J. Electroanal. Chem., sous presse

Discussion of the potential step method for the determination of the diffusion coefficients of guest species in host materials.

I. Influence of charge transfer kineticsand ohmic potential drop

C. Montella*

École Nationale Supérieure d'Électrochimie et d'Électrométallurgie de Grenoble,

Laboratoire d’Électrochimie et de Physicochimie des Matériaux et Interfaces, UMR 5631 CNRS-INPG,

associé à l’UJF, Domaine Universitaire, B.P. 75, 38402 Saint Martin d'Hères, France

Abstract

Theoretical expressions are given for the output response of ion-insertion electrodes to a

potential step assuming linear diffusion, restricted (blocking) diffusion conditions and possible

limitations by insertion reaction kinetics. The effects of ohmic potential drop are also investigated.

It is shown that slow interfacial charge transfer cannot be distinguished from ohmic drop effects, in

contrast to impedance diagrams where ohmic drop and charge transfer effects can be separated.

The influence of potential step amplitude is discussed. Chronocoulometric analysis is dealt with

considering diffusion controlled processes as well as mixed control conditions. The error in the

determination of the chemical diffusion coefficient of a guest species from chronoamperometric

data, when using the limiting Cottrell equation in the short-time range or the exponential decay of

current in the long-time domain, is evaluated in relation to insertion reaction kinetics and ohmic

potential drop. Determination of the diffusion coefficients by curve fitting is also envisaged using

the current vs. time and charge vs. time relationships. Finally previous results in the electrochemical

literature are discussed in the light of the theoretical derivations proposed in this paper.

Keywords: Potential step; chronoamperometry; chronocoulometry; absorption; intercalation;

diffusion coefficient.

* Member of the Institut des Sciences et Techniques de Grenoble

Tel.: + 33-4-76826526; fax: + 33-4-76826630; e-mail: Claude.Montella@lepmi.inpg.fr

1. INTRODUCTION

The potential step chronoamperometry (PSCA) method is widely used for the determination of

the diffusion coefficients of guest species in host materials [1] based on the theoretical derivation

presented initially by Wen et al. [2] for diffusion controlled processes. Moreover, the potentiostatic

intermittent titration technique (PITT) [2, 3], which is based on a staircase potential signal, is an

extension of this method.

First, the diffusion model, the different graphical representations of the current transient used in

the electrochemical literature to characterize insertion processes and the principles for the

determination of the diffusion coefficients of guest species from PSCA data are reviewed. Next,

some examples of deviation from the diffusion model are quoted and finally the aim of this paper

is presented in the introduction.

1.1. Diffusion model

A restricted (or blocking) diffusion condition is satisfied for a planar electrode, i.e. the

diffusion flux is zero at some distance L from the electrolyteelectrode interface, either considering

linear diffusion in a thin film of host material, of thickness L, deposited on a substrate impermeable

to the diffusing species (Fig. 1A) or linear diffusion in a material foil or in platelet particles, of

thickness 2L, symmetrically submitted to insertion on both sides (Fig. 1B). In the latter case,

restricted diffusion is due to the symmetry for the concentration profile with respect to the middle

plane at the abscissa L.

Assuming diffusion control (very fast insertion reaction kinetics), linear diffusion and restricted

diffusion conditions, neglecting the effects of ohmic potential drop and double-layer charging

current, finally disregarding phase transition processes, equivalent expressions of the output

response to a potential step have been derived in the electrochemical literature. The following series

which converges rapidly in the long-time domain ( t > τ d ) can for example be used [2, 4, 5]:

I tQ

nt

n

dd d

( ) exp – ( – )=

=

∞

∑2 2 14

22

1

∆τ

πτ

(1)

or alternatively:

I tQ

tn t

n

dd

nd( ) – exp –= + ( ) ( )

=

∞

∑∆π τ

τ1 2 1 2

1

(2)

which is well suited to calculation for short times ( t < τ d ). In the above equations, Id is the

- 2 -

diffusion current, t the elapsed time from the beginning of the step, τ d the diffusion time constant

and ∆Q the total amount of Faradaic charge passed following a potential step, with:

τ d = L D2 , ∆ ∆Q I t t FAL c= ( ) = −∞

∫ d d0

(3)

where L is the diffusion length, D the chemical diffusion coefficient of guest species, F the Faraday

constant, A denotes the electrochemically active surface area and ∆c the variation of guest species

concentration in the host material due to the applied potential step. Equilibrium is assumed for the

insertion reaction both at the initial and final times, and positive and negative values of ∆c

correspond to insertion and de-insertion processes, respectively. Minus sign in Eq. (3) applies to

cation-insertion reactions ( H , Li , Na , K etc.+ + + + , ), so the Faradaic current and the related charge

are both negative (reduction reaction) for insertion processes ( ∆E < 0, ∆Q < 0 and ∆c > 0) in

accordance with the IUPAC convention.

Neglecting all exponential terms in Eq. (2), the well known Cottrell relationship is obtained for

short times (st):

I t I tQ

t

FA D c

td st Cottrelld

( ) = ( ) = = −∆ ∆π τ π

(4)

while an exponential decay of diffusion current with respect to time is predicted for long times (lt),

considering the leading term in Eq. (1), as:

I tQ t

FAD

Lc

Dt

Ld ltd d

( ) exp – exp –=

= −

24

24

2 2

2

∆ ∆τ

πτ

π(5)

1.2. Representations of the current transient

From Eqs. (1) and (2), the current is plotted vs. time in Fig. 2A. The current is divided by the

scale factor, 2 2∆ ∆Q FA D L cτ d = − ( ) , and the time divided by the diffusion time constant, τ d , to

obtain dimensionless quantities in the figure for the sake of generality. The current variation

shows a rapid decay, as usually observed for diffusion controlled processes.

The ratio of the diffusion current to the limiting Cottrell relationship in Eq. (4) is shown in

Fig. 2B. The theoretical representation in Fig. 2B is the dimensionless form of the Cottrell function

plot, I t t t( ) vs. log , used by Levi and Aurbach [6] to represent their experimental data for lithium

ion insertion into graphite electrodes. Deviation from semi-infinite linear diffusion, i.e. deviation

from a horizontal straight line in a I t t t( ) vs. log plot, can be visualized easily by plotting such a

- 3 -

graph.

In addition, the usual Cottrell plot (current vs. time–1/2) is given in Fig. 2C in dimensionless

form and compared to the Cottrell straight line (dashed line) plotted from Eq. (4). Finally, the

decimal logarithm of current is plotted vs. time in Fig. 2D, using dimensionless notation, and

compared to the asymptotic straight line (dashed line) predicted from Eq. (5).

Less than 1 % deviation from the limiting Cottrell equation is observed in Fig. 2B for

t τ d < 0 19. , i.e. 1 2 3t τ d > . in the Cottrell plot of Fig. 2C, while the linear variation in Fig. 2D is

satisfied (less than 1 % deviation from the limiting current in Eq. (5)) when t τ d > 0 23. . Hence, the

transition between the two limiting expressions for the diffusion current, respectively semi-infinite

linear diffusion current in Eq. (4) and exponential decay in Eq. (5), is observed within a narrow

time range near t = 0 2. τ d for diffusion controlled processes and restricted linear diffusion

conditions.

Some authors noted that ‘the change in shape of the current transient is of greater importance

than the absolute current level, and therefore it is very effective to introduce the derivative of the

current transient or logarithmic current with respect to time or logarithmic time’ [7, 8].

The logarithm of diffusion current is plotted in Fig. 3B vs. logarithm of time, using

dimensionless notation. The transient due to the diffusion process shows a linear dependence,

log current vs. log time, with a slope of (– 1/2) characteristic of semi-infinite linear diffusion,

followed by a steep exponential decay due to restricted diffusion. The derivative of logarithmic

current with respect to logarithmic time is plotted in Fig. 3C with the constant value (– 1/2) in the

short-time domain. Such plots are used in the electrochemical literature to discriminate between

diffusion in single-phase materials, as studied in this paper, and diffusion coupled with phase

transition processes in two-phase materials, such as lithium-ion insertion into Li CoO1 2−δ

electrodes [7, 8].

Finally, the graphical representation of Fig. 3D has not yet been used in electrochemical

literature, to the best of our knowledge. Nevertheless, it is well suited to the characterization of

diffusion controlled processes and restricted linear diffusion conditions because of the two limiting

expressions respectively valid for short and long times:

d dd stlog ( ) log /I t t = − 1 2 (6)

and:

- 4 -

d dd lt dlog ( ) logI t t t= − ( )π τ2 4 (7)

The validity domains for the above expressions are separated by a narrow time range near

t = 0 2. τ d , which is characteristic of restricted linear diffusion.

1.3. Determination of the diffusion coefficient of a guest species

The diffusion coefficient of a guest species is generally determined either from the short-time

expression of the diffusion current in Eq. (4) where I t tCottrell( ) is the characteristic time-invariant

function [6] and also the slope ( Sl ,stI ) of the Cottrell straight line in a current vs. time 1/2− plot,

from which D is obtained as:

DI t t

QL

QL

FA cI I= ( )

=

=

π π πCottrell ,st ,stSl Sl

∆ ∆ ∆

2 2 2

(8)

provided the diffusion length L is known independently, or from the long-time domain where an

exponential decay of current is predicted. Taking the decimal logarithm of both sides in Eq. (5) and

plotting log current vs. time, the diffusion coefficient can be obtained from both the intercept

( Int ,ltI ) and slope (Sl ,ltI ) of the portion of straight line, according to:

DL

Q

L

FA cI I= =

2

210

210

∆ ∆Int Int,lt ,lt (9)

and:

DL

I= 4102

2

lnπ

Sl ,lt (10)

More generally, using a curve fitting procedure, experimental chronoamperograms can be

compared to Eqs. (1) and (2) over a large time range provided ohmic potential drop is disregarded

and the double-layer charging current is negligible compared to the Faradaic current. The total

Faradaic charge ∆Q passed following a potential step is obtained by integration of the current

transient with respect to time, so only one adjustable parameter, τ d = L D2 , is needed to compare

experimental chronoamperograms to Eqs. (1) and (2).

1.4. Deviation from the diffusion model

Nevertheless, deviations from the theoretical relationships in Eqs. (1) to (10) and related graphs

in Figs. 2 and 3 are frequently observed in experimental works focused on host materials and

- 5 -

insertion reactions. Recent examples include studies on lithium-ion insertion in thin graphite,

Li NiOx 2 and Li Co Ni Ox 0.2 0.8 2 electrodes by Levi et al. [4, 6, 9] and Li CoO1- 2δ electrodes by Shin

and Pyun [7, 8], potassium-ion insertion in thin films of prussian blue (ferric ferrocyanide, PB) by

Garcia-Jareño et al. [10], as well as other works not discussed in this paper.

Potassium-ion insertion in thin PB films deposited on ITO substrate was thoroughly examined

by Garcia-Jareño et al. using PSCA [10], cyclic voltammetry [11] and the EIS method [12].

Considering chronoamperometric results, deviation from the Cottrell equation was analyzed by the

above authors in terms of ohmic potential drop in the electrolytic solution and the ITO substrate.

Numerical simulations were carried out by these authors, taking ohmic drop into consideration, and

the theoretical model was consistent with experimental data.

However, the impedance diagrams plotted for the same materials [12] clearly show the presence

of a large interfacial charge transfer resistance compared to the solution and substrate resistances,

which denotes slow ion-transfer kinetics at the solutionPB interface and/or slow electron-transfer

kinetics at the PBITO interface. Hence, the question to be answered is whether or not kinetic

limitations by interfacial charge transfer can be neglected when analyzing the current transient due

to a potential step for potassium-ion insertion in PBITO electrodes.

On the other hand, Levi et al. [4, 6, 9] studied lithium-ion insertion in thin graphite, Li NiOx 2

and Li Co Ni Ox 0.2 0.8 2 electrodes using cyclic voltammetry, chronoamperometry (PITT), current

pulses and the EIS method. Considering potential step experiments, these authors always observed

that the Cottrell region (Eq. (4)) is narrow with respect to time given that the short-time domain is

affected by ohmic potential drop, lithium-ion migration through superficial passivation layers on

particles of host material and slow ion-transfer at the particle surface, while the long-time deviation

from Eq. (5) is connected with particle size distribution. Similarly, deviation from the restricted

(or finite space) diffusion impedance was observed by these authors in the low frequency domain,

considering the impedance diagrams plotted for the same materials [6, 9].

Finally, Shin and Pyun [7, 8] observed that the log current vs. log time plot for lithium-ion

insertion in Li CoO1- 2δ electrodes and potential stepping in the single α phase domain deviates

from the theoretical prediction in Fig. 3B and ‘shows a monotonic increase in its slope from an

almost flat value to the infinite one in absolute value’. These authors interpreted their

chronoamperometric data in terms of ‘purely cell-impedance controlled diffusion’ of lithium-ion in

Li CoO1- 2δ [8].

- 6 -

1.5. This work

The aim of this work is to point out and discuss theoretically the effects of insertion reaction

kinetics and ohmic potential drop on the output response of ion-insertion electrodes to a potential

step and to demonstrate the advantage of the simultaneous application of potential step and EIS

methods for the determination of the chemical diffusion coefficients of guest species in host

materials.

The theoretical response to a potential step depends on the geometric model used to represent

insertion and diffusion processes in host materials. In this paper we are concerned with planar

films of host material, of thickness L, deposited on a substrate impermeable to the diffusing species

(Fig. 1A), as generally assumed for thin films obtained by electrodeposition, sputtering or spray

deposition. Diffusion of guest species starts at the electrolytefilm interface, while electron transfer

is observed simultaneously at the substratefilm interface. Ideal thin films (isotropic material, no

structural defects) are assumed, so that the diffusion coefficient of intercalated species takes on a

simple meaning in this case due to one-dimensional diffusion.

The model used below for insertion and diffusion processes also applies to linear diffusion in

material foils, of thickness 2L, symmetrically submitted to insertion on both faces (Fig. 1B). Levi

and Aurbach [6] used the same geometric model to represent lithium-ion insertion into platelet

particles of graphite in thin film composite electrodes.

Two-phase interfaces are considered in the above model. A reaction front advancing from the

three-phase junction, electrolytesubstrateactive material, was envisaged alternatively by Schröder

et al. [13] and Lovric and Scholz [14, 15] for modeling the propagation of a redox reaction through

microcrystals (mixed ionic-electronic conductor) immobilized at an electrode surface and in contact

with an electrolyte solution. The influence of crystal shape and crystal size on the output response

to a potential step was thoroughly examined in these papers using algebraic as well as numerical

methods.

A three-phase junction, electrolyteactive particleelectronic conductor, may also be considered

in composite electrodes where additives, such as carbon black, are generally used to improve

electronic conduction in the electrode, which results in a complicated geometry for coupled mass-

transport of ions and electrons in electroactive particles.

Two- or three-dimensional diffusion occurring from a three-phase junction is disregarded in

this paper because of the simple geometric model assumed for the thin film electrodes and particles

- 7 -

in Fig. 1. Moreover, to focus on insertion reaction kinetics and ohmic drop effects, linear diffusion

of guest species in single phase materials and/or potential domains (no phase transformation) is

assumed below. Other effects related to the geometry of diffusion in host materials and the

distribution of diffusion lengths in particles of composite electrodes will be dealt with in the second

and third parts of this work [16].

First the theoretical response to a potential step is derived in this paper and the possible shapes

for the Faradaic current transient discussed, next chronocoulometric analysis is dealt with and

finally the conditions for the determination of the diffusion coefficients of guest species are

analyzed. Previous results in the electrochemical literature are also discussed in the light of the

theoretical derivations given in this paper.

2. CHRONOAMPEROMETRY

To our knowledge, the first theoretical work focusing on the response of insertion electrodes to

a potential step and taking into account kinetic limitations by surface processes is that of Krapvinyi

et al. [17, 18] who studied hydrogen extraction from metallic sheets using large-amplitude potential

steps. More recently, Chen et al. [19] given the theoretical response to a small-amplitude potential

step for hydrogen absorption in thin metal films or foils, under restricted linear diffusion

conditions, assuming no hydride formation and taking kinetic limitations at the metalelectrolyte

interface into consideration.

Using the metallic-type potential distribution in host materialelectrolyte systems, as discussed

by Levi and Aurbach [6], the model in Ref. [19] also applies to ion-insertion processes in thin

films, foils or platelet particles of host material with planar geometry. This model is extended below

to cover the case where ohmic potential drop is present.

The insertion material is assumed to be initially ( t < 0) in the equilibrium state corresponding

to an electrode potential E0 and a uniform concentration of guest species, c c E0 0= ( ), given by the

insertion isotherm c(E), which is the reverse function of the coulometric titration curve E c( ) generally used in the literature. A potential step from E0 to E E E1 0= + ∆ , where ∆E can be either

positive or negative, which corresponds respectively to de-insertion and insertion processes for

cations, is imposed on the electrode at t = 0. After the current transient, the electrode tends towards

the new equilibrium state corresponding to potential E1 and uniform concentration of guest species

c c E1 1= ( ) , with c c c1 0= + ∆ .

- 8 -

2.1. Laplace transform of Faradaic current

Assuming small-signal (linearity) conditions for both potential step and EIS methods, the

Laplace transform of the current transient due to a potential step from E0 to E E0 + ∆ , with

∆E RT nF<< ( ) where the number (n) of electron involved in the reaction is equal to one in this

paper and the other symbols have their usual meaning, is related to the electrode impedance Z s( )

measured at the steady-state potential E0 through the following equation where s denotes the

Laplace complex variable:

I sE

sZ s( ) =

( )∆

(11)

Using the usual equivalent circuit for the impedance of electrochemical systems [20], Z s( ) can

be written as:

Z s R sC Z s( ) = + + ( )[ ]Ω 1 1dl F (12)

where Z sF ( ) is the Faradaic impedance, Cdl the differential (double-layer) capacitance and RΩ the

sum of ohmic resistances in the electrolyte, bulk material and material substrate or current collector.

More complicated equivalent circuits are sometimes proposed for insertion processes to include the

effects of ion migration through superficial passivation layers when considering lithium-ion

insertion into graphite electrodes [4, 9] for example.

Provided the differential (double-layer) capacitance, as well as the possible capacitances of

superficial layers on the host material (e.g. passivation layers on graphite particles), are sufficiently

low with respect to the insertion capacitance of the bulk material, the electrode impedance can be

approximated, except in the high frequency (or s) domain, by:

Z s R Z s( ) ≈ + ( )Ω F (13)

Assuming a uniformly accessible electrode, the Faradaic impedance for insertion reactions is

well known. Neglecting limitations by mass transport in the electrolyte, considering a direct (one-

step) insertion reaction, assuming linear diffusion of guest species in a thin film or foil of host

material or in platelet particles of a composite electrode (in the latter case the potential gradient in

the electrode is neglected), with restricted diffusion conditions (Fig. 1), finally disregarding phase

transition processes, Z sF ( ) takes on the following expression [19, 21, 22]:

Z s R R s sF ct d d d( ) = + coth τ τ , τ d = L D2 (14)

- 9 -

where the interfacial charge transfer resistance, Rct , and the diffusion resistance, Rd , are written in

terms of thermodynamic, kinetic and diffusion parameters [19] as:

R FAvEct = − ( )1 (15)

R FAm c Ed d d= − ( )[ ]1 , m D L= (16)

In these equations, m is the diffusion constant, τ d the diffusion time constant and d dc E the

insertion isotherm slope which satisfies:

d dc E v vE c= − (17)

because of equilibrium state at the initial potential of potential step. vE and vc are the partial

derivatives of insertion reaction rate (v) with respect to potential and concentration of guest species,

both evaluated at the initial potential E0 :

v v EE c= ∂ ∂( ) , v v cc E

= ∂ ∂( ) (18)

The quantities d dc E , vE and vc are all negative in Eqs. (15) to (18) due to the assumption of

cation-insertion reactions in this paper. Moreover, considering small signals (linearity condition),

the concentration variation in the host material is related to the potential step amplitude by the

linearised relationship:

∆ ∆c c E E= ( )d d (19)

Introducing the dimensionless parameter:

Λ =+

=+

=+

R

R R

v m

R R m v R Rc

c

d

ct ct dΩ Ω Ω11

(20)

where all quantities are evaluated at the initial potential E0 , the Laplace transform of Faradaic

current is derived from the above equations as:

I sE R R

s s s( ) =

+( )+( )

∆ Ω ct

d d1 Λ coth τ τ(21)

Setting s → ∞ in Eq. (21), the term coth τ τd ds s tends towards zero, and, using the inverse

Laplace transform, the Faradaic current at the initial time is given by the equivalent expressions:

- 10 -

Ι Λ Λ Λ0( ) =+

= = = −∆ ∆ ∆ ∆Ω

E

R R

E

R

QFA

D

Lc

ct d dτ(22)

The initial current depends linearly on the concentration variation in the host material, ∆c , the total

charge passed following a potential step, ∆Q , and the potential step amplitude ∆E . Hence, the

Faradaic current vs. potential relationship follows Ohm’s law for short times ( t → 0) provided

∆E RT F<< .

From Eqs. (21) and (22), we obtain finally:

I sI

s s s( ) = ( )

+( )0

1 Λ coth τ τd d

(23)

2.2. General expressions for the Faradaic current

Inverse Laplace transform of Eq. (23) gives the Faradaic current transient caused by a small-

amplitude potential step as the infinite series expansion:

I t Ib

b tn

n

n

( ) exp –= ( )+ ( )

=

∞

∑2 0 2 22

1

ΛΛ Λ +

τ d (24)

where bn is the nth positive root of the equation:

b btan – Λ = 0 (25)

Equivalent formulations of I t( ) are easily derived from Eqs. (22) and (24) as:

I tE

R R bb t

nn

n

( ) exp –=+ + ( )

=

∞

∑2 2 22

1

∆

Ω ctd

ΛΛ Λ +

τ (26)

and:

I tE

R bb t

nn

n

( ) exp –=+ ( )

=

∞

∑22

2 22

1

∆

dd

ΛΛ Λ +

τ (27)

Eqs. (26) and (27) give the current transient in terms of electrical components of the equivalent

circuit for the electrode impedance. These equations can also be written in terms of thermodynamic,

kinetic, diffusion and ohmic parameters, from Eq. (22), as:

I t FAD

Lc

bb

Dt

Lnn

n

( ) exp –= −+ +

=

∞

∑22

2 22

21

∆ ΛΛ Λ

(28)

- 11 -

where minus sign applies to cation-insertion reactions ( H , Li , Na , K etc.+ + + + , ). A in Eq. (28) is

the total surface area (both faces) when using the electrode or particle geometry of Fig. 1B, while it

is the area of the materialelectrolyte interface (single face) for the thin film in Fig. 1A.

Finally, using the diffusion time constant and the total amount of Faradaic charge passed

following a potential step, as defined in Eq. (3), an alternative formulation of the current transient is

derived from Eq. (28) as:

I tQ

bb t

nn

n

( ) exp –=+ + ( )

=

∞

∑22

2 22

1

∆τ

τd

dΛ

Λ Λ(29)

The above relationship is valid for both electrode geometries in Fig. 1 and generalizes the well

known Eq. (1) which is obtained as a limiting case setting Λ → ∞ (diffusion control) in Eq. (29)

with b nn = ( )2 1 2– π from Eq. (25).

Eq. (24) and Eqs. (26) to (29) give five equivalent formulations of the electrode response to a

small-amplitude potential step for cation-insertion processes studied by the PSCA method under

restricted linear diffusion conditions, in the absence of phase transition processes. Due to the use

of a small signal amplitude, the theoretical derivation is valid whatever the insertion isotherm type

(Langmuir, Frumkin or other isotherm) and reaction rate expression (Butler-Volmer or other

relationship).

The relationship between the chemical diffusion coefficient of a guest species and the insertion

isotherm was derived first by Armand [23], to the best of our knowledge, and discussed more

recently by Chen et al. [24] and Levi and Aurbach [6]. However, Eqs. (24) to (29) apply whether

the diffusion coefficient is constant or depends on the guest species concentration in the host

material because of (i) a small signal amplitude, (ii) restricted diffusion conditions and therefore

(iii) uniform concentration of guest species in the host material and equilibrium for the insertion

reaction under steady-state conditions [25].

The theoretical expressions for the current transient caused by a small-amplitude potential step

have been derived in this Section assuming that the capacitive current is negligible compared to the

Faradaic current. Therefore, these equations cannot be used to fit experimental data obtained for

very short times where measurements are corrupted due to the presence of the double-layer

capacitance. In particular, the initial current in Eq. (22) cannot be measured directly and

extrapolation techniques should be used to determine I 0( ).

2.3. Parameter ΛΛΛΛ: a key factor for the kinetics of insertion processes

- 12 -

Clearly, the dimensionless parameter Λ is a key factor for the kinetics of insertion processes

studied by PSCA. In Eq. (20), v mc compares the diffusion constant, m D L= , and the reaction

rate constant which is included in the partial derivative vc . In addition, the influence of ohmic

potential drop is characterized by the resistance ratio R RΩ ct or R RΩ d .

The general expression for the parameter Λ, irrespective of insertion isotherm type and rate

control conditions, is obtained from Eqs. (15) and (20), in terms of partial derivatives of reaction

rate, as:

Λ =+

v L D

R FA vc

E1 Ω

(30)

while equivalent formulations can be written using the insertion isotherm slope, d dc E , and the

derivative calculus rule in Eq. (17).

The Λ value for PSCA experiments depends on the insertion isotherm type, the initial potential

of potential step, the reaction rate constant, the diffusion coefficient of guest species, the film, foil

or particle thickness in Figs. 1A and B, and the presence of ohmic drop. For example, assuming

Langmuir isotherm conditions for the insertion process (formation of a solid solution with no

interaction in the host material) and a cation concentration of 1mol L 1− (standard concentration) in

the electrolyte solution, Λ takes on the following expression derived in the Appendix A:

Λ =°( ) −( ) +( )

+ ° −( )[ ] +( )k L D

R fFAk c

exp exp

exp exp

α ξ ξα ξ ξ

r

max r

0 0

0 0

1

1 1 1Ω

(31)

where k° is the standard rate constant for the insertion reaction, α r the symmetry factor of

interfacial charge transfer in the direction of reduction, cmax the maximal (saturation) concentration

of guest species in the host material, ξ0 the initial potential of potential step written in

dimensionless form, ξ0 0= − °( )f E E with f F RT= ( ), and E° denotes the standard potential, i.e.

the equilibrium potential corresponding to the standard concentration in the solid solution,

c c° = max 2.

On the other hand, whatever the insertion isotherm, Λ can be estimated from Eq. (20) using

experimental values of the diffusion resistance, charge transfer resistance and electrolyte and/or

substrate resistances measured on an impedance diagram plotted for the same potential range.

- 13 -

2.4. Parameter ΛΛΛΛ and concentration profile in the host material

The influence of the parameter Λ is well illustrated by the concentration profile vs. time

variation plotted in Figs. 4 to 6. Solving the diffusion equations together with the boundary

conditions at the material surfaces (continuity condition at x = 0 and blocking diffusion condition

at x = L), the Laplace transform of guest species concentration in the host material is given by:

c x sc

s

I x L

FAms,

sinh( ) = −

( ) −( )[ ]+[ ]

00 1cosh s

cosh s s sd

d d d

ττ τ τΛ

, 0 ≤ ≤x L (32)

where x is the distance from the electrolytehost material interface. Using inverse Laplace

transforms of the different terms, we obtain, after some reorganization, the normalized

concentration vs. space and time dependence due to potential stepping:

c x t c

c c b

b x L

bb t

n

n

nn

n

,cos

exp –( ) −

−= −

+ +( )−( )[ ] ( )

=

∞

∑0

1 02 2

2

1

1 21Λ

Λ Λcos

dτ (33)

This relationship can alternatively be obtained from the theoretical expression given by Crank [26]

for similar problems of diffusion in a plane sheet coupled with surface evaporation processes,

because of the same diffusion equations and boundary conditions once written in dimensionless

form.

Large Λ values are characteristic of diffusion controlled processes due to very fast insertion

reaction kinetics and negligible ohmic drop or IR compensation. A concentration step from c0 to

c1 is imposed at the electrolytehost material interface ( x = 0) when Λ → ∞ , due to the applied

potential step from E0 to E1, and the mass transport process varies from semi-infinite diffusion

conditions for short times to restricted diffusion conditions for long times (Fig. 4). The related

limiting expression for the concentration is derived, setting Λ → ∞ in Eqs. (25) and (33), as:

c x t c

c c nn

x

Ln

t

n

,sin exp –

( ) −−

= −−( )

−( )

−( )

→∞

=

∞

∑Λ 0

1 0

22

1

14 1

2 12 1

22 1

4ππ π

τ d

(34)

In opposition, considering small Λ values due to slow charge transfer kinetics at the interface

and/or large ohmic drop in the different phases, we have b1 = Λ and b nn = π (for n > 1) from

Eq. (25), so all exponential terms, except the first, can be neglected in Eq. (33) when Λ tends

towards zero. It follows that:

c x t c

c ct

,exp

( ) −−

= − −( )→Λ Λ0 0

1 0

1 τ d (35)

c depends only on the time variable under the above conditions and quasi-uniform concentration of

- 14 -

guest species (negligible concentration gradient in the host material) is predicted over the whole

time domain, as shown in Fig. 5.

Finally, intermediate values of the parameter Λ in Eqs. (20) and (30) are characteristic of mixed

control conditions, i.e. control by diffusion and interfacial charge transfer kinetics and/or ohmic

drop. An example for such a concentration profile vs. time variation is presented in Fig. 6. The

mass transport process varies from semi-infinite diffusion conditions for short times to restricted

diffusion conditions for long times, but, in contrast to Fig. 4, the interfacial concentration of guest

species is time dependent, which is generally the case for most real systems.

2.5. Influence of capacitive current and ohmic drop

In Ref. [19] focused on hydrogen absorption in thin metal films, we wrote: ‘We have not taken

into consideration the electrolytic solution resistance Rs in the preceding calculation as the

influence of such a resistance and of the differential (double-layer) capacitance Cdl can be

neglected for times longer than the time constant, τ = +( )[ ]R R R R Cs ct s ct dl , whose value ...’.

This is incorrect and should be replaced by: under restricted diffusion conditions, the current

transient caused by a small potential step is not modified, due to the presence of the differential

(double-layer) capacitance, for times longer than the time constant, τ = +( )[ ]R R R R Cs ct s ct dl ,

provided Cdl takes on a very low value with respect to the insertion capacitance of the host material,

i.e. C FAL c Edl d d<< , which is generally observed for most real systems. However, if the latter

condition is not satisfied, the double-layer capacitance influences the response to a potential step

also in the long-time range due to the blocking condition, J L t,( ) = 0, for the diffusion process in

the host material.

On the other hand, whatever the kinetic and diffusion conditions, uncompensated ohmic drop

affects the current transient over the whole time domain, in opposition to the assertion in Ref. [19].

2.6. Influence of insertion reaction kinetics and ohmic drop

The shape of the current vs. time curve calculated from Eqs. (25) and (29) depends only on the

dimensionless parameter Λ given that the ratio ∆Q τ d and the diffusion time constant τ d are scale

factors for current and time, respectively. Looking at Eq. (20), we note that the effects of slow ion-

transfer kinetics at the electrolytehost material interface, as well as slow electron-transfer kinetics

at the host materialsubstrate (or current collector) interface, cannot be distinguished from ohmic

drop effects. Due to slow interfacial charge transfer (high Rct value compared to Rd) and/or large

ohmic drop (high RΩ value), the Λ value decreases in Eq. (20) and the current transient due to a

- 15 -

potential step is affected over the whole time domain, as indicated by Eqs. (25) and (29). Hence, the

effects of surface reaction kinetics and ohmic potential drop should be discussed at the same time

for the PSCA method. In contrast, ohmic drop effects and kinetic limitations by interfacial charge

transfer can be discriminated using the EIS method and the Nyquist representation of impedance

diagrams (see Fig. 13B below).

Hereafter, we assume that the double-layer charging current is negligible compared to the

Faradaic current over the time range considered and we use Eqs. (20), (25) and (29) to model the

current transient. The current is plotted in Fig. 7 using the same dimensionless representations as

in Fig. 2. Large Λ values (∞ in the figure) give the same theoretical dependences as in Fig. 2 due to

diffusion control.

The limiting response observed for short times (st) depends on Λ. Diffusion control and semi-

infinite linear diffusion (which corresponds to the concentration profile of Fig. 4 for short times),

i.e. the Cottrell equation, are satisfied when Λ → ∞ and t → 0, as shown in Figs. 7B and C:

I t I tQ

t

FA D c

t( )st, Cottrell

dΛ→∞ = ( ) = = −∆ ∆

π τ π(36)

In contrast, interfacial charge transfer and/or ohmic drop control ( Λ → 0 and therefore

b1 = Λ from Eq. (25)), which corresponds to quasi-uniform insertion conditions (Fig. 5), yields

the exponential dependence over the whole time domain:

I tQ

tE

R Rt

FA v c

R FA v

v t L

R FA vc

E E

( ) exp exp exp –Λ Λ Λ Λ→ = −( ) =+

−( ) = −+ +

0 1 1

∆ ∆ ∆

Ω Ω Ωττ τ

dd

ctd

c

(37)

which can be linearised for short times as:

I tQ

tE

R Rt

FA v c

R FA v

v t L

R FA vc

E E

( ) – –st,d

dct

dc

Λ Λ Λ Λ→ = ( ) =+

( ) = −+

−+

0 1 1

11

1∆ ∆ ∆

Ω Ω Ωττ τ

(38)

In the above equations, the partial derivative ( vc < 0) is replaced by ( − vc ), so minus sign is

apparent for cation insertion processes (reduction reaction, with ∆E < 0, ∆Q < 0 and ∆c > 0).

Finally, intermediate Λ values, which correspond to mixed control by diffusion and insertion

- 16 -

reaction kinetics and/or ohmic drop, lead for short times (semi-infinite linear diffusion in Fig. 6 for

t << τ d) to the relationships:

I tQ

t tE

R Rt t( ) exp expst

dd d

ctd derfc erfc= ( ) ( ) =

+ ( ) ( )Λ Λ Λ Λ Λ∆ ∆

Ωττ τ τ τ2 2

= −+ +( )

+

FA v c

R FA v

v t D

R FA v

v t D

R FA vc

E

c

E

c

E

∆

Ω Ω Ω1 1 1

2

2exp erfc (39)

which can be simplified for very short times (vst), using Maclaurin series expansions of

exponential function and error function complement, as:

I t Q tE

R Rt( ) – –vst d d

ctd= ( ) ( )[ ] =

+( )[ ]Λ Λ Λ∆ ∆

Ωτ πτ πτ1 2 1 2

= −+

−( )

+

FA v c

R FA v

v t D

R FA vc

E

c

E

∆

Ω Ω11 2

1

π(40)

In contrast, Eq. (39) becomes equivalent to (less than 1 % deviation from) the Cottrell relationship

in Eq. (36) when Λ2 48 5t τ d > . .

Setting RΩ = 0 (no ohmic drop or IR compensation) and H v Dc= =Λ τ d , Eqs. (39)

and (40) are similar in shape to the relationships given by Bard and Faulkner [27] for one-step

redox reactions with semi-infinite linear diffusion in the electrolyte and kinetic limitations at the

electrolyteelectrode interface. Nevertheless, the influence of ohmic drop is taken into

consideration in Eqs. (39) and (40), in contrast to the derivation by the above authors.

Deviation from diffusion controlled processes, due to limitations by insertion reaction kinetics

and/or ohmic potential drop, is well characterized by the presence of the maximum of the function

in Fig. 7B, which is equivalent to a maximum of the Cottrell function plot, I t t t( ) vs. log , from

the experimental point of view. No portion of straight line is present in the Cottrell plot of Fig. 7C

under the above conditions. This is observed, for example, considering the experimental results

obtained by Levi et al. [6, 9] for lithium-ion insertion into thin graphite electrodes.

The asymptotic behavior observed in the long-time domain is illustrated in Fig. 7D. For long

times (large t τ d ), restricted diffusion is satisfied in Figs. 4 to 6 whatever the Λ value. Only the

first term of the series in Eq. (29) and the equivalent expressions of Faradaic current is considered,

so the absolute value of current shows an exponential decay vs. time. It follows that a log I t( ) vs.

time plot should be linear for long times whether diffusion control ( Λ → ∞ = and b1 2π ):

I tQ t E

R

tFA

D

Lc

Dt

L( ) exp – exp – exp –lt,

d d d dΛ→∞ =

=

= −

24

24

24

2 2 2

2

∆ ∆ ∆τ

πτ

πτ

π

(41)

- 17 -

or control by interfacial charge transfer and/or ohmic drop ( Λ Λ→ =0 1 and b ):

I tQ

tE

R Rt

FA v c

R FA v

v t L

R FA vc

E E

( ) exp exp exp –lt,d

dct

dc

Λ Λ Λ Λ→ = −( ) =+

−( ) = −+ +

0 1 1

∆ ∆ ∆

Ω Ω Ωττ τ

(42)

or mixed control by diffusion and interfacial charge transfer and/or ohmic drop (intermediate Λ

value and b1 given by the first positive root of Eq. (25)) is satisfied:

I tQ

b

b tFA

D

Lc

bb

Dt

L( ) exp – exp –lt

d d

=+ +( )

= −+ +( )

2 2

2

212

12 2

212 1

22

∆ ∆τ τ

ΛΛ Λ

ΛΛ Λ

(43)

Levi et al. [6, 9] noted that the short-time response of ion-insertion electrodes to a potential step

is modified due to slow ion-transfer kinetics at the host material surface. More generally, slow

interfacial charge transfer and ohmic potential drop affect the current transient over the whole time

domain as shown in Fig. 7. Moreover, an important feature of Eqs. (41) to (43) is that an

exponential decrease of current with respect to time, in the long-time range, is not a sufficient

condition for predicting a diffusion controlled rate for insertion processes in thin films, foils or

particles of host material.

The logarithm of current is plotted vs. logarithm of time in Fig. 8B, using the same

dimensionless representation as in Fig. 3B, and the derivative of this curve is given in Fig. 8C. The

influence of insertion reaction kinetics and ohmic drop is clearly shown in the short-time domain

where d dlog ( ) logI t t varies from (– 1/2) for diffusion controlled processes to zero due to

constant current at short times for interfacial charge transfer and/or ohmic drop control.

The effects of insertion reaction kinetics and ohmic drop on the current transient are also

illustrated in Fig. 8D. Diffusion control ( Λ → ∞ ) gives the same characteristic curve as in Fig. 3D

with the limiting value (– 1/2) for short times and the limiting slope, − ( )π τ2 4 d , for long times in

a d d vs. timelog ( ) logI t t plot. In contrast, d dlog ( ) logI t t tends towards zero when t → 0

due to control by insertion reaction kinetics and/or ohmic drop, while a portion of straight line is

predicted in the long-time range, whatever the Λ value, according to:

d dlt dlog ( ) logI t t b t= − 12 τ (44)

- 18 -

Hence, deviation from diffusion controlled processes, due to limitations by interfacial charge

transfer kinetics and/or ohmic drop, is well characterized by the value ( d dlog ( ) logI t t > −1 2)

noted in the short-time domain. This is observed, for example, in the papers by Shin and Pyun [7,

8] for lithium-ion insertion in Li CoO1- 2δ electrodes and potential stepping in the single α phase

domain. The experimental results in Ref. [8] were interpreted by these authors assuming ‘purely

cell-impedance controlled diffusion’ of lithium-ion in Li CoO1- 2δ , i.e. assuming that the current-

potential relationship is ‘purely ohmic’. This model is an alternative to the theoretical derivation

presented in this work. The two models will be compared in a separate paper.

As a first conclusion, it should be emphasized that the use of the classical diffusion laws in thin

film materials, i.e. Eqs. (1) and (2) above, is only valid when considering very fast insertion

reaction kinetics, negligible ohmic drop or IR compensation, and therefore diffusion controlled

processes. In contrast, Eqs. (24), (28) and (29) derived in this paper apply for kinetic, diffusion and

ohmic drop control, as well as mixed control conditions. In addition, the current transient can be

modeled alternatively in terms of electrical components of the equivalent circuit for the electrode

impedance in Eqs. (26) and (27). Whatever the expression considered for the current transient, the

key factor for the kinetics of insertion processes studied by the PSCA method is the dimensionless

parameter Λ in Eqs. (20) and (30)

2.7. Potential step method vs. EIS method

As indicated above, the Laplace transform of the current transient caused by a potential step and

the electrode impedance measured at the same initial potential are related through Eq. (11) provided

a sufficiently small potential step is considered. Under the above conditions direct correspondences

can be stated between the results obtained by the two methods. The limiting Cottrell behavior

observed in the short-time domain corresponds to the Warburg impedance satisfied for high

frequencies, the exponential decay of current noted for long-time chronoamperometric experiments

is related to the low frequency capacitive behavior in EIS data, etc. Hence, the same theoretical

model should be used to fit experimental chronoamperograms and EIS data collected for the same

material.

Correspondences between PSCA and EIS data were noted in the experimental work by Levi

and Aurbach [6] focused on lithium-ion insertion in thin graphite electrodes. These authors noted

that chronoamperometric data deviate from Eq. (1) in the long-time range and EIS data are not

fitted well by the restricted diffusion impedance in the low frequency domain, due to particle size

distribution. Levi and Aurbach used the so-called Frumkin and Melik-Gaykazyan (FMG)

impedance to fit their EIS data. However, the FMG impedance is given by the series combination

of a bounded diffusion impedance [28] (or finite length diffusion impedance in Ref. [6]) and the

- 19 -

insertion capacitance; in contrast, restricted diffusion (finite space diffusion according to Ref. [6])

is generally assumed for insertion materials. Moreover, these authors propose no theoretical

expression for the output response of a composite electrode to a potential step that is consistent

with the FMG impedance. Therefore, the use of the FMG impedance cannot be recommended to fit

experimental data collected for ion-insertion materials.

2.8. Influence of potential step amplitude

A small-amplitude potential step was assumed in the above Sections of this paper in order to

linearise the kinetic equations. The situation is more complicated when considering non-linear

conditions due to a large potential step amplitude, as used experimentally by Garcia-Jareño et al.

[10] for studying potassium-ion insertion in PBITO electrodes. These authors imposed a

potential step from the initial potential, E0 = 0.6 V vs. Ag/AgCl/KCl 1M reference electrode,

corresponding to the prussian blue domain, to the final potential, E1 = – 0.2 V vs. the same

reference, which corresponds to the prussian white (or Everitt salt) domain. The same remark

applies to the experiments of Shin and Pyun [7, 8] focusing on lithium-ion insertion in Li CoO1- 2δ

electrodes. These authors imposed potential steps with amplitudes of several hundred of mV.

No general analytical solution of the diffusion and kinetic equations is possible under the above

conditions and numerical calculation, using a finite difference method for example, is needed to

simulate the theoretical behavior of the insertion electrode. However, a particular situation can be

treated theoretically in closed form to understand the influence of potential step amplitude.

Assuming that the diffusion coefficient of guest species does not depend on its concentration

inside the host material, and therefore on the electrode potential, that Langmuir isotherm conditions

(formation of an ideal solid solution with no interaction in the host material) are satisfied for the

insertion process, and disregarding ohmic drop and double-layer charging current effects, the

theoretical derivation presented for small-amplitude potential steps in Eqs. (24), (28) and (29)

above can be generalized to large potential steps [19] provided (i) the expression for the

dimensionless parameter Λ in Eq. (20) is replaced by:

Λ = v mc (45)

because of the assumption of a negligible ohmic drop or IR compensation, and (ii) the partial

derivative of reaction rate with respect to guest species concentration is evaluated at the applied

potential E1 rather than the initial potential E0 . Under the above conditions, the parameter Λ takes

on the expression:

- 20 -

Λ = °( ) −( ) +( )k L D exp expα ξ ξr 1 11 , ξ1 1= − °( )f E E (46)

and Eqs. (24), (28) and (29) remain valid for large potential step. In contrast, the linear dependence

of the concentration variation ∆c on the potential step amplitude ∆E in Eq. (19) is not satisfied

due to a large signal amplitude and therefore Eqs. (26) and (27) cannot be used to fit experimental

data for large potential steps.

Disregarding ohmic drop effects, large Λ values (diffusion control) can be achieved for thin

films, foils or platelet particles of host materials, and therefore the limiting relationships in Eqs. (1)

and (2) satisfied, either considering a small-amplitude potential step and a very high insertion rate

constant, irrespective of the insertion isotherm type and the dependence or not of the diffusion

coefficient on the electrode potential, or using a sufficiently large potential step amplitude,

irrespective of insertion reaction kinetics, provided D is constant.

In contrast, in the presence of ohmic drop, Garcia-Jareño et al. [10] neglected kinetic limitations

by interfacial charge transfer for potassium-ion insertion in PBITO electrodes submitted to a

large potential step despite the fact that the impedance diagrams plotted by these authors [12] for

the same electrodes in the PB region (which corresponds to the initial potential of potential step)

clearly show the presence of a large charge transfer resistance compared to the electrolyte and

substrate resistances. Pure ohmic drop control was assumed by these authors to fit their

experimental data. This should be discussed using a numerical method to solve the diffusion

equations.

- 21 -

3. CHRONOCOULOMETRY

The electrical charge variation (potential step chonocoulometry (PSCC) method) can also be

used to characterize insertion processes and determine the diffusion coefficients and kinetic

parameters from experimental data. The charge is obtained by integration of the current with

respect to time:

Q t It

( ) = ( )∫ τ τ0

d (47)

and, using dimensionless notation, Q t( ) is divided by the total amount of charge passed following

a potential step, ∆Q Q= ∞( ) , and the time divided by the diffusion time constant τ d .

3.1. Diffusion controlled processes

The Faradaic charge vs. time relationship is derived from Eq. (1), for diffusion controlled

processes and restricted linear diffusion conditions, as:

Q t

Q nn

t

n

d

d

( )–

–exp – ( – )

∆=

( )

=

∞

∑18 1

2 12 1

42 22

2

1π

πτ

(48)

or, which is equivalent, from Eq. (2):

Q t

Q

tn t

tn

n

tn n

nn

d

dd

d d

erfc( )

exp∆

= + −( ) −( ) − −( )

=

∞

=

∞

∑∑2 1 2 12

12

11πτ

τ πτ τ

(49)

The dimensionless charge is plotted vs. dimensionless time in Fig. 9A. The short-time

(Cottrell-domain) expression is obtained either setting t τ d → 0 in Eq. (49) or integrating the

current with respect to time in Eq. (4):

Q t

Q

tQ t Q t FA

Dtcd st

dd st Cottrell

( )( ) ( )

∆∆= ⇔ = = −2 2

πτ π(50)

as indicated by the dashed lines in Figs. 9B and C using the Q t Q t td Cottrell dvs.( ) ( ) ( )log τ and

Q t Q td dvs.( ) ∆ τ graphical representations. Less than 1 % deviation from the limiting Eq. (50)

is observed in Eqs. (48) and (49) when t < 0 32. τ d .

Under the above conditions, the diffusion time constant and therefore the diffusion coefficient

can be determined from the slope of the portion of straight line observed for short times in a

Q t t( ) vs. plot.

On the other hand, the dimensionless charge in Eq. (48) tends towards unity for long times,

- 22 -

according to:

Q t

Q

td lt

d

( )exp

∆= − −

18

42

2

ππτ

(51)

and a log 1 − ( )[ ]Q t Qd ∆ vs. time plot gives a portion of straight line (Fig. 9D) with intercept,

log 8 2π( ) , and slope, − ( )π τ2 4 10d ln , from which the diffusion coefficient can be obtained

provided t > 0 14. τ d (less than 1 % deviation in Eq. (48) from Eq. (51)).

3.2. Influence of insertion reaction kinetics and ohmic drop

The influence of insertion reaction kinetics and ohmic drop on the Faradaic charge variation is

shown in Fig. 10 using the same dimensionless representations as in Fig. 9. The theoretical curve

shape depends only on the parameter Λ in Eqs. (20) and (30), according to the following

expression derived from Eq. (29) by integration of the current with respect to time:

Q t

Q b bb t

n nn

n

( )– exp –

∆=

+ +( ) ( )=

∞

∑1 22

2 2 22

1

ΛΛ Λ

τ d (52)

Large Λ values (diffusion control, ∞ in the figure) give the same theoretical dependences as in

Eq. (48) and Fig. 9. In contrast, small Λ values corresponding to control by interfacial charge

transfer kinetics and/or ohmic drop give the following limiting form of Eq. (52):

Q t Q t( ) – exp –Λ Λ→ = ( )0 1∆ τ d (53)

which can be linearised for short times as:

Q t Q t( )st, dΛ Λ→ =0 ∆ τ (54)

A linear dependence is therefore predicted for the charge vs. time variation in the short-time range

(Fig. 10A) due to constant current for short times when insertion reaction kinetics and/or ohmic

drop control is satisfied.

Deviation from diffusion controlled processes and related Eqs. (48) and (49), due to limitations

by interfacial charge transfer kinetics and/or ohmic drop, is well characterized by the presence of

the maximum of the function plotted in Fig. 10B, which is equivalent to a maximum of the Cottrell

function plot for the charge, Q t t t( ) vs. log , from the experimental point of view. Such a

deviation can also be characterized by the inflection point observed in the graphical representation

(Cottrell plot for the electrical charge) of Fig. 10C.

- 23 -

Finally, from Eq. (52), the long-time variation of Faradaic charge follows the equation:

Q t

Q b bb t

( )explt

d∆= −

+ +( ) −( )1 22

12 2

12 1

2ΛΛ Λ

τ (55)

as illustrated by the semi-logarithmic plot of Fig. 10D. Whatever the Λ value, a portion of straight

line is predicted for long times in a log 1 − ( )[ ]Q t Q∆ vs. time plot with slope, − ( )b12 10τ d ln , and

intercept, log 2 212 2

12Λ Λ Λb b+ +( )[ ] , from which the diffusion coefficient can be obtained if Λ is

not too small. Less than 1 % deviation from Eq. (55) is predicted in Eq. (52) when t > 0 14. τ d for

diffusion controlled processes ( Λ → ∞ ), while a straight line is observed over the whole time

domain when Λ → 0, according to Eq. (53), due to rate control by insertion reaction kinetics

and/or ohmic drop.

As indicated above for the current transient, the theoretical predictions for the Faradaic charge

in Eqs. (48) to (55) and Figs. 9 and 10 are valid either considering small-amplitude potential steps,

irrespective of the insertion isotherm type, the reaction rate expression, the presence or not of

ohmic drop in the electrolyte, bulk material, etc., and the dependence or not of the diffusion

coefficient on the guest species concentration (here the partial derivatives vc and vE in Eq. (30) are

calculated at the initial potential E0 ), or using larger potential steps provided the diffusion

coefficient is constant, the Langmuir isotherm satisfied for the insertion process, ohmic drop

effects can be neglected and the partial derivatives vc and vE are evaluated at the applied potential

E1.

4. DETERMINATION OF THE DIFFUSION COEFFICIENT

OF A GUEST SPECIES

As indicated in the introduction of this paper, the chemical diffusion coefficients of guest

species in host materials are generally determined either from the limiting Cottrell equation or the

exponential decay of current for short- or long-time experiments, respectively.

The error in the determination of D due to limitations by surface reaction kinetics and/or ohmic

potential drop is discussed in this Section. Hereafter we assume that the double-layer charging

current is negligible compared to the Faradaic current over the time range considered and we use

Eqs. (25) and (29) to model the current transient where the parameter Λ takes on the expressions in

Eqs. (20) and (30). Determination of the diffusion coefficients of guest species by curve fitting is

also envisaged using the current vs. time and charge vs. time relationships.

- 24 -

4.1. Determination of D from a Cottrell plot (current vs. time–1/2)

Less than 1% deviation from the limiting Cottrell relationship (Eq. (4)) is predicted in Eq. (29)

when the condition, Λ2 48 5t τ d >( ). I t τ d <( )0 19. , is satisfied, as indicated in Fig. 11. Λ > 16

(very fast insertion kinetics and negligible ohmic drop) is therefore required to observe the Cottrell

domain in PSCA experiments for the diffusion geometry of Fig. 1.

The above condition corresponds to the time domain, 48 5 0 192 2. .R R R Dt Lct d+( )[ ] < <Ω ,

written in terms of characteristic resistances, or alternatively:

48 5 0 191 2

2. .D v R FA c E t L Dc

− +[ ] < <Ω d d (56)

written in terms of thermodynamic, kinetic, diffusion and ohmic parameters. Hence, determination

of the diffusion coefficient from the slope of a Cottrell straight line, using Eq. (8), is only possible

within a limited time range for very fast interfacial charge transfer kinetics (high vc value) and

negligible ohmic drop or IR compensation (low RΩ value).

In contrast, a typical Cottrell plot (current vs. time– /1 2 ) for the current transient caused by a

small-amplitude potential step is given in Fig. 12A, taking kinetic limitations by surface processes

and ohmic drop into consideration (Λ = 3). The plot of I t( ) vs. t– /1 2 (thick solid line) does not

show a linear dependence for short times (no Cottrell straight line for large t– /1 2 values), in

agreement with the predictions in Fig. 11, and therefore Eq. (8) cannot be used to determine

accurately the diffusion coefficient of the guest species.

However, Garcia-Jareño et al. proposed (see Fig. 3 in Ref. [10]) to evaluate an apparent

diffusion coefficient ( Dap ) from the slope of the tangent line (dashed line) in Fig. 12A, using

Eq. (4) and setting the slope of the tangent line equal to ∆Q D Lap π . The Cottrell straight line,

with slope ∆Q D Lπ , is also plotted in Fig. 12A (thin solid line) for the sake of comparison.

On the other hand, Levi et al. noted in Figs. 6(b) and 7(c) of Ref. [9] that the Cottrell function

plot, I t t t( ) vs. log , is expressed experimentally by a shallow peak rather than a horizontal

straight line in the short-time range and taken the peak value of I t t( ) equal to ∆Q D Lap π as

indicated in Fig. 12B of this paper where the characteristic time-invariant function,

I t t Q D LCottrell( ) = ∆ π , is also plotted for comparison.

- 25 -

Despite the different graphical representations in the papers by Levi et al. [9] and Garcia-Jareño

et al. [10], the approximation used by the above authors to determine the apparent diffusion

coefficients of guest species is the same. Moreover, the diffusion coefficient is underestimated

using Eq. (8) and replacing the characteristic time-invariant function, I t tCottrell( ) , by the slope of

the tangent line in Fig. 12A or the peak value of the function in Fig. 12B. The ratio of the apparent

diffusion coefficient to the exact value of D is given by:

D D I t t I t tap max Cottrell= ( )( ) ( )( )[ ]2(57)

This ratio depends only on the dimensionless parameter Λ and is plotted vs. log Λ in Fig. 13A.

Less than 5 % relative error on D is predicted provided Λ > 11 (fast charge transfer kinetics). In

contrast, the determination error taken in absolute value increases rapidly when Λ decreases due to

slow insertion reaction kinetics and/or large ohmic drop.

The Λ value for a given electrode depends on the insertion isotherm type (Langmuir, Frumkin

or other isotherm), the initial potential of potential step, the insertion reaction rate constant, the

diffusion coefficient of guest species, the film, foil or particle thickness in Figs. 1A and B, and the

presence of ohmic drop in the electrolyte, bulk material, etc. However, considering Eq. (20), Λ can

be estimated from experimental values of the diffusion resistance, charge transfer resistance and

electrolyte and/or substrate resistances measured on an impedance diagram plotted for the same

potential range, as shown in Fig. 13B. The corresponding error for the diffusion coefficient, when

using the approximation proposed by Levi et al. [9] and Garcia-Jareño et al. [10], is then given in

Fig. 13A. For example, Λ = 3, which corresponds to the dashed line in Fig. 13A and the impedance

diagram of Fig. 13B, gives a relative error of ε = – 26 % for the diffusion coefficient.

4.2. Determination of D from a logcurrent vs. time plot

The diffusion coefficient of a guest species can be determined alternatively from the slope and

intercept of the straight line observed when plotting log current vs. time for sufficiently long

times. The intercept ( Int ,ltI ) and slope (Sl ,ltI ), for this line, are given from Eq. (43) by:

Int ,ltd

IQ

b=

+ +( )

log 22

212

∆τ

ΛΛ Λ

(58)

and:

Sl ,ltd

Ib= − 1

2

10τ ln(59)

- 26 -

Eqs. (58) and (59) should be solved together with the equation:

b b1 1 0tan − =Λ (60)

to determine Λ , τ d and therefore D, provided the diffusion length is known independently and Λ

is not too small. In contrast, if Λ takes on a very low value, due to slow interfacial charge transfer

kinetics and/or large ohmic drop, we have b1 = Λ and b v L R FA v12 1τ d c E= ( ) +( )Ω from

Eqs. (25) and (30), respectively, so the diffusion coefficient cannot be obtained from Eqs. (58) to

(60).

A portion of straight line is predicted for a log current vs. time plot, in the time domain

indicated in Fig. 11 (less than 1 % deviation in Eq. (29) from Eq. (43)), whether diffusion control

(Eq. (41)), charge transfer and/or ohmic drop control (Eq. (42)) or mixed control conditions

(Eq. (43)) are satisfied for insertion processes. Hence, Eq. (43) and Eqs. (58) to (60) derived in

this paper should be used rather than the usual Eqs. (5), (9) and (10) to fit PSCA data in the long-

time domain. The use of the latter equations give only apparent values of the diffusion coefficient

and the ratio of Dap to the exact value of D is calculated as:

D D bap = 4 12 2π (61)

and:

D D bap = + +( )Λ Λ Λ2 212 (62)

considering respectively the slope and intercept of the straight line. The diffusion coefficient ratio

depends only on the dimensionless parameter Λ and is plotted vs. log Λ in Figs. 14A and B. Less

than 5 % relative error on D is predicted provided Λ > 38 (or 21) using the slope (or intercept) of

the straight line, which corresponds to very fast insertion kinetics and very low ohmic drop. The

error increases rapidly in absolute value when Λ decreases. For example, the relative error on D

predicted for Λ = 3, which corresponds to the dashed lines in Fig. 14 and the impedance diagram

of Fig. 13B, is equal to ε = – 42 % and ε = – 33 % from Figs. 14A and B, respectively.

The problem in the determination of D from the slope of a log I t( ) vs. time plot in the long-

time range was pointed out first by Chen [29] for hydrogen absorption in thin foils of Pd and Pd

alloys, to the best of our knowledge. Chen observed that the diffusion coefficient of hydrogen

determined from Eq. (10) disagreed with the values obtained from EIS data. This was the starting

point of our previous theoretical work [19]. Lundqvist and Lindberg [30] discussed more recently

- 27 -

the same problem for diffusion of hydrogen inside spherical particles of hydrogen absorbing

alloys.

4.3. Determination of D by current vs. time curve fitting

The more general way to determine the diffusion coefficients of guest species from

experimental chronoamperograms is to use Eqs. (25) and (29) and a curve fitting procedure with

adjustable parameters Λ and τ d = L D2 , from which D can be obtained. Such a procedure was

used by Chen [29] to study hydrogen absorption in α phase Pd-H and Pd alloy-H electrodes. The

hydrogen concentration and the diffusion coefficient of this species determined by curve fitting

were in very good agreement with the values obtained from EIS data.

Levi et al. [9] indicated that lithium-ion insertion into Li Co Ni Ox 0.2 0.8 2 in the potential range of

3.65 to 4.05 V vs. Li+Li proceeds via the formation of a solid solution. They calculated the

apparent diffusion coefficient of lithium-ion vs. electrode potential from PSCA data using Eq. (8)

above where the time-invariant Cottrell function, I t tCottrell( ) , is replaced by the peak value

I t t( )max

as indicated in Fig. 12B. Under the above conditions, Dap is underestimated with

respect to D as discussed in Section 4.1. We therefore decided to check whether or not the

experimental results of Levi et al. can be fitted by Eqs. (25) and (29) derived in this work.

Experimental data taken from Fig. 6(b) in Ref. [9] are plotted in Fig. 15 of this paper for a

potential step from 3.70 to 3.73 V vs. Li+Li, using the Cottrell function representation of the

current transient, I t t t( ) vs. log . Fig. 15A shows that the experimental results obtained by Levi et

al. are not fitted very well by the theoretical model assuming restricted linear diffusion conditions

and limitations by insertion reaction kinetics and/or ohmic drop. Therefore, from Eq. (11), we

predict that EIS data collected for the same material cannot be fitted in the low frequency range to

the theoretical expression for the restricted (finite space) diffusion impedance in Eqs. (13)

and (14). This was observed by Levi and Aurbach for lithium-ion insertion in thin graphite

electrodes (see Figs. 13 and 14 in Ref. [6]).

Despite the fact that the potential step amplitude, ∆E = 30 mV in the work by Levi et al. [9] and

in Fig. 15 of this paper, does not satisfy the small-signal condition, ∆E RT F<< , and therefore the

diffusion coefficient of lithium-ion cannot be considered strictly constant in this potential range,

deviation from Eq. (29) is due mainly to long-time experimental data (compare Figs. 15A and B)

and is related probably to the geometry of diffusion in graphite particles and/or the distribution of

- 28 -

particle size and therefore diffusion length in the composite electrode. This will be investigated in

the second and third parts of this work [16].

4.4. Determination of D from chronocoulometric data

Considering diffusion controlled processes, due to very fast insertion reaction kinetics and

negligible ohmic drop or IR compensation, the diffusion coefficient of a guest species can be

determined (i) from the slope ( Sl stQ, ) of the straight line predicted for short times ( t < 0 32. τ d) in a

Q t t( ) vs. plot (Cottrell plot for the charge), derived from Eq. (50) as:

DQ

LFA c

Q Q=

=

π πSl Slst st, ,

2 2

2 2

∆ ∆(63)

(ii) from the slope ( Sl ltQ, ) of the portion of straight line observed for long times ( t > 0 14. τ d) in a

log 1 − ( )[ ]Q t Q∆ vs. time plot, according to Eq. (51):

D L Q= 4 102 2ln ,Sl lt π (64)

and, more generally, over a larger time range (iii) using Eq. (48) or Eq. (49) and a curve fitting

procedure with adjustable parameter τ d = L D2 .

On the other hand, under mixed control conditions, D could be obtained from the slope and

intercept of the portion of straight line predicted for long times in a log 1 − ( )[ ]Q t Q∆ vs. time plot.

From Eq. (55), the intercept and slope, for this line, are given by:

Int ltQb b

, log=+ +( )

2 2

12 2

12

ΛΛ Λ

(65)

and:

Sl ltQ b D L, ln= − ( )12 2 10 (66)

Eqs. (65) and (66) should be solved together with Eq. (60) to determine Λ and D, provided Λ is not

too small (no charge transfer and/or ohmic drop control).

More generally, chronocoulometric data could be compared with Eq. (55) using a curve fitting

procedure with adjustable parameters Λ and τ d .

5. CONCLUSION

In this paper, we have given five equivalent theoretical expressions for the Faradaic current

transient caused by a potential step, assuming linear diffusion of guest species in thin films, foils or

- 29 -

platelet particles of host material (in the latter case the potential gradient is neglected in the

composite electrode), restricted (blocking) diffusion conditions (Figs. 1A and B) and kinetic

limitations at the host material surface. Provided a direct (one-step) insertion reaction is assumed,

restricted linear diffusion satisfied, phase transition processes disregarded and ohmic potential

drop as well as double-layer charging current effects neglected, the current vs. time relationship can

be written as:

I tQ

bb t

nn

n

( ) exp –=+ + ( )

=

∞

∑22

2 22

1

∆τ

τd

dΛ

Λ Λ(A)

where bn is the nth positive root of the equation: b btan – Λ = 0 . The above expression includes

only two adjustable parameters: the diffusion time constant, τ d = L D2 , and the dimensionless

parameter, Λ = v mc , which compares the reaction rate constant and diffusion constant. A large Λ

value means a diffusion controlled rate; in contrast a small value of Λ is characteristic of control by

interfacial charge transfer kinetics. Mixed control conditions are achieved for intermediate values of

this parameter. Finally, the total amount of Faradaic charge passed following a potential step is

obtained by integration of the current with respect to time, assuming equilibrium for the insertion

reaction both at the initial and final times, as ∆Q I t t= ( )∞

∫ d0

. Alternative formulations of the above

equation have been derived in this paper in terms of electrical components of the equivalent circuit

for the electrode impedance in Eqs. (26) and (27), as well as in terms of thermodynamic, kinetic

and diffusion parameters in Eq. (28).

Eq. (A) applies either considering small-amplitude potential steps, irrespective of the insertion

isotherm type (Langmuir, Frumkin or other isotherm), the related kinetic equation (Butler-Volmer

or other relationship) and the dependence or not of the diffusion coefficient on the guest species

concentration and therefore on the electrode potential, or using larger potential steps provided the

diffusion coefficient is constant and the Langmuir isotherm applies for the insertion process. The

formation of an ideal solid solution, with no interaction in the host material, is assumed in the latter

case. The only difference between the two situations lies in the electrode potential where the partial

derivative vc is calculated. For a small potential step from E0 to E E E1 0= + ∆ , with ∆E RT F<<, the partial derivative is considered for the initial potential of potential step. In contrast, vc is

calculated for the applied potential when considering larger potential steps.

Setting Λ → ∞ in Eq. (A), we obtain the well-known diffusion law in thin film materials for

diffusion controlled processes and restricted linear diffusion conditions [2, 4]:

I tQ

nt

d ( ) exp – ( – )=

∞

∑2 2 1 22∆

τπτ

(B)

- 30 -

and the two limiting relationships satisfied respectively for short and long times respectively:

I tQ

td std

( ) = ∆π τ

(C)

and:

I tQ t

d ltd d

( ) exp –=

24

2∆τ

πτ

(D)

Accurate determination of the diffusion coefficient of a guest species from the linear part of a

Cottrell plot (current vs. time–1/2) in the short time domain or a semi-logarithmic plot

( log current vs. time) in the long time domain, using respectively Eqs. (C) and (D) to fit

experimental data, requires (i) very fast charge transfer kinetics at the interfaces, (ii) very low

double-layer charging current compared to the Faradaic current, (iii) negligible ohmic drop or IR

compensation, (iv) no or negligible film resistance (or more generally impedance) on the electrode

surface due to passivation layers (on graphite doped by lithium for example), and finally (v) no

phase transition processes.

Under the above conditions, the electrode impedance is close to the diffusion impedance which

is characteristic of restricted linear diffusion. Experimental observation of more complicated

impedance diagrams for insertion materials, due to ohmic drop, charge transfer limitations,

presence of superficial passivation layers, radial diffusion with cylindrical or spherical symmetry,

particle size distribution in composite electrodes, porous electrode effects or phase transition

processes should indicate that the diffusion model in Eqs. (B) to (D), which is widely used in the

electrochemical literature to fit experimental data, cannot accurately represent potential step

chronoamperometric data obtained for the same materials. Hence, there is an advantage to the

simultaneous application of PSCA and EIS methods for the characterization of ion-insertion

materials.

In contrast to Eq. (B), Eq. (A) applies whether diffusion, interfacial charge transfer or mixed

control conditions are satisfied experimentally. In addition, considering small-amplitude potential

steps, typically when ∆E RT F<< , this equation can be extended to cover the case where ohmic

drop is present by setting Λ = +( )R R Rd ctΩ where the charge transfer resistance Rct , the

diffusion resistance Rd , and the resistance RΩ related to ohmic drop in the electrolyte, the host

material, etc., can be obtained from an impedance diagram plotted at the initial potential of potential

step.

Using small potential steps and considering single-phase materials and/or potential domains,

- 31 -

the dimensionless parameter Λ in Eq. (A) is the key factor for the kinetics of insertion processes

studied by the PSCA method. The general expression for this parameter is written in terms of

partial derivatives of insertion reaction rate with respect to electrode potential and interfacial

concentration of guest species, both evaluated at the initial potential E0 , as:

Λ =+

v L D

R FA vc

E1 Ω

(E)

The Λ value for PSCA experiments depends on the insertion isotherm type (Langmuir, Frumkin or

other isotherm), the initial potential of potential step, the reaction rate constant which is included in

the partial derivatives, the diffusion coefficient of guest species, the film, foil or particle thickness in

Figs. 1A and B, and the presence of ohmic drop in the electrolyte, bulk material, etc.

An important feature of Eq. (A) is that additional resistances, i.e. electrolyte, bulk material and

substrate resistances, superficial film resistances and interfacial charge transfer resistance,

depending on the electrochemical system considered, severely distort the shape of the current

transient as well as the Cottrell function plot, I t t t( ) vs. log , in the short-time domain, thus

masking the expected responses for diffusion controlled processes, as envisaged by Levi et al. [6,

9], but the current is also largely affected in the long-time domain by slow charge transfer kinetics

and/or ohmic drop. Under the above circumstances, the effects of slow ion-transfer kinetics at the

host materialelectrolyte interface, as well as slow electron-transfer kinetics at the host

materialsubstrate (or current collector) interface, cannot be distinguished from ohmic drop effects.

In contrast, ohmic drop effects and kinetic limitations by interfacial charge transfer can be

discriminated using the EIS method and the Nyquist representation of impedance diagrams.

For long-time experiments, only the first term of the series in Eq. (A) is considered, so the

current taken in absolute value decreases exponentially with increasing time according to:

I tQ

bb t( ) exp –lt

dd=

+ +( ) ( )22

212 1

2∆τ

τΛΛ Λ

(F)

Hence, experimental observation of an exponential decay of current with respect to time, in the

long-time range, is not a sufficient condition for predicting a diffusion controlled rate for insertion

processes in thin films, foils or particles of host materials. It follows that a log I t( ) vs. t plot

should be linear for long times whether diffusion, charge transfer, ohmic drop or mixed control by

diffusion and interfacial charge transfer and/or ohmic drop is satisfied experimentally.

The validity domains for the Cottrell current in Eq. (C) and the exponential decay of current vs.

time in Eq. (F) are visualized in Fig. 11 of this paper in terms of the dimensionless parameter Λ

and dimensionless time t τ d .

- 32 -

Considering experimental PSCA or PSCC data obtained under restricted linear diffusion

conditions, deviation from diffusion controlled processes and the usual Eq. (B), due to limitations

by insertion reaction kinetics and/or ohmic drop, can be characterized from (i) the presence of a

maximum in a I t t t t( ) vs. or log plot as well as in a Q t t t t( ) vs. or log plot, i.e. using the

Cottrell function representation for current and charge, respectively, (ii) the absence of a portion of

straight line in a I t t( ) vs. 1 plot (usual Cottrell plot) for short times (large 1 t values), (iii) the

presence of an inflection point in a Q t t( ) vs. plot (Cottrell plot for the charge) and finally, (iv)

from the value of the slope of a log logcurrent vs. time plot, i.e. − < ( ) <1 2 0d dlog logI t t ,

measured for short times.

We also predicted in this work the errors in the determination of the diffusion coefficients of

guest species due to limitations by surface reaction kinetics and/or ohmic drop when using the

methods proposed in the electrochemical literature and based on the slope of the tangent line in a

Cottrell plot, I t t( ) vs. – /1 2, the peak value of the Cottrell function plot, I t t t( ) vs. log , and finally

the slope and intercept of a semi-logarithmic plot, log .I t t( ) vs . In all cases, the apparent diffusion

coefficient obtained using the limiting Eqs. (C) and (D) instead of the more general equations

given in this paper is underestimated with respect to the exact value of D. In addition, we envisaged

the possibility for determining D from chronocoulometric data using a Cottrell plot for the charge,

Q t t( ) vs. , in the short-time domain, or a semi-logarithmic plot, log 1 − ( ) ∞( )[ ]Q t Q vs. time, in

the long-time domain.

The more general way to determine the diffusion coefficients of guest species from expe-

rimental chronoamperograms obtained for small-amplitude potential steps (as well as large

potential steps if D is constant with respect to the electrode potential, Langmuir isotherm conditions

are satisfied for the insertion process and ohmic drop effects can be disregarded) is to use Eq. (A)

and a curve fitting procedure with adjustable parameters Λ and τ d = L D2 , from which D can be

obtained provided the diffusion length L is known independently. Alternatively, the electrical

charge can be measured vs. time and compared to the following expression with the same

adjustable parameters, Λ and τ d , as above:

Q t Qb b

b tn n

n

n

( ) = ∞( )+ +( ) ( )

=

∞

∑1 22

2 2 22

1

– exp –Λ

Λ Λτ d (G)

It should be noted that the theoretical derivations in this paper are based on a simple geometric

model for insertion and diffusion processes in thin film electrodes and particles schematized in

Fig. 1. Real cases will certainly be more complex and the derivation of diffusion coefficients more

difficult. Complications with respect to ideal conditions may be encountered for insertion

- 33 -

processes in relation to the geometry of diffusion in host materials and the distribution of diffusion

lengths in electroactive particles of composite electrodes. This will be dealt with respectively in the

second and third parts of this series [16].

Finally, the ‘cell-impedance controlled diffusion’ model presented recently by Shin and Pyun

[8] and used by Shin et al. [31] to fit experimental data for lithium-ion insertion in transition metal

oxides is an alternative to the theoretical derivation in this work. The two models will be compared

in a separate paper [32].

Acknowledgments

Thanks are due to J.-P. Diard for his help in using Mathematica software.

- 34 -

APPENDIX A

The insertion reaction for cations can be written formally as:

M e M , e+ − + −+ + ↔ (A.1)

where M+ , and M , e+ − denote respectively a cation in the electrolyte solution, the insertion

site in the host material and the intercalated species. Assuming Langmuir isotherm conditions

(formation of an ideal solid solution) for the insertion process and a constant concentration ( cM+ )

in the electrolyte solution, the reaction rate is given by:

v t K t c c c t K t c t( ) = ( ) − ( )[ ] − ( ) ( )r M max o+ 0 0, , (A.2)

where c t0,( ) is the guest species concentration at the interface (x = 0, Fig. 1), cmax the maximal

(saturation) concentration, and K to( ) and K tr ( ) denote the rate constants in the direction of

oxidation and reduction, respectively, with:

K t k fE to o o( ) = ( )[ ]exp α , K t k fE tr r r( ) = − ( )[ ]exp α (A.3)

where E t( ) is the electrode potential, f F RT= ( ) and the symmetry factors for charge transfer

satisfy α αo r+ = 1.

E* denotes below the equilibrium potential corresponding to the constant concentrations cM+

in the electrolyte and cmax 2 in the host material. The formal rate constant, k* cms 1− , is given by:

k k fE k c fE* * *exp exp= ( ) = −( )o o r M r+α α (A.4)

and the reaction rate can be rewritten from Eqs. (A.2) to (A.4), setting:

ξ t f E t E( ) = ( ) −[ ]* (A.5)

as:

v t k c c t t c t t( ) = − ( )[ ] − ( )[ ] − ( ) ( )[ ] * , exp , expmax r o0 0α ξ α ξ (A.6)

The partial derivatives of reaction rate with respect to interfacial concentration of guest species

and electrode potential, both evaluated at the initial potential E0 , are calculated as:

- 35 -

v kc = − −( ) +( )* exp expα ξ ξr 0 01 (A.7)

v f k c c cE = − −( ) −( ) +[ ]* exp expα ξ α α ξr r max o0 0 0 0 (A.8)

where:

ξ0 0= −( )f E E* (A.9)

and, due to the assumption of Langmuir isotherm conditions:

c c0 01= +( )max expξ (A.10)

From Eqs. (A.8) and (A.10), we obtain:

v f k cE = −−( )[ ]

+* exp

expmaxr1

10

0

α ξξ

(A.11)

and the theoretical expression for the dimensionless parameter Λ, under Langmuir isotherm

conditions for the insertion process, is finally derived from Eqs. (30), (A.7) and (A.11) as:

Λ =( ) −( ) +( )

+ −( )[ ] +( )k L D

R f FAk c

*

*

exp exp

exp exp

α ξ ξα ξ ξ

r

max r

0 0

0 0

1

1 1 1Ω

(A.12)

Taking cM

1+ mol L= −1 (standard concentration in the electrolyte solution) for the sake of

simplification, E* is the standard (formal) potential E°, k* in Eq. (A.4) is the standard rate

constant k°, and Eq. (31) in the text is obtained from Eq. (A.12).

- 36 -

APPENDIX B. NOMENCLATURE

Abbreviations

EIS Electrochemical impedance spectroscopy.

Int Intercept.

PITT Potentiostatic intermittent titration technique.

PSCA Potential step chronoamperometry.

PSCC Potential step chronocoulometry.

Sl Slope.

Subscripts

ct Related to charge transfer.

d Related to diffusion.

dl Double layer.

F Faradaic.

I Related to current.

lt Long time.

Q Related to charge.

st Short time.

vst Very short time.

Ω Related to ohmic drop.

Roman letters

A Interfacial surface area.

bn nth positive root of the equation: b btan − =Λ 0 .

c Guest species concentration.

c0 Equilibrium concentration at the initial potential E0 .

c1 Equilibrium concentration at the applied potential E1.

cmax Maximal (saturation) concentration in the host material.

c° Standard concentration in the solid solution: c c° = max 2 .

Cdl Differential (double-layer) capacitance.

D Diffusion coefficient.

Dap Apparent diffusion coefficient.

- 37 -

E Electrode potential.

E0 Initial potential.

E1 Applied potential.

E° Standard potential.

E* Formal potential.

f f F RT= ( ).

F Faraday constant.

I Faradaic current.

Id Diffusion current.

ICottrell Semi-infinite linear diffusion current.

J Diffusion flux of guest species.

k° Standard rate constant.

k* Formal rate constant.

L Film thickness (Fig. 1A), or foil or particle half-thickness (Fig. 1B).

m Diffusion constant for linear diffusion: m D L= .

Q Faradaic charge.

QCottrell Faradaic charge related to the Cottrell current.

Qd Faradaic charge related to the diffusion current.

R Perfect gas constant.

Rct Charge transfer resistance.

Rd Diffusion resistance.

Rs Electrolyte solution resistance.

RΩ Sum of ohmic resistances.

s Laplace complex variable.

t Time variable.

T Absolute temperature.

v Insertion reaction rate.

vc Partial derivative: v v cc E= ∂ ∂( ) .

vE Partial derivative: v v EE c= ∂ ∂( ) .

x Space variable.

y Insertion level: y c c= max .

Z Electrode impedance.

Zd Diffusion impedance.

ZF Faradaic impedance.

Greek letters

α α phase material.

- 38 -

αo Symmetry factor for charge transfer in the direction of oxidation.

αr Symmetry factor for charge transfer in the direction of reduction: αo + αr = 1.

∆c Variation of guest species concentration due to a potential step: ∆c c c= −1 0 .

∆E Potential step amplitude: ∆E E E= −1 0.

∆Q Total amount of Faradaic charge passed following a potential step: ∆Q Q= ∞( ) .

∆Qd ∆Q for diffusion controlled processes.

Λ Key parameter for the kinetics of insertion processes in Eqs. (20) and (30).

τ d Time constant for linear diffusion: τ d = L D2 .

ξ Dimensionless potential: ξ = − °( )f E E .

ξ0 Dimensionless initial potential: ξ0 0= − °( )f E E .

ξ1 Dimensionless applied potential: ξ1 1= − °( )f E E .

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[25] C. Montella, J. Electroanal. Chem. 462 (1999) 73.

- 40 -

[26] J. Crank, The Mathematics of Diffusion, 2nd edition, Clarendon Press, Oxford, 1975, p. 60.

[27] A.J. Bard, L.R. Faulkner, Électrochimie, Principes, Méthodes et Applications, Masson, Paris,

1983, p. 185.

[28] IUPAC, M. Sluyters-Rehbach (ed.), Pure & Appl. Chem. 66 (1994) 1831.

[29] J.S. Chen, Thesis, Grenoble, 1992.

[30] A. Lundqvist, G. Lindbergh, J. Electrochem. Soc. 145 (1998) 3740.

[31] H.-C. Shin, S.-I. Pyun, S.-W. Kim, M.-H. Lee, Electrochim. Acta 46 (2001) 897.

[32] C. Montella, in preparation.

Figure captions

Fig. 1: Schematic representation of ion-insertion and diffusion processes, and examples of

concentration profiles, c x t x,( ) vs. at a time t, in thin film electrodes, under restricted linear

diffusion conditions. (A) Thin film, of thickness L, deposited on a substrate impermeable to the

diffusing species. (B) Thin foil or platelet particle, of thickness 2L, symmetrically submitted to

insertion on both sides. Arrows indicate the directions for ion transfer through the interfaces.

Fig. 2: Dimensionless representations of the current transient caused by a potential step for

diffusion controlled processes and restricted linear diffusion conditions. (A) Current vs. time curve

calculated from Eqs. (1) and (2). (B) Ratio of the diffusion current to the limiting Cottrell Eq. (4)

plotted vs. the decimal logarithm of time (solid line). (C) Cottrell plot of the current transient (solid

line) and limiting Cottrell relationship (dashed line) plotted from Eq. (4). (D) Decimal logarithm of

current plotted vs. time (solid line) and asymptotic straight line (dashed line) plotted from Eq. (5).

Fig. 3: Dimensionless representations of the current transient caused by a potential step for

diffusion controlled processes and restricted linear diffusion conditions. (A) Current vs. time curve

calculated from Eqs. (1) and (2). (B) Logarithm of current plotted vs. logarithm of time (solid line)

and same representation of the Cottrell relationship from Eq. (4) (dashed line). (C) Derivative of

the curve in (B) plotted vs. logarithm of time (solid line). (D) Derivative of the curve in (B) plotted

- 41 -

vs. time (solid line). The asymptotic straight line (dashed line) satisfied for long times is plotted

from Eq. (7).

Fig. 4: Evolution of normalized concentration profile vs. time calculated from Eqs. (25) and (33)

for Λ = 103 (diffusion controlled processes) and t τ d = 0.002 (a), 0.01, 0.025, 0.05, 0.1 (e), 0.2,

0.3, 0.4, 0.5, 0.7, 1 (k).

Fig. 5: Evolution of normalized concentration profile vs. time calculated from Eqs. (25) and (33)

for Λ = −10 2 (rate control by insertion reaction kinetics and/or ohmic drop) and t τ d = 10 (a), 30,

50, 70, 90 (e), 110, 130, 150, 170, 190, 210, 230 (1).

Fig. 6: Evolution of normalized concentration profile vs. time calculated from Eqs. (25) and (33)

for Λ = 3 (mixed control by diffusion and insertion reaction kinetics and/or ohmic drop) and

t τ d = 0.002 (a), 0.01, 0.025, 0.05, 0.1 (e), 0.2, 0.3, 0.4, 0.5, 0.7, 1, 1.5 (l).

Fig. 7: Influence of interfacial charge transfer kinetics and/or ohmic potential drop on the Faradaic

current transient caused by a potential step, under restricted linear diffusion conditions, using the

same dimensionless representations as in Fig. 2. The current is calculated from Eqs. (25) and (29)

and plotted in dimensionless form for Λ = 0.25, 0.5, 1, 2, 5 and very large values (∞). (A) Current

vs. time curve. (B) Ratio of the Faradaic current to the limiting Cottrell Eq. (4) plotted vs. the

decimal logarithm of time (solid lines). (C) Cottrell plot of the current transient (solid lines) and

limiting Cottrell equation (dashed line) plotted from Eq. (4). (D) Decimal logarithm of current

plotted vs. time (solid lines) and asymptotic straight lines (dashed lines) plotted from Eqs. (25) and

(43).

Fig. 8: Influence of interfacial charge transfer kinetics and/or ohmic potential drop on the Faradaic

current transient caused by a potential step, under restricted linear diffusion conditions, using the

same dimensionless representations as in Fig. 3. The current is calculated from Eqs. (25) and (29)

and plotted in dimensionless form for Λ = 0.25, 0.5, 1, 2, 5 and very large values (∞). (A) Current

vs. time curve. (B) Logarithm of current plotted vs. logarithm of time. (C) Derivatives of the curves

in (B) plotted vs. logarithm of time. (D) Derivatives of the curves in (B) plotted vs. time (solid

lines). The asymptotic straight lines (dashed lines) are plotted from Eqs. (25) and (44).

Fig. 9: Dimensionless representations of the Faradaic charge variation due to a potential step for

diffusion controlled processes under restricted linear diffusion conditions. (A) Charge vs. time

curve plotted from Eq. (48) or (49). (B) Charge referred to the Cottrell relationship in Eq. (50) and

- 42 -

plotted vs. the decimal logarithm of time (solid line). (C) Cottrell plot of the Faradaic charge (solid

line) and limiting straight line (dashed line) plotted for short times from Eq. (50). (D) Logarithm of

1 − ( )Q t Qd ∆ plotted vs. time (solid line) and asymptotic straight line (dashed line) plotted from

Eq. (51).

Fig. 10: Influence of interfacial charge transfer kinetics and/or ohmic potential drop on the

Faradaic charge variation caused by a potential step, under restricted linear diffusion conditions,

using the same dimensionless representations as in Fig. 9. The charge is calculated from Eqs. (25)

and (52) and plotted in dimensionless form for Λ = 0.1, 0.2, 0.5, 1, 2, 5, 10 and very large values

(∞). (A) Charge vs. time curve. (B) Charge referred to the Cottrell relationship in Eq. (50) and

plotted vs. the decimal logarithm of time (solid lines). (C) Cottrell plot of the Faradaic charge (solid

lines) and limiting straight line (dashed line) for short times plotted from Eq. (50). (D) Logarithm

of 1 − ( )Q t Q∆ plotted vs. time (solid lines) and asymptotic straight lines (dashed lines) plotted

from Eqs. (25) and (55).

Fig. 11: Validity domains for (less than 1 % deviation from) the Cottrell current in Eq. (4) and the

exponential decay of current vs. time in Eq. (43), determined from Eqs. (25) and (29).

Fig. 12: Principles for the determination of the apparent diffusion coefficients of guest species in

host materials used in Refs. [9, 10]. (A) Cottrell plot of the Faradaic current obtained from

Eqs. (25) and (29) (thick solid line), tangent line (dashed line) and Cottrell straight line (thin solid

line) plotted for Λ = 3, ∆Q = 50 mC and τ d s= 500 . (B) Cottrell function plot for the current

transient (thick solid line) vs. the decimal logarithm of time, using the same parameter values as

above. The maximal value of this function is compared to the characteristic time-invariant function

I t tCottrell( ) (thin solid line).

Fig. 13: (A) Error in the determination of the diffusion coefficient of a guest species due to slow

charge transfer kinetics and/or ohmic potential drop when using Eq. (8) and the slope of the

tangent line in Fig. 12A or the peak value of the function in Fig. 12B instead of the characteristic

time-invariant function I t tCottrell( ) in Eq. (4). The ratio of the apparent diffusion coefficient to the

exact value of D is plotted vs. the decimal logarithm of the dimensionless parameter

Λ = +( )R R Rd ctΩ . (B) Example of impedance diagram plotted using the Nyquist representation

for R R RΩ = =ct d 6 and therefore Λ = 3, assuming that the differential (double-layer) capacitance

is very low compared to the insertion capacitance of the host material. Characteristic angular

frequencies and resistances are given in the graph. Additional impedances due to superficial films

on the host material (e.g. for lithium-ion insertion into graphite) are omitted in the impedance

diagram for the sake of simplicity.

- 43 -

Fig. 14: Ratio of the apparent diffusion coefficient to the exact value of D given as a function of

log Λ when using the slope (A) and intercept (B) of the straight line ( log .I t t( ) vs ) observed in the

long-time range and the usual relationships in Eqs. (5), (9) and (10) instead of the more general

Eq. (43) and Eqs. (58) to (60) derived in this paper.

Fig. 15: Experimental data (•) taken from Fig. 6(b) in Ref. [9] for lithium-ion insertion in

Li Co Ni Ox 0.2 0.8 2 and a potential step from 3.70 to 3.73 V vs. Li Li+ , plotted using the Cottrell

function representation of the current transient ( I t t t( ) vs. log ) and compared to the model in

Eqs. (25) and (29) of this paper for restricted linear diffusion conditions with limitations by

surface reaction kinetics and/or ohmic drop. Best fit parameter ( Λ and dτ ) values correspond to the

solid lines. (A) Data fit over the whole time domain. (B) Data fit limited to the short-time range.

A B

0

elec

trol

yte

elec

trol

yte

L 2L

c(x, t)elec

trol

yte

0

subs

trat

e

L

c(x, t)

hostmaterial

host material

Figure 1

- 44 -

0 2 40

.5

1

0 1-1

0

0 1 20

1

2

-1 0 10

.5

1

t τ d

t τ d

log t τ d( )

1 t τ d

It

Qd

d2

()(

)∆

τI

tQ

dd

2()

()

∆τ

It

It

dC

ottr

ell

()()

log

It

Qd

d2

()(

)[

]∆

τ

A

0.5

B

C D

Figure 2

- 45 -

-2 -1 0 1

0

-1

0 0.25 0.5

0

-1

0 1 20

1

2

-2 -1 0 1

0

-1

-2

t τ d

t τ d

log t τ d( )

It

Qd

d2

()(

)∆

τ

log

It

Qd

d()

()

[]

2∆τ

log t τ d( )

dd

dlo

glo

gI

tt

()

dd

dlo

glo

gI

tt

()

–1/2 –1/2

A B

C D

Figure 3

- 46 -

0 10

1

x L

cx

tc

cc

,(

)−[

]−

()

01

0

a

e

k

Figure 4

0 10

1

x L

cx

tc

cc

,(

)−[

]−

()

01

0

a

e

l

Figure 5

0 10

1

x L

cx

tc

cc

,(

)−[

]−

()

01

0

a

e

l

Figure 6

- 47 -

0 2 40

.5

1

0 1

-1

0

0 1 20

1

2

-1 0 10

.5

1

t τ d

t τ d

log t τ d( )

1 t τ d

It

Q()

()

2d

∆τ

It

Q()

()

2d

∆τ

It

It

()()

Cot

trel

llo

gI

tQ

()(

)[

]2

d∆

τ

0.250.25

0.25 0.25

∞

11

1

0.5

∞

∞ ∞

C D

A B

Figure 7

- 48 -

-2 -1 0 1

0

-1

0 .5 1

0

-1

0 1 20

1

2

-2 -1 0 1

0

-1

-2

t τ d

t τ d

log t τ d( )

It

Q()

()

2d

∆τ

log

It

Q()

()

[]

2∆τ d

log t τ d( )

dd

log

log

It

t()

dd

log

log

It

t()

–1/2 –1/2

0.25

0.25

0.25

0.25

∞

1

1

1

∞

∞ ∞

A B

C D

Figure 8

- 49 -

0 1 2 30

.5

1

0 0.5 1

0

-1

0 1 2 30

.5

1

-1 0 1 20

.5

1

t τ d

Qt

Qd()

∆

Qt

Qt

dC

ottr

ell

()()

t τ d log t τ d( )

Qt

Qd()

∆

t τ d

log

1−

()[

]Q

tQ

d∆

A B

C D

Figure 9

- 50 -

0 1 2 3 40

.5

1

0 0.25 0.5

0

.2

.4

0 1 2 3 40

.5

1

-1 0 1 20

.5

1

t τ d

Qt

Q()

∆

Qt

Qt

()()

Cot

trel

lt τ d log t τ d( )

Qt

Q()

∆

0.1

0.1

∞

1

1

∞

∞

1

10

0.1

0.1

1

10∞

t τ d

log

1−

()[

]Q

tQ∆

A B

C D

Figure 10

- 51 -

-3 -2 -1 0 1-2

-1

0

1

2

3Cottrell domain

Eq. ( )4

Exponentialdecay

Eq. ( )43

b b1 1tan = Λ

Eq. ( )42 b1 ≈ Λ

Eq. ( )41

b1 2≈ π

log t τ d( )

log

Λ

Figure 11

0 0.2 0.40

0.2

0.4

0 1 2 3 40

0.5

1.0

1.5

log t s( )1 t s

It ()

mA

It

t()

mA

s12/

I t tCottrell ( )

I t t( )max

I tCottrell ( )

A B

Figure 12

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1.0

log Λ

DD

ap

Re Z

–Im

Z

RΩ Rct Rd 3

3 88. τ d

1 R Cct dl( )

A B

Figure 13

- 52 -

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1.0

log Λ

DD

ap

log Λ

DD

ap

A B

Figure 14

-1 0 1 2 30

0.5

1.0

-1 0 1 20

0.5

1.0

log t s( ) log t s( )

It

t()

mA

s1/2

A B

It

t()

mA

s1/2

Figure 15

- 53 -