KINETICS AND DYNAMICS OF ION INSERTION INTO HOST ELECTRODES
FOR RECHARGEABLE Li AND Mg BATTERIES
M.D. Levi, Y. Gofer, E. Levi and D. Aurbach
Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel
Abstract
Using cyclic voltammetry and electrochemical impedance techniques, Li and Mg
ions insertion into the Chevrel phase MxMo6S8 (M = Li, Mg) has been studied. The
drastic difference, both in the cyclic voltammetric responses, as well as in the impedance,
is ascribed to a relatively fast kinetics of Li ion insertion. This comes in contrast to very
slow initial insertion kinetics (and final deinsertion) of Mg ions due to trapping effect.
When the kinetics of Mg ion insertion is improved drastically, it comes with an increase
in the intercalation level. The model of ion insertion, considering two non-equivalent
lattice sites with two different energy barriers describes semi-quantitatively the measured
impedance characteristics (semicircle at low-frequencies and Gerischer-type impedance
at high frequencies). Li ion insertion is not inhibited by ion trapping and is better
reproduced by a single-site model (a particular case of two-site model). Qualitative
arguments referring to the unusual crystallographic structure of Chevrel phase were used
to substantiate the application of the two-sites model for Mg ions insertion.
Understanding the origin of multicharged ions trapping and the conditions at which this
effect generates are of crucial importance in developing rechargeable Mg battery.
Introduction
The operation of high-energy density, rechargeable Li and Mg batteries is based
on the phenomenon of electrochemical insertion of Li and Mg ions into appropriate host
of the cathode materials (positive electrode). for anode, Li metal can be replaced by some
appropriate hosts capable of Li-ion insertion at a potential not far from the equilibrium
potential of Li electrode, e.g. graphite and some other carbonaceous materials. Until now,
there were no reports in literature concerning Mg-ion insertion anodes, thus Mg metal
remains the single anode to Mg-ion insertion cathode.
Li-ion intercalation (deintercalation) processes occurring in a rechargeable
batteries can be understood by considering the following reactions. Fully lithiated (fully
charged) graphite is a stoichiometric compound LixC6, which release Li-ions during
charge (i.e. applying anodic current):
LixC66C + xLi+ +xe-
(A)
The released Li-ions and electrons are inserted into the positive electrode host (usually, a
transition metal (M) oxide or sulfide, Li1-xMX2, M = O or S) according to the following
reaction:
xLi+ + xe- + Li1-xMO2→LiMO2
(B)
In case of Mg battery, Mg ions are discharged to the solution by Mg metal
electrodissolution:
xMgxMg2+ + 2xe-
(C)
which, together with the charge-balancing electrons, are inserted into the positive
electrode (e.g. Mo cluster chalkogenide, the so-called Chevrel phase, Mg2xMo6Y8(Y=S,Se)):
xMg2+ + 2xe- + Mg2-xMo6S8 → Mg2Mo6S8
(D)
High specific capacity of Li or Mg ion insertion electrode implies both high value
of x and a relatively low molecular weight of the host material. A search for a proper
candidate host material should normally include good knowledge of their crystallographic
features and electronic structure as well as the character of ion-electron interactions.
High-energy density of the battery necessitates, in addition to high specific
capacity, that the difference in equilibrium potentials between positive and negative
electrodes to be large. This potential difference should be, of course, within the limits of
electrolyte solution stability (both with respect to oxidative and reductive
decomposition).
One more important characteristic of a battery based on the ion-insertion
electrodes, is its rate capability, i.e. the rate at which battery charge can be consumed or
recharged by an external device. The rate capability of practical batteries can be
substantially improved by proper design, e.g. (i) using highly porous electrode masses
(e.g. porous powdery composite electrode instead of a compact one), (ii) good
impregnation with the electrolyte solutions, (iii) use of low-resistance contact between
the active electrode mass and current collector, and low ion-resistance membrane,
separating the two electrodes, keeping them as close as possible. All these requirements
become understandable, considering reactions A - D above, and keeping in mind that
low-resistance paths should be ensured for both ionic and electronic fluxes from one
electrode to the other. Rechargeability of the battery means that reactions A – D may
occur in backward direction, i.e. from right to left.
However, in addition to the above factors i-iii, determining an improved rate
capability of the battery as a whole, one should consider the dynamics of electrons and
ions movements within the bulk of the electrode particles, e.g. their electronic and ionic
conductivities, determining, in turn, the related mobilities and diffusion coefficients.
Recently, we reported that a reversible cathode can be realized with the Chevrel
phase compound (reaction D) using simple Mg2+-containing salts in organic carbonate
solutions (1). Further studies resulted in an increase of the specific capacity of the
cathode owing to a proper choice for the preparation of the initial Chevrel compound
(leaching procedure (2)). Furthermore, we have shown improvement of the electrode
cycleability by replacing Mg(ClO4)2/propylene carbonate electrolyte solution for
alkylmagnesium chlorides of the general formula Mg(AlCl(4–n)Rn)2 (with R = alkyl group)
possessing a relatively high anodic stability. Their preparation and properties were
already reported (3). Mg(AlCl2-EtBu)2, abbreviated here as DCC, was used as 0.25 M
solution in THF. Note also that DCC/THF is the only electrolyte solution, in which Mg
anode behaves reversibly (reaction C)
First studies of Mg ions insertion into Chevrel compound, when compared with
that for Li ions insertion, revealed their specific features that are probably typical for the
2
intercalation of multicharged ions in host materials: (i) The mobility of Mg ions appeared
to be extremely low at ambient temperatures at the beginning of the insertion process,
and at the end of the deinsertion. (ii) The ions mobility drastically increases from certain
intercalation level. (iii) Estimation of the coulombic efficiency of Mg insertion revealed a
considerable difference between the first and the subsequent cycles: about 15 – 20% of
the ions inserted into the host in the course of the first discharge cannot be withdrawn
from it at ambient temperatures at the same rate of charge. They remain in the host as
trapped ions during the subsequent discharges. All these features were not observed for
Li ion insertion into the same Chevrel phase electrodes, which exhibited the characteristic
of highly reversible process (4). Taking into account that the host possesses high
electronic conductivity, the drastic difference in the electrode behavior could be
unequivocally ascribed to the difference in the ionic dynamics of the single and doublecharged ions as they enter the electrode bulk.
In this paper, we summarize our early reports (4-8) concerning the origin of the
different kinetics of Li and Mg ions during insertion into Chevrel phase electrode.
Without proper understanding the nature of the above difference, one can hardly avoid
the effect of trapping Mg ions in the host electrodes.. Special attention is paid throughout
this work to the impedance characterizations of Li and Mg ion insertion systems (6-8),
which can be translated into the difference of the related dynamics of these ions.
Results and Discussion
The voltammograms of Li and Mg insertion into Mo6S8 Chevrel phase (a and b,
respectively) are presented in Fig. 1a and b, respectively. It is seen from this figure that
whereas intercalation of Li ions occurs reversibly via three first-order phase transitions
(note 3 reversible redox-peaks a/a', b/b' and c/c' in Fig. 1a), Mg ions intercalation was
found to occur via two first order phase transitions (two redox-peaks a/a' and b/b' in Fig.
1b). The coulometric analysis for the first cycle shows that the initial magnesiation (and
final demagnesiation) involves considerable charge-trapping at ambient temperature
(25°C). During the subsequent cycling, although the amount of the trapped ions does not
change, the voltammetric behavior is typical of very slow insertion kinetics. The
reversibility of Mg insertion, however, drastically increases at higher intercalation levels,
so that the second Mg-ions transfer (i.e. the transfer from Mg1Mo6S8 to Mg2Mo6S8)
proceeds under quasi-equilibrium conditions. We earlier suggested (6-8) that the slow
initial kinetics of the Mg-ion insertion is due to trapping of the divalent cations whithin
the host crystal.
The structure of this phase can be presented as a three-dimensional array of Mo6S8
units, consisting of distorted Mo6-octahedra, which, in turn, are surrounded by S8 cubes
(see Fig. 2) (9). Intersecting, three-dimensional channels crossing the crystal structure of
the Chevrel phase make possible the transport of the guest ions to the vacant sites during
intercalation. It is known that small guest ions (below 1 Å) do not occupy the exact center
of the S8 – vacant site, but rather shift away from it (9). Moreover, due to a characteristic
thermal motion of these ions in site, small cations are delocalized. Li ion has a radius
close to 0.68 Å, and thus belong to the class of small cations. They were proven (9) to be
distributed randomly within the 6 inner and 6 outer ring-like sites (see Fig. 2). The
stoichiometry of the Li-insertion reaction in the chevrel phase has been previously
3
studied by Schollhorn (10) using chronopotentiometry and 7Li NMR. Combination of
these techniques allowed for a reasonable conclusion with regard to the nature of the sites
for Li-accommodation and the character of their bonding in the host as a function of the
intercalation level. Neutron diffraction analysis of lithiated Mo6S8, revealed (11) that
there are two types of sites for small ions accommodation in this host, namely, a six-sites
inner and six-sites outer rings.
Mg-ion has the radius of 0.67 Å, similar to that of Li-ion. It has completely filled
2s22p6 sublevels, rendering a small cation that exhibit high charge density since it is
divalent. This high charge density of Mg ion, in combination with relatively low
polarizability of the sulfide-anion framework is assumed to be the major reason for the
high diffusion barriers during the first stage of Mg ions intercalation (occupation of the
inner-ring site). Each of the two rings containing 6 sites is capable to host a single Mg ion
(totally 2 Mg ions per a Mo6S8 unit). As intercalation level x in MgxMo6S8 increases, one
should expect appearance of strong repulsion between neighboring double-charged Mg
ions within each site. Due to this repulsion, the first intercalated Mg ion switches position
from the inner ring to the outer. Since the diameter of the outer ring is larger, the distance
between two neighboring sites in the later one is larger than that in the inner ring, thus
relaxing some of the repulsion. Furthermore, the distance between two outer rings from
two neighboring unit-sites is small enough to provide easy path for transport of the
divalent Mg ions. Therefore, the above scenario predicts an increase in the Mg ions
mobility with an increase in the intercalation level. the lower charge density of the
monovalent Li ions, in comparison with that of Mg ions, makes their transition from
inner to outer ring sites easier (lower energy barrier), so that Li ions can be completely
removed from the inner-ring sites via transitional outer-ring sites.
Thus, the difference between the dynamics of Li and Mg ions transport in the Chevrel
phase can be formulated as follows: For Mg ion insertion one should distinguish between
two types of sites in the host, namely, between the shallow and the deep sites with
different diffusion barriers. For Li ion insertion, the difference in site energy and the
related diffusion barriers is not as large as that for Mg ion. Electrochemical impedance
spectroscopy (EIS) study, combined with modeling of ion insertion into a host with the
energetically non-equivalent sites would be the best diagnostic tool for revealing the
difference of dynamics of both ions and their rational explanation.
Fig. 3 shows typical Nyquist plots related to Li insertion into Li xMo6S8 (0 < x < 1)
Chevrel phase electrodes. The spectra comprise high frequency semicircle (HFS) which
relates to charge transfer (no surface films or other complicated surface phenomena in
this case), and a low-frequency Warburg-type element which reflects the solid-state
diffusion. Finally, at the very low frequencies, the -Z'' vs. Z' plots become steep and
reflect the differential intercalation capacitance (Cdif = -1/ Z'',  →0). These spectra
reflect the rapid intercalation of Li ions into this host, which occurs with no
complications as charge trapping, slow diffusion or pronounced surface resistance.
At the beginning of Mg-ions intercalation (see the relevant Nyquist plots for Chevrel
phase electrode at the potentials 1.4 and 1.3 V in Fig. 4) the diameter of the LFS is so
large that practically only an arc can be obtained down to the lowest frequency of 5 mHz
(for details see ref. (6)). While the initial Mg intercalation involves the full capacity of
the host (2 Mg ions per Mo6S8 unit), the reversible intercalation level of Mo6S8 with Mgions at ambient temperature (25º C) is around x = 1.6 – 1.7 (4), i.e. by 15 – 20 % less than
4
the theoretical value x = 2. At 1.25 V the host accommodates x = 0.6 – 0.7 of Mg-ions
(4), so that starting from this potential only, a well-developed LFS is formed, which
transforms into a purely capacitive vertical line in the limit of the lower frequencies (see
Fig. 4). A characteristic feature is observed as the frequency increases: the LFS
transforms rather abruptly to a line, which can be approximately described by a semiinfinite Warburg element (see Fig. 4 and ref. (6-8) for details). Thus, two characteristic
frequencies (shown in Fig. 4 as solid circles), one related to the above border between the
LFS and the Warburg domain, and the other (at lower frequencies) located between the
LFS and the limiting capacitive line) can be experimentally distinguished.
Further decrease in potential (i.e. an increase in the intercalation level) results in a
continuous decrease in the LFS diameter, while the higher frequency domain becomes
more and more Gerischer-type like impedance (see Fig. 4 dotted curly brace and refs. (68)),. The Gerischer domain is located between the HFS and the LFS. The former
semicircle relates to the interfacial charge transfer, complicated by the chemical
equilibria in the DCC/THF solution and possible involvement of adsorption processes. It
will be further ignored because herein we focus on the solid-state diffusion and ions
entrapment). It is of great interest that as the potential approaches the value of 0.72 V,
which is close to the limit of Mg-ions insertion into the Chevrel phase electrode at
ambient temperature (x = 1.6 – 1.7), the high-frequency limit of the impedance does not
include the semicircles anymore (neither the LFS nor the curved Gerischer-type
impedance): the impedance behavior is rather similar to a semi-infinite Warburg line,
transforming into a vertical capacitive line at lower frequencies (Fig. 4). Within this
intercalation domain, the impedance spectra for Mg ions insertion become qualitatively
similar to that for Li (compare with Fig. 3).
The above features of the impedance behavior for Mg-ions insertion into Mo6S8
Chevrel phase electrodes can be rationalized and semi-quantitatively described by a twostate model elaborated recently by Bisquert and Vikhrenko (12). Originally, this model
considers two energetically different types of sites: deep (or hollow) and shallow ones. At
the beginning of the intercalation, the hollow sites with limited ion mobility (because of
the higher diffusion barrier) are occupied. As intercalation level increases, the shallow
sites (with a lower diffusion barrier) start to fill with the guest ions. It is easy to see that
such two-state model can be applied for description of the Mg ions intercalation into
Mo6S8 taking into account the two types of occupation sites for the divalent ion insertion
as outlined above. In this case, the overall complex plane impedance of the intercalation
system containing two types of sites (Z(s)) can be conveniently presented as a function of
the angular frequency () in the form (see ref. 12):
1
2
1
2
Z ( s)  Ro ( ( s) / s) coth[( s /  ( s)) ] ,
*
n
*
n
(1)
where s  j , and the diffusion resistance Ro (related to the motion of the intercalated
ions between the shallow sites) relates to the corresponding diffusion coefficient, Do,
through the intercalation capacitance of the shallow sites, Co, and the characteristic
frequency, o:
o 
Do
1

.
2
L
Ro Co
5
(2)
Here L is the electrode thickness. According to the two state model (12), the intercalated
ions move by ordinary diffusion mechanism between the shallow sites, interacting, at the
same time, with the deep sites and becoming immobilized for a lapse of time. The
characteristic frequency related to ion trapping and release at the deep sites, t,, defines
an effective, frequency dependent diffusion coefficient, Dn* ( s ) , through the ratio of the
intercalation capacitances describing the occupation of the hollow and the shallow sites,
Ctrap and Co, respectively (12):
Dn* ( s ) 
 o L2
1

s  Ctrap
1  1  
  t  Co
.
(3)
Note also that the diffusion motion coupled with ions trapping in the deep sites can be
presented by a characteristic frequency in a conventional form:
 n* ( s )  Dn* ( s ) / L2 ,
(4)
with Dn* ( s ) obtained from Eq. 3. Thus introduced  n* ( s ) should be inserted in Eq. 1 for
final calculation of the electrode impedance in case of the diffusion coupled with ion
trapping.
The impedance obtained with the use of Eq. 1 can be better understood in terms of
the transmission line representation (see Fig. 5). Here, the two transverse branches in
parallel, comprising the intercalation capacitance of the shallow sites, Co, and the
intercalation capacitance of the deep sites, Ctrap, in series with the relevant kinetic
parameter, namely, the trapping resistance, Rtrap. An obvious equation expresses the
additive (cumulative) character of the total capacitance observed at very low frequencies:
Ctot = Co + Ctrap.
(5)
Application of Eqs. 1 - 5 for the modeling of the impedance spectra of an
intercalation electrode with two different energy sites require four independent
parameters: o, t, C trap, and Co. One of the essential problems of the modeling is the
knowledge of the dependencies of these parameters on the electrode potential (or
intercalation level, x). A general property of the two states models (e.g. considered by
Bisquert and Vikhrenko (12), or Chvoj et al (13)) is that the chemical diffusion
coefficient (D) of the inserted ions increases sharply at a certain intercalation level related
to the end of the occupation of the deep sites and the beginning of occupation of the
shallow ones. We assume that the impedance spectrum measured at 1.25 V (see Fig. 4)
approximately fulfills this condition. In order to reproduce the impedance spectra in the
vicinity of this transition, the only parameter, which changes with potential, is believed to
be the characteristic frequency of the trapping process, t. This is because Mg-ions are
doubly-charged, and as a result, an increase in the intercalation level beyond the above
limit promotes ion-hopping from the deep (inner ring) to the shallow site (outer ring)
owing to a drastically increasing electrostatic repulsion between the Mg ions. For the
purpose of a qualitative comparison of the model with the experimental curves, we
assumed an augmentation in the t by more than 3 orders of magnitude as the
intercalation level increases (see captions to Fig. 4). Notice that the intercalation of singly
6
charged Li-ions into the same Chevrel phase electrode did not reveal ion trapping (see
Fig. 3) and should be described by a single-energy state model.
Fig. 6 (“a” refers to the whole frequency domain, whereas “b” enlarges, for
clarity, the high frequency domain) shows theoretical impedance spectra calculated with
the use of Eqs. 1-5 as a function of the parameter t (or, equivalently, as a function of
the charge-trapping resistance, Rtrap, obtained from the equation: Rtrap = 1/ (tCtrap) (12)
and indicated in the figure). The curve related to Rtrap = 500 cm2 shows three
characteristic frequencies, namely, o, t and n (the later was found as the boundary
frequency separating the LFS and the low-frequency, capacitive line on the related
impedance spectrum as indicated). This case corresponds to a very slow trappingdependent kinetics of the initial Mg-ions insertion (t << o). The high-frequency
impedance shows a pattern of a classical semi-infinite Warburg behavior (see Fig. 6b and
the location of o for Rtrap = 500 cm2) since in this high frequency domain the
diffusion of ions via the shallow sites is uninterrupted by trapping kinetics (12). At lower
frequencies, the Mg-ions trapping appears as LFS, formed by parallel combination of Co
and Rtrap (see the equivalent circuit in Fig. 5).
A decrease in the values of the parameter Rtrap (i.e. an increase in t) obviously
results in a considerable decrease in the diameter of the LFS (see Fig. 6a and b). As a
result, the impedance spectra acquire a new feature of the Gerischer-type in the highfrequency domain followed by a kind of the semi-infinite Warburg behavior, and finally,
vertical capacitive line as the frequency decreases. These features, first obtained
theoretically by Bisquert (14), can be easily understood from the transmission line
properties shown in Fig. 5 and from the values of the characteristic frequencies t and
o.: as the operative frequency is high as well as Ctrap, the reactance due to Ctrap is small
compared to Rtrap. In this case, the later quantity is much less than Ro, hence t >>o
(see the values indicated in Fig. 6b), and the concentration profile of the intercalated ions
decay before the species reach the boundary at the current collector (this feature is typical
of the Gerischer-type impedance (14)). The impedance spectrum has two semi-infinite
Warburg regions having different prefactors related to o and n at high and medium
frequencies, respectively. When Rtrap decreases virtually to zero (i.e. the traps disappear),
the impedance response will be reflected by the classical finite-space Warburg element
(FSW) with a single characteristic frequency o (see Fig. 6b). Comparison between the
model curves in Figs. 6a and b and the experimental ones obtained for the Mg-ions
insertion into the Chevrel electrode (Fig. 4, compare also the limiting case: FSW vs. plot
measured at 0.72 V) reveal their qualitative similarity.
Conclusion
Considerable difference in the shape of impedance spectra has been observed for
Li and Mg ion insertion into a Chevrel phase. Together with the features of the cyclic
voltammetry responses, this difference was ascribed to a relatively fast kinetics of Li ion
insertion, in contrast to a very slow initial insertion kinetics (and kinetics of final
deinsertion) of Mg ions. The dynamics of Mg ion insertion is increased drastically with
increasing intercalation level from certain level. It appeared that the model of Bisquert
and Vikhrenko (12) considering two non-equivalent lattice sites with two different energy
barriers describes semi-quantitatively the measured impedance characteristics (semicircle
at low-frequencies and Gerischer-type impedance at high frequencies). Li ion insertion is
7
free of ion trapping and is better reproduced by a single-site model (which is a particular
case of the two-sites model). Understanding the origin of multicharged ions trapping and
the conditions at which this effect can become less pronounced are of crucial importance
in developing rechargeable Mg battery.
References
1. Aurbach, D.; Lu, Z.; Schechter, A.; Gofer, Y.; Gizbar, H.; Turgeman, R.; Cohen,
Y.; Moshkovich, M.; Levi, E. 'Prototype systems for rechargeable magnesium
batteries'. Nature (London) (2000), 407(6805), 724-727.
2. Lancry, Eli; Levi, Elena; Gofer, Yossi; Levi, Mikhael; Salitra, Gregory; Aurbach,
Doron. 'Leaching Chemistry and the Performance of the Mo6S8 Cathodes in
Rechargeable Mg Batteries'. Chemistry of Materials (2004), 16(14), 28322838.
3. Chusid, Orit; Gofer, Yosef; Gizbar, Haim; Vestfrid, Yulia; Levi, Elena; Aurbach,
Doron; Riech, Israel. 'Solid-state rechargeable magnesium batteries'. Advanced
Materials (Weinheim, Germany) (2003), 15(7-8), 627-630.
4. Levi, M. D.; Lancry, E.; Gizbar, H.; Lu, Z.; Levi, E.; Gofer, Y.; Aurbach, D.
'Kinetic and thermodynamic studies of Mg2+ and Li+ ion insertion into the Mo6S8
chevrel phase'. Journal of the Electrochemical Society (2004), 151(7), A1044A1051.
5. Levi, M. D.; Lancry, Eli; Gizbar, H.; Gofer, Y.; Levi, E.; Aurbach, D. 'Phase
transitions and diffusion kinetics during Mg2+- and Li+-ion insertions into the
Mo6S8 Chevrel phase compound studied by PITT'. Electrochimica Acta (2004),
49(19), 3201-3209.
6. Levi, M. D.; Gizbar, H.; Lancry, E.; Gofer, Y.; Levi, E.; Aurbach, D. 'A
comparative study of Mg2+ and Li+ ion insertions into the Mo6S8 Chevrel phase
using electrochemical impedance spectroscopy'. Journal of Electroanalytical
Chemistry (2004), 569(2), 211-223.
7. Levi, M. D.; Aurbach, D. 'Distinction between Energetic Inhomogeneity and
Geometric Non-Uniformity of Ion Insertion Electrodes Based on Complex
Impedance and Complex Capacitance Analysis'. Journal of Physical Chemistry
B (2005), 109(7), 2763-2773.
8.
Levi, M.D.; Aurbach, D. 'A Comparison between Intercalation of Li and Mg Ions
into the Model Chevrel Phase Compound (MxMo6S8): Impedance Spectroscopic
Studies'. J. Power Sources (in press).
9. K. Yvon, 'Bonding and Relationships between Structure and Physical Properties
in Chevrel-Phase Compounds MxM6X8 (M=metal, X=S,Se,Te)', in Current Topics
in Material science, vol. 3, E. Kaldis, Editor, p. 53, North-Holland Publishing
Company, Amsterdam (1979).
10. Gocke, E.; Schoellhorn, R.; Aselmann, G.; Mueller-Warmuth, W. 'Molybdenum
cluster chalcogenides Mo6X8: intercalation of lithium via electron/ion transfer'.
Inorganic Chemistry (1987), 26(11), 1805-12.
8
11. Ritter, C.; Gocke, E.; Fischer, C.; Schoellhorn, R. 'Neutron diffraction study
on the crystal structure of lithium intercalated Chevrel phases'. Materials
Research Bulletin (1992), 27(10), 1217-25.
12. Bisquert, Juan; Vikhrenko, Vyacheslav S. 'Analysis of the kinetics of ion
intercalation. Two state model describing the coupling of solid state ion diffusion
and ion binding processes'. Electrochimica Acta (2002), 47(24), 3977-3988.
13. Chvoj, Z.; Conrad, H.; Chab, V. 'Thermodynamics of adatoms diffusing on a
surface with two different sites: a new type of phase transition'. Surface Science
(1997), 376(1-3), 205-218.
14. Bisquert, Juan. 'Theory of the Impedance of Electron Diffusion and
Recombination in a Thin Layer'. Journal of Physical Chemistry B (2002),
106(2), 325-333.
9
b'
a
12
a'
c'
Cdif / CV
-1
4
c
-4
a
-12
b
-20
1.6
1.4
1.8
2.6
2.4
2.2
2.0
2.8
+
E / V (vs . Li/Li )
12
b'
b
a'
Cdif / CV
-1
4
-4
a
-12
b
Fig. 2 Basic elements of crystal
structure of Chevrel phase
compound. Mo6 – octahedra
are inscribed in a slightly
distorted
cubic
anionic
framework (X are shown by
green spheres). 6-sites inner
and
outer
rings
for
accommodation of small
cations are shown by blue
dots
between
the
neighboring cubic anionic
elements (9)
-20
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
E / V (vs . Mg)
Fig. 1 SSCV curves measured at 25
Vs-1 for the insertion of Li
(a) and Mg ions (b) from 1 M
LiClO4/propylene carbonate
and 0.25 M Mg(AlCl2EtBu)2
(DCC) / THF solutions,
respectively. Phase transitions
occur along the redox-peaks
a/a', b/b', c/c' for Li ions
insertion, and along the peaks
a/a' and b/b' for Mg ions
insertion.
10
1.40 V
1.30 V
1.400 V
5 mHz
150
1.300 V
300
1.250 V
1.150 V
Low-frequency capacitive lines
1.110 V
1.090 V
1.06 V
0.723 V
Potential
decreases
50
HFS
200 Hz
4 Hz
Warburg region
/
- -ZZ'' ''/ 
- Z '' / 
100
2.53 V
2.50 V
2.47 V
2.45 V
2.43 V
2.41 V
2.40 V
2.39 V
2.38 V
200
Low-frequency,
capacitive line
5 mHz
0.72 V
100
140 Hz
125
175
1.06 V
1.15 V
1.11 V
50 HFSGerischer150
225
impedance
Z' / 
Fig.3 Nyquist plots for Li ions insertion into a
Mo6S8 electrode, for the entire range of
intercalation potentials (the direction of
decreasing the potential is indicated by the
arrow) (8).
1.25 V
1.09 V
800 Hz
0
75
fn = 17 mHz
LFS
f0 = 1Hz
0
25
0.12 Hz
250
350
' / / 
Z Z'
Fig. 4 Impedance spectra measured from a
Mo6S8 Chevrel electrode for Mg
ions insertion at different potentials
as indicated. Solid black circles
mark two different experimental
characteristic frequencies, fo and fn,
in the high and in the low-frequency
domain, respectively (8).
11
b
R’m
R’m
R’trap
R’m
R’trap
R’m
R’trap
C’shallow
C’shallow
C’trap
C’trap
C’shallow
C’trap
Fig. 13b
Fig. 5 Transmission line representation of coupling between the diffusion
and ions trapping in the host material of an intercalation electrode
(14).
6
0.4
a
b
Low-frequency,
capacitive lines
5
Rtrap(cm2):
fn = 1 Hz
4
ft = 0.2 Hz
No Warburg
impedance
here!
3
LFS
2
- (Z ''/R o)
- (Z ''/R o)
0.3
fn = 0.1 Hz
Rtrap(cm2):
fo = 10 Hz
(Rtrap=
500cm2)
0.2
fo = 10 Hz
LFS
ft = 550 Hz
0.1
1
Gerischer
impedance
Gerischer
impedance
fo = 10 Hz
0
0
0
1
2
3
4
5
6
0
0.1
0.2
0.3
0.4
(Z '/R o)
(Z '/R o)
Fig. 6. Theoretical impedance spectra calculated with the use of Eqs. 1 – 5 for
the whole frequency domain from 320 kHz to 0.1 mHz. (a) and for
the high-frequency domain (b). The following parameters were used: Ro =
100 cm2, Rtrap (as indicated), Co and Ctrap = 0.001 and 0.01 mFcm-2, fo = 10
Hz. ftrap and fn are also indicated (8).
12