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Density Functional Theory Based Study of the Electron Transfer
Reaction at the Lithium Metal Anode in a Lithium−Air Battery with
Ionic Liquid Electrolytes
Saeed Kazemiabnavi, Prashanta Dutta, and Soumik Banerjee*
School of Mechanical and Materials Engineering, Washington State University, Pullman, Washington 99164-2920, United States
ABSTRACT: Room temperature ionic liquids, which have unique properties such as a
relatively wide electrochemical stability window and negligible vapor pressure, are
promising candidates as electrolytes for developing lithium−air batteries with enhanced
performance. The local current density, a crucial parameter in determining the performance
of lithium−air batteries, is directly proportional to the rate constant of the electron transfer
reaction at the surface of the anode that involves the oxidation of pure lithium metal into
lithium ion (Li+). The electrochemical properties of ionic liquid based electrolytes, which
can be molecularly tailored on the basis of the structure of their constituent cations and
anions, play a crucial role in determining the reaction rate at the anode. In this paper, we
present a novel approach, based on Marcus theory, to evaluate the effect of varying length
of the alkyl side chain of model imidazolium based cations on the rates of electron transfer
reaction at the anode. Density functional theory was employed for calculating the necessary
free energies for intermediate reactions. Our results indicate that the magnitude of the
Gibbs free energy of the overall reaction decreases linearly with the inverse of the static dielectric constant of the ionic liquid,
which in turn corresponds with an increase in the length of the alkyl side chain of the ionic liquid cation. Nelsen’s four-point
method was employed to evaluate the inner sphere reorganization energy. The total reorganization energy decreases with
increase in the length of the alkyl side chain. Finally, the rate constants for the anodic electron transfer reaction were calculated in
the presence of varying ionic liquid based electrolytes. The overall rate constant for electron transfer increases with increase in
the static dielectric constant. The presented results provide important insight into identification of appropriate ionic liquid
electrolytes to obtain enhanced current densities in lithium−air batteries.
■
INTRODUCTION
Over the past several years, researchers have studied coupled
electrochemical reactions that have the potential to achieve
gravimetric energy densities that are significantly greater than
that of lithium ion cells with two intercalation electrodes.1
State-of-the-art Li ion cells come nowhere close to the target of
1700 (W h)/kg.2 New chemistries are therefore required to
attain this goal. Li−air batteries, which employ a lithium anode
electrochemically coupled with oxygen, possess much greater
theoretical gravimetric energy storage density compared to
other battery technologies.3 In particular, Li−air batteries have
received significant scientific attention in recent times due to
their potential use in long-range electric vehicles, where the
gravimetric energy density, the volumetric energy density, and
safety are important factors.2,4 Li−air batteries may also play
important roles in other applications, such as in powering
consumer electronics and remote sensors.1
Since lithium is highly reactive,5 one of the principal
concerns in all the above-mentioned applications of lithium
batteries is safety during operation.6,7 Lithium metal has a very
high chemical reactivity in the presence of polar aprotic and
protic organic solvents, which are commonly used in lithium
batteries, causing electrode−electrolyte side reactions.8 In
addition, organic solvents typically have a high vapor pressure,
leading to increased flammability. Therefore, Li−air batteries
© 2014 American Chemical Society
that employ organic liquid electrolytes could be potentially
dangerous for use in aerospace and automobile applications.9
Therefore, it is extremely important to identify novel
electrolytes, beyond the conventional organic solvents, that
lead to high safety standards in Li−air batteries. In addition to
flammability, an ideal electrolyte for a Li−air battery needs to
be tailored to enhance the performance of these batteries. The
cyclic performance strongly depends on the electrochemical
stability and other physicochemical properties of the electrolyte,
such as ionic conductivity, static and optical dielectric
constants, and the ability to dissolve various chemical species
that are generated.4 Many efforts have been made to develop
safe electrolytes for use in lithium ion and lithium−sulfur
batteries.10−13 Room temperature ionic liquids (RTILs), which
have a high electrical conductivity, a wide electrochemical
stability window, and also a low vapor pressure, potentially
provide a much safer substitute to conventional lithium battery
electrolytes.14−16 Table 1 presents a qualitative comparison of
ionic liquids with organic solvents.17
Room temperature ionic liquids are usually quaternary
ammonium salts such as tetralkylammonium, [R4N]+, or are
Received: July 1, 2014
Revised: October 29, 2014
Published: October 30, 2014
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properties, such as static and optical dielectric constants, of
these ionic liquids have been characterized in several
studies.21,25 The commonly used TFSI− was chosen as the
anion due to its wider electrochemical stability window and
lower viscosity compared to other anions such as BF4− and
PF6−.22,26−34 The chemical structures of a representative ionic
liquid cation for n = 3 and the chosen anion are shown in
Figure 1.
Table 1. Qualitative Comparison of Ionic Liquids with
Organic Solvents17
property
organic solvents
ionic liquids
number of solvents
applicability
vapor pressure
flammability
solvation
thermal stability
electrochemical stability window
>1000
single function
usually high
usually flammable
weakly solvating
low
narrow
>1000000
multifunction
very low
mostly nonflammable
tunable solvation
high
wide
based on cyclic amines, both aromatic (pyridinium, imidazolium) and saturated (piperidinium, pyrrolidinium). Lowtemperature molten salts based on sulfonium, [R3S]+, as well
as phosphonium, [R4P]+, cations are also known. Therefore, by
combining these cations with both inorganic (halides [BF4]−,
[PF6]−, and [AsF6]− or amide [N(CN)2]−) and organic
([C4F9SO3]−, trifluoroacetic [CF3CO2]−, triflate [CF3SO3]−,
imides [N(CF3SO2)2]− and [CF3CONCF3SO2]−, or methide
[C(CF3SO2)3]−) anions, various ionic liquids with a wide
variety of tunable physicochemical and electrochemical properties can be synthesized.15 It is possible to tune these properties
by choosing from various combinations of cations and anions as
well as by changing the chemical structure of these ions. While
these unique characteristics of ionic liquids have made them
promising substitutes for conventional solvents in a wide
variety of energy storage applications, it is important to identify
specific ionic liquid structures that lead to the desired
properties of electrolytes for enhanced performance of
batteries.
One of the important factors that determine the performance
of lithium−air batteries is the local current density. A higher
current density leads to enhanced battery performance. In
particular, the current density at the anode can be expressed as
a function of the rate constant for the electron transfer reaction
at the surface of the electrode within the framework of the
Butler−Volmer equation.18,19 During the discharge cycle of Li−
air batteries, pure lithium metal gets oxidized at the anode,
producing positively charged Li+ ions:20
Li → Li+ + e−
Figure 1. Chemical structures of model ionic liquid (a) 1-propyl-3methylimidazolium (C3MIM+) cation and (b) bis[(trifluoromethyl)sulfonyl]imide (TFSI−) anion.
Chemical reaction rates are generally calculated using the
transition-state theory (TST)35 and other theories based in
whole or in part on the fundamental assumptions of TST or
some quantum mechanical generalization of these assumptions.36−38 However, these theories are not applicable for
electron transfer reactions since characterizing the transition
state in this type of reaction is not feasible. In the present study,
we employed the Marcus theory39 formulation to calculate the
rate constant using relevant thermodynamic parameters. The
Marcus theory has been widely employed to calculate the rate
constants for several electron transfer reactions between donor
and acceptor species, and they show excellent agreement with
experimental results.39,40 While purely experimental studies
have been done to determine the rate constant of electron
transfer reactions in solution,41 the literature lacks data on the
kinetics of electron transfer reactions at the electrode−
electrolyte interface. We calculated the electron transfer rate
constant for the oxidation of lithium metal in various
imidazolium based ionic liquids as the electrolyte. The Gibbs
free energy, inner and outer sphere reorganization energies, and
electronic coupling energy were calculated to evaluate the rate
constant based on Marcus theory. The solvation energy for
separated ions and atoms was calculated to investigate the effect
of the solvent on the driving force and reaction rate. The
calculated thermodynamic parameters, including the first
ionization energy and vaporization energy of lithium, match
the experimental values very well, with the largest relative error
being only 5.8%. The results obtained relate the rate constant
and thermodynamic driving force for this reaction to the static
dielectric constant of the electrolyte, which directly depends on
the length of the alkyl side chain of the imidazolium based
cation.
(1)
The reverse reaction happens during the charge cycle. The
electrodeposition and dissolution of the metal-like lithium
occur in a single step.2,20 However, the literature lacks detailed
studies that relate the effect of the molecular structure of liquid
electrolytes on the reaction kinetics at the anode, which has a
crucial impact on the performance of batteries. The main
purpose of this study was to investigate the effect of ionic liquid
based electrolytes on the reaction rate and obtain a trend for
the variation of the electron transfer reaction rate constant as a
function of the dielectric constant. In an effort to assess the
effect of the molecular structure of ionic liquids on the reaction
kinetics, we calculated the electron transfer rate constant for the
oxidation of lithium metal in the anode side in contact with
ionic liquids with varying structures. In particular, in this study,
1-alkyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide (CnMIM+TFSI−) ionic liquids with varying number of
carbon atoms in the alkyl side chain, n, were chosen as the
model electrolyte. The choice was justified due to the relatively
wide electrochemical stability window, low viscosity, and higher
ionic conductivity of imidazolium based ionic liquids compared
to other ionic liquids.15,21−24 Additionally, relevant physical
■
THEORETICAL FRAMEWORK
Marcus theory39 describes the rates of electron transfer
reactions, where an electron moves or jumps from an electron
donor to electron acceptor species. The mathematical
expression for the Marcus theory, which relates the rate
constant to various energy parameters, is given as
ket =
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2π
|VRP|2
ℏ
⎛ (λ + ΔG°)2 ⎞
1
exp⎜ −
⎟
4λkBT ⎠
⎝
4πλkBT
(2)
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where ket is the rate constant for electron transfer, |VRP| is the
electronic coupling energy between the initial and final states, λ
is the total reorganization energy, ΔG° is the total Gibbs free
energy change for the electron transfer reaction, kB is the
Boltzmann constant, ℏ is the reduced Planck’s constant h/2π,
and T is the absolute temperature. A qualitative plot of the
energy parameters, mentioned in Marcus theory, is shown in
Figure 2 to get a qualitative sense of how these parameters are
related to each other.
(VTZPD). Since there is no vibrational frequency associated
with a single atom, the vibrational contributions were neglected
to avoid any inconsistency in the energy of single atomic
systems with those of other systems. The optimization of the
lithium lattice was performed using periodic boundary
condition DFT (PBC-DFT) with the revised Perdew, Burke,
and Ernzerhof46 (revPBE) GGA functional and 6-31++G*47,48
basis set. All calculations were done using the NWChem 6.1
computational chemistry software package.49
The conductor-like screening model50 (also known as the
COSMO solvation model), which is an approach to account for
dielectric screening in solvents, was implemented to investigate
the effect of the solvent on the reaction rates. The primary
inputs to the model are the dielectric constant of the solvent
and the radii of the solvated species and solvent molecules.
However, for all calculations, the Li+ ions with a radius of 0.76
Å51 are surrounded solely by the TFSI− anions, which have a
constant radius of 3.801 Å.21 Therefore, the only parameter that
determines the solvation energy is the dielectric constant of the
solvent. Figure 3 shows the variation of the static dielectric
Figure 2. Energy diagram for the electron transfer reaction, including
the reorganization energy and electronic coupling energy, which are
used to calculate the reaction rate constant in the framework of
Marcus theory.
Figure 3. Static dielectric constant, ε, of some imidazolium based ionic
liquids with TFSI− as the anion.21 The only difference in these ionic
liquids is the number of carbons in the side alkyl chain, which results in
different physical properties, such as the static dielectric constant. The
static dielectric constant decreases as the length of the side chain
increases.
Marcus theory formulation is analogous to the traditional
Arrhenius equation for the rates of chemical reactions and
provides a detailed framework to evaluate the rate constant in
the case of electron transfer reactions. First, it provides a
mathematical expression for the activation energy, based on the
reorganization energy and the Gibbs free energy of the electron
transfer reaction:
ΔG⧧ =
(λ + ΔG°)2
4λ
constant of the imidazolium based ionic liquids considered in
this study with respect to the number of carbon atoms in the
side alkyl chain.21 The static dielectric constant of the
imidazolium cation based ionic liquid decreases with an
increase in the number of carbons in the alkyl side chain.
(3)
Second, it provides a mathematical relationship, as shown in eq
2, to determine the pre-exponential factor in the Arrhenius
equation. The pre-exponential factor is expressed as a function
of the electronic coupling between the initial and final states of
the electron transfer reaction, i.e., the overlap of the electronic
wave functions of the two states.
The reorganization energy is defined as the energy required
to “reorganize” the structure of the system from initial to final
coordinates, after the charge is transferred, and has two parts:42
the inner sphere and outer sphere reorganization energies. The
inner sphere reorganization energy, λin, is due to the structural
changes of the reacting species, while the outer sphere
reorganization energy, λout, results from solvent relaxation.42,43
RESULTS AND DISCUSSION
The principal objective of the present study is to determine the
reaction rate at the anode−electrolyte interface of lithium−air
batteries in the presence of imidazolium cation based ionic
liquids. The goal is to relate the structure of the cation of the
ionic liquid solvent to the overall reaction rate. The oxidation
process of lithium metal in the anode, due to electron transfer,
can be represented as
COMPUTATIONAL METHODOLOGY
In this study, all energy calculations were done using density
functional theory (DFT) with the Becke, three-parameter,
Lee−Yang−Parr44,45 (B3LYP) exchange-correlation functional
and 6-311++G** as the basis set, which is a valence triple-ζ
basis set with polarization and diffuse functions on all atoms
As discussed earlier, in the framework of Marcus theory in eq
2, the reaction rate is a function of the Gibbs free energy and a
kinetic prefactor. The Gibbs free energy for this reaction, in the
case of various ionic liquid solvents, provides an estimate of the
thermodynamic driving force for the anode reaction. To
calculate the Gibbs free energy of the reaction, the overall
oxidation process of lithium metal in the anode is assumed to
■
Li(s) → Li+(solv) + e−
■
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(4)
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be the cumulative effect of the following intermediate
reactions:52
Li(s) → Li(g)
(5)
Li(g) → Li+(g) + e−
(6)
Li+(g) → Li+(solv)
(7)
The initial reaction is the vaporization of lithium metal followed
by the first ionization of lithium and finally the solvation of Li+
ion. The Gibbs free energy of the complete reaction can be
calculated by adding the corresponding free energies of the
constituent reactions:
ΔG° = ΔGvap(Li) + ΔGion(Li) + ΔGsolv (Li+)
Figure 4. Total solid-phase energy of each atom in a system
comprising 18 lithium atoms in the solvated phase (Es). The lines
joining the data points are provided to guide the eye. The maximum
difference between the solid-phase energies is 0.0856 kJ mol−1, which
is merely 0.000433% of the average value.
(8)
where ΔGvap(Li) is the free energy of vaporization of lithium
metal, ΔGion(Li) is the first ionization energy of lithium, and
ΔGsolv(Li+) is the solvation energy of Li+ ion.
To calculate the vaporization energy of lithium metal, the
optimized lattice structure of lithium metal was determined
using PBC-DFT, and the total energy of the system was then
calculated using nonperiodic DFT to determine the energy of
each lithium atom in the lattice (Es). Then a single-point energy
calculation was performed to find the energy of one free lithium
atom in the gas phase (Eg). The difference between these two
energies is used to evaluate the vaporization energy:
ΔGvap(Li) = Eg − Es
the total solid-phase energy of the neutral lithium atoms in the
solvent will be less negative, and therefore less stable, in ionic
liquids with longer alkyl side chains. However, it should be
noted that the maximum difference between the presented
solid-phase energies is 0.0856 kJ mol−1, which is only
0.000433% of the average solid-phase energy. Therefore, the
solid-phase energy of neutral lithium atoms in different ionic
liquids is almost constant.
To evaluate the vaporization energy of lithium, the gas-phase
energy of a free lithium atom needs to be determined. Since
there is no electrostatic field interacting with lithium atoms in
the gas phase due to the absence of solvent molecules, the gasphase energy of lithium atoms is a constant value. The
calculated gas-phase energy for a neutral lithium atom is
−19668.921 kJ mol−1. Using these values, the vaporization
energy of lithium can be calculated from eq 9.
On the basis of eq 10, calculation of the Gibbs free energy
also requires the evaluation of the free energy of Li+ ion in the
gas phase. Since the electrostatic solvent effect is not present,
this remains constant at the calculated value of −19126.611 kJ
mol−1. Hence, using eq 10, the obtained value of the first
ionization energy of lithium was 542.31 kJ mol−1.
The free energy of solvation of Li+ ion, which is the
difference between the energies of a solvated ion and an ion in
the gas phase, is an important driving force in the overall anode
reaction and is dependent on the solvent properties. The
calculated values of the energy of Li+ ion in the solvated phase
as well as in the gaseous phase are presented in Figure 5 as a
function of the length of the alkyl chain in the ionic liquid’s
cation. As shown in Figure 3, increasing the number of carbon
atoms in the alkyl side chain leads to a decrease in the dielectric
constant of the ionic liquid. The diminished magnitude of the
dielectric constant leads to an increase in the total energy of a
single solvated Li+ ion, which has less negative energy values.
The average rate of increase in the solvated-phase energy is 12.3
kJ mol−1 per carbon atom, considering all data points shown on
the plot. As expected, the energy of the ion in the gas phase,
where the electrostatic solvent effect is not present, remains
constant.
Figure 6 shows the variation of the magnitude of the
solvation free energy with respect to the inverse of the dielectric
constant, calculated from the results presented in Figure 5 using
eq 11. Tjong and Zhou53 showed that the dependence of the
electrostatic solvation energy of a charged species on the
dielectric constant of the solvent can be expressed by the
following equation:
(9)
where Es is the total solid-phase energy of each lithium atom in
a system comprising 18 neutral lithium atoms as a cluster using
nonperiodic DFT. The ionization energy of lithium was then
determined by calculating the energy of a Li+ ion in the gas
phase (Eion) and finding the difference between this energy and
the energy of one free lithium atom in the gas phase (Eg):
ΔGion(Li) = E ion − Eg
(10)
+
Finally, to determine the solvation energy of Li ion, a singlepoint energy calculation using DFT was performed on Li+ ion
in the solution phase to evaluate the energy of Li+ ion in the
solvated phase, Esolv:
ΔGsolv (Li+) = Esolv − E ion
(11)
In accordance with the aforementioned method for
calculation of the Gibbs free energy, it is necessary to
determine the vaporization energy, which in turn requires
evaluation of the energy of each lithium atom in the optimized
lattice structure in solvated phase, Es. Figure 4 shows the total
solid-phase energy of each lithium atom in a system comprising
18 neutral lithium atoms in a body-centered cubic (bcc) lattice
structure in different ionic liquids as the solvent medium. The
results are shown as a function of the number of carbon atoms
in the side alkyl group of the ionic liquid’s cation. The presence
of polar solvents induces electrostatic interactions between
lithium atoms and the solvent molecules. Although the lithium
lattice is neutral, the lithium atoms in the bcc lattice structure
possess partial charges, resulting in electrostatic interactions
with the electrical field caused by the solvent dipoles. As shown
in Figure 3, the static dielectric constant of the ionic liquid
decreases with an increasing number of carbons in the alkyl side
chain. This decrease in the static dielectric constant causes a
decrease in the magnitude of the solvation energy of lithium
atoms as the length of the side chain increases.53 Consequently,
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Figure 5. Total energies of a single Li+ ion in the gas phase (Eion) and
solvated phase (Esolv) in imidazolium based ionic liquids of varying
side chain length. The gas-phase energy remains constant since there is
no electrostatic interaction due to the absence of ionic liquid solvent.
An increase in the number of carbons in the alkyl side chain leads to a
decrease in the dielectric constant of the ionic liquid, thus resulting in
an increase in the total energy of a single solvated Li+ ion. The average
rate of increase in the solvated-phase energy is 12.3 kJ mol−1 per
carbon atom added to the side chain.
Figure 7. Variation of the vaporization (ΔGvap), ionization (ΔGion),
magnitude of solvation (|ΔGsolv|), and magnitude of the total Gibbs
free energy (|ΔG°|) of the anode reaction with the static dielectric
constant of the solvent. The difference in appearance of the Gibbs free
energy and solvation energy lines, with approximately the same slope,
is due to the different y-axis scales.
solvent cation. As previously discussed, the vaporization energy
remains almost constant since the change in the energy of solidphase lithium in different ionic liquids is negligible. Comparing
the calculated value of the vaporization energy with the
experimental value of 118 kJ mol−1 shows a 5.8% relative
error.52,54 The ionization energy of lithium is independent of
the solvent since the reactant and product of the ionization
reaction are in the gas phase. The calculated value of the
ionization energy has a 4.2% relative error compared to the
experimental value of 520.23 kJ mol−1.55 The magnitude of the
solvation energy of Li+ ion, however, decreases with decreasing
dielectric constant of the solvent. As a consequence, the Gibbs
free energy of the oxidation of lithium metal into Li+ ion is less
negative, and hence lower in magnitude, in solvents with a
smaller dielectric constant. Figure 7 shows that the magnitude
of the overall Gibbs free energy for the oxidation reaction,
|ΔG°|, is inversely proportional to the inverse of static dielectric
constant of the solvent. Therefore, the thermodynamic driving
force for the oxidation of lithium metal is weaker in ionic
liquids with longer alkyl side chains. As can be seen from the
results presented in Figure 7, the slopes of the Gibbs free
energy and the solvation energy lines are almost equal, which is
expected since the vaporization energy of lithium is almost
independent of the dielectric constant of the solvent compared
to the solvation energy of Li+ ion.
While the reaction rate constant for the electrochemical
reaction at the anode is a strong function of the thermodynamic
driving force, the Gibbs free energy, it is also a function of the
inner and outer sphere reorganization energies for the electron
transfer reaction in the framework of Marcus theory as shown
in eq 2. The inner sphere and outer sphere reorganization
energies are independent, and the cumulative value gives the
total reorganization energy (= λin + λout), which may be used to
calculate the reaction rate constant on the basis of eq 2.
The most common method for calculating the inner sphere
reorganization energy is Nelsen’s four-point method of
separating oxidants and reductants, which can be expressed as
follows:
Figure 6. Relationship between the magnitude of the solvation energy,
|ΔGsolv|, of positively charged Li+ ions and the corresponding dielectric
constant for the imidazolium based ionic liquid solvents with varying
alkyl side chain length. The magnitude of the solvation energy
decreases as the dielectric constant of the solvent decreases and shows
a linear trend with the inverse of the dielectric constant.
ΔG(εi , εs) =
−Q 2 ⎛ 1
1⎞
⎜ − ⎟
εs ⎠
R ⎝ εi
(12)
where R is the radius of the solute molecule, ΔG(εi, εs) is the
electrostatic solvation energy, Q is the net charge on the solute
molecule, and εi and εs are the static dielectric constants of the
solute and solvent, respectively. The solvation model, presented
in eq 12, indicates that the magnitude of the electrostatic
solvation energy decreases with decreasing static dielectric
constant of the solvent. In the present study, the solvation
energy of a Li+ ion in ionic liquids was calculated using the
COSMO solvation model. The trend shown in Figure 6,
whereby the magnitude of the solvation energy of Li+ ion
decreases linearly with the inverse of the dielectric constant of
the ionic liquid, is consistent with that predicted by eq 12.
The Gibbs free energy, which is the driving force for the
oxidation of lithium metal, is obtained by adding the
vaporization energy and the first ionization energy of lithium
to the solvation energy of Li+ ion. Figure 7 shows the variation
of the vaporization energy, ionization energy, and solvation
energy as a function of the inverse of the dielectric constant,
which is in turn a function of the length of the alkyl chain in the
λ in = [E(D+|D) − E(D+|D+)] + [E(A−|A) − E(A−|A−)]
(13)
where E(a|b)is the energy of state “a” calculated at the
equilibrium structure of state “b”, D designates the donor
species, A designates the acceptor species, and the +/−
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superscript designates the charge on the species. Figure 8
illustrates the model system that was used to calculate the inner
Figure 9. Total solvated energy of the system containing (i) 17 lithium
atoms and a single Li+ ion (Ew/ion) and (ii) 17 lithium atoms and a
dummy charge (Ew/dummy) in different ionic liquids as the solvent. The
average increase rates for the system with the Li+ ion and the one with
the dummy charge are 3.45 and 3.27 kJ mol−1 per carbon atom,
respectively.
Figure 8. Configuration of the model system, which comprises 17
neutral lithium atoms and 1 Li+ ion in each unit, used in the DFT
calculation for inner sphere reorganization energy: yellow, Li+ ion;
pink, neutral lithium atom.
the imidazolium based ionic liquid. This effect is well correlated
with the decrease in the static dielectric constant of the ionic
liquids with an increase in the length of the side chain. The
average rates of decrease in magnitude for the system with Li+
ion and the one with the dummy charge are 3.45 and 3.27 kJ
mol−1 per additional carbon atom, respectively. Therefore, the
dielectric constant of the solvent has a slightly greater effect on
the energies of the system with Li+ ion compared to those of
the system with the dummy charge. Moreover, the energy of
the system containing Li+ ion is greater in magnitude compared
to that of the system with a dummy charge. This is due to the
electronic energy of Li+ ion, which is not present in a dummy
charge.
Figure 10 shows the energy of Li+ ion in the optimized
structure of neutral lithium, E(Li+|Li), solvated in ionic liquids
sphere reorganization energy. Out of 18 lithium atoms in the
lattice structure, 1 lithium atom in the lattice acts as the donor
(D) while the remaining 17 lithium atoms act as the acceptors
(A) on which the electron released by the donor is distributed
as an extra negative charge (A−), resulting in the formation of a
single positively charged Li+ ion (D+). We assume that the
lattice structure and the corresponding lattice parameters of
lithium metal in the anode side do not change during the
electron transfer reaction and only the atoms on the free
surface are oxidized. Consequently, since A− and A have
identical lattice structures, E(A−|A) is assumed to be equal to
E(A−|A−). Therefore, the contribution to the inner sphere
reorganization energy due to the differences in energy between
the states of the acceptor species is negligible. Therefore, for
the oxidation of lithium metal at the anode, the simplified
expression for the inner sphere reorganization energy is given
as
λ in = [E(Li+|Li) − E(Li+|Li+)]
(14)
In this equation, the term E(Li+|Li+) is equal to the energy of
Li+ ion in the solvated phase as shown in Figure 5. According to
the definition of the reorganization energy, the above equation
is equal to the energy required to “reorganize” the system
structure from initial to final coordinates, after the charge is
transferred.
The term E(Li+|Li), which is the energy of Li+ ion in the
optimized structure of lithium metal, is the energy of a
positively charged Li+ ion in a neutral lattice of lithium atoms.
To find this energy term, we first determined the optimized
lattice structure of lithium metal comprising 18 sites.
Subsequently, we replaced 1 of the lithium atoms with a Li+
ion and calculated the energy of a system containing 17 neutral
lithium atoms and 1 Li+ ion (E1). Finally, we replaced that Li+
ion with a positively charged dummy center (ghost ion) and
calculated the energy again (E2). The difference between the
two energies, E1 and E2, provides the energy E(Li+|Li):
E(Li+|Li) = E1 − E2
Figure 10. Energy of the Li+ ion in the optimized structure of the
neutral lithium lattice, E(Li+|Li), for varying length of the alkyl side
chain on the imidazolium cation.
with varying structure of the cation side chains. On the basis of
eq 15, E(Li+|Li) is calculated from the difference between the
energies of the two systems shown in Figure 9. Since the rates
of increase of the energies with a change in the length of the
alkyl side chain are nearly identical, the quantity E(Li+|Li) is
almost constant for varying length of the alkyl side chain.
The outer sphere reorganization energy was calculated using
the classical electrostatics model based on Marcus theory:39
(15)
Figure 9 shows the total energy of the system comprising 17
neutral lithium atoms and 1 Li+ ion located in the center of the
top layer as well as the system in which the Li+ ion is replaced
with a positively charged dummy center in the presence of an
ionic liquid as the solvent. As expected, the total energy of the
solvated species in both systems decreases in magnitude with
an increase in the number of carbon atoms in the side chain of
λout =
Δe 2NA ⎛ 1
1 ⎞⎛⎜ 1
1⎞
− ⎟⎟
⎜ −
⎟⎜
εs ⎠
8πεo ⎝ r
R e ⎠⎝ εop
(16)
where r is the ionic radius, Re is twice the distance from the
surface of the electrode at which the electron transfer reaction
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takes place, εs and εop are the static and high-frequency optical
dielectric constants of the solvent, Δe is the amount of charge
transferred, NA is Avogadro’s number, and εo is the permittivity
of a vacuum. Since we assumed that the electron transfer
reaction happens when the donor lithium atom is still in its
optimized position in the lattice, Re will be equal to the lattice
constant, 3.507 Å. Referring to eq 16 for calculating the outer
sphere reorganization energy, λout is inversely proportional to
−εs. Therefore, an increase in the length of the alkyl side chain,
which results in a decrease in the dielectric constant, leads to a
decrease in the outer sphere reorganization energy. This trend
in outer sphere reorganization energy is quantified and
presented in Figure 11. The results show that the decrease in
reorganization energy is significantly more dominant compared
to the effect of changes in the outer sphere reorganization
energy.
The method of corresponding orbital transformation was
utilized to calculate the electronic coupling energy using the
initial and final wave functions, Ψa and Ψb:57,58
Va,b = (1 − Sa,b 2)−1 Ha,b −
1
Sa,b(Ha,a + Hb,b)
2
(17)
where Sa,b = ⟨Ψa|Ψb⟩ is the reactant−product overlap, H is the
total electronic Hamiltonian, excluding nuclear kinetic energy
and nuclear repulsion terms of the system,57 Ha,b = ⟨Ψa|H|Ψb⟩
is the total interaction energy, also referred to as the “electronic
coupling matrix element”, and Ha,a = ⟨Ψa|H|Ψa⟩ is the electronic
energy of the reactants or products. Usually Va,b is very weak,
on the order of a few kilojoules per mole, in electron transfer
reactions. The calculated parameters are shown in Table 2.
Table 2. Calculated Parameters for Determining the
Electron Transfer Coupling Energy
param
calcd value
param
calcd value
Sa,b
Ha,a
Hb,b
−3.55 × 10−4
−774996.272 kJ mol−1
−764864.005 kJ mol−1
Ha,b
Va,b
275.125 kJ mol−1
1.645 kJ mol−1
Since in the corresponding orbital transformation method the
electronic energies of the reactants and products, the reactant−
product overlap, and the total interaction energy are not
calculated in the presence of an external potential energy such
as the one created by the electrostatic field of solvent
molecules, the presence of a solvent does not affect these
energies and the electronic coupling energy will be invariant
with the solvent.
Finally, in an effort to determine the effect of the length of
the alkyl side chain of imidazolium ion on the overall anode
reaction, all the calculated parameters, including the electronic
coupling energy, reorganization energies, and Gibbs free energy
of the reaction, were employed to calculate the electron transfer
rate constant, ket, using Marcus theory. The effects of the Gibbs
free energy and total reorganization energy on the reaction rate
constant for electron transfer at the anode are compared in
Figure 12. As shown in the linear fits included in the plot, the
Figure 11. Variation of the inner sphere (λin), outer sphere (λout), and
total reorganization (λ) energies with the inverse of the static dielectric
constant. The decrease in the inner sphere reorganization energy is
more significant than that of the outer sphere reorganization energy
since the slope of its line is 3.5 times larger than that of the line for the
outer sphere reorganization energy. The average inner sphere
reorganization energy is almost 10 times greater than the average
outer sphere reorganization energy.
the inner sphere reorganization energy is more significant than
that of the outer sphere reorganization energy. The slope of the
linear variation of the inner sphere reorganization energy with
the inverse of the dielectric constant is 3.5 times larger than that
of the outer sphere reorganization energy. As mentioned earlier,
the outer sphere reorganization energy results from the solvent
relaxation due to the change in the orientation of the solvent
molecules during the electron transfer reaction. On the other
hand, the inner sphere reorganization energy is a consequence
of the structural changes of the reactant species. The results
presented in Figure 11 imply that the effect of a structural
change during the electron transfer reaction is more dominant
compared to that of solvent relaxation. This effect is reasonable
since after the electron transfer reaction is complete, a lithium
atom has left the lattice and become an isolated solvated ion
experiencing a totally different environment due to a phase
change from solid to solvated. Therefore, there is a major
structural change resulting in a large inner sphere reorganization energy compared to a comparatively small outer sphere
reorganization energy resulting from the solvent relaxation. The
effect of solvent relaxation is minor due to the relatively low
dielectric constant of these solvents compared to other organic
solvents.56
Moreover, the average inner sphere reorganization energy is
almost 10 times greater than the average outer sphere
reorganization energy. As a consequence, the effect of changes
in the inner sphere reorganization energy on the total
Figure 12. Effects of the variation in the Gibbs free energy and total
reorganization energy on the reaction rate constant for electron
transfer at the anode. The labels with values of n represent the number
of carbons in CnMIM+TFSI−. Purely empirical fits to the data with
corresponding mathematical relations are also shown in the plots.
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electron transfer at the anode in the presence of solvents
based on varying imidazolium cations. Our results indicate that
the total reorganization energy associated with the electron
transfer reaction decreases linearly with the inverse of the
dielectric constant. The decrease in the inner sphere
reorganization energy with the inverse of the dielectric constant
is much more significant than that of the outer sphere
reorganization energy. Additionally, the average inner sphere
reorganization energy is almost 10 times greater than the
average outer sphere reorganization energy. The calculated
Gibbs fee energy, total reorganization energy, and coupling
energy values were used to calculate the reaction rate constant.
The results demonstrate that a decrease in the thermodynamic
driving force for the reaction, due to an increase in the length of
the alkyl side chain of the imidazolium cation, results in a
decrease in the reaction rate constant for oxidation reaction
involving electron transfer at the anode. The decrease in the
logarithm of the reaction rate constant follows a linear trend
with respect to the inverse of the dielectric constant of the ionic
liquid medium. The presented trend can be further employed
to evaluate the rate constant of the oxidation of lithium metal in
ionic liquids with disparate cations and the TFSI− anion if the
dielectric constant of the ionic liquid varies within the range
pertinent to the present study.
slope of the linear variation with respect to the Gibbs free
energy is 27% greater than that of the total reorganization
energy. Therefore, the effect of changes in the Gibbs free
energy dominates the effect of changes in the reorganization
energy. Moreover, the results presented in Figure 12 show that
with decreasing number of carbon atoms in the alkyl side chain,
n, the magnitudes of the Gibbs free energy, which is the
thermodynamic driving force of the reaction, and the
reorganization energy increase, resulting in an increase in the
electron transfer rate constant. Consequently, in this reaction,
the thermodynamic and kinetic effects augment each other.
However, it should be noted that the Gibbs free energy and
reorganization energy are both a function of the dielectric
constant of the solvent. Therefore, these two are not
independent parameters, and by varying the dielectric constant
of the solvent, both parameters change simultaneously,
ultimately changing the rate constant.
Figure 13 shows that the electron transfer rate constant on
the logarithmic scale is inversely proportional to the inverse of
■
AUTHOR INFORMATION
Corresponding Author
*Phone: 509-335-0294. E-mail: [email protected].
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
We acknowledge the use of Washington State University’s highperformance computing cluster for carrying out the simulations.
We also acknowledge funding from the Joint Center for
Aerospace Technology Innovation (JCATI) sponsored by the
State of Washington.
Figure 13. Variation of the rate constant for the electron transfer
reaction with the static dielectric constant of the ionic liquid solvent.
the static dielectric constant of the solvent. Consequently,
increasing the static dielectric constant of the solvent increases
the electron transfer reaction rate. Therefore, it is concluded
that, in imidazolium based ionic liquids with TFSI− as the
anion, the electron transfer reaction at the anode happens faster
for cations with shorter alkyl side chains. However, since the
dielectric constant is the most important solvent parameter, the
fitted expression log ket = −19.265(1/ε) + 0.5465 can be used
to evaluate the reaction rate constant of the oxidation of lithium
metal in other ionic liquids with a dielectric constant that varies
within the studied range and that have TFSI− as the anion.
■
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In an effort to study the kinetics of the electrochemical reaction
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