Examination of methods to determine free-ion ... - APS Link Manager

PHYSICAL REVIEW E 87, 042308 (2013)

Examination of methods to determine free-ion diffusivity and number density from analysis of electrode polarization Yangyang Wang,1,* Che-Nan Sun,2 Fei Fan,3 Joshua R. Sangoro,1 Marc B. Berman,4 Steve G. Greenbaum,4 Thomas A. Zawodzinski,2,5 and Alexei P. Sokolov1,3 1

Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 3 Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, USA 4 Department of Physics and Astronomy, Hunter College of the City University of New York, New York, New York 10065, USA 5 Department of Chemical and Biomolecular Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA (Received 11 October 2012; published 18 April 2013) 2

Electrode polarization analysis is frequently used to determine free-ion diffusivity and number density in ionic conductors. In the present study, this approach is critically examined in a wide variety of electrolytes, including aqueous and nonaqueous solutions, polymer electrolytes, and ionic liquids. It is shown that the electrode polarization analysis based on the Macdonald-Trukhan model [J. Chem. Phys. 124, 144903 (2006); J. Non-Cryst. Solids 357, 3064 (2011)] progressively fails to give reasonable values of free-ion diffusivity and number density with increasing salt concentration. This should be expected because the original model of electrode polarization is designed for dilute electrolytes. An empirical correction method which yields ion diffusivities in reasonable agreement with pulsed-field gradient nuclear magnetic resonance measurements is proposed. However, the analysis of free-ion diffusivity and number density from electrode polarization should still be exercised with great caution because there is no solid theoretical justification for the proposed corrections. DOI: 10.1103/PhysRevE.87.042308

PACS number(s): 66.10.Ed, 77.22.Jp

I. INTRODUCTION

One of the fundamental problems impeding the understanding of ionic conductors is the inability to unambiguously estimate the free-ion diffusivity and number density from the measured dc conductivity. One possible way to determine these quantities is to analyze the electrode polarization (EP) effect [1–20] observed by dielectric spectroscopy. In a typical dielectric measurement, cations and anions migrate under the applied electric field and accumulate on the surface of electrodes. The resulting space-charge polarization gives rise to a dramatic increase of the dielectric permittivity at low frequencies. The EP phenomenon is ubiquitous in all ionic conductors, including aqueous solutions, polymer electrolytes, ionic liquids, and superionic glasses. Recent years have evidenced a renewed interest in evaluating the free charge carrier mobility and number density of ionic conductors from the EP effect. In particular, various versions of Poisson-NernstPlanck model of electrode polarization [1,2,5,7,21] have been applied to evaluate the ion mobility of polymer electrolytes, hydrogels, and ceramics [13,22,23]. Despite these efforts, the diffusivities obtained by EP analysis have not been carefully compared with the results from other experimental techniques. One of the current theoretical models of electrode polarization, proposed by Macdonald [1] and Trukhan [2], is based on the Debye-Hückel type of approach, where the Nernst-Planck equation for ionic motion is combined with the Poisson equation and linearized with respect to the electric field. Other models [4,9,10,16,19] are typically based on similar methods. It is not obvious that the EP analysis will hold at high ion concentrations, when the ionic interaction strength exceeds the thermal energy. However, almost all ionic *

Corresponding author: [email protected]

1539-3755/2013/87(4)/042308(9)

conductors of practical interests are concentrated systems. A closer examination of the EP model, especially in the cases of high ion concentrations, seems necessary. In the present study, we report the results of dielectric measurements on a wide variety of ionic conductors. The freeion number density and diffusivity in these materials are analyzed according to the current theoretical model (MacdonaldTrukhan model) of electrode polarization. Although the EP analysis yields reasonable ionic dissociation energy, the obtained ionic diffusivities are not in agreement with pulsed-field gradient (PFG) NMR measurements. Possible origins of the failure are discussed and a successful empirical correction method is proposed. Despite the agreement of the estimated diffusion with NMR data after corrections, we suggest that the analysis of ion diffusivity and number density with the current EP model should be exercised with great caution. II. MATERIALS AND METHODS A. Materials

To acquire a general understanding of the electrode polarization phenomenon, different types of ionic conductors were investigated, including polymer electrolytes, aqueous and nonaqueous solutions, and ionic liquids. The details of these samples are summarized as follows: (a) PPG-LiTFSI (polymer electrolyte): Polypropylene glycol (PPG) is commonly used as a model system in polymer electrolytes studies. For the present investigation, a PPG with an average molecular weight of 4000 g/mol was purchased from Scientific Polymer Products. The polymer electrolytes were prepared by dissolving the polymer and the bis(trifluoromethane) sulfonimide lithium salt (LiTFSI) from Sigma-Aldrich in methanol and subsequently removing the solvent in a vacuum oven. The vacuum procedure was

042308-1

©2013 American Physical Society

YANGYANG WANG et al.


first carried out at room temperature for 2 weeks and then at 60 ◦ C for another 2 weeks. Here, we report the results of three different salt concentrations: 1.6 wt.%, 5.8 wt.%, and 11 wt.%. (b) LiCl-H2 O (aqueous solution): A lithium chloride (LiCl) aqueous solution was prepared using deionized water and anhydrous ultradry lithium chloride powder (99.995%) from Alfa Aesar. The molar ratio of H2 O and LiCl was 7.3. The details of sample preparation can be found in the earlier study [24]. (c) LiCF3 SO3 -ethanol (nonaqueous solution): A lithium trifluoromethanesulfonate (LiCF3 SO3 ) ethanol solution was prepared using ethanol and LiCF3 SO3 powder from SigmaAldrich. The salt concentration was 0.1 mol/L. (d) NaCl–(glycerol-H2 O mixture) (nonaqueous solution): Glycerol (Fisher) and ultrapure H2 O (Alfa Aesar) were mixed at a molar ratio of 3:7. Puratronic sodium chloride (NaCl, Alfa Aesar) was then dissolved in the glycerol-H2 O mixture with the aid of gentle heating. The weight percentage of NaCl was 0.1%. (e) [OMIM][TFSI] (ionic liquid): We reanalyzed the previously published data of an ionic liquid, 1-octyl3-methylimidazolium (OMIM) bis(trifluoromethylsulfonyl) imide (TFSI) [18]. The broadband dielectric measurement of this sample was carried out in F. Kremer’s group at the University of Leipzig, using a Novocontrol alpha analyzer (0.1 Hz–10 MHz) and an HP impedance analyzer (1 MHz–1.8 GHz).

2. PFG-NMR measurements

Pulsed-field gradient nuclear magnetic resonance (PFGNMR) measurements of PPG-LiTFSI were performed at CUNY-Hunter College. The PPG-LiTFSI samples were slowly added to 5-mm NMR tubes up to 4 cm in height and then transferred to a vacuum oven to remove the residual moisture. After overnight vacuum, the sample tubes were rapidly capped and sealed with Parafilm. PFG-NMR measurements were carried out on a Varian Direct Drive spectrometer in conjunction with a JMT superconducting magnet of field strength 7.1 T. The spectrometer equipped with Doty pulse-field-gradient gradient probe could yield a maximum gradient strength of 10 T/m for the diffusion measurements. Temperature control was achieved by a Doty temperature module. The samples were equilibrated for 30 min before each experiment. III. RESULTS AND DISCUSSIONS A. Method of EP analysis

The analysis of the electrode polarization is based on the simplified Macdonald-Trukhan model [13,21,23,25]. It has been shown that for a 1:1 electrolyte solution, where cations and anions carry the same amount of charges, by assuming (1) all the ions have equal diffusivity, and (2) the sample thickness L is much larger than the Debye length LD , the complex dielectric function of electrode polarization in a plane capacitor can be effectively described by the Debye relaxation function, ε∗ (ω) = εB +

B. Methods 1. Dielectric measurements

Broadband dielectric measurements were carried out at various temperatures using a Novocontrol Concept 80 system. Standard parallel plate sample cell (Novocontrol), liquid parallel plate sample cell (BDS 1308, Novocontrol), and a home-made parallel plate sample cell were used in this study. The surfaces of the first two sample cells were gold-plated, whereas the homemade cell was made of brass. The electrode surfaces were carefully polished and cleaned before each experiment. Cooling and heating of the samples were achieved by using a Quatro N2 cryostat. The dielectric measurements were all performed in the linear response regime with voltage no more than 0.1 V. The detailed information about the electrode material, sample thickness, and applied voltage is provided in Table I. TABLE I. Electrode material, sample thickness, and applied voltage.

Sample PPG-LiTFSI (1.6%) PPG-LiTFSI (5.8%) PPG-LiTFSI (11%) LiCl-H2 O LiCF3 SO3 -ethanol NaCl–(glycerol-H2 O) [OMIM][TFSI]

Electrode material

Sample thickness (mm)

Applied voltage (V)

Gold Gold Gold Gold Gold Brass Gold

0.054 0.054 0.054 0.9 0.5 0.5 0.155

0.1 0.1 0.1 0.1 0.1 0.03 0.05

εEP , 1 + iωτEP

(1)

where εB is the bulk permittivity, εEP = (L/2LD − 1)εB , and τEP is the characteristic relaxation time of electrode polarization. τEP is defined by the bulk resistance RB and the interfacial capacitance CEP : τEP = RB CEP . In the MacdonaldTrukhan model, CEP is simply determined by the Debye length LD and the bulk permittivity εB as CEP = ε0 εB

A , 2LD

(2)

with A being the surface area of the electrode. The total number density of free ions (n), which is the sum of the number densities of free cations (n+ ) and anions (n− ), can be related the Debye length by the following equation: ε0 εB kB T ε0 εB kB T = . (3) LD = (n+ + n− )q 2 nq 2 Here q is the amount of charges carried by an ion and T is the absolute temperature. One can, therefore, obtain the freeion number density and diffusivity by analyzing the dielectric spectra of electrode polarization. It has been shown [7,23,25] that the ion diffusivity can be calculated as D=

2πfmax L2 , 32(tan δ)3max

(4)

where (tanδ)max is the maximum value of ε /ε in the frequency range of EP and fmax is the frequency at the tanδ maximum. The free-ion number density can be obtained from the dc 042308-2

EXAMINATION OF METHODS TO DETERMINE FREE-ION . . .

density and diffusivity from the interfacial capacitance due to electrode polarization (CEP ), which can be essentially captured by the Eq. (1). Here, the low-frequency response (below the frequency of σ maximum) does not affect our analysis. It is necessary to clarify the relation of our current approach to the model of electrode polarization with generation and recombination (dissociation and association) effect (denoted as EP-GR model) [27–29]. First, we explicitly assume that the dissociation of salt is generally incomplete. In this sense, the generation and recombination effect is already considered in our model. On the other hand, we assume that the dissociation and association are sufficiently fast and, therefore, they do not directly contribute to the low-frequency electrical response, i.e., electrode polarization. In the limit of fast dissociationassociation dynamics (characteristic dissociation and association time τEP ), the two models would be essentially identical. This situation is demonstrated using simulated spectra in Fig. 2(a). When the dissociation and association dynamics are slow, our model will qualitatively differ from the EP-GR model. The EP-GR model predicts an additional “relaxation” due to ion dissociation and association in the low-frequency end of the spectrum [Fig. 2(b)]. However, this situation requires the rate of ion dissociation and association to be

log10 ε', ε"

7 NaCl-(glycerol-H2O)

5

ε"

ε'

3

log10 σ', σ" [S/cm]

(a)

1 -5 Full development of EP

-6 -7

σ"

-8 Onset of EP

-9 80

tan δ

σ'

(b)

(tanδ)max

60 40 20

(c) fmax

0 -1

1

3

5


7

log10 f [Hz] FIG. 1. (Color online) Dielectric spectrum of NaCl–(glycerolH2 O) at -5 ◦ C. (a) Complex permittivity. (b) Complex conductivity. (c) Tanδ. Here, the main “relaxation” is due to electrode polarization. Solid lines denote fits by Eq. (1). The onset and full development of electrode polarization are indicated by the arrows.

log10 ε', ε"

5

conductivity and ion diffusivity according to the definition of dc conductivity and the Einstein relation, σ kB T . Dq 2

(5)

It should be noted that in our current approach the dissociation of salts is not assumed to be complete, but the dissociationassociation dynamics are assumed to be much faster than the macroscopic electrode polarization. A representative dielectric spectrum of our samples is shown in Fig. 1, where the ε , ε , and tanδ of NaCl–(glycerolH2 O) are plotted as a function of frequency. On the highfrequency side, the real part of permittivity is independent of frequency while the imaginary part increases with decreasing frequency as ε ∼ f −1 , exhibiting the normal behavior of dc conduction. On the low-frequency side, the complex permittivity shows a Debye-like shape, due to the electrode polarization effect. The corresponding fmax and (tanδ)max can be used to analyze the free-ion number density and diffusivity using the method outlined above. One can define the onset of EP as the minimum in σ and the “full development” of EP as the maximum in σ . The solid lines in Fig. 1 represent the fits using Eq. (1). The shape of the spectrum can be clearly approximated by the Debye function up to the “full development” of electrode polarization. In reality, no electrodes can be perfectly blocking, even in the case of gold. As a result, substantial deviation from Eq. (1) occurs at low frequencies. In order to describe the spectra in the whole frequency region, more sophisticated models [22,26] have to be considered. However, it should be emphasized that the main goal of this study is to see if one can evaluate free-ion number

4

ε"

3

ε'

2 1

(a)

5

log10 ε', ε"

n=

Fast Dissociation/ Association

Slow Dissociation/ Association

4 3

1/τA

1/τEP

1/τ

ε'

2 1

ε"

(b)

-6

-4

-2

0

log10 ωτ FIG. 2. (Color online) Demonstration of representative spectra from the EP-GR model [29] in the limit of (a) fast and (b) slow dissociation and association dynamics. ε∗ = εB (1 + iωτ )/(iωτ + tanh Y /Y ), where εB is the bulk permittivity, τ is the conductivity relaxation time, and Y = (ϒ)1/2 M(1 + iωτ )1/2 . M = L/2LD , with the Debye length LD defined by Eq. (3). ϒ = (iωτ + 2rA τ )/(iωτ + rA τ ), where rA is the rate of association. For simplicity, the term of dissociation has been neglected, under the assumption that the rate of association is much larger than the rate of dissociation (incomplete dissociation). The bulk permittivity is assumed to be 10, and the magnitude of electrode polarization M is assumed to be 104 . Therefore, τEP = Mτ = 104 τ . In the case of “fast” dissociation or association, the characteristic association time τA = 1/rA is assumed to be 102 τ . In the slow case, τA = 106 τ .

042308-3



NaCl-(glycerol-H2O) PPG-LiTFSI (1.6%)

-3

log10 n [cm ]

18

[OMIM][TFSI]

PPG-LiTFSI (5.8%)

17

PPG-LiTFSI (11%)

LiCl-H2O LiCF3SO3-ethanol

(a) 16 2

3

4

5

6

7

-1

1000/T [K ]

0

log10 n/ntot

NaCl-(glycerol-H2O)

PPG-LiTFSI (1.6%) PPG-LiTFSI (5.8%) LiCF3SO3-ethanol

-4

[OMIM][TFSI]

PPG-LiTFSI (11%)

(b) LiCl-H2O

-6 0

2

4

-1

6

1000/T [K ] FIG. 3. (Color online) Temperature dependence of (a) free-ion number density n and (b) free-ion fraction n/ntot , evaluated from the electrode polarization effect using the Macdonald-Trukhan model. Solid lines denote Arrhenius fits. n = n0 exp(−Edis /kB T ). The horizontal dashed line indicates the limit of 100% dissociation.

smaller than the characteristic relaxation rate of electrode polarization, which is typically several orders of magnitude smaller than the conductivity relaxation rate. As a result, such ultraslow ion dissociation and association dynamics should not occur in most real physical systems. At least, it is not observed in any of the samples in the present study. Moreover, our estimated energy for dissociation (see below) is rather low and cannot lead to very slow dissociation or association process. B. Failure of the EP analysis

Figure 3(a) presents the free-ion number density, determined from the EP analysis, as a function of 1000/T for all the samples in this study. As a first observation, the free-ion number density n decreases with decrease of temperature in all samples except LiCl-H2 O, where n is almost a constant. The temperature dependence of ion number density can be described by the Arrhenius equation, n = n0 exp(−Edis /kB T ),

cαcα α2 [A+x ][B−x ] = =c ≈ cα 2 , (7) [AB] c(1 − α) 1−α where c is the salt concentration and α is the free-ion fraction. It follows that α decreases with increase of salt concentration c as α ∼ c−1/2 , whereas the total free-ion concentration cα increases with c as cα ∼ c1/2 . On the other hand, Fig. 3(a) shows that in PPG-LiTFSI, the free-ion concentration decreases with increasing salt content. This result clearly contradicts the general understanding of electrolyte solutions. As we shall see later, this abnormal behavior of free-ion number density is due to the failure of EP analysis. The temperature dependence of free-ion fraction n/ntot is shown in Fig. 3(b), where the Arrhenius fits are extrapolated to infinitely high temperature, i.e., 1000/T = 0. The total ion number density ntot is calculated from the total amount of salt in the solution, under the assumption of complete dissociation. If the electrode polarization analysis could indeed quantitatively capture the free-ion concentration, one would expect n0 ≈ ntot . In other words, log10 (n/ntot ) should be zero at 1000/T = 0. The physical picture is that the ions should become fully dissociated at sufficiently high temperature. Strictly speaking, this na¨ıve picture cannot be true since the thermodynamics of ion solvation is not considered. Nevertheless, the n/ntot → 1 limit should still serve as a good guide for checking the validity of the electrode polarization analysis. In PPG-LiTFSI (1.6%) and NaCl–(glycerol-H2 O), the n0 in the Arrhenius fit is fairly close to the total ion number density ntot in the sample. However, such agreement is not found in all the other samples. In particular, the free ions in LiCl-H2 O are only about 30 ppm of the total ions in the solution. Since aqueous solutions are generally considered as strong electrolytes where salts fully dissociate into free cations and anions, the result of LiCl-H2 O is obviously at odds with the current understanding of electrolyte solutions. Furthermore, while recent studies of room temperature ionic liquids have demonstrated that most of the ions exist as free ions [15], the EP analysis suggests that the free-ion fraction is only on the order of 100 ppm. It is, therefore, quite evident that the EP analysis underestimates the free-ion concentration in PPG-LiTFSI (5.8%), PPG-LiTFSI (11%), LiCl-H2 O, LiCF3 SO3 -ethanol, and [OMIM][TFSI]. A problem interconnected with underestimation of free-ion concentration is the overestimation of ion diffusivity, which is a natural consequence of the definition of ionic conductivity. Kdis =

100% Dissociation

-2

where Edis is the dissociation energy and n0 is the number density in the high-temperature limit. These features are in accordance with earlier investigations on polymer electrolytes using similar approaches [13,25,30]. It is important to mention that the temperature dependence of free-ion concentration from EP analysis is at odds with the studies by radiotracer diffusion [31], Raman spectroscopy [32], and computer simulations [33,34]. As pointed out earlier [25], this apparent contradiction possibly arises from the fact that different experimental techniques are sensitive to only certain populations of ions and different timescales of ion association [30]. Moreover, ions at high concentration can form aggregates that will complicate analysis of all the experimental data. These discussions are out of the scope of the present study. For a simple chemical equilibrium between free ions and ion pairs for a weak electrolyte solution, AB

A+x + B−x ,

(6)

042308-4

-3

log10 n [cm ]


2

log10 D [cm /s]

-4

PPG-LiTFSI series does not vary significantly with the salt concentration in the studied range. This result is in agreement with the general expectations for polymer electrolytes and further confirms the EP analysis can qualitatively capture the change of free-ion concentration with temperature. It is worth noting that Runt and coworkers [13] proposed a different fitting procedure by forcing n0 to be the stoichiometric value ntot in the Arrhenius fit. It is obvious from the Fig. 3(b) that this approach cannot render reasonable fits for samples other than PPG-LiTFSI (1.6%) and NaCl–(glycerol-H2 O). In particular, the dissociation energy in the polymer electrolytes would change significantly with salt concentration—a result that does not seem to be consistent with our general understanding of electrolyte solutions.

22 Complete Dissociation

20 18 16

-6

0

2

4

6 -1

1000/T [K ]

-8 EP NMR

-10 4.5

5.0

5.5

6.0

6.5

7.0


7.5

-1

C. Corrections to the EP analysis

1000/T [K ] FIG. 4. (Color online) Temperature dependence of ion diffusivity for LiCl-H2 O, determined from electrode polarization and PFGNMR. Inset: Temperature dependence of free-ion number density (uncorrected). The horizontal dashed line indicates the level of total ions in the sample.

This issue is highlighted in Fig. 4, where we compare the diffusivity from EP analysis with that from PFG-NMR measurement for the LiCl-H2 O sample. The diffusion coefficient from EP analysis is about four orders of magnitudes higher than the values obtained from NMR experiments. The inset of Fig. 4 shows that corresponding free-ion number density in LiCl-H2 O is considerably lower than the number density at complete dissociation. Similar problems are also found in PPG-LiTFSI (5.8%), PPG-LiTFSI (11%), LiCF3 SO3 -ethanol, and [OMIM][TFSI]. Overestimating ion diffusivity seems to be a generic problem in the electrode polarization analysis of ionic mobility. Macdonald and coworkers recently analyzed a polymer electrolyte containing polyethylene imine (PEI) and LiTFSI with a N:Li molar ratio of 400:1 [26]. The estimated ion mobility is 4.87 × 10−4 cm2 /(Vs) at 20 ◦ C, corresponding to a diffusion coefficient of 1.23 × 10−5 cm2 /s. This value is typical for aqueous solutions at room temperature and is too high for a polymer. The diffusivity of Li+ in infinitely dilute aqueous solution at 25 ◦ C is 1.029 × 10−5 cm2 /s [35]. In general, Walden’s rule should apply at high temperature, i.e., the ion diffusivity should be controlled by the viscosity of the surrounding medium. Since the (microscopic) viscosity of PEI is significantly higher than water, the EP analysis clearly overestimates the ion diffusivity. Despite the problem of underestimating free-ion concentration and overestimating ion diffusivity, the electrode polarization analysis does seem to be able to reasonably describe the temperature dependence of free-ion concentration. The dissociation energy Edis from the Arrhenius fit is in the range of 0.002–0.08 eV, which is commonly seen in these ionic conductors. In addition, Edis seems to correlate with the dielectric constant of the electrolytes: While near zero Edis is found in the strong electrolyte LiCl-H2 O, the Edis of weak electrolyte PPG-LiTFSI is considerably higher. It is important to indicate that the dissociation energy in the

It is well known that electrode polarization can be affected by the roughness and chemical composition of electrodes [15,17,36]. These effects are not considered in the MacdonaldTrukhan model of electrode polarization. As a result, it is quite likely that the model may only qualitatively describe the dielectric response but fail to yield quantitative values. We have seen that the EP analysis can produce reasonable dissociation energy, but the absolute level of free-ion number density can be off by orders of magnitude. Based on these observations, we propose that the free-ion number density and diffusivity from EP analysis can be simply corrected by using a proportionality constant. If salts fully dissociate in the infinitely high temperature limit, then the true free-ion number density should be described by the Arrhenius equation, ntot ntot n˜ = n= [n0 exp(−Edis /kB T )] n0 n0 = ntot exp(−Edis /kB T ), (8) where n˜ is the free-ion number density after correction, n is the free-ion number density before correction, ntot is the total ion number density at complete dissociation, and Edis is the dissociation energy from the original EP analysis. Here, we rescale the original free-ion concentration n by a factor of ntot /n0 , assuming complete dissociation at infinitely high temperature limit. Similarly, we need to correct the diffusivity by a factor of n0 /ntot , n0 D˜ = D , (9) ntot with D˜ being the corrected diffusivity and D the original diffusivity from the EP analysis. It should be emphasized that possible entropic contribution to Edis is not considered in our correction. In Figs. 5–7, the corrected and uncorrected free-ion diffusivities are compared with the results of PFG-NMR measurements, along with the diffusivities Dσ evaluated from the Nernst-Einstein relation, assuming 100% ion dissociation. Here, we include the examples of three polymer electrolytes, one aqueous solution, and one ionic liquid. In all samples, the uncorrected diffusivities from EP analysis are much higher than the values obtained from NMR measurements. The difference is as large as four orders of magnitudes in the case of LiCl-H2 O. It is worth pointing out that PFG-NMR

042308-5



-4

-4 LiCl-H2O Before correction 2

-8

(a)

After correction

-6

log10 D [cm /s]

2

log10 D [cm /s]

-6

PPG-LiTFSI (1.6%)

Before correction

-10

After correction

-12

DEP, Corrected DEP, Uncorrected

-10

DEP, Corrected DEP, Uncorrected Dσ

-12

Dσ

-14

-8

DF, PFG-NMR

D, PFG-NMR, Averaged

DLi, PFG-NMR

-16

-14 4.5

5.0

5.5

6.0

6.5

7.0

3.0

3.2

3.4

-1

3.6

3.8

4.0

4.2

4.4

4.6

-1

1000/T [K ]

1000/T [K ] PPG-LiTFSI (5.8%) Before correction

-8

After correction

-10

DEP, Corrected

-12

Dσ

DEP, Uncorrected DF, PFG-NMR DLi, PFG-NMR

-14 3.0

3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

-1

1000/T [K ] -4 PPG-LiTFSI (11%) -6

D. Possible origins of the failure

Before correction

(c)

-8 After correction

-10

DEP, corrected DEP, uncorrected Dσ

-12

In the preceding discussion, we show that the EP analysis fails for many ionic conductors but can be corrected by rescaling free-ion number density and diffusivity by a certain factor. This correction is somewhat expected, as the effects of surface roughness and electrode materials have not been taken in account in the current theoretical model of electrode polarization. In addition to the known effects of the roughness and chemical composition of electrodes, it has been suggested that nonlinear dielectric response could also lead to the failure of the EP analysis [37]. However, this is not the case in our measurements. Figure 8 presents the dielectric spectra of PPG-LiTFSI (11%) at different voltages. In the frequency

(b)

2

log10 D [cm /s]

-6

2

experiments measure the average diffusivities of both free ions and ion pairs, whereas the EP phenomenon in theory only reflects the diffusivity of free ions. However, there is no physical model to support the idea that the free-ion diffusivity can differ from the average value by orders of magnitude. In the case of aqueous LiCl solution the majority of ions are expected to be dissociated, and even if we have only 10% of free ions, the average diffusivity cannot be more than 10 times smaller than the diffusivity of free ions. After the proposed correction, reasonable agreement between EP analysis and PFG-NMR measurement is found in all samples, although there is still a slight difference between diffusivities determined by these two methods. This discrepancy might be due to either the over-simplification in the EP analysis (e.g., by assuming cations and anions have equal diffusivity) or the real difference in diffusivity of free ions and ion pairs.

-4

log10 D [cm /s]

FIG. 5. (Color online) Temperature dependence of diffusivity for LiCl-H2 O. Filled (black) circles denote diffusivity from the EP analysis, after correction. Unfilled (black) circles denote diffusivity before correction. Half-filled (black) circles denote diffusivity from the Nernst-Einstein relation. (Red) Diamonds denote averaged diffusivity of lithium and chloride from PFG-NMR [24]. Here, the chloride diffusivity is calculated from the lithium diffusivity according to the Stokes-Einstein relation.

DF, PFG-NMR DLi, PFG-NMR

-14 3.0

3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

-1

1000/T [K ] FIG. 6. (Color online) Temperature dependence of diffusivity for three PPG-LiTFSI samples: (a) PPG-LiTFSI (1.6%); (b) PPGLiTFSI (5.8%); and (c) PPG-LiTFSI (11%). Filled (black) circles denote diffusivity from the EP analysis, after correction. Unfilled (black) circles denote diffusivity before correction. Half-filled (black) circles denote diffusivity from the Nernst-Einstein relation. Up- (red) triangles denote fluoride diffusivity from PFG-NMR. Down- (blue) triangles denote lithium diffusivity from PFG-NMR.

042308-6


that in the linear regime. Our data analysis protocol, described in Sec. II, relies only on the high-frequency response. The low-frequency nonlinear behavior will not play any role in our EP analysis. It is perhaps useful to indicate that Colby and coworkers have proposed a dimensionless parameter [37], defined as 2 QλB qV nλ3B = , (10) Aq kB T 16π

-4 [OMIM][TFSI] Before correction

2

log10 D [cm /s]

-6

-8 After correction

-10

DEP, Corrected

to characterize the degree of nonlinearity in their dielectric measurement of single-ion conductors. Here Q is the charge accumulated on the electrode surface, A is the electrode surface area, q is the charge carried by the ions, V is the applied voltage, and λB is the Bjerrum length. It has been argued that the EP analysis should remain valid as long as Qλ2B /Aq is

DEP, Uncorrected Dσ

-12

D, PFG-NMR, Average

3.0

3.3

3.6

3.9

4.2

4.5


4.8

-1

1000/T [K ] 2

range of 1 Hz–10 MHz, the dielectric response below 0.3 V is essentially independent of the applied voltage, i.e., there is a well-defined linear response regime. Since our dielectric measurements were all carried out at voltages no larger than 0.1 V, the failure of the EP analysis is not related to the nonlinear response. In additional, although the dielectric behavior becomes nonlinear at 1.0 V, the spectrum between 106 Hz (the onset of electrode polarization) and 103 Hz (the full development of electrode polarization) is still identical to

[OMIM][TFSI]

LiCF3SO3-ethanol PPG-LiTFSI (11%)

-4

(a)

LiCl-H2O

-6 19

20

21

22

-3

log10 ntot [cm ] 2

o

NaCl-(glycerol-H2O)

PPG-LiTFSI (1.6%)

0

4 (a)

log10 (n0/ntot)

log10 ε"

PPG-LiTFSI (5.8%)

-2

PPG-LiTFSI (11%), 59 C

6

2

0

4 log10 tan δ

2

2

-1

1

2 4 6 log10 f [Hz]

2

3

4

5

6

LiCl-H2O

-1

0 0

PPG-LiTFSI (11%)

-6

(b) 0

LiCF3SO3-ethanol

(b)

1 0

PPG-LiTFSI (5.8%) [OMIM][TFSI]

-2

-4

1.0 V 0.3 V 0.1 V 0.03 V 0.01 V

6

log10 ε'

NaCl-(glycerol-H2O) PPG-LiTFSI (1.6%)

0

log10 (n0/ntot)

FIG. 7. (Color online) Temperature dependence of diffusivity for ionic liquid [OMIM][TFSI]. Filled (black) circles denote diffusivity from the EP analysis, after correction. Unfilled (black) circles denote diffusivity before correction. Half-filled (black) circles denote diffusivity from the Nernst-Einstein relation. (Red) Diamonds denote diffusivity from PFG-NMR [18].

7

log10 f [Hz] FIG. 8. (Color online) Frequency dependence of imaginary (a) and real (b) permittivity for PPG-LiTFSI (11%) at various applied voltages. Inset: Corresponding tan δ = ε /ε .

0

log10 (λB/LD)

1

2

FIG. 9. (Color online) Correction factor n0 /ntot as a function of (a) ntot and (b) λB /LD . n0 is the prefactor in the free Arrhenius fit of free-ion concentration n(T ) : n = n0 exp(−Edis /kB T ). ntot is the total ion number density in the material. The degree of correction to the experimental ion number density n from the EP analysis is, therefore, determined by the ratio of n0 and ntot . λB is the Bjerrum length and LD is the Debye length. The horizontal dashed lines indicate the limit of no correction, i.e., log10 (n0 /ntot ) = 0.

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much less than 1. The physical interpretation is that nonlinear electrode polarization starts to occur when the ion separation on the electrode surface is smaller than the Bjerrum length. For the PPG-LiTFSI (11%) sample, Fig. 8 shows that nonlinear electrode polarization occurs at approximately 0.3 V. On the other hand, the above criterion predicts that the critical voltage is 0.25 V, when Qλ2B /Aq = 1. The linearity criterion proposed by Colby and coworkers seems to be in good agreement with our experimental observation. The current theoretical model of electrode polarization [1,13] is based on a Debye-Hückel-type approach, where the ionic interaction strength is assumed to be much smaller than the thermal energy kB T . Therefore, the model is expected to fail at high salt concentrations, when strength of ionic interaction increases. The ratio of Bjerrum length λB and Debye length LD can be used to roughly determine whether an electrolyte is in the dilute state. When λB /LD 1, the system typically can be considered as a dilute solution, and Debye-Hückel-type approaches should apply. We recall that the free-ion concentration from the EP analysis (without correction) can be described by the Arrhenius equation: n = n0 exp(−Edis /kB T ). In general, n0 is not the same as the total ion number density ntot in the material. The initial free-ion concentration n from the EP analysis therefore needs to be corrected by a factor of ntot /n0 : n˜ = n(ntot /n0 ). In other words, the ratio of n0 and ntot reflects the degree of correction that needs to be introduced in the EP analysis. In the case of no correction, n0 /ntot = 1. Figure 9 shows n0 /ntot as a function of (a) ntot , and (b) λB /LD . The goal of this presentation is to explore a possible correlation between the degree of correction in the EP analysis and the extent to which the system deviates from the dilute limit. It is clear from Fig. 9 that the original Macdonald-Trukhan model fails progressively with increasing salt concentration ntot and λB /LD . Significant correction has to be made in concentrated electrolytes. Interestingly, the degree of correction, n0 /ntot , seems to correlate better with the total

ion concentration ntot rather than λB /LD . The linear fit of log10 (n0 /ntot ) and ntot has a R 2 of 0.80, whereas the R 2 for log10 (n0 /ntot ) and λB /LD is only 0.31. This observation suggests that the failure of EP model may also be related to the structure of the compact double layer, where the bulk permittivity plays a much less significant role.

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IV. CONCLUSIONS

The electrode polarization analysis is used to estimate the ion diffusivity and number density of a wide variety of ionic conductors. Although the analysis does yield reasonable estimates in a few cases, it fails for most of the materials. For systems with high ion concentration, the EP analysis overestimates the free-ion diffusivity while underestimates the free-ion number density. Since the original EP model is based on the Debye-Hückel approach, it is expected to fail at high ion concentrations. An empirical method is proposed to correct the results at high ion concentrations, and ion diffusivities that are in close agreement with PFG-NMR measurements are obtained. Despite this success, the analysis of ion diffusivity and number density from electrode polarization should still be exercised with great caution, because there is no solid theoretical justification for the proposed corrections. ACKNOWLEDGMENTS

The authors thank A. L. Agapov and M. Nakanishi for fruitful discussions. This research was sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the US Department of Energy. F.F. thanks the NSF Polymer Program (DMR-1104824) for funding. J.R.S. and A.P.S. acknowledge the financial support from the DOEBES Materials Science and Engineering Division. The NMR program at Hunter College is supported by a grant from the US DOE-BES under Contract No. DE-SC0005029.

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