3 Lithium-ion transport in LLTO perovskites 84
4.8 Dynamic crossover of relaxation times in glass-forming materials
4.8.2 Does a “magic” relaxation time at crossover temperature
glass-forming liquids?
Recently [231], Novikov and Sokolov have analyzed the structural
-relaxation time at crossover temperature Tc. They have found that in most glass-formers studied, log(Tc) varies in a narrow range. Strengthened by the analysis of the data presented by Beiner and coworkers [313], they have introduced the concept of the universal “magic” relaxation time at crossover temperature, i.e. ~s. In order to shed the light on this universality, in the next paragraphs, we will investigate the fragility index dependence of the
-relaxation time at crossover temperature, in the light of the BSCNF model.
An interesting paper is that for Fragiadakis and coworkers [341] who investigated segmental and local dynamics of lithium-ion transport in poly(ethylene oxide)-based single-ion conductor PE600-x%Li, x being the lithium content. They observed three different types of -relaxation processes. The high frequency process, namely -process, corresponds to the usual primary relaxation, whereas 2-process (ion mode) occurs in the ionomers at frequencies nearly two orders of magnitude lower than that of the -process. The relaxation strength of 2-process increases roughly proportionally to lithium content, in contrary to that of -process which does not change significantly. For neutral
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polyester as well as for 6% and 11% lithium content, a weak third process, namely
3-process, at an even lower frequency than the 2-process has been observed.
The origin of 3-process still remains unclear insofar as it is not present in all glass-forming materials.
Recently [292], we have carried out a comparative study between the Bond-Strength-Coordination Number Fluctuation (BSCNF) model and the Random Walk (RW) model of viscosity. We have found that both models show excellent agreement with experiments. However, for the RW model, two equations corresponding to two temperatures regimes (low-T and high-T) separated by the crossover temperature Tc are needed to describe the temperature dependence of the viscosity of a “fragile” system. At the same time, for the BSCNF model, a single equation with clear physical meaning describes the temperature dependence of the viscosity of both the “fragile” and “strong” systems.
In the light of the BSCNF model, we have proposed the following equation [291,292]
ln( )
)
ln( C B
B B m
T T
g
, (4.34)
where
RTg
Z
C E and Tg
R Z B ( E) ( )
. (4.35)
T0 is the ideal glass transition temperature, usually considered similar to the Kauzmann temperature [305]. The BSCNF model characteristic parameters B* and C* are interrelated by the following equation
ln( / ) ln( )
) (
)
( B
B
C B Tg
,
Z Z
E E
/
/ . (4.36)
The parameters have usual meanings (see for instance sections 4.3.5 and 4.6). m is
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the fragility index which has been derived as [291,226,342]
) (
) ln(
ln )
ln( B
B C
B m
Tg
. (4.37)
In Fig. 4.10, we have shown the composition dependence of the -relaxation time at crossover temperature Tc, namely log(Tc). It is noteworthy to mention that as far as we are informed, the values of Tc for the materials in consideration here are not available in the literature. Nevertheless, recently [291], we have collected some experimental evidences leading to a correlation between the crossover temperature and the glass transition temperature. For most polymeric materials, such a relationship is approximated, within experimental errors, as [343]
Tc (1.200.05)Tg. (4.38) Eq. (4.38) has been used to evaluate the values of Tc for poly(ethylene oxide)-based single-ion conductor PE600-x%Li (Fig. 4.10), for neat and plasticized ionomers, and for silicate and borate glasses (Table 4.1).
The results shown in Fig. 4.10 clearly indicate that log(Tc) varies with lithium content, between 2.7 and 6.2 ((Tc) is expressed in seconds).
Experimental data are taken from Ref. [341]. The apparent decrease of the values of )log(Tc with the increase of lithium content is due to the fact that the strength parameter D increases with the increase of lithium content, as reported recently [341]. If we refer to Eq. (4.32), this situation leads to the increase of the values of the -relaxation time at Tc, i.e. (Tc). Therefore, the material becomes less “fragile”.
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Ph.D. Thesis – Physics 7 6 5 4 3 2 1 0 -log (Tc)(s)
100 80
60 40 20 0
Li+content x(%)
-process
-process
PE600-x%Li
At Tc, by using Eqs. (4.32) and (4.34), we obtain
ln( ) ln( ) ln( )
) ln(
) ln(
exp )
(
B B C
m B T m
T
B B C
m B D T
g c
c
. (4.39)
Insofar as the values of the fragility index for the materials in consideration here are not available in the literature, in order to cope with this problem, we have used the following VFT equation
0 0exp 0
T T
DT
, (4.40)
where the symbols have usual meanings. Eqs. (4.24) and (4.40) lead to Figure 4.10. -relaxation time at crossover temperature as a function of the lithium content.
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2
0 0
1 ) 10 ln(
1
g g
T T T DT
m . (4.41)
The behavior expressed by Eq. (4.39) is shown in Fig. 4.11. We can observe that by increasing the fragility index, the quantity log(Tc) increases.
Let’s note here that for the theoretical results obtained from the BSCNF model, we have used the typical values of the strength parameter D and the frequency at high temperature limit f0 obtained after a careful analysis of the data presented by Fragiadakis et al. [341]. The values of the fragility index have been calculated from Eq. (4.41). These values of the fragility index enabled us to evaluate the characteristic parameters B and C for each material sample, based on Eqs. (4.36) and (4.37). To improve the reliability and increase our
Figure 4.11. -relaxation time at crossover temperature as a function of the fragility index.
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understanding of Fig. 4.11, we have considered the values of the characteristic parameters B and C by taking into account the scatter range of the values obtained for each material sample. As it can be observed in Fig. 4.11, our model describes quite well the experimental data [341].
In addition to the results for poly(ethylene oxide)-based lithium-ion conductors primarily discussed in this section, we have collected some experimental data obtained by fitting VFT equation. Then, we have calculated the values of log(Tc) for each material sample, thanks to Eq. (4.32). It should be noted however that the values of 0 as indicated in Table 4.1 are not available in the literature. For neat and plasticized ionomers, Klein and coworkers [344]
argued that at a temperature Tref, referred here as a more accurate indicator of the glass transition temperature, the VFT equation (Eq. (4.32)) takes the value of 1 s.
We will also use this tool to determine the values of 0 for glasses collected in Ref. [345]. Some authors consider, without altering the meaning of 0, that at glass transition temperature, the -relaxation time takes the value of 100 s [27,30,32,308]. All other fitting parameters of Eq. (4.32) are taken from Ref.
[344] for neat and plasticized ionomers, and from Ref. [345] for silicate and borate glasses. For the sake of brevity, only some typical values for the materials studied by Nascimento et al. [345] are shown in Table 4.1.
At this point, let’s pause to reflect upon the implications of the behavior displayed in Fig. 4.11, as well as the data summarized in Table 4.1 for
) ( log Tc
. First and foremost, it is of interest to mention that Casalini and coworkers [229,230] have carried out dielectric spectroscopy measurements on four glass-forming liquids having simple molecular structures. They found that for phenolphthalein-dimethyl-ether (PDE), the relaxation time at Tc is s, while for cresolphthalein-dimethyl-ether (KDE) no crossover within the frequency range of measurements has been observed. However, for the latter (KDE), it has been reported that the characteristic relaxation time at which change of dynamics occurs at ambient pressure is ~s [346]. For polychlorinated biphenyls having 42% and 62% by weight of chlorine (PCB42 and PCB62, respectively),
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the corresponding relaxation times at Tc have been found to coincide at the same value of s [229].
The values of log(Tc) reported in Table 4.1 spread between 1.2 and 5.1, with )(Tc expressed in seconds. It is remarkable to observe that log(Tc) increases with the fragility index (not shown, see Refs. [344] and [345]), as for PE600-x%Li. In this sense, the lowest value of log(Tc) is for the “strong”
silicate glass-forming liquid SiO2 [24,345] while the highest value corresponds to the plasticized form of “fragile” triethylamine [344].
Sample Tc(K) 0(s) log[(Tc)(s)]
Neat and Plasticized Ionomers
Neat 308.4±12.8(*) 1.058×10-8 3.852 +Dimethyl sulfoxide 265.2±11.0(*) 1.671×10-8 3.941 + N,N-dimethylformamide 302.4±12.6(*) 1.020×10-8 4.011 + Dioctyl phthalate 297.6±12.4(*) 6.778×10-9 4.423 + Propylene carbonate 273.6±11.4(*) 8.266×10-9 4.516 + Triethylamine 271.2±11.3(*) 9.599×10-9 5.089 + Ethylene glycol 255.6±10.6(*) 6.843×10-9 3.756
Glasses
SiO2 1740.0(*) 9.103×10-8 1.275 Li2O · SiO2 711.6(*) 1.350×10-8 2.142 Li2O · 2SiO2 872.4(*) 6.323×10-8 2.125 Li2O · 3SiO2 880.8(*) 3.539×10-8 1.747 B2O3 648.0(*) 8.319×10-8 1.930 Li2O · B2O3 831.6(*) 4.423×10-8 3.519 Li2O · 2B2O3 915.6(*) 4.196×10-8 3.757 Li2O · 3B2O3 921.6(*) 4.743×10-8 3.487
(*) Calculated from Eq. (4.38)
Table 4.1. -relaxation times at crossover temperature for some glass-forming materials.
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From our point of view, we believe that the behavior displayed in Fig. 4.11 and the data summarized in Table 4.1 for log(Tc) can be understood as follows. It is well known elsewhere in the literature that the fragility index derives from an energy landscape interpretation of the dynamics near the glass transition temperature [27,31,225]. By increasing the fragility index, the glass-formers become more “fragile” and exhibit a rapid change in liquid structure (local packing and positioning) with changing temperature. Therefore, the relaxation time decreases. This behavior is more marked near Tg which is related to the crossover frequency Tc by Eq. (4.38).
On the other hand, as mentioned above, by increasing lithium content, the fragility index decreases and the corresponding strength parameter D increases.
In this course, log(Tc) would decrease if we refer to Eq. (4.32). It is instructive to mention that a decrease in fragility index with the increase in lithium content has also been observed for other PEO-based electrolytes [347].
The origin of this behavior has been attributed to the lithium-ions which increase the glass transition temperature by acting as intra-chain, rather than inter-chain, leading to the increase of the rigidity of polymer chains.
The results obtained and the explanations given above allow us to assert that the -relaxation time at crossover temperature Tc depends not only on the type of glass-forming materials, but also on their compositions. Therefore the conclusions given by Novikov and Sokolov [231] according to the so-called universal “magic” relaxation time at Tc turn out to be inconsistent and are probably based on a combination of accidental circumstances [348].