TimeâTemperature Scaling and Dielectric Modeling of Conductivity

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Time–Temperature Scaling and Dielectric Modeling of Conductivity Spectra of Single-Ion Conducting Liquid Dendrimer Electrolytes Sudeshna Sen, Haijin Zhu, Maria Forsyth, and Aninda Jiban Bhattacharyya J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b08985 • Publication Date (Web): 05 Dec 2018 Downloaded from http://pubs.acs.org on December 6, 2018

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The Journal of Physical Chemistry

Time–Temperature Scaling and Dielectric Modeling of Conductivity Spectra of Single-Ion Conducting Liquid Dendrimer Electrolytes Sudeshna Sen*, Haijin Zhu¶, Maria Forsyth¶ and Aninda J. Bhattacharyya*

*Solid

State and Structural Chemistry Unit, Indian Institute of Science, Bengaluru 560012, India

¶Institute

for Frontier Materials, Deakin University, Waurn Ponds, VIC3216, Australia

Abstract

We discuss here the time-temperature scaling and dielectric modeling of the variation of singleion conductivity with frequency of first generation (G1) liquid dendrimer electrolyte, viz. (propylether-imine) (PETIM): Li-salt. The PETIM: Li-salt electrolyte exhibits cation/anion transference number close to unity in the liquid state. On switching from an ester (G1-COOR) to cyano (G1CN) peripheral group, keeping constant the linker (ether) and branching groups (amine), an interesting transformation from cationic (t+ ~1) to an anionic conductor (t- ~1) takes place. The

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switch in the nature of predominant charge carrier is directly related to the change in the magnitude of anion diffusion (D-), which increases by one order in magnitude from D- = 1.1×10-12 m2 s-1 (at 30 oC) in G1-COOR to D- = 1.3×10-11 m2 s-1 (at 30 oC) in G1-CN. This intriguing ion transport mechanism is probed comprehensively using ac-impedance spectroscopy. The frequency dependent ionic conductivity of G1-CN/G1-COOR, comprising of distinct frequency regimes, is analyzed using time−temperature superposition scaling principle (TTSP) based on Summerfield and Baranovski scaling methods. To gain insight in to the electrical polarization (EP) phenomenon, the relevant frequency regime is converted from conductivity to dielectric versus frequency. The dielectric versus frequency data is modelled using Macdonald and Coelho. The combined approach of TTSP and dielectric modeling provide explicitly the extent of the influence of ion-dendrimer, ion-ion interactions and also the mobile charge carrier density on the effective ion transport in the homogeneous single ion conducting dendrimer electrolytes. The combined analysis suggests that ion transport in PETIM-COOR is only due to enhanced ion mobility whereas in PETIM-CN it is due to both mobile charge carrier concentration and ion mobility. To the best of our knowledge, the scaling and modeling approaches employed here constitute a rare example for validation of such concepts in the context of dendrimer electrolytes.

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Single-ion, especially single-cation conductors (tM+ ≈ 1) are very promising and crucial for efficient operation of rechargeable metal-ion batteries, actuators and sensors. One school of thought has been to develop single-ion polymer-based electrolytes. The cation transference number is the key factor influencing specific power and average specific energy of batteries. The depletion of electrolyte on the porous electrode usually limits the battery operation at high discharge rates (as required for electrical vehicles). However, this can be overcome easily if the electrolyte has a transference number equalling unity. Simultaneously, concentration polarisation which largely depends on the transference number can also be minimised. Despite the extensive efforts over the last several decades, overwhelming majority of solid polymer electrolytes are binary conductors i.e. both mobile cations and anions. As a result, the tM+ is often low (tM+= 0.2-0.5)1. In this respect, several polyelectrolytes2, 3 have been designed via anchoring or tethering of the counter anion to polymer backbone which results in the cations being the only mobile charge carriers leading to tM+= 1. This design strategy however, has often been found to be not very useful due to poor effective ion conductivity. To the best of our knowledge, reports of unity transference number in liquids especially in viscous liquids are rare. Previously few reports demonstrated potential of branched dendrimers as alternative lithium conducting electrolytes.4-6 We have recently reported first generation poly(propyl-ether-imine) (G1-PETIM) dendrimer electrolytes7 which exhibit lithium transference number near to unity (tLi+= 1) even in the liquid state. The type of predominant charge carrier i.e. cation or anion could be precisely tuned via suitable choice of the peripheral chemical functionality (tLi+ = 0.9 for G1ester; tPF6- = 0.8 for G1-cyano), keeping constant the linker and branching moieties. In spite of the near unity transference number for cations, interestingly the diffusive behaviour of both cations and anions are observed in such dendrimers. In other words, the ion transport properties of dendrimer system are intermediate to poly(electrolyte)s (selective cation/anion mobility) and the conventional liquid electrolyte (mobile cation and anion). Such intriguing nature of ion conduction prompted us to undertake theoretical studies for obtaining a deeper understanding of the experimental observations. In the present study, we conclusively probe the ion transport of the G1-PETIM dendrimer electrolytes via time-temperature scaling ACS Paragon Plus Environment

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and dielectric analysis of the various regimes of the frequency (ν) dependent ionic conductivity, σ(ν) obtained from ac-impedance spectroscopy. In comparison to polymer-salt complexes, which displays spatial heterogeneity and binary-ion transport, homogeneity and single-ion conductivity of the liquid dendrimer electrolytes using the TTSP and dielectric analysis will provide an accurate evaluation of the various solvation parameters affecting ion transport. Over the past few decades, scaling concepts have been extensively applied to account for the ac-charge transport in various classes of disordered solids which include polycrystalline and amorphous semiconductors, organic semiconductors, glasses, viscous melts, non-stoichiometric crystals, ion or electron conducting polymers, metal cluster compounds, transition metal oxides.8-14 A typical scaling strategy implies investigation of frequency dependent behaviour of the real part of ac-impedance over a range of experimental temperature and merge the responses into a single ‘master curve’ by scaling both the conductivity and frequency axes. If the frequency dispersion of conductivity obeys time-temperature superposition principle this would imply that the spectral shape is independent of temperature. The majority of conducting glasses have been shown to obey the time–temperature superposition principles (TTSP). The TTSP follow the general expression of scaling law:10

𝜎′(𝜈) 𝜎𝑑𝑐

( ), where σ

=𝐹

𝜈 𝜈0

dc, and

ν0 are dc-

conductivity and characteristic onset frequency and F is a scaling function which is independent of frequency. The ν0 is later expressed as ν* and defined as σʹ(ν*) = 2σdc. Different types of scaling approaches have been so far reported depending on the choice of F. For the first time, the validity of TTSP in ion conducting glass was confirmed by Taylor15. Taylor and Israd found that the master curves are very similar in shape for different glasses. In 1991, Kahnt et al utilized scaling parameters σdc in conductivity axis and Cole-Cole parameter (1/τ) in frequency axis to establish the scaling relation σʹ(ν*) = 2σdc. Kahnt employed this relation to study a variety of silicate, borosilicate, and germanate glasses and the shape of the master curve is found to be independent of the ion concentrations and glass structure16. In 1985 Summerfield proposed more general scaling law by using (σdcT) as a scaling parameter rather than using ν0 in the scaling function. Although this approach is initially proposed for amorphous semiconductors,17 ACS Paragon Plus Environment

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several ion conducting glasses viz. Na2O.B2O3, Li2O.B2O3, Na2O.GeO2 are found to obey this scaling law.18-20 Later Baranovskii and Cordes proposed a more general scaling law:21


(

=𝐹

𝜈

). The

𝜎𝑑𝑐𝑇𝑇 ―𝛼

term α signifies the Coulomb interaction between the mobile ions. Schrøder, and Dyre22 and Roling et al13 demonstrated that positive value of α implies coulomb interaction between ions and α=0 for absence of any interaction. Later, both the Summerfield and BC scaling laws have been successfully applied to cross-linked polyelectrolyte (PSS-PDADMAC) by Cramer et al23 and for lithium-salt doped plastic crystalline liquid electrolyte by Das et al.24 The conduction mechanism as a function of structure, crosslinks, salt concentration is accounted on the basis of number density of mobile charges and their temperature dependent behavior with these scaling approaches. Following on the literature reports, we have attempted here to comprehend the intriguing ion transport mechanism in the PETIM-dendrimer electrolytes using the Summerfield and BC scaling laws. In addition to the investigation of bulk ion conductivity which gives macroscopic insight into the ion transport process, fundamental understanding of electrode-electrolyte properties has become one of the important issues in the realm of electrolyte research. Significant number of reports focus on optimization of materials and cell design parameters to improve interface property which further influences local current density, associated charge transfer overpotential and power efficiency of devices.25 The complex behavior of interface is generally demonstrated well by widely known equivalent circuit model to understand resistive, capacitive behavior of electrochemical devices. Apart from such quantitative estimation, many other bulk properties such as mobility of ions and number density of charges can also be directly estimated from the frequency dependent behavior of electrical response originating from interface. While measuring impedance spectra, electrode polarization (EP) is widely observed in the dielectric spectra of ion conductors. A significant drop in conductivity is observed when ion accumulates at the electrode surface forming a space charge layer. Although EP hampers dielectric property determination, it has strong relevance in the design of fuel cell or double layer capacitors. The motivation behind modeling of electrode polarization in the lower frequency region is twofold.26 Firstly, the EP can ACS Paragon Plus Environment

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provide valuable information regarding mobile ion concentration and ion mobility as demonstrated in case of single ion conducting glasses and polymer electrolytes19 and secondly, formation of ion accumulating/depletion layer can be used to quantify energy storage in supercapacitors. The dielectric modeling of EP has been proposed specifically for polyelectrolytes (polyphosphagene ionomers, PEObased cation exchange membranes), and single ion glass electrolytes (Na-Ca-phosphosilicate glasses of compositions).26-29 The quantitative description of the number density of mobile charges and ion mobility and their individual contribution in determining transport processes of polyelectrolytes are described successfully by this method.26-29 As ion mobility and free charge carrier density play a key role on the effective ion transport properties in the G1-dendrimer Li-salt electrolytes, dielectric modelling of EP is employed for additional understanding and revalidation of the TTSP findings. Materials and Methods The detailed synthesis procedure of single cation conductor (G1–COOR) and single anion conductor (G1– CN) has already been described in Ref. [5,6]. Briefly, first generation cyano terminated PETIM dendrimer (G1-CN) is synthesized from bis-nitrile (G0-CN) through alternative Michael addition and reduction reactions. The tert-butyl ester terminated dendrimer (G1-COOR, R= tButyl) is synthesized by Michael addition of tert-butyl acrylate to G0-NH2. The dendrimer electrolytes are prepared by dissolving requisite amount of LiPF6 salt into pristine G1-CN and G1-COOR dendrimers to obtain a salt concentration as 0.1 M for both the dendrimers. The ionic conductivity is estimated using ac–impedance spectroscopy (Novocontrol Alpha-A; frequency range: 1 to 1×106 Hz, signal voltage: 0.05 V). The electrolyte is sandwiched between two stainless steel electrodes in home-built glass cells for ionic conductivity measurements. The conductivity cell is inserted in to a home–built glass jacket for temperature dependent conductivity measurements under vacuum. The glass jacket with the conductivity cell is inserted into thermostat (FP50MC) containing ethylene glycol (EG)-water mixture to measure temperature dependent ionic conductivity. All measurements are performed in the temperature range (0-60) oC at an interval of 5 oC for both heating and cooling cycles. The electrodes are polished well and dried under vacuum and ACS Paragon Plus Environment

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Ar flow successively prior to cell assembly. All sample preparations and cell assemblies are carried out in Argon filled glove box (MBraun, MB 20G LMF, pressure: 3 mbar, H2O < 0.5 ppm, O2 < 0.5 ppm). The cell constants measured are 0.13 cm-1 and 0.05 cm-1 for G1–COOR-0.1 M LiPF6 and G1–CN–0.1 M LiPF6 electrolyte respectively. For the study of the electrical polarization, the ionic conductivity as a function of frequency is converted to dielectric and the data is analyzed using theoretical models as discussed later. The ionic transference numbers are measured from temperature dependent self-diffusion coefficients of 7Li, 19F

nuclei by using pulsed field gradient NMR. The PFG-NMR are carried out on a Bruker Advance

III 300 MHz wide bore spectrometer (with proton Larmor frequency of 300.13 MHz) equipped with a 5 mm diff50 probe. The pulse-field gradient stimulated echo (PFG-STE) pulse sequence is used with maximum gradient strength as 29.454 T/m, gradient pulses (Δ): 5-10 ms, length of gradient pulse (δ): 14 ms, and the g in the suitable strength range: 0.3 - 29.4 T/m. Recycle delays are set to 5 s for all the diffusion experiments. Transference numbers for Li+ and PF6- ions are calculated by using the equation, -1 t+ = (1 – t-) = D+ (D+ + D-) , where the D+ and D- are the cationic and anionic diffusion coefficients,

respectively. Fourier transform infrared (FTIR) spectra of the dendrimer and the dendrimer-0.1M LiPF6 salt concentrations are recorded on a Perkin Elmer Spectrum 2000 Spectrometer at a spectral resolution of 4 cm-1 in the transmission mode within frequency range from 500-4000 cm-1. Results and Discussion The chemical structures of the single ion conductor G1-PETIM dendrimers with the peripheral functional groups, -COOR and –CN are shown schematically in Figure 1a. In both cases, the linker (ether) and branching point (tertiary- amine) groups are the same. The scheme demonstrates the change in the majority mobile ions as a result of switch from ester (cationic) to cyano (anionic) peripheral group. The temperature dependent ionic conductivity and transference numbers (estimated from the self-diffusion ACS Paragon Plus Environment

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coefficients of 7Li and

19F)

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of single-cation (G1-COOR-LiPF6) and single-anion (G1-CN-LiPF6)

dendrimer in the temperature range from 30-60o C are shown in Figure 1b. Due to large difference in the observed anion diffusion coefficient values in G1-COOR-LiPF6 and G1-CN-LiPF6, the majority mobile charge carrier changes significantly as a function of the peripheral group. This results in a very high cation transference (t+) number, t+ = 0.9 in G1-COOR. On the other hand, G1-CN displays a high anion transference number (t-), t- = 0.8. The high transference number with regard to the majority mobile ions makes them very similar to solid poly(electrolyte) membranes. On the other hand, the non-zero transference number value (t-/+ = 0.1 to 0.2) of the minority charge carriers suggests that the PETIMdendrimer electrolytes also retain characteristics of conventional liquid electrolytes. The interesting switch in nature of predominant charge carriers in dendrimer electrolytes suggests a unique ion transport property, which was previously investigated in detail via the combination of various spectroscopic and electrochemical techniques in Ref. [7]. As mentioned earlier, the significant difference in t+ between the two dendrimers is correlated to the differences in values of the DPF6- or anion mobility. The DPF6- for G1CN-LiPF6 is found to be nearly one order higher (1.3×10-11 m2s-1 at 30 oC) compared to that of DPF6- for the G1-COOR (1.1×10-12 m2s-1 at 30 oC) (Figure S1). It is proposed that, the high viscosity (ηG1-COOR (= 0.3 Pa.s) > ηG1-CN (= 0.15 Pa.s)) and steric hindrance to bulkier peripheral group in G1-COOR significantly decreases the mobility of the bulkier anion. Thus, bulkier ester peripheral groups not only can solvate Li+ ion but also can trap the bulkier hexafluorophosphate anion. Such trapping effect is not expected in the case of linear -CN groups of G1-CN dendrimer. The higher value of the estimated stoke radius RPF6(= DH/ DF) for 19F in G1-COOR (RPF6- = 0.8) as compared to G1-CN (RPF6- = 0.6) further suggests -

a stronger solvent correlated PF6 motion in G1-COOR compared to G1-CN. As a result of the hindered anion mobility in G1-COOR, extremely high cation (Li) transference number (t+= 0.9) is ACS Paragon Plus Environment

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observed in spite of the lower conductivity in G1-COOR-0.1M LiPF6 electrolyte (1.9 ×10-6 Ω-1cm-1 at 25 oC)

as compared to that of G1-CN-0.1M LiPF6 (1.9 ×10-5 Ω-1cm-1 at 25 oC). On the other hand, the

peripheral functional group and viscosity does not affect Li+ mobility resulting in nearly similar DLi+ in both the electrolytes (average: 2.010-12 m2s-1 - 510-11 m2 s-1 between 30 to 60 oC). In addition to the ionic mobility, the solvating ability of functional groups and polarity also differs between the dendrimers. The extent of coordination between Li+-ion and -COOR group is expected to be stronger than in the -CN group. This ion-ligand interaction and polarity of the solvating media influences the extent of salt dissociation in the dendrimer electrolytes. The extent of salt dissociation in both the G1-CN-LiPF6 and G1-COOR-LiPF6 dendrimers are investigated via FTIR spectroscopy as shown in Figure S2. In the -

IR spectra of G1-CN-LiPF6, an additional band at 875 cm-1 appears in addition to the free PF6 anion band at 848 cm-1.7 This suggests presence of the tri-dentate ligand pair of LiPF6 salt in G1-CN-LiPF6. On the other hand, complete salt dissociation in G1-COOR-LiPF6 is evident from the absence of any ion pair bands. Thus, spatial distribution influences ion mobility and polarity of functional groups affects the ion solvation to a significant extent in the dendrimer electrolytes. Such intriguing ion transport mechanism in dendrimer electrolytes encouraged us further to investigate in detail the ion conduction behavior in the dendrimer electrolytes.


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Figure 1. (a) Schematic depiction of the nontrivial ion transport characteristics in PETIM-dendrimer electrolyte. The chemical constituents of G1-PETIM dendrimers (G1-CN and G1-COOR) are shown below the scheme. (b) Ionic conductivity (red symbols) and transference numbers (blue symbols) of G1-COOR+

0.1 M LiPF6 (open symbols) and G1-CN-0.1 M LiPF6 (filled symbols). The t+ (Li ) for G1-COOR-0.1 M -

LiPF6 (open blue triangles) and t- (PF6 ) for G1-CN-0.1 M LiPF6 (filled blue squares).


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Figure 2. (a) Conductivity spectra at different temperatures and (b) Summerfield scaling of conductivity spectra for single-cation conductor: G1–COOR–LiPF6.

Figure 2a exhibits the real part (σʹ) of the complex conductivity as a function of frequency (ν) in the temperature range (0–60) oC for G1-COOR-LiPF6. The conductivity spectra exhibit three distinct regions. The low frequency (10 mHz < f < 1 Hz) is marked by a strong frequency dispersion in ionic conductivity which is due to the effect of EP occurring at the blocking electrode. The frequency independent regime at moderate frequencies (1 Hz < f < 105 Hz) marks the long-range diffusion regime from which the dcconductivities can be ascertained. Finally, the frequency dispersion at high frequencies (f > 105 Hz) signifies the fast ion dynamics13,14 which is characterized by a back–and–forth motion marking the onset of the host dynamics dependent conductivity with a characteristic cross over frequency, ν*. ν* is defined as, σʹ(ν*) = 2σdc following the Jonscher-type power law.9 The temperature dependent conductivity spectra can be suitably scaled such that all σ′(ν) versus ν curves superimpose to a single Master curve. This method ACS Paragon Plus Environment

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of scaling, as discussed earlier, is referred to as the time-temperature superposition principle (abbreviated as TTSP). Successful application of TTSP implies that the underlying mechanism of temperature dependent ion conduction is similar, irrespective of the system characteristics9-13, 23. The TTSP as per the 𝜎′(𝜈) 𝜎𝑑𝑐

Summerfield scaling law is as follows:10,23

( ). Assuming a linear response theory, the

=𝐹

𝜈

𝜎𝑑𝑐T

Summerfield scaling law describes the mean square displacement (r2(t)) of the mobile ions as9: r2(t)= f (𝜎𝑑𝑐T, t). The Summerfield scaling law is employed to merge all conductivity spectra into a single Master curve, as shown in Figure 2b. As observed clearly in Figure 2b, the TTSP is not valid in the low frequency in the EP regime (f < 0.5 Hz) and hence this regime is omitted for analysis using TTSP. The validation of the data using Summerfield scaling has twofold significance. Firstly, it reflects the temperature independent behavior of the microscopic dynamics. Secondly, temperature dependent ionic conductivity is a function of number density of mobile ions and ionic mobility. The Summerfield scaling assumes ν*  σdcT, at a constant number density of mobile ions. This is valid only for dilute solutions of electrolytes with degree of salt dissociation tending to unity. Thus, increase in the ionic mobility contributes to the enhancements in ionic conductivity with increasing temperature for G1–COOR–LiPF6, at constant number density of mobile ions (Figure S3). Figure 3a shows the conductivity spectra of G1–CN–0.1 M LiPF6. The Summerfield scaling is also investigated for the anion conductor, G1–CN–LiPF6, as shown in Figure 3b. The inadequacy of the Summerfield scaling in G1–CN–LiPF6 is evident from it’s inability to generate a single Master curve in the frequency regime 109-1012 Hz. This leads us to employ the scaling law of Baranovskii and Cordes:21


(

=𝐹

𝜈

). The Master curve (Figure 3c) is generated from Figure 3a

𝜎𝑑𝑐𝑇𝑇 ―𝛼

using α = 0.1. The positive value of α signifies that the mean square displacement r2(t) increases with temperature. The increase in r2(t) with temperature can be either due to increase in number density of mobile ion or decrease in accessible pathways for ion transport.24,23 The latter is not valid here as the ion transport in the G1-dendrimer is not guided by segmental mobility, as is applicable for conventional polymer electrolytes. Absence of infinite chains similar to polymers should necessarily eliminate this ACS Paragon Plus Environment

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possibility. Additionally, Arrhenius behavior of temperature dependency of conductivity and selfdiffusion coefficients of Li+ and PF6- ions (Figure 1b and S1 respectively) further supports this proposition.7 The more convenient explanation would be that the improvement in ionic conductivity is influenced by the increase in number density of free charge carriers with temperature in G1–CN–LiPF6. This is only possible if the ion dissociates with increase in temperature for G1–CN–LiPF6. Thus, in the CN PETIM-dendrimer both the number density of mobile charges and ionic mobility are important factors to determine the variation of conductivity as a function of temperature. The scaling laws employed here validates conclusively the temperature dependent ionic conductivity in terms of free ion carriers in both the cation (PETIM-COOR) and anion (PETIM-CN) dendrimer electrolytes. This phenomenon is also supported by the measurement of ionicity, expressed by the ratio Λimp/ΛNMR. Here Λimp and ΛNMR (ΛNMR = (Nq2/kT)∑xiDi for the ith ion; Di is the diffusion coefficient, xi is the mole fraction) are the molar conductivities estimated from impedance spectroscopy and PFG-NMR respectively. The measured ionicity of G1CN-LiPF6 is 0.8 at 30 oC. The ionicity value < 1 for G1-CN dendrimer suggests the presence of ion pair in -CN dendrimer. In this way diffusion coefficient measurement is useful for comprehending the ion solvation in G1-CN dendrimer. This is in good agreement with other spectroscopic measurement. In the case of ester dendrimer, the ionicity value is ~0.3 which is even lower due to a significant difference in σLi (1.1 × 10-5 Ω-1cm-1) and σPF6 (3.9 ×10-6 Ω-1cm-1) value due to the anion trapping effect. So direct prediction of salt association for this particular case (G1-COOR) may not be straight forward from the ionicity measurement. The solvent correlated anion transport is predictable in the case of G1-COOR dendrimer [7] from the estimated RPF6_ (= DH/DF) in G1–COOR (=0.8). Thus, the contribution of mobile charge density in G1-COOR dendrimer is negligible and hence the ion mobility term is the guiding factor as observed by self-diffusion coefficient behavior.


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Figure 3. (a) Conductivity spectra and master curves using (b) Summerfield and (c) Baranovskii and Cordes scaling laws for the single-anion conductor: G1–CN–LiPF6.

We now shift our focus to the dielectric analysis of the EP at the blocking electrodes observed at low frequencies. At low frequency, the accumulation or depletion of ions near the ion blocking electrodes results in the formation of space-charge layers, at which the voltage drops significantly. The build-up of EP and the drop in the electric field in the bulk are reflected in terms of an increase in the ac-permittivity and decrease in the ac-conductivity with decreasing frequency.27-31 The accumulation is limited by the concentration gradient, which opposes the Coulombic force of the electric field.27 When steady state is ACS Paragon Plus Environment

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reached, the statistical space charge distribution resembles as shown in Figure 4a. Two parallel-plate electrode of well-defined and equal area (A) are separated by distance L. LD is the Debye length, i.e. length scale of the electrostatic double layer. The spatial dimension of interest, x is bound at x = 0 and x = L by blocking electrodes. In this situation, the system essentially behaves as a macroscopic dipole. In the Coelho model, the EP is considered as a simple Debye relaxation, with complex dielectric constant expressed as29,30 : 𝜀𝐸𝑃 ∗ = 𝜀𝑠+

∆𝜀𝐸𝑃 1 + 𝑖𝜔𝜏𝐸𝑃

where, εs and τEP ( ≡

𝜀𝐸𝑃 𝜀0 𝜎𝑑𝑐

) are static dielectric constant and time

scale for complete polarization respectively. The term ΔεEP = εEP–εs, with εEP being the effective permittivity after polarization (timescale of conduction is given by: τσ ≡

𝜀𝑠𝜀0 𝜎𝑑𝑐

where ε0 and σdc are vacuum

permittivity and dc conductivity respectively). The loss tangent, tan δ (= εʹʹ/εʹ) of G1–COOR–LiPF6 (Figure 4b) and G1–CN–LiPF6 (Figure 4c) is fitted using the equation: 𝑡𝑎𝑛 𝛿 =

𝜔𝜏𝐸𝑃 1 + (𝜔𝜏𝐸𝑃)2/M

. The τEP is

related to the crossover frequency (fc) as τEP = 1/2πfc. The fc represents the cross-over frequency, below which ion builds up near the electrode and above implies long range transport. M is a fitting parameter and can be presented in terms of τEP = Mτσ26. τEP and τσ are the timescales of total polarization and conduction respectively. The estimation of number density of mobile ions (P0) from the fitting parameters, τEP and τσ is obtained by using the equation:29-31 𝑃0 =

4𝐾𝑇𝜎𝐷𝐶𝜏𝐸𝑃2 𝑒2𝐿2𝜏𝜎

, where σDC, e, L are dc-conductivity,

electronic charge and cell constant respectively. The dependence of P0 on temperature can be fitted using Arrhenius equation (P0 = P exp (-Ea/KT)) (for both G1–COOR and G1–CN, Figure 4d). The P and P0 are the number density of charges at near to infinite (1/T  0) temperature and at the experimental temperature T respectively. P is obtained from the extrapolation of Arrhenius plot till 1/T = 0. The P0 for G1-COOR (1.4–1.8 × 1016 cm-3) is 1-2 orders higher in magnitude than G1-CN (8.5–75 × 1014 cm-3) in the temperature range (0-60o C). While temperature dependency of P0 in G1-CN is much more significant compared to G1-COOR, invariance of P0 in G1-COOR suggests complete salt dissociation at ambient temperatures. Additionally, the close proximity of P0 to P in G1–COOR indicates that the total number


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density is close to that at T ∞ which suggests that the concentration of ion pairs is expected to be negligible at elevated temperatures.

Figure 4. (a) Schematic representation of charge density distribution near blocking electrodes under dc-electrical field at steady state. Fitting (solid lines) of tan δ versus frequency using Macdonald/Coelho model for G1–COOR–LiPF6 (b) and G1–CN–LiPF6 (c). (d) P0 versus temperature for G1–COOR–LiPF6 (red circle) and G1–CN–LiPF6 (blue circle).

On the other hand, the P is nearly 4-5 orders higher in magnitude than P0 for G1-CN–LiPF6 (Figure 4d). This signifies that at ambient temperature the salt is completely dissociated in G1-COOR, while ion pair is remnant in the G1-CN. In other words, the larger activation energy in temperature dependency of P0 in G1-CN as compared to that in G1-COOR electrolyte is attributed to the dissociation of ion pairs with increase in temperature. Notably, G1-CN exhibits much higher P (due to single-anion conduction: P = 2.6 × 1020 cm-3), nearly four orders higher than G1–COOR (due to single- cation conduction: P = 5× 1016 ACS Paragon Plus Environment

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cm-3). The nature of predominant charge carrier is completely different in both the cases. The observed P -

+

+

in G1–COOR is due to Li whereas that in G1–CN corresponds to PF6 . Thus, coordination of Li to peripheral –COOR groups in the cation conductor results in lower P in G1–COOR. On the other hand, solvation of the mobile anion in G1–CN electrolyte is negligibly small. Consequently, mobile anion density in G1–CN becomes significantly higher than the mobile cation density in G1–COOR. Thus, the -

Li+ ions are solvated via ester functional groups in G1-COOR and simultaneously mobility of the PF6

anion is hindered due to the bulkier tert-butyl group containing ester periphery. In the case of G1-CN dendrimers, the ion pairs are present due to lower solvating ability of -CN functional group to Li+ ion and anions are preferably mobile giving rise to single anion conduction. Such useful predictions drawn from the modelling of EP strongly supplement the findings from TTSP. The variation of ionic mobility with temperature as calculated from dielectric modeling of electrode polarization is shown in Figure S3. The ionic mobility for G1-COOR-LiPF6 electrolyte exhibits a significant temperature dependency over the temperature range 0 oC-60 oC (8.5×10-5 cm2s-1V-1- 1.8×10-3 cm2s-1V-1). This further supports the fact that the temperature dependency of conductivity is more guided by ionic mobility rather than the density of mobile charge due to complete salt dissociation in the ester dendrimer. Similar prediction is also drawn from TTSP scaling. On the other hand, it would be interesting to observe the insignificant variation of ionic mobility over the experimental temperature range for G1-CN-LiPF6 electrolyte. This behaviour suggests that the temperature dependency of ion conductivity is influenced by the number density of mobile charges significantly due to the presence of ion pair in the system. In the G1-CN-LiPF6, the anion is the predominant charge carrier and its mobility is not largely hindered by dendrimer peripheral group. Thus, the variation is not so significant as expected for correlated ion mobility in G1-COOR dendrimer. Notably, the transference number of G1-CN-LiPF6 is equal to 0.8 and Li-ion conduction cannot be completely neglected in the G1-CN dendrimers. This also explains the different behaviour of temperature dependency of ionic mobility and self-diffusion coefficients of anion as observed from NMR techniques. the validation of the Macdonald-Coelho model in this case (G1-CN dendrimer) may not be completely


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agreed upon due to the nontrivial nature of ion transport in dendrimers, which is neither a liquid nor a polyelectrolyte. Notably, in figure S3 the ionic mobility of lithium (as obtained from the MacdonaldCoelho model) in G1-COOR dendrimer increases non-linearly specially above 50 oC. Thus, there is a slope change near to 50 oC. The diffusion coefficients of the anion and cation for temperatures above 50degC is within a factor of two of each other. A slight deviation of non-linearity can’t be completely neglected in the DLi behaviour in figure S1 for the ester dendrimer. The plausible reasons behind such discrepancy can be due several factors. i) Firstly, the ester dendrimer has bulkier tertiary butyl group which can coordinate to lithium and simultaneously offering motional hindrance to larger anion. The lithium ion will prefer to diffuse from one coordinating site to another. This phenomenon is observed definitely below 50 oC. Above this temperature there can be some additional chain relaxation from ester peripheral group. In this case it is possible that the lithium diffusion can show non-linear behaviour above the 50 oC. ii) secondly, above this temperature it is difficult to predict whether any additional relaxation is arising from any structural changes of interface. iii) Thirdly, contribution of both lithium and anion -

mobility is present. Difference in diffusion coefficient of Li+ and PF6 in ester is much less at higher temperature. This means anion mobility is favourable at higher temperature (larger activation energy) in ester dendrimer. So, Macdonald theory can’t fully account for such possibility of diffusion of both the cation and anion as the theory was proposed for poly(electrolytes) (for selective ion transport). We have to consider such discrepancy and further modification of new theoretical models are encouraged for this interesting system specially for higher temperature regime. However, the analysis clearly exhibits the distinct difference in ion conduction process between the two dendrimer electrolytes as a function of their peripheral functional groups. Both the TTSP (at high frequency) and EP (at low frequency) analyses provide detailed enough insight into the nature of ion solvation and behaviour of the mobile charge density and the information obtained from both the analyses are correlate well with each other and findings from other characterization studies.


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Based on the findings from the above experimental techniques viz. impedance, NMR, scaling, dielectric modelling approach, the plausible ion transport mechanism in the dendrimer electrolytes can be summarised as follows: Lithium ion conduction: The most probable mechanism of lithium ion conduction in both the G1-COOR and G1-CN dendrimers could be Grotthus-type. Here, lithium ions bind to peripheral functional groups (ester or cyano) and hopping is from one coordinating site to another. For both cases, Li-ion is solvated by the dendrimers, similar to conventional liquid electrolyte. The thermally activated Arrhenius-type conduction is expected within the regime of measurement: 0-50 oC. For temperature above 50 oC, there can be some additional chain relaxation from ester peripheral group. In this case, it is possible that the lithium diffusion can show non-linear behaviour above the 50 oC. The lithium coordination is stronger with ester group in G1-COOR than that of -CN groups in G1-CN. This results in dissociation of the ion pairs in the ester dendrimer completely as predicted from the scaling and dielectric modelling approaches. With increase in temperature, ion pairs dissociate and increases the number density of mobile charges in G1-CN as observed from the dielectric modelling of electrode polarisation. Anion transport mechanism: The anion transport is completely different between the ester and cyano dendrimers. The ion mobility of PF6- anion is hindered by bulkier tert-butyl group in G1-COOR (R= tButyl)

dendrimer more as compared to the linear -CN groups. This leads to the significant difference,

nearly one order in magnitude in the anion diffusion value between G1-CN-LiPF6 and G1-COOR- LiPF6. The dendrimer correlated anion mobility leads to the nearly unity lithium transference number in the ester dendrimer i.e. lithium as the predominant charge carrier. At elevated temperature > 50 oC, the trapping effect of functional group is lesser and anion diffusion increases to a greater extent. However, in the ester dendrimer diffusion of both the cation and anion is observed. Unlike conventional polyelectrolytes, conductivity behaviour of dendrimer electrolyte is somewhat liquid-like (i.e. both ion types are mobile in the solvated state). But the unity transference number make them differ from conventional liquid electrolytes. We do not expect any electrostatic interaction between anion and ester dendrimer. In cyano


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dendrimer such anion trapping effect by peripheral -CN group is not observed. So, the anion is diffusing as predominant charge carrier (as a counter ion with lithium) in G1-CN. Conclusions: In conclusion, we have successfully demonstrated here the utility of TTSP and dielectric modelling in accounting for the ion transport mechanism of dendrimers, which exhibit nearly unity cation/anion transference numbers. The combined theoretical and experimental approaches successfully identify the role of the important solvation parameters on the ion transport of dendrimers. The scaling approach gives significant physical insights in to the contribution of number density of mobile charges towards conductivity behaviour. On the other hand, the EP modeling extracts quantitative description of contribution of number density of mobile charges and ion mobility terms in the dendrimers. The complete +

salt dissociation and coordination of Li to G1–COOR leading to cation conduction is analogous to cationconducting polyelectrolytes and inorganic solid electrolytes. However, incomplete salt dissociation in G1– CN has different implications and is well accounted by the EP analysis. The theoretical approaches discussed here, which apart from being extended to other homogeneous soft-matter systems, will also provide predictive structural insights in to the design novel liquid single-ion conductors. Moreover, this study necessitates proposition of newer theoretical approaches of modelling specially while considering electrode polarisation phenomena. ASSOCIATED CONTENT

Supporting Information The supporting information consists of temperature dependent self-diffusion coefficients and ionic mobility. This material is available free of charge via the Internet at http://pubs.acs.org.

Author Information Corresponding Author ACS Paragon Plus Environment

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*E-mail: [email protected]

Acknowledgement AJB acknowledges the AISRF (DST/INT/AUS/P-71/2016; Dt. 6.6.16) for financial support. Author acknowledge Prof. N. Jayaraman, Department of Organic Chemistry, Indian Institute of Science, Bangalore for his suggestions. We thank Dr. Rudresha B. Jayappa for helping in synthesis of the dendrimers.


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References (1) Appetecchi, G. B.; Zane, D.; Scrosati, B. PEO Based Electrolyte Membranes Based on LiBC4O8 Salt. J. Electrochem. Soc. 2004, 151, A1369-A1374. (2) Shubha, N.; Zhu, H.; Forsyth, M.; Srinivasan, M., Study of Lithium Conducting Single Ion Conductor Based on Polystyrene Sulfonate for Lithium Battery Application. Polymer 2016, 99, 748-755. (3) Tsuchida,

E.;

Kobayashi,

N.;

Ohno,

H.

Single-Ion

Conduction

in

Poly[(oligo(oxyethylene)methacrylate)-co-(alkali-metal methacrylates)]. Macromolecules, 1988,

21, 96-100. (4) Greene, G. W.; Ponzio, F.; Iranipour, N.; Zhu, H.; Seeber, A.; Forsyth, M.; Howlett, P. C., Enhanced Ionic Mobility in Organic Ionic Plastic Crystal – Dendrimer Solid Electrolytes.

Electrochimica Acta 2015, 175, 214-223. (5) Das, S.; Suresh, B. R.; Jayaraman, N.; Bhattacharyya, A. J., Ionic Conductivity of bis(2cyanoethyl) ether-Lithium Salt and Poly(propylether imine)-Lithium Salt Liquid Electrolytes. J.

Polym. Res. 2012, 19 (8), 9924. (6) Jain, S.; Kaur, A.; Puri, R.; Utreja, P.; Jain, A.; Bhide, M.; Ratnam, R.; Singh, V.; Patil, A. S.; Jayaraman, N.; Kaushik, G.; Yadav, S.; Khanduja, K. L. Poly propyl ether imine (PETIM)


22

Page 23 of 27 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


Dendrimer: A Novel Nontoxic Dendrimer for Sustained Drug Delivery. Eur. J. Med. Chem., 2010, 45, 4997-5005. (7) Sen, S.; Jayappa, R. B.; Zhu, H.; Forsyth, M.; Bhattacharyya, A. J. A Single Cation or Anion Dendrimer Based Liquid Electrolyte. Chem Sci. 2016, 7, 3390-3398. (8) Pas, S, J.; Banhatti, R, D. Funke, K. Conductivity Spectra and Ion Dynamics of a Salt-inPolymer Electrolyte. Solid State Ionics, 2006, 177, 3135–3139. (9) Dyre, J. C.; Maass, P.; Roling, B.; Sidebottom, D. L. Fundamental Questions Relating to Ion Conduction in Disordered Solids, Rep. Prog. Phys. 2009, 72: 046501, 1-5. (10) Taylor, H. E. The Dielectric Relaxation Spectrum of Glass. Trans. Faraday Soc. 1956, 52, 873-881. (11) Roling, B.; Martiny, C. Non-Universal Features of the Ac Conductivity in Ion Conducting Glasses. Phys. Rev. Lett 2000, 85 (6), 1274. (12) Murugavel, S.; Roling, B. Ac Conductivity Spectra of Alkali Tellurite Glasses: Composition Dependent Deviations from the Summerfield Scaling. Phys. Rev. Lett. 2002, 89, 195902−4. (13) Roling, B. Modeling of Ion Transport Processes in Disordered Solids: Monte Carlo Simulations of the Low Temperature Particle Dynamics in the Random Barrier Model. Phys.

Chem. Chem. Phys. 2001, 3, 5093−5098.


23


Page 24 of 27

(14) Summerfield, S. Universal Low-Frequency Behaviour in the Ac Hopping Conductivity of Disordered Systems. Philos. Mag. B 1985, 52, 9-22. (15) Isard, J. O., Dielectric Dispersion in Amorphous Conductors. Journal of Non-Crystalline Solids 1970, 4, 357-365. (16) Kahnt, H., Ionic Transport in Oxide Glasses and Frequency Dependence of Conductivity. Berichte der Bunsengesellschaft für physikalische Chemie 1991, 95 (9), 1021-1025 (17) Summerfield, S.; Butcher, P. N., Universal Behaviour of AC Hopping Conductivity in Disordered Systems. Journal of Non-Crystalline Solids 1985, 77-78, 135-138. (18) Roling, B.; Happe, A.; Funke, K.; Ingram, M. D., Carrier Concentrations and Relaxation Spectroscopy: New Information from Scaling Properties of Conductivity Spectra in Ionically Conducting Glasses. Physical Review Letters 1997, 78 (11), 2160-2163. (19) Zielniok, D.; Eckert, H.; Cramer, C., Direct Correlation between Nonrandom Ion Hopping and Network Structure in Ion-Conducting Borophosphate Glasses. Physical Review Letters 2008, 100 (3), 035901. (20) Sidebottom, D. L., Universal Approach for Scaling the Ac Conductivity in Ionic Glasses. Physical Review Letters 1999, 82 (18), 3653-3656. (21) Baranovskii, S. D.; Cordes, H. On the Conduction Mechanism in Ionic Glasses. J. Chem.

Phys. 1999, 111, 7546−7557. (22) Schrøder, T. B.; Dyre, J. C., Scaling and Universality of Ac Conduction in Disordered Solids. Physical Review Letters 2000, 84 (2), 310-313.


24

Page 25 of 27 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


(23) Imre, A.; Schönhoff, M.; Cramer, C. Unconventional Scaling of Electrical Conductivity Spectra for PSS-PDADMAC Polyelectrolyte Complexes. Phys. Rev. Lett. 2009, 102, 255901. (24) Das, S.; Bhattacharyya, A. J. Time-Temperature Scaling of Conductivity Spectra Organic Plastic Crystalline Conductors. J. Phys. Chem. Lett. 2012, 3, 3550–3554. (25) Bard, A. J.; Abruna, H. D.; Chidsey, C. E.; Faulkner, L. R.; Feldberg, S. W.; Itaya, K.; Majda, M.; Melroy, O.; Murray, R. W., The Electrode/Electrolyte Interface - a Status Report. The Journal

of Physical Chemistry 1993, 97 (28), 7147-7173. (26) Mariappan, C. R.; Heins, T. P.; Roling, B. Electrode Polarization in Glassy Electrolytes: Large Interfacial Capacitance Values and Indication for Pseudocapacitive Charge Storage. Solid

State Ionics, 2010, 181, 859-863. (27) Klein, R. J.; Zhang, S.; Dou, S.; Jones, B. H.; Colby, R. H.; Runta, J. Modeling Electrode Polarization in Dielectric Spectroscopy: Ion Mobility and Mobile Ion Concentration of Single-Ion Polymer Electrolytes. J. Chem. Phys. 2006, 124, 1-8. (28) Klein, R. J.; Welna, D. T.; Weikel, A. L.; Allcock, H. R.; Runt, J. Counter Ion Effects on Ion mobility and Mobile Ion Concentration of Doped Polyphosphazene and Polyphosphazene Ionomers. Macromolecules 2007, 40, 3990-3995.


25


Page 26 of 27

(29) Coelho, R. On the Static Permittivity of Dipolar and Conductive Media-an Educational Approach. J. Non-Cryst. Solids 1991, 131, 1136-1139. (30) Macdonald, J. R. Theory of Ac Space-Charge Polarization Effects in Photoconductors, Semiconductors and Electrolytes. Phys. Rev. 1953, 92, 1-17. (31) Wanga, J. H. H.; Colby, R. H. Exploring the Role of Ion Solvation in Ethylene Oxide Based Single-Ion Conducting Polyanions and Polycations. Soft Matter 2013, 9, 10275–10286.


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TimeâTemperature Scaling and Dielectric Modeling of Conductivity

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