It is interesting that the EW exponent appears to be not only temperature independent, but also system independent, i.e. γ ≈ 0.2. In order to further check this non-trivial result of the analysis, we focus next on glass-formers for which the β-contribution is minimal or, alternatively, fast and well separated from the α-peak. For these systems the type A characteristics should be less affected by the β-contribution and, within the light of the above results, the unspoiled EW exponent should appear as γ ≈ 0.2 in the temperature range below Tg.
10-3 10-1 101 103 105 107 10-3
10-2 10-1 100 101
2-picoline
125 116
131
107
ν
/ Hz137 139 133
129
97
ε ''
∝ν −0.19
(a)
Fig. IV.22 Dielectric spectra of 2-picoline in the temperature range139
K– 97 K from [26]. The solid line is a power law Aν-0.19.
The glass forming 2-picoline is a system which appears, at first glance, to show no curvature in its spectra close to and below Tg, cf. Fig. IV.22. This is easily proved by the good interpolation of the spectrum at 107 K extending over eight decades in frequency by using a simple power-low Aν-γ. The exponent γ of the power-law is in this case γ = 0.19, close to the one discussed above, and it appears as not changing with temperature below Tg.
Next we present in Fig. IV.23 the results for m-TCP investigated at temperatures close and below Tg. The measurements between 200 K and 160 K and are obtained as part of this work by employing an Alpha Analyzer∗ from Novocontrol [149]. The lower resolution limit of the Alpha spectrometer (almost one decade below the one of the Schlumberger spectrometer, cf. II.2) allows to monitor the evolution of the EW in the glass. In a broad (low) frequency range the spectra below 200 K can be interpolated by power-laws Aν-γ with a temperature independent exponent γ. The value of γ is again 0.2.
This system shows no β-peak in the spectra above Tg (cf. Fig. IV.2), however, a secondary peak appears at much lower temperatures, cf. Fig. IV.28. The maximum of the peak is revealed only if the AH2700 high precision bridge is applied, as shown later.
10-5 10-3 10-1 101 103 105 10-3
10-2 10-1 100
160 170 180
ν
/ Hz m-TCPε ''
247 203 207 217 225 231
200 190
Fig. IV.23 Dielectric spectra of m-TCP as obtained using the Schlumberger spectrometer for 247
K > T > 203 K from [43] and the Alpha spectrometer (this work) for 190 K > T > 160 K . Above Tg, few temperatures (in K) are indicated.
Solid line: a power-law with exponent -0.2.
∝ν−0.2
We finally present the TMP spectra measured below Tg. This system was already introduced in IV.I. For the intermediate temperature range below Tg
down to say 100 K the spectra contains two contributions. As seen in Figure IV.24 (a), one arises from the EW as a power-law Aν-γ exponent γ = 0.19 that does not change from 130 K to 100 K, and the second one from a fast β-process. Below 100 K the β-peak can be analyzed with the distribution Gβ (cf.
II.3.4). The resulting time constants τβ are plotted together with the results for the α-process in Fig. IV.24 (b). The β-process in TMP is fast, well separated
∗This spectrometer was only recently acquired by our group and used in this work only for m-TCP investigations.
from the α-peak (Ea = 15.5*Tg), thus favoring the investigation of the resolved EW.
10-2 10-1 100 101 102 103 104 105 106 10-2
10-1
85 K
100 K 70 K
ν
/ Hzε ''
130 K TMP
∝ ν-0.19
0.6 0.8 1.0 1.2 1.4 1.6 1.8
-10 -8 -6 -4 -2 0
Tg / T
lg(τ / s) α
β
Fig. IV.24 (a) Dielectric spectra of TMP in the temperature range below Tg down to 70 K together with Gβ fits for the β-process. Solid lines are power laws Aν-0.2 (b) Time constants
for the α- and β-processes as function of the reduced temperature Tg/T. The dotted line indicates Tg and the dashed line is an Arrhenius fit.
In order to compare the temperature dependence of the EW amplitude for the systems discussed above, namely 2-picoline, m-TCP and 4-TBP, the prefactor A of the power-law ε’’EW = Aν-γ is plotted in Fig. IV.25 as function of the reduced temperature T/Tg. Note that the parameter A is just the value of ε’’EW at 1 Hz. Here are also included the results for glycerol and m-FAN from Fig. IV.19, obtained within approach II analysis (model dependent).
0.4 0.6 0.8 1.0 1.2 1.4
10-3 10-2 10-1 100 101
glycerol m-FAN mTCP 2-picoline TMP
T / Tg A
∝ 5T/T
g
Fig. IV.25 The prefactor A of the EW power-law Aν-γ as function of the reduced
temperature T/Tg.
For all systems an exponential temperature dependence for A(T/Tg) is observed and, with the exception of m-TCP, the parameter A appears as
identical in this representation. As it will be shown later, this is a consequence of a similar molecular dipole moment of these systems.
As suggested by the dashed line, the slope of lgA(T/Tg) is close to 5 for all systems, thus one may write for the EW below Tg:
ε’’EW(ν,T)∝ν-0.2exp(5T/Tg) (IV.3) Further systems with fast β-process should be investigated to check if they reveal the same behavior as m-TCP or TMP: for temperatures below Tg, the spectra should consist of both the β-contribution and an EW with a power law exponent γ = 0.2 ± 0.01.
IV.4.2. The Nearly Constant Loss
According to previous investigations, type A glass-formers show similar relaxation features in the supercooled regime but also in the intermediate temperature range below Tg [26,75]. Here, the extremely broad spectra can be interpolated, in the first approximation, by a simple power-law, i.e. ε’’(ν) = Aν-γ with an small exponent γ ≈ 0.1 - 0.2, resembling the previously called nearly constant loss (NCL). In the previous investigated temperature range from Tg down to say, 50 – 70 K, exponent γ was found almost material and temperature independent, and the prefactor A revealed a similar exponential temperature dependence, i.e. A ∝ exp(T/TNCL) with TNCL ≅ 34 K for most of type A glass-formers investigated in the kHz regime. This relaxation behavior is revealed not only by dielectric data, but also by NMR and acoustic attenuation measurements [26].
Examples of NCL spectra can be depicted from Fig. IV.13 (Inset) for glycerol, Fig. IV.14 for PC, Fig. IV.22 for 2-picoline and Fig. IV.23 for m-TCP.
According to the discussion above, for 2-picoline and m-TCP this NCL is nothing else than the pure EW with the exponent γ ≈ 0.2. On the other hand, for glycerol or 4-TBP the NCL results from the interplay the overwhelming EW contribution and a weak β-process. For these systems the weak β-contribution may change the apparent exponent α from γ ≈ 0.2, as commonly observed at T ≈ Tg, to lower values (γ ≈ 0.1) in the glass, cf. Fig. IV.26 (a). For m-TCP the
power-law analysis at low temperatures is hampered by the presence of the
Fig. IV.26 (a) Power law exponent γ at temperatures close to and below Tg (indicated by an arrow for every system). (b) γ from (a) vs. reduced temperature T/Tg.
According to approach II, one expects a common exponent for all systems at Tg. In order to demonstrate this, γ is displayed as a function of T/Tg in Fig.
Fig. IV. 27 Temperature dependence of ε’’ at 1 kHz; the dashed
lines corresponds to an exponential dependence ε’’(T) ∝ exp(T/TNCL) with
TNCL= 34.
In type A systems the EW contribution is larger than the one of the β-process at temperatures above, as well below Tg, where the NCL is discussed. This difference in the amplitudes of the two processes can be depicted from the aging analysis for glycerol and 4-TBP (cf. IV.3.1 and Appendix D). As the EW dominates here, the temperature dependence of its amplitude below Tg (see Fig. IV.25), A ∝ exp(5T/Tg) is in agreement with the previous observed
exponential temperature dependence for the NCL. i.e. A ∝ exp(T/TNCL) with TNCL ≅ 34 K. This is justified by the fact that for most of the systems considered here Tg ≈ 5TNCL.
The amplitude of the EW decreases faster than the one of the β-process and the latter dominates the spectra at temperatures far below Tg: for example, the EW amplitude A (ε’’EW at 1 Hz) in the glycerol spectrum at 95 K is below 10-3 (cf. Fig. IV.19 a), while the amplitude of the β-process at 1 Hz is clearly above, cf. Fig. IV.13. If this is true, one should be able to scale the spectra attributed to the thermally activated β-process at such low temperatures, ending the NCL regime. As demonstrated later in V.3.2 this is indeed the case for glycerol.
IV.4.3 The influence of the molecular dipole moment on the