New Insight Into Ion Transport Through Dynamic Modulus Studies

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New Insight into Ion Transport through the Dynamic Modulus Studies Agnieszka Jedrzejowska, Kajetan Koperwas, Zaneta Wojnarowska, and Marian Paluch J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b06523 • Publication Date (Web): 17 Aug 2016 Downloaded from http://pubs.acs.org on September 11, 2016

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The Journal of Physical Chemistry

New Insight Into Ion Transport Through The Dynamic Modulus Studies Agnieszka Jedrzejowska, Kajetan Koperwas, Zaneta Wojnarowska and Marian Paluch* Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland Silesian Center for Education and Interdisciplinary Research, 75 Pulku Piechoty 1A, 41-500 Chorzow, Poland

Corresponding Author Email: [email protected]

Phone: +48 349 7637

ABSTRACT In this paper, the dynamic properties of classical imidazolium ionic liquid [C8MIM][NTf2] are examined in terms of new physical quantity “modulus of elasticity” (M), being defined as the ratio of activation energy (Eact) and activation volume (Vact) - parameters determined from isobaric and isothermal dc-conductivity data, respectively. By taking advantage of T-V version of Avramov entropic model we demonstrate that the dynamic modulus can be defined by means of thermodynamic quantities, i.e. isothermal compressibility and thermal expansion coefficient. From this point of view new connection between thermodynamic and dynamic properties of supercooled liquids, being an alternative to density scaling idea, is provided.

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INTRODUCTION Over the last couple of decades the importance of the ionic liquids (ILs) in the industrial processes continue to rise.1,2 The diversity of many potential applications results from their unique physicochemical features.3 For example, the ability to dissolve many chemical compounds, the high thermal stability, the low vapour pressure and the good biodegradability make ILs perfect organic solvents.4,5 Further, due to the ionic nature they possess very high electric conductivity (0.1–18 mS/cm), that combined with broad electrochemical window makes them attractive candidates for electrochemical applications.6,7 For instance, in batteries ILs become an alternative to traditional carbonate-based electrolytes.8,9 It is worth mentioning that cells made of ILs (Li-ion cells) are safer than the traditional ones because they are thermally, mechanically and electrically resistant to extreme conditions.10 Another very promising possibility is the potential usage of ILs as double-layer capacitors (EDLC).

11,12

In the view of these electric device applications, the conductivity of

ionic liquids and its temperature dependence are of the great importance. It is well known that the melting temperature of many ILs is located within the range of T=Troom(298K)±50K that could result in a phase transition (crystallization) and causing serious problems during the IL’s utilization process. However, there are also ionic materials that can be very easily supercooled.[13] When supercooled, they become denser and more viscous. Moreover, drastic slowing down of the molecular dynamics and thus dramatic decrease of dc-conductivity (σdc) is observed on lowering temperature toward the glass transition region. The changes of σdc can even exceed 12 decades of magnitude.14 The experimental method which enables direct monitoring of dc-conductivity variation in such extraordinary range is the broadband dielectric spectroscopy (BDS). Additionally, BDS can


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be easily adapted to conductivity studies at elevated pressures and, thus, provide new and relevant information about electric conduction mechanism.15-17 In the case of a simple thermally activated process, when ions hop over the energy barrier between potential well the ions diffusion is solely controlled by thermal fluctuations. However, in reality ions migration is driven not only by thermal, but also density changes. The significance of the density fluctuations can be assessed from isothermal experiments by changing pressure at constant temperature. Then the molecular crowding is modified without any change of thermal energy.18,19 Isobaric and isothermal dependencies of dc-conductivity are usually characterized by two fundamental quantities: the activation energy (Eq.1) and activation volume (Eq. 2),

 d log10 σ dc  Eact = 2.303R  −1  dT 

(1)

 d log10 σ dc  Vact = 2.303RT  , dP  

(2)

where R is a gas constant, Eact defines the energy value, which ions needs to gain to hop over the energy barriers, and Vact reflects the volume requirements for local molecular motion. More precisely, according to the transition state theory, the activation volume is the difference between the volumes occupied by an ion in activated and non-activated states. Experimentally, it is commonly observed that the value of Eact increases during isobaric cooling toward the glass transition temperature. This is due to the energy barriers growth related to the increase of the molecular packing. On the other hand, the increase in molecular packing implies the molecular volume occupied by molecules at non-activated states decreases with lowering temperature. Hence, the difference between molecular volumes at activated and non-activated states i.e., activation volume, will continuously increase when approaching the glass transition temperature.20 Since, both activation energy and activation



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volume are getting higher when lowering the temperature at constant pressure, the natural question about this process is the possible correlation between these two quantities arises. Recently, Ingram et al., based on unified treatment of ion transport demonstrated a direct relationship between the activation volume and activation energy parameters, Eact = M ⋅ Vact

(3)

for a few polymer electrolytes as well as CKN.21,22 These authors pointed out that the parameter M is constant in the first approximation and, consequently, they postulated this can be used to characterize and/or classify the dynamic properties of various materials. According to Ingram et al., M can be treated as a kind of “modulus of elasticity”, which can be strongly influenced by intermolecular forces, and reflects to some extent the rigidity of the molecular structure. However, the most intriguing finding obtained by the Ingram and co-workers is that values of dynamic modulus determined from analysis of structural relaxation and ion conductivity are basically the same for given material. Although values of Eact and Vact can significantly differ for both processes. These findings implies that both ion migration and structural relaxations are controlled by the same intermolecular forces.21 Thus, the elementary processes, which take place at short times and involve short-range interactions, govern “fast” and “slow” molecular dynamics. Consequently, Authors postulate that it should be possible to find unified description of ion transport in ionic conductors. In this paper, we focus on verifying the proportionality between Eact and Vact suggested by Ingram et al. for classical aprotic ionic liquid [C8MIM][NTf2] (1-octyl-3methylimidazolium bis[(trifluoromethyl)sulfonyl]imide). More importantly, we derived the relationships between the dynamic and thermodynamic bulk modulus using Avramov model, being one of the most popular entropic model commonly applied for description of the temperature and the pressure dependence of the transport (viscosity, electric conductivity) and relaxation properties (structural relaxation) of glass forming liquids.


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RESULTS & DISCUSSION Determination of the dynamic modulus requires knowledge of temperature as well as pressure dependences of the quantities characterizing transport properties or relaxation dynamics of matter, e.g., viscosity, electric conductivity, diffusion or structural relaxation time. In the case of ionic liquids the electric conductivity is a key quantity characterizing the charge transport. Therefore, in order to determine the dynamic modulus for aprotic ionic liquid [C8MIM][NTf2], we used the ambient and high pressure dc-conductivity data already published by Paluch et al. in Ref. 13. The temperature dependence of dc-conductivity, taken from Ref. 13, is depicted in Figure 1a (the values of dc-conductivity were listed in Table 1S in supplemental materials). To highlight the non-Arrhenius behavior of experimental points we plotted log10σdc versus inverse of temperature. These experimental points, well parameterized by Vogel-Fulcher-Tamman (VFT) equation, were next used to determine the activation energies at different temperatures according to Eq. 1. It can be noted from Figure 2a that Eact continuously increases on decreasing temperature approaching the value of ca. 250 kJ/mol at Tg. The temperature dependence of the Eact was also calculated using the fitting parameters of 2

VFT equation Eact

 T   and shown in Figure 2a and 2b as a solid line.22. = BR  T − T0 

On the other hand, the pressure dependences of dc-conductivity, necessary to determine the activation volume, are presented in Figure 1b. The derivative method, being a model independent analysis, was used to calculate Vact. In the first step the derivative of log10σdc(P) dependences were calculated. Then, we used equation 2 to calculate Vact. The results of such analysis are presented in Figure 2c. From this plot one can see that the value of Vact is nearly constant for given isotherm. Thus, it can be concluded that the simple volume activated equation,  PVact    RT 

σ dc (P ) = σ 0 exp

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is suitable for describing the conductivity experimental data. Moreover, it is evident from Figure 2c that the activation volume systematically increases with lowering temperature. Its temperature dependence is plotted in Figure 2d. The experimentally determined values of activation volumes and activation energies for [C8MIM][NTf2] are collected in Table 1. According to Eq. 3, the dynamic modulus is a ratio between activation energy and activation volume. To check if Eact(Vact) dependence is indeed linear we plotted Eact versus Vact. As can be seen in Figure 3, the data satisfy fairly well a linear regime with the value of the intercept equals to 0. The value of the slope, obtained from the simple linear regression, defines the dynamic modulus Mlin for [C8MIM][NTf2] and it is equal to 1.5±0.1 GPa. Approximately the same values are also obtained if the ratio Eact(Vact) is calculated directly at given temperature, i.e. from Eq. 3 (see Table 1). Interestingly, the values of dynamic modulus estimated for other classical ionic liquids [C4MIM][NTf2] M=1.87 and BMP-BOB (-butyl-1-methylpyrrolidinium bis[oxalate]-borate) M=1.82 are similar to that determined for [C8MIM][NTf2]. This result suggests the similarities between dynamic modulus parameter estimated for various aprotic ionic conductors. An alternative approach to analyze the combined temperature and pressure dependence of the electric conductivity is to use the entropic model proposed by Avramov. This model has been originally introduced to describe the viscosity of supercooled liquids in T-P thermodynamic space. However, it can be easily adopted to analyze dc-conductivity data (Eq. 5). β   Tr α  p  1 + ,   T   Π    

σ dc = σ dc ∞ exp  ε 

(5)

where σdc∞ - is the conductivity at high temperature, ε is equal to 30 (It results from fact that Tg usually appears at the temperature at which the viscosity is in the order of 1013P (≈e30P)),


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Tr – references temperature (e.g. Tg), P – pressure, T –temperature, Π – a constant with the dimension of pressure and α, β are parameters related to some thermodynamic quantities as discussed below. Having the mathematical expression for dc-conductivity, we are able to derive the formula for the dynamic modulus: M =∏

α β

(6)

However, in order to calculate the dynamic modulus from Eq. 6 we need to know the Π, α and β parameters. Their values can be easily found from the fitting of Avramov equation to the experimental electric conductivity data presented in Figure 1a and 1b. Applying this procedure the value of the dynamic modulus was found to be equal 1.6±0.1 GPa. This is in perfect agreement with what we have found previously when analyzing the data presented in Figure 3a. As mentioned above, the two parameters: α and β in the Avramov equation are related to the thermodynamic quantities as follow: α = 2C p / ZR and β = 2α 0Vm ∏ / ZR where Cp is a specific heat capacity at constant pressure, α0 is the thermal expansion coefficient and Vm denotes the molar volume. Substituting these two formulas into the Eq. 6, leads to the new expression that relates the dynamic modulus only to thermodynamic quantities: M =

Cp

α 0Vm

,

(7)

Thus, it should be possible to determine M based on thermodynamic measurements solely. To check the validity of equation 7 in the case of [C8MIM][NTf2] we have employed the temperature modulated DSC technique to determine the specific heat capacity for this compound (see inset in Figure 3 and supplementary materials for details about TMDSC measurements). On the other hand, the values of the two other parameters: α0 and Vm appearing in Eq. 6 were calculated from parameters of the equation of state reported in Ref. 13. As a result we found that the value of M is approximately equals to 2.7±0.2 and it is 7 ACS Paragon Plus Environment


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significantly larger than the value determined previously from Eact(Vact) analysis (Mlin=1.5±0.1). The discrepancy between the values of the dynamic modulus determined directly from dielectric data and thermodynamic parameters can result from the theoretical background of Avramov equation (Eq.5). In order to extend the model to a pressure variable, the Avramov assumed that the thermal expansion coefficient is inversely proportional to pressure. This simple assumption enabled him to derive the analytical expression for temperature and pressure dependence of viscosity. It has been demonstrated for many glass-forming liquids that the mentioned expression fits very well experimental data over a wide pressure and temperature range.23,24 Nevertheless, detailed analysis of the expression originally derived by Avramov reveals some shortcomings. The parameters obtained from fitting the experimental data differ from that calculated using the thermodynamic quantities, as it has been already pointed out for the dynamic bulk modulus. Moreover, which is more important, the temperature – pressure version of Avramov model leads to the constant value of the steepness index (isobaric fragility), defined as mP = ∂ logτ / ∂ (Tg / T )

T =Tg

that disagree with common

experimental observation revealing a decreasing trend of mP(P) for van der Waals liquids, polymers and ILs.25-28 These problems have become a motivation to develop the original Avramov model. As a consequence Casalini et al.2 derived a new formula that describes the temperature-volume dependence of viscosity:

 A  log10 η (T ,V ) = log10 η0 +  γ   TV 

D

(8)

where log10η0 denotes the viscosity at high temperature (T), V is volume, D=2Cv/ZR (Cv is an isochoric heat capacity, Z represents the degeneracy of the system, i.e., it is the number of available pathways for local motion of a molecule, R is a gas constant), A and γ are the fit parameters. 8 ACS Paragon Plus Environment

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Importantly Eq. 8 predicts appropriate behavior of mp, i.e. its drop with compression and thus avoids the previous limitations of Avramov model. It is also worth mentioning that the coefficient γ, being the material constant, is directly related to the exponent in the repulsive term of intermolecular potential. Moreover, the T-V version of the Avramov model predicts one of the most intriguing feature of molecular dynamics, i.e., density scaling, experimentally observed for more than 100 different glass-forming liquids. According to this idea, the viscosity as well as electric conductivity points measured both at isobaric and isothermal conditions can be scaled onto single master curve by plotting log10η or log10σdc versus new generalized variable TVγ. It has been established that the density scaling provides the direct connection between thermodynamics and dynamics of the supercooled liquids. Following the procedure used to estimate the dynamic modulus from the T-P version of the Avramov model, we can derive the equation for Eact and Vact directly from Eq. 8. Since the values of activation energy are calculated at constant pressure conditions, which has not been specified by the temperature-volume dependence of conductivity, we can exploit the following thermodynamic relation: ∂ log 10σ ∂T

= P

∂ log 10σ ∂T

+ V

∂ log 10σ ∂V ∂V T ∂T

(9) P

On the other hand, the pressure derivative of conductivity, that occurs in the definition of the activation volume, can easy be rewritten as follows:

∂ log 10σ ∂p

= T

∂ log 10σ ∂V ∂V T ∂P

(10) T

Using the above relationships one can easily derive the equation for the dynamic modulus:

M=

1 + Tγα P

(11)

γκT

where κ T =

− 1  ∂V    is a isothermal compressibility. V  ∂P T



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As can be seen in Table 1 the values of the dynamic modulus predicted from Eq. 11 are comparable to those obtained from Eq. 6. Herein, it should be noted that Ingram et al. postulated the existence of direct connection between the dynamic modulus and isothermal compressibility, M=1/κT. From Eq. 11 one can see there is indeed relationship between dynamic and thermodynamic bulk modulus, but it is much more complex than originally proposed. To shed more light on this problem, we can rewrite Eq. 11 to the following form Mκ T − α pT = 1 / γ and recall the wellP V known relation between isobaric ( Eact = Eact ) and isochoric ( Eact = [∂ ln σ / ∂ (T −1 )]V )

activation energy V P Eact / Eact = 1 /(1 + γTα P )

(12)

Combining Eqs. 11 and 12 we arrive to the following formula V P Eact / Eact = 1 /( Mγκ T )

(13)

Herein, it should be mention that the ratio between activation energies results directly from the analyzed entropic model, and it provides essential information about relative roles of the V P / Eact thermal and volume fluctuations to the molecular dynamics behavior. The ratio Eact

T =T0

(where T0 is an any reference temperature) approaches 0 when the isochoric activation energy decreases toward 0. Then, the intermolecular free volume is a main variable controlling the V P / Eact molecular dynamics. On the contrary, if the Eact

T =T0

approaches 1, i.e., when the

activation energy at constant volume and the activation energy at constant pressure are equal to each other, then the temperature plays a decisive role in the dynamic behavior. It means that the relaxation or transport process is thermally activated and the scaling exponent γ is V P equal to 1. In the case of purely thermally activated processes, i.e., when γ and Eact / Eact are

equal to 1, Eq. 11 takes very simple form: M=1/κT which express the simple connection between dynamic modulus and isothermal compressibility proposed by Ingram et al. Thus, it 10 ACS Paragon Plus Environment

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can be concluded that simple relationship between M and κT is generally not true and it can be satisfied only for systems in which the charge transport is purely thermally activated process. V P / Eact for examined ionic conductor was found to be lower than one Since the value of Eact

(0.79) one can look forward the discrepancy between the values of M determined using the isothermal compressibility parameter and Eact/Vact ratio. As expected, we have found that the dynamic modulus calculated for [C8MIM][NTf2] from 1/κT relation, being in the range 1.82.2±0.1, markedly differs from that estimated directly from isothermal and isobaric conductivity data (1.5±0.1) (see Figure 3b). Intuitively, the next step is to check the connection between scaling exponent γ, isothermal compressibility, thermal expansion coefficient and dynamic modulus for other glass-forming liquids. Therefore, we have used formula 11 as well as the relation 1/ κT and model-independent way to calculate the dynamic modulus for three systems: (i) verapamil hydrochloride being an ionic salt, (ii) PDE (phenylphthalein-dimethyl ether) that belongs to van der Waals liquid class and (iii) carvedilol – the non-ionic glass-former with week Hbonded network. The obtained results are presented in new Figure 4. As one can see, in all these three systems the values of M calculated from Eq. 11 are in excellent agreement with those determined directly from dielectric experimental data. At the same time the modulus V P data coming from 1/ κT relation are markedly higher. However, if the values of Eact / Eact are

taken into account (verapamil hydrochloride-0.69, PDE-0.53 and carvedilol-0.65)17,27, one can see that the dynamics of these three systems is not purely thermally activated. Thus, the simple relationship between M and κT cannot be satisfied. These results proof that Eq.11 is satisfied for glass forming systems in general, and consequently strongly confirm the idea that studies in terms of dynamic modulus provide new connection between thermodynamic and dynamic properties of supercooled liquids. CONCLUSIONS 11 ACS Paragon Plus Environment


In this paper we explored the predictions of Avramov entropic model in the context of new physical quantity – dynamic modulus (M). The direct analysis of isothermal and isobaric dc-conductivity data collected for classical ionic liquid clearly shows that at ambient pressure conditions the M parameter is constant, as it was suggested by Ingram et al.21,22 Moreover, the value of M is in perfect agreement with dynamic modulus predicted from T-P version of Avramov model. However, when the thermodynamic definition of Avramov fit parameters is taken into account the relation M=Cp/α0Vm is no longer valid. Therefore, herein we took advantage of modified T-V version of Avramov formula to provide new definition of dynamic modulus. As we demonstrated, the obtained relation Mκ T − α pT = 1 / γ gives M parameter much closer to that determined in model-independent way and thus provides the direct connection between thermodynamics and dynamics of supercooled liquids. Moreover, it is well visible that the relation between dynamic modulus and inversed isothermal compressibility is exceedingly complex than it was postulated by Ingram et al. FIGURES AND TABLES 14

A

B

[C8MIM][NTf2]

12

-log10σdc /Scm-1

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

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273 K 283 K 293 K 303 K 313 K

6 N

10

N

+

O F 3C

S

O N

O

8

-

S

CF3

5

O

6 4 4

Isobar at 0.1 MPa

2 2,3

2,8

3,3

3,8

4,3

1000T-1 /K-1

4,8

5,3

Isotherms

3 0

100

200

300

400

500

Pressure /MPa

Figure 1. Dependence of dc-conductivity of [C8MIM][NTf2] recorded in different temperature (A) and pressure (B) conditions. The isothermal data was parameterized by


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classical Avramov equation (Eq. 5) with the following fitting parameters log10σdc∞ = 1.79 Scm-1, α = 3.55, β = 0.98, Π = 453 MPa.

250

30

[C8MIM][NTf2]

A

C

273 K 303 K

283 K 313 K

293 K

28 +

N

N

Eact /kJmol

-1

O F3 C

150

S

O -

N

CF3

S

O

Vact /cm3mol-1

200

O

100 50

26 24 22 20

0

18 2.0

2.5

3.0

3.5

4.0 -1

4.5

5.0

5.5

0

100

-1

50

28

B

45

300

400

500

D

26

Vact /cm3mol-1

-1

200

Pressure /MPa

1000T /K

Eact /kJmol

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


40 35 30 25

283K

313K 303K 293K

3.1

3.2

3.3

3.4 -1

3.5 -1

1000T /K

3.6

3.7

22 20

273K

20 3.0

24

3.8

18 3.15

3.30

3.45

3.60

3.75

1000T-1 /K-1

Figure 2. The activation energy (A and B) and activation volume (C) calculated for isobaric and isothermal conductivity data, respectively. Temperature dependence of activation volume parameter is depicted in panel D.


-1

Heat capacity / Jg K

Eact /kJmol

-1

40

36

1.4

[C8MIM][NTf2]

1.2

Tg=186 K

1.0 0.8 140

∆Cp=0.96 Jg-1K-1 160

180

200

220

Temperature /K

M=1.5± ±0.1 GPa 32

A

28 18

20

22

3

24

Vact /cm mol 3

26

-1

From T-P Avramov parameters: M=CP/α0Vm

B

From: M=1/κT

M /GPa

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

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-1


From T-P Avramov −1 parameters: M=Παβ

2

1

Experimental data: M=Eact/Vact From T-V Avramov parameters: M=(1+TγαP)/γκT

0 270

280

290

300

310

320

Temperature /K

Figure 3. Activation energy (determined as presented in Figure2b) plotted as a function of activation volume for [C8MIM][NTf2] is depicted in panel A. The inset panel presents the thermogram of studied classical ionic liquid. Tg of [C8MIM][NTf2] determined in this paper is in good agreement with value reported in ref [30]. Panel B presents the comparison between values of dynamic modulus determined from for of each discussed models


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Verapamil 2.5 B hydrochloride

A

PDE 3.0

Carvedilol

C

2.5

M /GPa

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2.0 2.5 2.0 1.5 2.0

1.5 1.0

330

345

360

375

390

315

330

345

324

360

328

332

temperature /K

Figure 4. The comparison between values of dynamic modulus determined from Eq. 11 (●), Eq. 3 (■) and

(▲) for verapamil hydrochloride (A), PDE (B) and carvedilol (C).

Table 1. Activation parameters Eact(T), Vact(T) and value of “process moduli” determined for ion transport in [C8MIM][NTF2].

Eact

Vact

T [K]

Eq. 3 -1

3

-1

[kJ·mol ] [cm ·mol ]

Eq.6

Eq. 7

Eq. 11

[GPa]

273

40.7±2.1

26.7±0.1

1.5±0.1

2.8±0.2

1.3±0.1

2.2±0.1

283

36.9±3.0

24.1±0.1

1.5±0.1

2.8±0.2

1.3±0.1

2.1±0.1

293

34.0±2.3

22.4±0.1

2.7±0.2

1.2±0.1

2.0±0.1

303

31.4±0.8

20.5±0.1

1.5±0.1

2.7±0.2

1.2±0.1

1.9±0.1

313

29.3±1.7

19.0±0.1

1.5±0.1

2.7±0.2

1.2±0.1

1.8±0.1

1.5±0.1

1.5 ±0.1

1.6±0.1



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ACKNOWLEDGMENT The authors Z.W. and M.P. are grateful for the financial support of the National Science Centre within the Maestro2 project (Grant No. DEC 2012/04/A/ST3/00337).


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TOC GRAPHICS 4

[C8MIM][NTf2]

N

N

+

O

dynamic modulus (M) /GPa

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


3

F 3C

From T-P Avramov parameters: M=CP/α0Vm

S O

O N

-

S

CF3

O

From T-P Avramov parameters: M=Παβ−1

From: M=1/κT

2

1

Experimental data: M=Eact/Vact From T-V Avramov parameters: M=(1+TγαP)/γκT

0 270

280

290

300

310

320

temperature /K


New Insight Into Ion Transport Through Dynamic Modulus Studies

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