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 =dPP¼0:1 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#
ẳ
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