the diffusion in active particles. However, this assumption is usually unjustified. As exemplified by Fig. 10, the diffusion impedance in low frequency range can be dominated by the electrolyte phase diffusion, especially when the porous electrode is very thick. This caveat has been discussed in previous studies [56e58]. As a remedy approach, one may greatly reduce the thickness of the porous electrode to diminish the interference effect of the electrolyte phase diffusion. Otherwise, delicate models accounting for both solid phase and electrolyte phase diffusion should be employed. A frequently invoked justification for neglecting the electrolyte phase diffusion is that De is usually several orders higher than Ds . However, it is stressed that the diffusion coefficient is not an appropriate descriptor of a diffusion process in a certain system. Instead, we should use the dimensionless characteristic time defined as, tind ¼ 2puL2d;i =Di . In consequence, the diffusion length Ld;i is equally important in judging whether or not we can neglect a diffusion process at a given frequency. The electrolyte phase occurs at the electrode scale, which is usually two-to-three orders larger than the particle size at which scale the solid phase diffusion takes place, that is, Ld;e z102  103 Ld;s . In this scenario, tend ztsnd although De z104  106 Ds . A diffusion process can be neglected when tind ≪1. Therefore, the electrolyte phase diffusion can be neglected when the electrode thickness is reduced to be much smaller than pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi De =2pu.

6. Conclusions We have revisited the classical Warburg impedance, scrutinized its assumptions, and made a series of extensions to tackle diffusion in more complicated and realistic scenarios. The main results, findings and implications from this theoretical analysis are recapitulated below. The original Warburg impedance model is strictly applicable to semi-infinite, planar, purely concentration-gradient-driven, Fickian diffusion of neutral species in the dilute limit. The original Warburg impedance model is physically ill-posed when treating charged species, because the charge conservation law is violated (in other words, the current density becomes position-dependent). Other physical processes such as migration must be added into the picture, in order to fulfil the charge conservation law. The finite-length diffusion, also termed bounded diffusion, brings the boundary condition as a key player in determining the impedance response. Impedance expressions of the bounded diffusion with a general boundary condition have been derived. The effect of the boundary condition can be ‘sensed’ only in the lowfrequency range where the characteristic diffusion length is comparable with or larger than the length of the diffusion region. Diffusion impedance models in cylindrical and spherical geometries with a general boundary condition have been developed, which are useful to describe diffusion in battery materials. Moreover, the shape effect is also not ‘sensed’ in the high frequency range. Combined, the classical Warburg impedance can be applicable is high frequency range (the frequency range in which a 45
line can be observed) where the finite length effect, the boundary condition effect and the shape effect are nearly absent. Unfortunately, the 45
line is very short or even absent in many scenarios, hence, there may be not enough data points to fit the Warburg impedance with high confidence.

In electrolytic solutions, the concentration-gradient-driven diffusion is strongly coupled with the potential-gradient-driven migration. Moreover, the transport of cations and anions are coupled. Equations describing the mass transport in electrolytic solutions have been derived from simple fundamental laws. Regarding the classical semi-infinite planar diffusion case, a new diffusion impedance expression has been derived, in which three diffusion coefficients, namely the diffusion coefficient of cations, that of anions, and the ambipolar diffusion coefficient, are present. To measure the diffusion coefficient of ‘participating ions’ that are involved in electrochemical reactions, it is desirable that ‘spectating ions’ have a much higher diffusion coefficient. Double layer effects on the diffusion impedance are usually trivial. A useful parameter to judge the significance of double layer effects is the Debye frequency. In presence of convective diffusion, the stagnant Nernst diffusion layer approximation is usually a good approximation for the cases with a not very low kinematic viscosity. Homotopy perturbation method is introduced to treat the convective diffusion. A new approximation is obtained and improves over the Nernst diffusion layer approximation in the high frequency range in the case with a low kinematic viscosity. Although it is capable of treating the problem with an arbitrary Schmidt number in the full frequency range, an artificial inductive loop in the low frequency range appears in the new approximation, a remaining issue for future studies. Oblique lines with the angle deviating from 45
in the highfrequency part of the diffusion impedance can be rationalized by generalizing the continuity equation with fractional orders. The fractional order can be related with the fractal dimension of the diffusion media. It is a common practice to estimate the diffusion coefficient of active particles based on the diffusion impedance measured on a porous electrode. It is, however, very challenging due to the multiscale structure of and multi-physics processes involved in porous electrodes. Therefore, the original Warburg impedance model is usually insufficient when interpreting the diffusion impedance measured on a porous electrode. One should use complicated models considering multi-scale and multi-physics processes, or largely reduce the thickness of the porous electrode to decrease latent errors caused by using the original Warburg impedance model. Acknowledgement The author is indebted to Bin Wu at University of Michigan-Ann Arbor for suggesting to provide a physical interpretation of the bounded diffusion, Laisuo Su at Carnegie Mellon University, Hao Ge (with whom the discussion initiates this paper), Ying Sun, Yuncong Xiao, Shangshang Wang, Ruqing Fang and Dechun Si at Tsinghua University, and Yufan Zhang at Beihang University for critical reading and many useful comments. A major part of this work was done in early morning, hence I would like to record my special thanks to Lou Lu for her forbearance to the alarm clock ring. This work is financially supported by the startup fund (No. 502045001) for new faculty member of Central South University. References [1] A. Lasia, Electrochemical impedance spectroscopy and its applications, in: B.E. Conway, J.O.M. Bockris, R.E. White (Eds.), Modern Aspects of Electrochemistry, Springer US, Boston, MA, 2002, pp. 143e248. [2] J. Huang, Z. Li, B.Y. Liaw, J. Zhang, Graphical analysis of electrochemical impedance spectroscopy data in Bode and Nyquist representations, J. Power Sources 309 (2016) 82e98. [3] M.D. Levi, G. Salitra, B. Markovsky, H. Teller, D. Aurbach, U. Heider, L. Heider,

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