Density Functional Theory Based Study of the Electron Transfer

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Density Functional Theory Based Study of the Electron Transfer Reaction at the Lithium Metal Anode in Lithium-Air Battery with Ionic Liquid Electrolytes Saeed Kazemiabnavi, Prashanta Dutta, and Soumik Banerjee J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp506563j • Publication Date (Web): 30 Oct 2014 Downloaded from http://pubs.acs.org on November 12, 2014

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The Journal of Physical Chemistry

Density Functional Theory Based Study of the Electron Transfer Reaction at the Lithium Metal Anode in Lithium-Air Battery with Ionic Liquid Electrolytes Saeed Kazemiabnavi, Prashanta Dutta, and Soumik Banerjee∗ School of Mechanical and Materials Engineering, Washington State University Pullman, WA, 99164-2920, U.S.A ABSTRACT. Room temperature ionic liquids, which have unique properties such as relatively broad 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 reactions 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 based on the structure of their constituent cations and anions, plays a crucial role in determining the reaction rate at the anode. In this study, we present a novel approach, based on Marcus theory, to evaluate the effect of varying length of alkyl side chain of model imidazolium based cations on the rates of electron transfer reaction at the anode. Density functional theory (DFT) 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 ∗

Corresponding Author, Tel: +1 509 3350294, E-mail: [email protected]

1 Environment ACS Paragon Plus


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the static dielectric constant of the ionic liquid that in turn corresponds with increase in the length of alkyl side chain of the ionic liquid cation. The 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 reactions were calculated in presence of varying ionic liquid based electrolytes. The overall rate constants for electron transfer increases with increase in the static dielectric constant. The presented results provide important insight on identification of appropriate ionic liquid electrolytes to obtain enhanced current densities in lithium-air batteries. KEYWORDS. Lithium-air battery, density functional theory, Marcus theory, electron transfer, rate constant


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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 Wh/kg.2 New chemistries are therefore required to attain this goal. Lithium−air (Li-air) batteries, which employ 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 gravimetric energy density, 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 reactive5, 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 high vapor pressure leading to increased flammability. Therefore, Li-air batteries 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 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



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ionic conductivity, static and optical dielectric constant, and 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 high electrical conductivity, wide electrochemical stability window and also 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 Table 1. Qualitative comparison of ionic liquids with organic solvents17 Property

Organic Solvents

Ionic Liquids

Number of solvents

>1,000

>1,000,000

Applicability

Single function

Multifunction

Vapor pressure

Usually high

Very low

Flammability

Usually flammable

Mostly nonflammable

Solvation

Weakly solvating

Tunable solvating

Thermal stability

Low

High

Electrochemical stability window

Narrow

Wide

Room temperature ionic liquids are usually quaternary ammonium salts such as tetralkylammonium [R4N]+ or based on cyclic amines, both aromatic (pyridinium, imidazolium) and saturated (piperidinium, pyrrolidinium). Low temperature molten salts based on sulfonium [R3S]+ as well as phosphonium [R4P]+ cations are also known. Therefore, by combining these cations with both inorganic (halide, [BF4]−, [PF6]−, [AsF6]− or amide [N(CN)2]-) and organic ([C4F9SO3]−, trifluoroacetic [CF3CO2]−, triflate [CF3SO3]−, amides [N(CF3SO2)2]− and [CF3CONCF3SO2]−, 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


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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 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 ButlerVolmer equation

18, 19

. During the discharge cycle of Li-air batteries, pure lithium metal gets

oxidized at the anode producing positively charged Li+ ions20: → +

(1)

The reverse reaction happens during the charge cycle. The electrodeposition and dissolution of the metal-like lithium occurs 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 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(trifluoromethanesulphonyl)amide (CnMIM+ TFSI-) ionic liquids with varying number of carbon atoms in the side alkyl chain, n, were chosen as the model



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electrolyte. The choice was justified due to the relatively wide electrochemical stability window, low viscosity and higher ionic conductivity of imidazolium based cations compared to other ionic liquids.15,

21-24

Additionally, relevant physical properties, such as static and optical

dielectric constant, 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.

Figure 1. The chemical structures of model ionic liquid a) 1-propyl-3-methylimidazolium (C3MIM+) cation and b) bis(trifluoromethanesulphonyl)amide (TFSI-) anion are shown. 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 this assumptions

36-38

. However, these theories are not

applicable for electron transfer reactions since characterizing the transition state in this type of reactions 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 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 6 Environment ACS Paragon Plus

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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 the 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 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 is 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. 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: =

ℏ

| |

exp −

∆"°$%

&

(2)

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, ∆G° 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 in order to get a qualitative sense of how these parameters are related to each other. 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 case of



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electron transfer reactions. Firstly, it provides a mathematical expression for the activation energy, based on the reorganization energy and the Gibbs free energy of the electron transfer reaction: ∆' ‡ =

)*°$%

(3)

Figure 2. The energy diagram for electron transfer reaction, including reorganization energy and electronic coupling energy, which are used to calculate the reaction rate constant in the framework of Marcus theory, is presented. Secondly, it provides a mathematical relationship, provided in Eq. (2), to determine the preexponential 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 parts42: the inner sphere and outer sphere reorganization energy. 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. 8 Environment ACS Paragon Plus

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COMPUTATIONAL MEHODOLOGY In this study, all energy calculations were done using Density Functional Theory (DFT) with Becke, three-parameter, Lee-Yang-Parr44,45 (B3LYP) exchange-correlation functional and 6311++G** as the basis set, which is a Valence Triple Zeta basis set with polarization and diffuse functions on all atoms (VTZPD). Since there is no vibrational frequency associated with a single atom, the vibrational contributions were neglected in order to avoid any inconsistency in the energy of single atomic systems with other systems. The optimization of lithium lattice was performed using periodic boundary condition DFT (PBC-DFT) with 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 package49. Static Dielectric Constant (ε)

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12 11

n=3

10 9 8 7 6 1

2

3 4 5 6 7 8 + n in [CnMIM] [TFSI] ionic liquid

9

Figure 3. Static dielectric constant, ε, of some imidazolium based ionic liquids with TFSI- as the anion21. 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 static dielectric constant. The static dielectric constant decreases as the length of side chain increases. The Conductor-like Screening Model50 (also known as COSMO solvation model), which is an approach to account for dielectric screening in solvents, was implemented to investigate the



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effect of solvent on reaction rates. The primary inputs to the model are the dielectric constant of the solvent as well as the radii of solvated species and solvent molecules. However, for all calculations, the Li+ ions with a radius of 0.76 Å which have a constant radius of 3.801 Å.

21

51

are surrounded solely by the TFSI- anions,

Therefore, the only parameter that determines the

solvation energy is the dielectric constant of the solvent. Figure 3 shows the variation of static dielectric constant of the imidazolium based ionic liquids considered in this study with respect to the number of carbon atoms in the side alkyl chain21. The static dielectric constant of the imidazolium cation based ionic liquid decreases with increase in the number of carbons in the alkyl side chain. RESULTS AND DISCUSSION The principal objective of the present study is to determine the reaction rate at the anodeelectrolyte interface of lithium-air batteries in presence of imidazolium cation based ionic liquids. The goal is to relate the structure of the cation of the ionic liquid solvent on the overall reaction rate. The oxidation process of lithium metal in anode, due to electron transfer, can be represented as: +$ → +,-.$ +

(4)

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 pre-factor. The Gibbs free energy for this reaction, in case of various ionic liquid solvents, provides an estimate of the thermodynamic driving force for the anode reaction. In order to calculate the Gibbs free energy of the reaction, the overall oxidation process of lithium metal in anode is assumed to be the cumulative effect of the following intermediate reactions52: +$ → /$

(5)


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/$ → /$ + (6) /$ → +,-.$

(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: ∆'° = ∆'.01 $ + ∆'2,3 $ + ∆'+,-. $

(8)

where, ∆'.01 $ is the free energy of vaporization of lithium metal, ∆'2,3 $ is the first ionization energy of lithium and ∆'+,-. $ is the solvation energy of Li+ ion. In order to calculate the vaporization energy of lithium metal, the optimized lattice structure of lithium metal was determined using periodic boundary condition DFT (PBC-DFT) and the total energy of the system was then calculated using non-periodic 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: ∆'.01 $ = 4/ − 4+

(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 non-periodic 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): ∆'2,3 $ = 42,3 − 4/

(10)

Finally, in order to determine the solvation energy of Li+ ion, a single point energy calculation using DFT was performed on Li+ ion in solution phase to evaluate the energy of Li+ ion in the solvated phase, Esolv :



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∆'+,-. $ = 4+,-. − 42,3 (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 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 increasing number of carbons in the alkyl side chain. This decrease in static dielectric constant causes decrease in magnitude of the solvation energy of lithium atoms as the length of the side chain increases.53 Consequently, 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 noticed 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.


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-19779.97 Es (kJ.mol-1)

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-19780.01

-19780.05

-19780.09 1

2

3 4 5 6 7 8 n in [CnMIM]+[TFSI]-

9

Figure 4. Total solid phase energy of each atom in a system comprising 18 lithium atoms in solvated phase (Esolv) is shown. 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. In order 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 filed interacting with lithium atoms in the gas phase due to the absence of solvent molecules, the gas phase 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). Based on Eq. (10), calculation of the Gibbs free energy also requires the evaluation of 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 with that of an ion in 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 solvated phase as well as in 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 side alkyl chain leads to a decrease in the dielectric constant of the ionic liquid. The diminished magnitude of the dielectric constant leads to 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. Solvated phase Gas phase

-19920

-19086 -19106

-19940

-19126

-19960

-19146

-19980

-19166

Eg (kJ.mol-1)

-19900 ESolv (kJ.mol-1)

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1 2 3 4 5 6 7 8 9 n in [CnMIM]+[TFSI]Figure 5. Total energies of a single Li+ ion in gas phase (Eg) and solvated phase (Esolv), in imidazolium based ionic liquids of varying length of side chains, are shown. The gas phase energy remains constant since there is no electrostatic interaction due to the absence of ionic liquid solvent. Increase in the number of carbons in the side alkyl 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.


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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 Fig. 5 using Eq. (11). Tjong and Zhou53 showed that the dependence of electrostatic solvation energy of a charged species on dielectric constant of solvent can be expressed by the following equation: ∆' =2 , =+ $ =

? %

@ − @ & A

B

(12)

where, C is the radius of solute molecule, ∆' =2 , =+ $ is the electrostatic solvation energy, D is the net charge on solute molecule and =2 and =+ are the static dielectric constant of 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). 860 |ΔGsolv| (kJ.mol-1)

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|ΔGsolv| = -920.44 (1/ε) + 920.54 R² = 1

840 820 800 780 760 0.08

0.1

0.12 1/ε

0.14

0.16

Figure 6. The relationship between the magnitude of solvation energy, |∆Gsolv|, of positively charged Li+ ions and the corresponding dielectric constant for the imidazolium based ionic liquid 15 Environment ACS Paragon Plus


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solvents, with varying length of side alkyl chains, is shown. The magnitude of solvation energy decreases as the dielectric constant of the solvent decreases and shows a linear trend with the inverse of the dielectric constant. 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 dielectric constant, which is in turn a function of the length of the alkyl chain in the solvent cation. As previously discussed, the vaporization energy remains almost constant since the change in the energy of solid phase 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 5.8% relative error.52, 54 The ionization energy of lithium is independent of solvent since the reactant and product of the ionization reaction are in gas phase. The calculated value of ionization energy has 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 smaller dielectric constant. Figure 7 shows that the magnitude of overall Gibbs free energy for the oxidation reaction, |ΔG˚|, is inversely proportional to the static dielectric constant of the solvent. Therefore, the thermodynamic driving force for the oxidation of lithium metal is smaller in ionic liquids with larger length of alkyl side chain. As can be seen from the results presented in Figure 7, the slope of the Gibbs free energy and the solvation energy lines are almost equal which is expected since the vaporization energy


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of lithium is almost independent of the dielectric constant of solvent compared to the solvation

860

Solvation Ionization

Vaporization Gibbs free energy

|ΔGsolv| = -920.44 (1/ε) + 920.54

840 820

600 400

800

|ΔG˚| = -919.13 (1/ε) + 266.95

200

780 ΔGvap = -1.3081 (1/ε) + 111.27

760 0.08

0.13 1/ε

0

ΔGVap, ΔGion,|ΔG˚|(kJ.mol-1)

energy of Li+ ion.

|ΔGSolv| (kJ.mol-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.18

Figure 7. Variation of vaporization (∆Gvap), ionization (∆Gion), magnitude of solvation (|∆Gsolv|) and the magnitude of total Gibbs free energy (|∆G˚|) of the anode reaction with the static dielectric constant of the solvent is presented. The difference in appearance of the Gibbs free energy and solvation energy lines, with approximately the same slope, is due to the different yaxis scales. While the reaction rate constant for the electrochemical reaction at 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, = E23 + E,F , that may be used to calculate the reaction rate constant based on Eq. (2).



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The most common method for calculating the inner sphere reorganization energy is the Nelsen’s four point method of separating oxidants and reductants, which can be expressed as follows: E23 = G4 H |H$ − 4 H |H $I + G4 J |J$ − 4 J |J $I

(13)

where, 4 K|L$ is the energy of state “K” calculated at the equilibrium structure of state “L”, D designates donor species, A designates acceptor species and +/- superscript designate the charge on the species. Figure 8 illustrates the model system that was used to calculate the inner sphere reorganization energy. Out of 18 lithium atoms in the lattice structure, one 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 J and J have identical lattice structures, 4 J |J$ is assumed to be equal to 4 J |J $. Therefore the contribution to inner sphere reorganization energy due to the differences in energy between the states of the acceptor species is negligible. So, for the oxidation of lithium metal at the anode, the simplified expression for inner sphere reorganization energy is given as: E23 = G4 |$ − 4 | $I

(14)

In this equation, the term 4 | $ is equal to the energy of Li+ ion in the solvated phase as shown in Figure 5. According to the definition of 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 4 |$, 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. In order 18 Environment ACS Paragon Plus

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to find this energy term, we first determined the optimized lattice structure of lithium metal comprising 18 sites. Subsequently, we replaced one of the lithium atoms with a Li+ ion and calculated the energy of a system containing 17 neutral lithium atoms and one 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, 4 |$: 4 |$ = 4 − 4

(15)

Figure 8. The configuration of the model system, which comprises 17 neutral lithium atoms and one Li+ ion in each unit, used in the DFT calculation for inner sphere reorganization energy is shown. Yellow: Li+ ion. Pink: Neutral lithium atom. Figure 9 shows the total energy of the system comprising 17 neutral lithium atoms and one 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 ionic liquid as the solvent. As expected, the total energy of solvated species in both systems decreases in magnitude with increase in the number of carbon atoms in the side chain of the imidazolium based ionic liquid. This effect is well correlated with the decrease in the static dielectric constant of the ionic liquids with increase in the length of the side chain. The average rate of decrease in magnitude for the system with Li+ ion and the one with dummy charge are 3.45 kJ.mol-1 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 that with the dummy charge. Moreover, the energy of the system containing Li+ ion is slightly larger in magnitude compared to the system with a dummy charge. This is due to the electronic energy of Li+ ion

-355365

-336484

-355370

-336489

-355375

-336494

-355380

-336499

-355385

-336504

Ew/dummy (kJ.mol-1)

which is not present in a dummy charge.

Ew/ion (kJ.mol-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 2 3 4 5 6 7 8 9 n in [CnMIM]+[TFSI]With Li ion With dummy charge Figure 9. The total solvated energy of the system containing a) 17 lithium atoms and a single Li+ ion (Ew/ion) b) 17 lithium atoms and a dummy charge (Ew/dummy) in different ionic liquids as the solvent. The average increase rates for the system with Li+ ion and the one with dummy charge are 3.45 kJ.mol-1 and 3.27 kJ.mol-1 per each carbon atom respectively. Figure 10 shows the energy of Li+ ion in the optimized structure of neutral lithium, 4 |$, solvated in ionic liquids with varying structure of the cation side chains. Based on Eq. (15), 4 |$ is calculated from the difference between the energies of the two systems shown in Figure 9. Since the rate of increase of the energies with change in the length of the alkyl side chain is nearly identical, the quantity 4 |$ is almost constant for varying length of alkyl side chain.


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-18870 E(Li+|Li) (kJ.mol-1)

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-18874 -18878 -18882 -18886 -18890 1

2

3 4 5 6 7 n in [CnMIM]+[TFSI]-

8

9

Figure 10. The energy of Li+ ion in the optimized structure of neutral lithium lattice E(Li+|Li), for varying length of alkyl side chain on imidazolium cation, is shown. The outer-sphere reorganization energy was calculated using the classical electrostatics model based on Marcus theory39: E,F =

) % MN O @∘

Q − & S@ R

TU

−@ V B

(16)

where, W is the ionic radius, C is twice the distance from the surface of the electrode at which the electron transfer reaction takes place, ɛ+ and ɛ,1 are the static and high frequency optical dielectric constants of the solvent, Y is the amount of charge transferred, Z[ is the Avogadro’s number and ɛ˳ is the permittivity of vacuum. Since we assumed that the electron transfer reaction happens when the donor lithium atom is still in its optimized position in the lattice, C will be equal to 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 decrease in the dielectric constant, leads to 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 the inner-sphere reorganization energy is more significant than that of the outer-sphere reorganization energy. The slope in the 21 Environment ACS Paragon Plus


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linear variation of inner sphere reorganization energy with inverse of 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 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 structural change during the electron transfer reaction is more dominant compared to the 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 phase. 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 small dielectric constant of these solvents compared to other organic solvents56. 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 innersphere reorganization energy on the total reorganization energy is significantly more dominant compared to that of outer-sphere reorganization energy.


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200

1200 λ = -1165.6 (1/ε) + 1292.1

λin and λ (kJ.mol-1)

1150

150

1100

100 λout = -258.94 (1/ε) + 128.42

1050 1000

50 λin = -906.65 (1/ε) + 1163.6

0.08

0.13 1/ε

Inner-sphere

Total

λout (kJ.mol-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 0.18

Outer-sphere

Figure 11. The variation of inner sphere (λin), outer sphere (λout) and total reorganization energy (λ) with inverse of static dielectric constant is presented. 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 outer sphere reorganization energy. The average inner-sphere reorganization energy is almost 10 times greater than the average 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, ]0 and ]^ 0,^ = _1 − a0,^ b

cd0,^ − a0,^ _d0,0 + d^,^ bc

57, 58

: (17)

where, a0,^ = e]0 |]^ f is the reactant-product overlap, d is the total electronic Hamiltonian, excluding nuclear kinetic energy and nuclear repulsion terms of the system57, d0,^ = e]0 |d|]^ f is the total interaction energy also referred to as “electronic coupling matrix element” and d0,0 = e]0 |d|]0 f is the electronic energy of reactants or products. Usually 0,^ is very weak, of the order of a few kilojoules per mole, in electron transfer reactions. The calculated parameters are shown in Table 2. Since in the Corresponding Orbital Transformation method, the electronic energies of reactant and products, the reactant-product overlap and the total interaction energy



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are not calculated in the presence of external potential energy such as the one created by the electrostatic field of solvent molecules, the presence of solvent does not affect these energies and the electronic coupling energy will be invariant with the solvent. Table 2. Calculated parameters for determining the electron transfer coupling energy are provided. Parameter Calculated value a0,^

d0,0 d^,^

d0,^ 0,^

-3.55×10-4 -774996.272 kJ.mol-1 -764864.005 kJ.mol-1 275.125 kJ.mol-1 1.645 kJ.mol-1

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 electronic coupling energy, reorganization energies and the Gibbs free energy of the reaction were employed to calculate the electron transfer rate constant, ket, using Marcus theory. The effects of Gibbs free energy and total reorganization energy on the reaction rate constant for electron transfer at anode are compared in Figure 12. As shown in the linear fits included in the plot, the slope of the linear variation with respect to Gibbs free energy is 27% greater than that of the total reorganization energy. Therefore, the effect of changes in Gibbs free energy dominates the effect of changes in reorganization energy. Moreover, the results presented in Figure 12 show that with decreasing the number of carbon atoms in the side alkyl chain, n, the magnitude of the Gibbs free energy which is the thermodynamic driving force of the reaction, and reorganization energy increase, resulting in an increase in the electron transfer rate constant. Consequently, in this reaction, the


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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 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. 1100 -1.1

1120

λ (kJ.mol-1) 1140 1160

1180

1200

n=2 n=3

log ket = 0.0165 λ - 20.808

log ket

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


n=4

-1.6 n=5

-2.1

n=6 n=8 log ket = 0.021|ΔG˚|-5.0488

-2.6 100

120

140 160 180 200 |ΔG˚|(kJ.mol-1) Gibbs free energy Reorganization energy Figure 12. The effects of variation in Gibbs free energy and total reorganization energy on the reaction rate constant for electron transfer at the anode are presented. 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. Figure 13 shows that the electron transfer rate constant in logarithmic scale is inversely proportional to the inverse of 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 dielectric constant that vary within the studied range and have TFSI- as the anion. -1.0 log ket = -19.265 (1/ε) + 0.5465 R² = 0.9996

-1.5 log ket

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-2.0

-2.5 0.08

0.10

0.12 1/ε

0.14

0.16

Figure 13. The variation of the rate constant for the electron transfer reaction with static dielectric constant of the ionic liquid solvent is shown. CONCLUSIONS In an effort to study the kinetics of the electrochemical reaction at the anode side of the Li-air battery, the reaction rate constant for electron transfer from lithium metal in ionic liquid electrolytes with varying dielectric constants was calculated using density functional theory (DFT). Marcus theory formulation was used to evaluate the rate constant and COSMO solvation model was implemented to investigate the effect of solvent on these reaction rates. We calculated the Gibbs free energy for the individual steps for the overall anode reaction including vaporization, ionization and solvation. The results indicate that increasing the number of carbons in the alkyl side chain of the imidazolium based ionic liquids, which decreases the dielectric constant of the solvent, leads to decrease in the magnitude of the Gibbs free energy. We also determined the inner sphere and outer sphere reorganization energies associated with electron transfer at the anode in presence of solvents based on varying imidazolium cations. Our results


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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 ten 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 decrease in the thermodynamic driving force for the reaction, due to 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 TFSI- anion if the dielectric constant of the ionic liquid varies within the range pertinent to the present study.

AUTHOR INFORMATION Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding Sources This project was funded by the Joint Center for Aerospace Technology Innovation (JCATI) sponsored by the State of Washington.



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ACKNOWLEDGMENT The authors acknowledge the use of Washington State University’s high performance computing cluster for carrying out the simulations. The authors acknowledge funding from the Joint Center for Aerospace Technology Innovation (JCATI) sponsored by the State of Washington. REFERENCES 1. Christensen, J.; Albertus, P.; Sanchez-Carrera, R. S.; Lohmann, T.; Kozinsky, B.; Liedtke, R.; Ahmed, J.; Kojic, A., A Critical Review of Li/Air Batteries. J. Electrochem. Soc. 2012, 159, R1-R30. 2. Girishkumar, G.; McCloskey, B.; Luntz, A. C.; Swanson, S.; Wilcke, W., Lithium - Air Battery: Promise and Challenges. J. Phys. Chem. Lett. 2010, 1, 2193-2203. 3. Rahman, M. A.; Wang, X.; Wen, C., A Review of High Energy Density Lithium-Air Battery Technology. J. Appl. Electrochem. 2014, 44, 5-22. 4. Imanishi, N.; Yamamoto, O., Rechargeable Lithium-Air Batteries: Characteristics and Prospects. Mater. Today 2014, 17, 24-30. 5. Kamienski, C. W.; McDonald, D. P.; Stark, M. W.; Papcun, J. R., Lithium and Lithium Compounds. John Wiley & Sons, Inc.: 2004. 6. Furr, A. K., CRC Handbook of Laboratory Safety. Boca Raton: CRC Press: 2000. 7. Roth, E. P.; Orendorff, C. J., How Electrolytes Influence Battery Safety. Electrochem. Soc. Interface 2012, 21, 45-49. 8. Nazri, G.-A.; Pistoia, G., Lithium Batteries: Science and Technology. Springer: Boston, 2003; p 610-611. 9. Marsh, R. A.; Vukson, S.; Surampudi, S.; Ratnakumar, B. V.; Smart, M. C.; Manzo, M.; Dalton, P. J., Li Ion Batteries for Aerospace Applications. J. Power Sources 2001, 97, 25-27. 10. Azimi, N.; Weng, W.; Takoudis, C.; Zhang, Z., Improved Performance of Lithium-Sulfur Battery with Fluorinated Electrolyte. Electrochem. Commun. 2013, 37, 96-99. 11. Markevich, E.; Baranchugov, V.; Aurbach, D., On the Possibility of Using Ionic Liquids as Electrolyte Solutions for Rechargeable 5 V Li Ion Batteries. Electrochem. Commun. 2006, 8, 1331-1334. 12. Borgel, V.; Markevich, E.; Aurbach, D.; Semrau, G.; Schmidt, M., On the Application of Ionic Liquids for Rechargeable Li Batteries: High Voltage Systems. J. Power Sources 2009, 189, 331-336. 13. Croy, J. R.; Abouimrane, A.; Zhang, Z., Next-Generation Lithium-Ion Batteries: The Promise of Near-Term Advancements. MRS Bull. 2014, 39, 407-415. 14. Lewandowski, A.; Swiderska-Mocek, A., Ionic Liquids As Electrolytes for Li-Ion BatteriesAn Overview of Electrochemical Studies. J. Power Sources 2009, 194, 601-609. 15. Galinski, M.; Lewandowski, A.; Stepniak, I., Ionic Liquids As Electrolytes. Electrochim. Acta 2006, 51, 5567-5580.


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Density Functional Theory Based Study of the Electron Transfer

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