Overview of Lithium Ion Battery Electrode Development

4.1 Candidate Cathode Material with High Voltage for Lithium Ion Batteries

4.1.1 Introduction

• Early Battery Development (Mid-1970s):
  • Whittingham proposed TiS2 for non-aqueous secondary batteries due to its high specific and power density.
  • TiS2 has good metallic properties and allows reversible lithium intercalation.
  • The discharge voltage was around $2V$ with current densities of $1-10 mAcm^{-2}$.
  • Commercialization was limited to button-size cells due to issues with practical non-aqueous batteries and reversible lithium deposition.
• LiCoO2 Introduction (1980):
  • Mizushima et al. suggested LiCoO2 with a-NaFeO2 structure as a cathode material.
  • LiCoO2 is structurally more stable than TiS2.
  • The open-circuit voltage of LiCoO2/Li cell was approximately $3.6V$, about twice that of LiTiS2.
  • The theoretical energy density was $1.1 kWhkg^{-1}$.
  • It took almost 10 years to commercialize LiCoO2 in batteries.
• Current Demands and Challenges:
  • Advancements in microelectronics and electric vehicles (EVs, HEVs) require low-cost, environmentally friendly, and thermally stable lithium-ion batteries.
  • These batteries need high power and energy density.
• LiMn2O4 as a Potential Cathode Material:
  • Lithium manganese spinel (LiMn2O4) is considered attractive due to these demands.
  • LiMn2O4 can be cycled at a $4V$ range but suffers from gradual capacity degradation, especially above $50°C$.
  • Capacity fading is attributed to Mn dissolution into the electrolyte via the disproportionation reaction: $2Mn^{3+}(solid) \rightarrow Mn^{4+}(solid) + Mn^{2+}(liquid)$.
• Strategies to Improve LiMn2O4 Stability:
  • Bittihn et al. explored Co substitution for Mn in LiMn2O4, which improved cycleability and formed LiCoMn2O4.
  • Research group systematically studied LiMyMn2-yO4 (M=Co,Cr,Ni) in the $4V$ range, finding better cycleability than LiMn2O4, even with small substitutions.
  • Substitution enhances spinel stability by strengthening M-O bonds over Mn-O bonds in the MO6 octahedra.
  • Thermodynamic stability of MO6, including MnO, in spinels was presented using a Born-Haber cycle with a simple ionic model.
  • Molecular dynamics (MD) calculations, considering a partially ionic model, further supported the structural stability of spinels.
• Development of 5V Class Cathodes:
  • Extensive charge-discharge studies have been conducted using partially substituted spinel oxides in the $5V$ class.
  • For large-scale EV batteries, operating voltage is typically $300V-400V$, making the number of single cells critical.
  • Higher single-cell discharge voltage reduces the required cell stack number and improves quality control.
• Overcoming Electrolyte Limitations:
  • The discovery of highly oxidation-resistant electrolytes has enabled the study of intercalation compounds above $5V$.
  • Electrolytes like 1M LiPF6 or LiClO4 in EC and DMC or EC and DEC mixtures are used.
• High Voltage Spinel Materials:
  • Several research groups have reported transition-metal substituted spinel materials, $LiMyMn<em>{2-y}O</em>4$, (M=Cr³+, Fe³+, Ni²+, Co³+) with plateaus around $5V$.
  • These spinels typically exhibit two plateaus at around $4V$ and $5V$.
  • The $4V$ plateau corresponds to $Mn^{4+}$ reduction to $Mn^{3+}$, while the $4.9V$ plateau is due to $Cr^{4+}$ to $Cr^{3+}$ or $Co^{4+}$ to $Co^{3+}$ reduction.
  • In $LiNiyMn<em>{2-y}O</em>4$, plateaus are present at around $4.3V$ and $4.7V$, corresponding to the $Mn^{3+}/Mn^{4+}$ and $Ni^{2+}/Ni^{4+}$ redox couples, respectively.
• LiNi0.5Mn1.5O4 Composition:
  • The ideal composition $LiNi<em>{0.5}Mn</em>{1.5}O_4$ is of special interest because Mn remains at +4 oxidation state.
  • This results in only the $Ni^{2+}/Ni^{4+}$ redox couple being active during charge/discharge.
  • However, synthesis difficulties exist, as high calcination temperatures above $800°C$ can partially reduce Mn from $Mn^{4+}$ to $Mn^{3+}$ .
  • Impurities like NiO and $Li_xNiO$ can also form during solid-state reactions.
  • Myung et al. synthesized pure $LiNi<em>{0.5}Mn</em>{1.5}O_4$ using an emulsion drying method.
  • Kim et al. synthesized $LiNi<em>{0.5}Mn</em>{1.5}O<em>{4-\delta}(Fd3m)$ using a molten salt method at $900°C$ and obtained a single cubic phase of $LiNi</em>{0.5}Mn<em>{1.5}O</em>4(P4_332)$ after annealing at $700°C$ in air.
• Structural Order-Disorder Effects: These two compositions exhibit distinct differences in charge-discharge properties. These differences appear to come from structural order-disorder, which affect Li diffusion.

4.1.2 Crystal Structure of Spinel Type Phase

• Spinel Structure of LiNi0.5Mn1.5O4:
  • Blasse first reported the structure of stoichiometric $LiNi<em>{0.5}Mn</em>{1.5}O_4$.
  • Gryffroy et al. proposed it was ordered-type (P4332) using ND and IR measurements, supporting Blasse's proposal.
  • Similar results were later obtained from Li MAS-NMR measurements.
  • Kim et al. determined the structure of $LiNi<em>{0.5}Mn</em>{1.5}O<em>4$ and nonstoichiometric $LiNi</em>{0.5}Mn<em>{1.5}O</em>{4-\delta}$ by careful XRD and electron diffraction measurements.
  • Their results on $LiNi<em>{0.5}Mn</em>{1.5}O<em>4$ were consistent with others, while $LiNi</em>{0.5}Mn<em>{1.5}O</em>{4-\delta}$ belongs to cubic spinel(Fd3m).
• Synthesis of LiNi0.5Mn1.5O4 in Research Group:
  • Stoichiometric $LiNi<em>{0.5}Mn</em>{1.5}O<em>4$ was synthesized by mixing $Li</em>2CO<em>3$, NiO, and $Mn</em>2O_3$ with a molar ratio of Li : Ni : Mn = 1.0 : 1.5 : 1.5 in ethanol.
  • NiO was obtained by decomposing $NiC<em>2O</em>4 \cdot 2H_2O$ in air at $700°C$ for 24h.
  • $Mn<em>2O</em>3$ was obtained by heating $MnCO<em>3$ at $600°C$ for 48h to prevent composition deviation due to hygroscopic properties of $MnCO</em>3$.
  • After evaporating ethanol, condensed materials were heated at $820°C$ in $O_2$ atmosphere for 4 days and slowly cooled to room temperature at a rate of $0.5°Cmin^{-1}$.
• Nonstoichiometric LiNi0.5Mn1.5O4-δ Synthesis:
  • Nonstoichiometric $LiNi<em>{0.5}Mn</em>{1.5}O_{4-\delta}$ was synthesized by heating the stoichiometric compound at $820°C$ under controlled partial pressure of oxygen.
  • The pressure was controlled by changing the mixing ratio of $O<em>2$ and $N</em>2$ using $O<em>2$ and $N</em>2$ - $O_2$ balanced gas.
  • δ values were evaluated from the weight change using TG (Seiko Denshi, EXSTAR6000TG/DTA).
  • To obtain data for electrochemical and structural properties at various δ values, nonstoichiometric samples were quenched from $820°C$ to $0°C$ by controlling partial pressures of oxygen in the same manner.
  • Slight amount of impurity $Li<em>xNi</em>{1-x}O$ coexists.
  • Disordered-type cubic spinel (Fd3m) is observed in the range of δ values between 0 and 0.31.
  • Lattice parameter changes with lithium intercalation.
  • Ordered $LiNi<em>{0.5}Mn</em>{1.5}O<em>4$ has three cubic phases, while disordered $LiNi</em>{0.5}Mn<em>{1.5}O</em>{4-\delta}$ forms only two phases.
• Neutron Diffraction (ND) Patterns:
  • ND patterns were obtained for $LiNi<em>{0.5}Mn</em>{1.5}O_{4-\alpha-\delta}$ (δ=0.03 and 0.21).
  • ND patterns of $LiNi<em>{0.5}Mn</em>{1.5}O<em>4$ (P4332) and $LiMn</em>2O_4$ (Fd3m), estimated by diffraction simulation, were included as references.
  • Neutron diffraction was carried out at KEK, Neutron Scattering in Tsukuba, Japan, using the TOF method with the Vega ND apparatus.
  • Simulation of the diffraction data was performed.
  • As δ values increase, the peaks belonging to ordered $LiNi<em>{0.5}Mn</em>{1.5}O_{4-\alpha-\delta}$ (α≈0) gradually diminished, supporting electron diffraction data.
  • This is attributed to increased disorder due to an increase in defects.
• Defects in Spinel Oxides:
  • Defects in spinel oxides has been discussed by many researchers.
  • Density measurements of $LiMn<em>2O</em>{4-\delta}$ proposed that the defective structures synthesized under specific oxygen partial pressure are metal excess structures rather than oxygen defects.
  • Metals in the octahedral 16d sites may migrate to 16c octahedral sites, which are usually vacant in stoichiometric spinels, and the same would happen in $LiNi<em>{0.5}Mn</em>{1.5}O_{4-\delta}$.
  • Koyama et al. evaluated the formation energy for $LiMn<em>2O</em>{4-\delta}$, including oxygen-vacancy and metal excess types, using first principles plane-wave pseudopotential calculation, which showed that metal excess defects had the lowest formation energy and were consistent with density measurement results.

4.1.3 Electrochemical Properties of LiNi0.5Mn1.5O4 and LiNi0.5Mn1.5O4-δ

• Charge-Discharge Curves:
  • First charge-discharge curves for $LiNi<em>{0.5}Mn</em>{1.5}O_{4-\delta}$ (δ=0-0.31) at $40\mu Acm^{-2}$ current density were obtained.
  • Even at δ=0, a small shoulder at around $4.0V$ is observed, which increases with δ values.
  • This shoulder is due to the $Mn^{4+}/Mn^{3+}$ redox reaction, indicating small amounts of $Mn^{3+}$ exist even in stoichiometric $LiNi<em>{0.5}Mn</em>{1.5}O_4$.
  • The plateau at around $4.7V$ corresponds to the $Ni^{2+/4+}$ redox reaction.
  • Similar charge-discharge profiles have been reported in literature.
  • The $4.7V$ discharge voltage is 15% higher than $4.0V$ range cathodes.
  • If a proper stable electrolyte enduring oxidation around $5V$ is developed, $LiNi<em>{0.5}Mn</em>{1.5}O_{4-\delta}$ could be a good high-voltage cathode candidate.
• Rate Performance:
  • Rate performance of $LiNi<em>{0.5}Mn</em>{1.5}O_{4-\delta}$ (δ=0 and 0.31) shows nonstoichiometric samples have excellent high-rate property.
  • Lithium diffusion in disordered spinels is easier than in ordered ones.
  • This may depend on the lithium diffusion path.
• Coulombic Potential Calculation: Site-potential energy for lithium migration in orderd and disorded spinels is caluculated using a simple Coulombic potential calculation

4.1.4 Coulombic Potential Calculation of Diffusion Path for Ordered (P4332) and Disordered (Fd3m) Spinels

• Lithium Diffusion Paths:
  • In the disordered spinel (Fd3m), lithium at the tetrahedral 8a site moves to a vacant octahedral 16c site, resulting in an 8a-16c diffusion path.
  • In the ordered spinel (P4332), the octahedral vacant 16c sites are split into ordered 4a and 12d sites (1:3 ratio), forming 8c-4a and 8c-12d diffusion paths.
• Coulombic Potential Calculation:
  • Electrostatic potential calculation uses the Coulombic potential equation:
    $U<em>{ij} = \frac{Z</em>iZ<em>je^2}{r</em>{ij}}$, where $Z<em>i$ and $Z</em>j$ are ion charges, and $r_{ij}$ is the distance between ions i and j.
  • Z values are set as: $O^{2-}$ (-2), $Mn^{4+}$ (+4), $Ni^{2+}$ (+2), $Li^{+}$ (+1).
  • Coulomb potentials for lithium diffusion paths in ordered and disordered spinels were calculated and normalized at lithium sites.
• Potential Curves and Diffusion:
  • Potential curves for each diffusion path are similar, with a maximum at the vacant site (saddle point).
  • The potential at each bottleneck is ordered as: 8c-4a (order) < 8a-16c (disorder) < 8c-12d (order).
  • In the ordered spinel, both the easiest (8c-4a) and most difficult (8c-12d) diffusion paths coexist.
  • Disordered-type containing 8a-16c enables easier diffusion.
  • Data suggests that lithium diffusivity may increase with increasing δ values.
• Relationship to Electrochemical Properties:
  • These calculations support the experimental charge-discharge property, where disordered spinels maintain higher capacity than ordered spinels.
  • Introducing defects to $LiNi<em>{0.5}Mn</em>{1.5}O_{4-\delta}$ improves the rate property of charge-discharge.

4.1.5 Conclusions

• $LiNi<em>{0.5}Mn</em>{1.5}O<em>4$ and $LiNi</em>{0.5}Mn<em>{1.5}O</em>{4-\delta}$ were synthesized via solid-state reactions at $820°C$.
• ND data confirmed that the stoichiometric spinel oxide had an ordered structure (P4332), while the nonstoichiometric oxides formed a disordered structure (Fd3m).
• The disordered spinel showed better rate property than the ordered spinel.
• Calculated Coulombic potential for diffusion paths suggested that diffusivity is higher in the disordered spinel.
• The calculation results aligned well with the experimental electrochemical properties.

4.2 Electrode/Electrolyte Interfaces in All-Solid-State Lithium Ion Batteries

4.2.1 Introduction

• Background: Solid electrolytes have gained attention for all-solid-state lithium-ion batteries due to their potential advantages.
• Advantages of Solid Electrolytes: Safety (non-flammable compared to liquid electrolytes) and high energy density for small portable and large-scale batteries.
• Focus of the Chapter: Two types of solid electrolytes: inorganic glass electrolytes and polymer electrolytes.
• Inorganic Glass Electrolytes:
  • SiS2-Li2S-Li4SiO4 glass electrolyte exhibits high lithium-ion conductivity (up to $10^{-3} S cm^{-1}$), excellent ionic conductivity (lithium ion transport number of 1.0), and a wide electrochemical window.
• Polymer Electrolytes:
  • Polyethers complexed with lithium salts are used, but they suffer from low ionic conductivity and transport number due to:
    • Low dissociation degree of lithium salts: low concentration of carrier ions.
    • Strong interactions between lithium ions and polymer chains (e.g., ether oxygen atoms), leading to low ionic mobility.
  • However, polymer electrolytes are flexible and easily make good contact between the electrode and electrolyte.
• Research Focus: While many studies target improving bulk properties of electrolytes, few focus on charge transfer reactions at the electrode/solid electrolyte interface.
• Importance of Interfacial Reaction Rate: Enhancing the interfacial reaction rate is crucial for fabricating high-power-density batteries, as battery performance is related to charge-transfer rates at the interface and the electrolyte's bulk conductivity.

4.2.2 Control and Charge Transfer Process at Electrode/Electrolyte Interface

• Three Main Topics:
  • (i) Factors determining the charge transfer reaction rate at polyether-based electrolyte/electrode interfaces.
  • (ii) Enhancement of ionic conductivity and lithium-ion transport number in polymer electrolytes by adding PEG-borate ester (a Lewis acid).
  • (iii) Enhancement of the reaction rate by adding PEG-borate ester.

(i) Study of Factors Influencing the Charge Transfer Reaction Rate at the Polyether-Based Electrolyte/Electrode Interface

• Goal: Clarify the reaction rate at interfaces and the ionic conduction mechanism.
• Method: Investigated the electrokinetics of the $Li^+/Li$ couple reaction on lithium metal electrodes in poly(ethylene glycol) dimethyl ether (PEGDME)-based electrolytes.
  • PEGDME solutions with $LiCF<em>3SO</em>3$ are used as a model for polymer electrolytes, resembling the amorphous conducting phase in high-molecular-weight PEO.
• Experiment: Measured charge transfer reaction rate using PEGDME (average molecular weight 500, PEGDME500) with varying amounts of higher molecular weight polyether, PEGDME1000.
• Conclusion: Viscosity is a major factor influencing the charge transfer reaction rate, explained by Marcus microscopic theories.
• Technique:
  • Double potential step chronoamperometry was used to study lithium deposition and dissolution on the microelectrode.
  • Coulombic efficiency of lithium deposition and dissolution was over 90%, due to fresh deposition and immediate dissolution on the microelectrode.
• Analysis:
  • Steady-state currents (i) at various overpotentials (E) were analyzed using the Allen-Hickling equation:
    $ln(\frac{i}{exp(\frac{\alpha FE}{RT}) - exp(\frac{(1-\alpha)FE}{RT})}) = ln i_0 - \frac{\alpha F}{RT}E$
  • Where F is the Faraday constant, R is the gas constant, T is the absolute temperature, α is the transfer coefficient, and $i_0$ is the exchange current density.
  • A linear relationship was observed in Allen-Hickling plots in the range of overpotentials around -150 to 170 mV vs. $Li^+/Li$.
  • Exchange current density and transfer coefficient were estimated from the intercept and slope of the Allen-Hickling plots.
  • The exchange current densities decreased with increasing amounts of PEGDME1000.
• Exchange Current Density Equation:
  • The exchange current density $i<em>0$ can be expressed as: $i</em>0 = K<em>r a</em>{Li^+}^{(1-\alpha)} a_{Li}^{\alpha}$
  • Where $a<em>{Li^+}$ is the activity of lithium ion, $a</em>{Li}$ is the activity of lithium metal, and $K_r$ is the standard rate constant.
  • According to advanced theory: $K<em>r = \frac{A}{T</em>L} exp(-\frac{\Delta G^*}{RT})$
    • A is the pre-exponential factor,
    • $\Delta G^*$ is the Gibbs activation energy of the reaction,
    • $T_L$ is the longitudinal relaxation time of the solvent.
  • Since all values of the transfer coefficient are close to 0.5 and $a<em>{Li}$ is unity, the equation for the exchange current density is simplified to $i</em>0 = \frac{A}{\sqrt{a_{Li^+}}} exp(-\frac{\Delta G^*}{RT})$ which indicates that the exchange current density depends on the activity of lithium ions, the Gibbs activation energy, and the longitudinal relaxation time of the solvent.
• Raman Spectroscopy:
  • Raman spectra were used to investigate the activity of lithium ions.
  • The $SO<em>3$ symmetrical stretching mode of $CF</em>3SO<em>3$ ($\nu</em>s(SO_3)$) at around $1040 cm^{-1}$ was used.
    • The band consists of three components:
      • Free ions (~$1034 cm^{-1}$).
      • Ion pairs (~$1042 cm^{-1}$).
      • Multiply aggregated ions (~$1052 cm^{-1}$).
  • The fraction of free ions (dissociated ions) corresponds to the ratio of free ion area to the areas of all components.
  • The fractions of free ions were almost the same (around 20%) even with increasing amounts of PEGDME1000, indicating constant lithium ion activity.
  • The solvation state of lithium ions in PEGDME should be similar, since ether oxygens act as donor polar groups.
  • The ratio of free ions was almost constant.
• Gibbs Activation Energies:
  • Gibbs activation energies were calculated from the slope of linear relationship plots. The electrokinetics of the $Li^+/Li$ couple reaction involves a desolvation/solvation process where the Gibbs activation energy may depend on desolvation/solvation energy.
  • The Gibbs activation energy depends on the solvation state of lithium ions.
  • Since lithium ions are incorporated into the helix structures of EO chains, activation energy is independent of the amount of PEGDME1000.
  • $i<em>0 \propto \frac{1}{T</em>L}$
  • Where $T_L$ is the longitudinal relaxation time of the solvent.
• Solvent Effects:
  • The obtained results clarified that the activity of lithium ions and the Gibbs activation energy do not depend on the ratio of PEGDME500/PEGDME1000 in the electrolyte solutions, while the exchange current densities decreased with increasing amounts of PEGDME1000.
  • The Solvent effect on charge transfer kinetics have elucidated that the reaction rate involves the frequency of solvent reorientation in the desolvation/solvation process of solvent dynamics where this frequency is inversely proportional to the longitudinal relaxation time.
  • $T<em>L = T</em>D \frac{\epsilon<em>{\infty}}{\epsilon</em>s}$
    • $T_D$: Debye relaxation time.
    • $\epsilon_s$:static dielectric constant.
    • $\epsilon_{\infty}$: infinite frequency dielectric constant.
  • $T_D \propto viscosity, \eta$
  • Therefore, $T_L \propto viscosity, \eta$, where viscosity is related to the frequency of solvent reorientation in the desolvation/solvation process.
• Viscosity Relationship:
  • Viscosity of PEGDME500-based solutions increased with increasing amounts of PEGDME1000.
    $i_o \propto \frac{1}{viscosity}$
  • The relationships between the exchange current densities and the viscosities of the solutions showed that the exchange current density was inversely proportional to the viscosity.
  • Therefore, the viscosity of the solvent is an important factor in determining the exchange current density of the polyether-based electrolytes, where high rates of charge transfer reactions at the interfaces is difficult when using high-viscosity electrolytes and especially polymer electrolytes.
• Conclusion:
  • Exchange current densities in PEGDME500-based electrolytes decreased with increasing amounts of PEGDME1000.
  • Raman spectroscopic studies of the electrolytes revealed that the activity of lithium ions was almost the same even with the addition of PEGDME1000.
  • Gibbs activation energies for charge-transfer reaction did not change with the amount of PEGDME1000, suggesting that solvation states of lithium ions were similar in the electrolytes to those in PEGDME whose molecular weight range was 500 to 1000.
  • The viscosity of the electrolytes increased with increasing amounts of PEGDME1000.
  • Inversely proportional relationships were found between the exchange current densities and the viscosities of the electrolytes.
  • Therefore, the viscosity of the electrolyte solution is key for the charge transfer reaction rate at the polyether-based electrolyte/electrode interfaces.
  • It will be difficult to get high rates of charge transfer at the interfaces when using polymer electrolytes, or other highly viscous electrolytes.

(ii) Enhancement of Ionic Conductivity and Transport Number of Lithium Ions in Polymer Electrolytes by the Addition of PEG-Borate Ester as a Lewis Acid into the Electrolytes

• Objective: To enhance ionic conductivity and transport number of lithium ions in solid polymer electrolytes for practical electrochemical devices.
• Strategy: Enhancement of the dissociation of supporting salts and decreasing the mobility of the counter anions.
• Lewis Acid Addition: By adding Lewis acid compounds one can enhance the transport number of lithium via interacting with Lewis base anions as the Lewis acid compounds trap the counter anions.
• Boron Compounds: Reportedly act as anion receptors due to Lewis acid character.
• Investigation Focus: The effect of the Lewis acid character of borate ester groups, which are fixed to the chains of the matrix polymer, on ionic conductivity and transport number of lithium ions in the polymer electrolytes. The relationships between the concentration of borate ester groups and ionic conductivity or transport number of lithium ions in the polymer electrolytes were examined.
• Polymer Preparation:
  • Backbone polymer prepared from polyethylene glycol and boric acid anhydride.
  • Polyethylene glycols were with various molecular weights (PEG150, PEG200, PEG400, PEG600, whose average molecular weight was 150, 200, 400, 600, respectively).
  • LIN(CF3SO2)2, LiCF3SO3 and LiClO4 were used for supporting salts without further purification.
  • Polymer electrolyte composition of polymer electrolyte, for example, prepared from PEG200, boric acid anhydride and LiN(CF 3 SO2)2, is represented as PEG200-B2O3 + LIN(CF3SO2)2.
• Computational Methods
  Ab initio Hartree-Fock (HF) self-consistent field molecular orbital calculations and density functional theory (DFT) calculations were performed with Gaussian98.
  Calculations for geometry optimizations were carried out at the HF level of the theory using the standard 3-21G basis set. Subsequently, single point calculation to investigate the energies of the optimized geometries was performed by using DFT methods with the B3LYP form for the exchange-correlation function and the 6-311G** basis set.
• Glass Transition Temperature:
  • $T_g$ of the polymer electrolytes decreased with increasing molecular weight of PEG, resulting in decreasing the concentration of the borate ester groups.
  • The mobility of the polymer chains increases due to decreesing concentration of crosslinking points.
    Ionic motion increased with descreased crosslinking
• Ionic Conductivity:
  • Increase in ionic conductivity of the polymer electrolytes is observed with increasing molecular weight of PEG.
  • The highest ionic conductivity was found for the polymer electrolyte prepared from PEG600, to be $5.9 \times 10^{-5}scm^{-1}$ at $30°C$ and $3.0 \times 10^{-4}Scm^{-1}$ at $60°C$.
• Free volume Theory: The free volume theory of polymers states which leads to the temperature dependence for segmental motion of polymer chains expressed by the Williams-Landel-Ferry (WLF) relationship:
  $\log \frac{\sigma(T)}{\sigma(T<em>g)} = \frac{C</em>1 (T - T<em>g)}{C</em>2 + (T - T_g)}$

Where σ(T) and σ(Tg) are the conductivity at temperatures T and Tg, respectively, and C1, C2 are the WLF parameters for the temperature dependence of ionic conductivity. However, the ionic conductivity at TÅ, σ(Tg), is difficult to measure in the present experiments because σ(Tg) is too low to measure with a complex impedance measurement, and therefore, 10 °C was selected as the reference temperature, To. The equation is rewritten as follows:
$\log \frac{\sigma(T)}{\sigma(T<em>o)} = \frac{C'</em>1 (T - T<em>o)}{C'</em>2 + (T - T<em>o)}$ The parameters in equation (1) are calculated as follows: $C'</em>1 = \frac{C<em>1 C</em>2}{[C<em>2 + (T</em>o - T<em>g)]}$$C'</em>2 = C<em>2(T</em>o - T_g)$

• WLF Plots:

  • The WLF plots for the ionic conductivity of the polymer electrolytes show that [log(σ(T)/σ (To))] is varies linearly with$\frac{1}{T - T<em>o}$, which indicates that the temperature dependence of the ionic conductivity for the polymer electrolytes follows a WLF-type equation as the temperature dependence of the ionic conductivity for the polymer electrolytes was dominated by that of the segmental motion of polymer chains. Therefore from the WLF plots and Tå values, the increase in ionic conductivity with increasing molecular weight of PEG is due to the increase in ionic mobility, where parameters were found to be close to the universal values of WLF parameters, $C'</em>1$ = 17.4 and $C'<em>2$ = 51.6/K47 $t</em>{Li^+} = \frac{I(0)R<em>b(0)(\Delta V - I(0)R</em>e(0)}{I (0)R<em>b(0)(\Delta V - R</em>e(\infty))}$

  I is the current, AV is the applied potential, R₁ is the bulk resistance,
  Re is the interface resistance, and 0 and ∞ refer to the initial and steady states, respectively.

• It was found that the transport number of the polymer electrolytes containing LiCF3SO3 or LiClO4 was higher than that of the electrolyte with LIN(CF3SO2)2, showwing that the Lewis acidity of the borate ester groups in polymer electrolytes affects the anions, ionic conductivity and Tg of the polymer electrolytes containing LiCF3SO3 or LiClO4 were examined.

• The HOMO exists around the oxygen atoms, where electron density around the oxygen atoms apart from the boron atoms is higher than that around the oxygen atoms next to the boron atom, where lithium ions interact more strongly with the oxygen atoms that are apart from the boron atom than to other oxygen atoms.
  On the other hand, the LUMO exists perpendicular to the BO3 plane, which indicates a location for interaction between the boron atom and a Lewis basic anion.The order of both X and 7 of the anions is CF3SO3 > C104 > N(CF3SO2)2. Therefore, CF3S03 or C10 should interact more strongly with the PEG-borate ester than N(CF3SO2)2 does as The boron atoms of the backbone polymer act as a