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3 Thermal and Density Fluctuations to the Temperature Dependence of σdc at Ambient and Elevated Pressure

3 Thermal and Density Fluctuations to the Temperature Dependence of σdc at Ambient and Elevated Pressure

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P¼0:1 MPa can be easily

calculated as the ratio of k1/k3. It is well known that the value of dTg/dP in the limit

of ambient pressure depends strongly on the material and it usually falls in the range

30–300 K/GPa. The low value of this coefficient reflects the high resistance of Tg

on the pressure changes and it is typical for hydrogen-bonded systems and metallic

glasses [17–19]. At the opposite extreme, there are van der Waals liquids with quite

strong pressure sensitivity of molecular dynamics and thus high values of dTg/dP

parameter [20–22]. On the other hand, the group of ionic materials is quite

diversified. The data collected in Table 4.1 indicate that the pressure sensitivity of

Tg raises beginning with inorganic glasses to typical ionic liquids and finishing with

protic materials.

Herein, it should be stressed that in the literature one can find the correlation

between both coefficients defining the pressure sensitivity of glass-forming liquids.

This relationship was derived by Paluch et al. in the following form:

DV #

ẳ 2:303R mP

dTg =dP



4:5ị



where mP ≡ d log(x)/d(Tg/T)PǀT=Tg, called isobaric fragility, is material-specific

property that describes the deviation of x (τσ, σdc, τα or η) from the Arrhenius

behavior when the Tg is approached. According to the recent paper of

Lunkenheimer et al. [32] among the ionic systems there are materials characterized

by small as well as high value of steepness index mP, i.e., those with weak and

significant deviation of σdc(T) from the linear dependence, respectively. In the first

group, commonly known as strong glass-formers one can find e.g., C10C1Im BF4

(mP = 55) [33] or C1C4Im SCN (mP = 56) [34]. On the other hand, 1,5-Bis

(3-benzyl-2-methylimidazolium)pentane di-bis(trifluoromethanesulfonyl)imide or

1,5-Bis(3-methyl-2-phenylimidazolium)pentane di-bis(trifluoromethanesulfonyl)

imide with mP = 173 and 168, respectively, belong to fragile systems [35]. The

difference between the conductivity behavior of strong and fragile ionic glasses

becomes particularly evident when σdc or τσ are plotted as a function of Tg/T. Such

prepared graph, usually called Tg-normalized Arrhenius plot, is presented in

Fig. 4.12. Although fragility of ionic glass-formers is usually measured at ambient

conditions, the pressure behavior of mP index has been also reported [36]. Similarly

as for van der Waals liquids the high-pressure results available for ionic compounds

reveal a drop in steepness index at elevated pressure.

Coming back to the relation between dTg/dP and DV # expressed by Eq. 4.5 one

have to note that it has been derived for the viscosity data. Since in accordance with

the fractional Walden rule the electric conductivity is controlled by fluidity,



4 High-Pressure Dielectric Spectroscopy …



85



Table 4.1 The values of Tg and dTg/dP determined for various ionic glass-formers



Method

Reference

Material

Tg(K) at

dTg =dP P¼0:1 MPa

0.1 MPa

(K/GPa)

Inorganic salts

215.7

43

DTA

Ca(NO3)2Á4H20

181.2

36

Ca(NO3)2Á8H20

259

44

Mg(OAc)2Á4H20

178

8.3

Li(OAc)Á6H20

174

4.8

Na(OAc)Á10H20

LiClÁR = 10

139

40

LiClÁR = 3

162

56

141

50

AgCl3ÁR = 30

CKN

335

66

Aprotic ionic systems

182*

115

BDS

C4C1ImÁNTf2

185*

116

C8C1ImÁNTf2

BMP-BOB

228

125

215

109

SiMIm BF4

Lidocaine docusate

299*

132

Protic ionic systems

Lidocaine hemisuccinate

274*

127

BDS

Sumatriptan

309.6*

144

hemisuccinate

Procainamide HCl

316

150

Carvedilol HCl

330

152

346

170

Carvedilol H2PO4

Lidocaine HCl anhydrous

311*

170

Verapamil HCl

328

208

Conducting polymers

219

196

N/A

PPG+NaCF3SO3

204

90

N/A

PEG+NaCF3SO3

317

90

BDS

Poly-BuVImÁNTf2

The Tg values marked with (*) were determined by means of DSC technique



[23]

[23]

[23]

[23]

[23]

[24]

[24]

[24]

[38]

[25]

[3]

[26]

[27]

[13]

[13]

[28]

[14]

[8]

[8]

[10]

[29]

[30]

[30]

[31]



σdc Á η−k = const, [37] formally, Eq. 4.5 can be also implemented for studies of

ionic systems. However, in practice three different cases should be considered.

The first one concerns ionic systems for which the exponent k is equal to unity

and it does not depend on thermodynamic variables, T–P. Then by differentiating

the logarithmic form of Walden rule with respect to Tg/T and taking into account

Eq. 4.5 supplemented by definition of mP the straightforward relation is obtained:



86



Z. Wojnarowska and M. Paluch



Fig. 4.12 Tg-normalized

Arrhenius plot of selected

ionic liquids. To avoid the

data extrapolation Tg was

defined as temperature at

which σdc reaches 10−10 S

cm−1. Data were taken from

Ref. [32]



DVr#

DVg#







mrP

ẳk

mgP



4:6ị



According to this formula if k = 1 and k ≠ f(T, P), there is no difference between

the apparent activation volume and isobaric fragility portraying both structural and

ion dynamics. As a consequence, the relation between dTg/dP and DV # parameters

is satisfied. Such scenario occurs for ionic materials for that the charge diffusion

fully mimics to viscosity behavior (pure vehicle mechanism). The second picture

refers to ionic glass-formers characterized by the exponent k lower than one but still

being insensitive to temperature and pressure changes. It means that σdc(T, P) and

η(T, P) are decoupled from each other, however the separation between the time

scale of charge and mass diffusion is constant regardless of T–P conditions. As a

result Eq. 4.5 is valid despite the fact that physical parameters describing

η(T, P) dependence are higher than those determined from conductivity measurements. An example of ionic system that fulfills this rule is [Ca(NO3)2]0.4Á[KNO3]0.6

commonly known as CKN [38]. On the other hand, the third group includes

materials characterized by the fractional and pressure dependent value of the

exponent k. Consequently, σdc(T, P) dependence does not mimic η(T, P) data and

the liquid–glass transition of given ionic material is no longer isochronal, i.e., Tg (or

Pg) = T or P(σdc ≠ const). Since, the isochronal definition of glass transition

temperature/pressure was a fundamental assumption of Eq. 4.5 this formula as well

as Eq. 4.6 are not satisfied for conductivity data. Simultaneously, the conductivity

activation volumeDVr# is considerably lower than DVg# determined at the same

T–P conditions and DVr# =DVg# 6¼ k. The case described herein involves ionic

systems in which the charge transfer is governed by the Grotthuss mechanism (fast

proton hopping). An experimental verification of above statements comes from



4 High-Pressure Dielectric Spectroscopy …



87



high-pressure studies of carvedilol dihydrogen phosphate, for which

DVr# ðTg Þ; DVg# ðTg Þ and k were found to be equal 486, 85 and 0.6, respectively [8].



4.3



Thermal and Density Fluctuations to the Temperature

Dependence of σdc at Ambient and Elevated Pressure



As already mentioned, many of ILs show a remarkable tendency of supercooling

and consequently glass formation. It gives an opportunity to study the temperature

evolution of the ionic dc-conductivity in an extraordinary wide range. On cooling

toward the glass transition, the dc-conductivity decreases from values of order 10−2

[Ω−1 cm−1] that are characteristic for the normal liquid state of typical IL to values

of 10−15 [Ω−1 cm−1] (conventionally taken as a hallmark of the liquid–glass transition), over a temperature range of a few tens of K. This enormous and continuous

drop of σdc is accompanied by the dramatic slowing down of the mobility/diffusion

of ions. It is well known that temperature can control the ions diffusion through two

different mechanisms: changing both kinetic energy (thermal energy) and the

packing density of ions. Consequently, in recent years, some efforts have been

made to clarify the role played by the density and thermal effects [3, 39, 40]. In

order to separate the contributions of these effects it is necessary to measure both

temperature and pressure dependence of ionic conductivity [12].

According to previous discussion, there are a numerous examples that the

temperature dependence of ionic conductivity significantly deviates from the

Arrhenius behavior upon approaching the glass transition. Although all ILs reveal

basically the same pattern of the behavior, the degree of the deviation from the

Arrhenius law is not the same for each material [32]. An intriguing aspect of this

behavior is also understanding whether or not the packing density of ions affect the

deviation from the Arrhenius law? If yes, then to what extent?

There are two simple approaches that are generally used to asses qualitatively the

contributions of thermal and density fluctuations to the temperature dependence of

σdc at ambient pressure. The first one is based on the comparison of isobaric and

isothermal data by plotting σdc against density or alternatively specific volume. An

example of such comparison for ionic liquid [C8MIM][NTf2] is depicted in

Fig. 4.13.

It can be easily noted that the increment of conductivity for the same volume

change is radically different at isobaric and isothermal conditions. The decrease of

the specific volume from 0.76 to 0.705 leads to the decrease of σdc about 9 and 1

decades at constant pressure and temperature, respectively, indicating that the

thermal effect is more dominating in this case. In fact, as will be discussed below,

such behavior seems to be a common feature of all ILs.

A second approach for estimating the relative importance of thermal and density

effects requires comparison between isobaric and isochoric data as shown in

Fig. 4.14.



88



Z. Wojnarowska and M. Paluch



Fig. 4.13 Isothermal and

isobaric data for [C8MIM]

[NTf2] plotted as a function of

volume. Data taken from

Ref. [3]



Fig. 4.14 The temperature

dependence of conductivity

relaxation times at constant

volume condition. The open

symbols represent the

dependence of relaxation

times at ambient pressure.

Data taken from Ref. [36]



In addition to the experimental data denoted by symbols, we also plotted two

isochoric curves (solid lines) corresponding to two limiting cases, i.e., when ion

diffusion is controlled solely by (1) thermal energy fluctuations (red line) or

(2) local density fluctuations (blue line). In the first case, isobaric and isochoric data

should collapse onto a single line and such result would be a rigorous conformation

that the purely thermally activated models are valid. The second limiting case,

manifested by the lack of any temperature dependence of σdc at constant volume,

would mean that free volume models are suitable. There is no doubt that the

experimental data for [C8MIM][NTf2] do not support fully any of these two



4 High-Pressure Dielectric Spectroscopy …



89



discussed limiting cases. However, it can be only concluded that ion mobility is

much more sensitive to changes in the thermal energy than changes in density,

indicating that the energy barriers for ion jumps are only slightly modified by

compression. Moreover it is obvious from Fig. 4.14 that the dependence of log σdc

on 1/T shows a very pronounced curvature at isochoric conditions. Thus, the

non-Arrhenius behavior can occurs even if the ion packing density remains

unchanged.

In the above-mentioned two cases, we needed to know explicitly how the ionic

conductivity depends on volume, σdc(V). Unfortunately, σdc(V) cannot be determined from a single experiment. Instead of this one has to perform two different

types of experiments: the high-pressure dielectric studies to determine σdc(T, P) and

dilatometry measurements of the temperature–pressure dependence of the specific

volume (PVT data). Having both these sets of data, it is possible to analyze the

ionic conductivity at all thermodynamic states. Since the temperature and pressure

ranges covered by these two experiments usually differ from each other, it becomes

necessary to interpolate and/or extrapolate PVT data using an equation of state

(EOS). The most popular one used for this purpose is the empirical equation of Tait.

More recently, a new equation of state has been formulated by Grzybowski et al.

[41] which has been experimentally verified for a wide variety of liquids.



À1=cEOS

cEOS

VðT; PÞ ẳ VT; P0 ị 1 ỵ

P P0 ị

BT P0 Þ



ð4:7Þ



BT ðP0 Þ ¼ BT ðT; P0 Þ ¼ b0 expb2 T T0 ịị



4:8ị



V0 ẳ VT; P0 ị ẳ



k

X



Al T T0 ịl



4:9ị



lẳ0



where b0 ẳ BT0 P0 ị, b2 ¼ b2 ðP0 Þ ¼ À@ ln BT ðT; P0 Þ=@TjT¼T0 , A0 ¼ VðT0 ; P0 Þ,



and Al ¼ 1=l!ị@ l VT; P0 ị=@T l TẳT0 for l = 1, 2,…, and the reference temperature

T0 and pressure P0 are arbitrarily chosen. Its fundamental advantage relies on its

derivation from the effective intermolecular potential [42]

Vrị ẳ



A

B

r 3cEOS



4:10ị



where A and B are constant. Above equation is limited to the density range where

the isothermal bulk modulus BT(P) linearly depends on pressure. However, this

limitation can be removed if the exponent γEOS is replaced by a density dependent

function, γEOS(ρ) as suggested by the General Density Scaling Law [43].

Obviously, a more quantitative assessment of the relative importance of different

thermodynamics variables is needed if we want to catch any differences among

various ILs. It is possible by analyzing the data from Fig. 4.14 in terms of the



90



Z. Wojnarowska and M. Paluch



apparent enthalpy, HP, at constant pressure and the apparent activation energy, EV,

at constant volume [44]:





@ ln r

HP ¼ R

@T 1 P



4:11ị







@ ln r

EV ẳ R

@T 1 V



4:12ị



where R is the gas constant. As discussed by Naoki et al. [45], the relative dominance of thermal and volume effects is reflected in the ratio of EV to HP which can

vary from 0 to 1. Note that values 0 and 1 correspond to two limiting cases (2) and

(1), respectively, from Fig. 4.14. This ratio should be determined at the point where

isobaric and isochoric curves meet together. For [C8MIM][NTf2] at temperatures

close to Tg the value of the ratio EV/HP is nearly constant and equal to 0.79 ± 0.02.

Similar but not the same values were also reported for other ILs. Thus, it is evident

that the temperature is the dominant control parameter in the vicinity of glass

transition temperature for ILs.

It will be value at this stage to review briefly the results of EV/HP obtained for

various types of materials [12, 46]. Literature data for van der Waals liquids shows

that the ratio EV/HP lies usually in the range 0.4–0.6, indicating that the effects of

thermal and volume fluctuations are comparable. This is in contrast to associated

liquids which have values of the ratio EV/HP frequently close to unity. It means that

pure temperature effects on molecular rearrangements are dominant. The reason is

that the associated liquids usually form extensive hydrogen-bonded structures with

strong hydrogen-bonding interactions. And consequently, the thermal fluctuations

provide necessary energy to break H-bonds and to facilitate the molecular diffusion

across the sample. Thus, clear pictures emerge that the magnitude of the ratio EV/HP

reflects the degree and type of intermolecular interactions (van der Waals,

H-bonded, Culombic).

Another interesting aspect worth to be mentioned is pressure dependence of the

ratio EV/Hp. In a few cases it was found the ratio slightly increases with compression indicating the growing role of the thermal fluctuations [3].

For the completeness of the discussion on methods for assessing the relative

contributions of the thermal and volume fluctuation in the temperature dependence

of ionic conduction, we should also mention the method proposed by Ferrer et al.

[47]. This requires calculating the coefficient of isobaric expansivity, aP ẳ

V 1 @V=@T ịP and the coefcient of isochronal (i.e., constant viscosity η or

structural relaxation time, τα) expansivity, ag ¼ V À1 ð@V=@T Þg . The second one can

be also determined at constant value of σdc provided that the fractional DSE law is

satisfied at considered T and P range. As pointed out by Ferrer et al. the ratio of

ag =ap ) 1 means that the ion transport is governed by thermal fluctuations and the

energy barrier for ion hopping does not change with compression. On the other

hand, the ratio of ag =ap approaches to zero when volume fluctuations alone control



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