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5 Temperature dependence of the parameters

[page 1287, §1]    [1287.1.1] Because the data have been fitted at different temperatures we are able to observe the temperature dependence of the fitting parameters. [1287.1.2] For τ1 and τ2 we perform Vogel-Tammann-Fulcher fits provided by equation (13). [1287.1.3] From the fits we obtain the Vogel-Fulcher temperatures TVF1 and TVF2 as well as the fragility parameters D1 and D2 for the relaxation times τ1 and τ2 for model A and model B (see Table 2).

material model TVF1 TVF2 D1 D2 τ01 τ02
5-methyl-2-hexanol A 89.2⁢K 92.1⁢K 25.5 22.1 3.53×10-13⁢s 7×10-14⁢s
5-methyl-2-hexanol B 88.3⁢K 101.6⁢K 26.3 15.3 5.21×10-13⁢s 5.77×10-12⁢s
methyl-m-toluate A 71.2⁢K 71.2⁢K 93.2 93.2 4.35×10-28⁢s 1.54×10-28⁢s
methyl-m-toluate B 67.2⁢K 85.6⁢K 102.2 53.6 9.0×10-28⁢s 1.13×10-22⁢s
glycerol A 127.8⁢K 131.5⁢K 17.1 14.9 3.7×10-14⁢s 3.23×10-14⁢s
glycerol B 152.7⁢K 137.8⁢K 6.68 11.9 2.9×10-10⁢s 2.15×10-13⁢s
Table 2: List of the fit parameters TVF, D and τ0 for various materials.

[1287.2.1] For all fits we see a temperature dependence of the relaxation times τ1 and τ2 (Fig. 4 - Fig. 6) that follows the Vogel-Tammann-Fulcher fitting function remarkably well. [1287.2.2] The relaxation times also show a clear downward trend as the temperature increases, which confirms that τ, τ1 and τ2 are physically meaningful and can be interpreted as relaxation times even tough they appear with a non-integer power in equations (11) and (12).

[1287.3.1] The parameters α, α1 and α2 also show a temperature dependence. [1287.3.2] In the case of 5-methyl-2-hexanol (Fig. 4) there is an increase of α with temperature until a plateau near α=1 is reached. [1287.3.3] This effect comes from the decreasing slope of the excess wing with increasing temperature. [1287.3.4] In the fitting function of model A this behavior can be achieved by increasing α. [1287.3.5] For the same material there is an apparent increase of α2 between 154⁢K (6.49⁢K-1) and 287⁢K (3.48⁢K-1) which has the same origin as the increase in α in model A. [1287.3.6] By increasing α2 the excess wing becomes less steep. [1287.3.7] The plateau at 190⁢K (5.26⁢K-1) and above comes from the fact that the fits at those temperatures are done mainly for the α-peak, because the excess wing is not visible.

[1287.4.1] For methyl-m-toluate and glycerol there is also a clear temperature dependence of α, α1 and α2 (Fig. 5 and Fig. 6). [1287.4.2] The trend is however reversed in comparison to 5-methyl-2-hexanol. [1287.4.3] This comes from the increasing slope of the excess wing with increasing temperature. [1287.4.4] This behavior can be achieved in the fit functions by decreasing α, respectively α2.

Figure 4: Temperature dependence of the fitting parameters τ1 (crosses), τ2 (stars) and α for 5-methyl-2-hexanol for model A (left two panels). Temperature dependence of the fitting parameters τ1 (crosses), τ2 (stars), α1 (crosses) and α2 (stars) for model B (right two panels). The solid lines are Vogel-Tammann-Fulcher fits (see eq. (13)) whose parameters are listed in Table 2.
Figure 5: Temperature dependence of the fitting parameters τ1 (crosses), τ2 (stars) and α for methyl-m-toluate for model A (left two panels). Temperature dependence of the fitting parameters τ1 (crosses), τ2 (stars), α1 (crosses) and α2 (stars) for model B (right two panels). The solid lines are Vogel-Tammann-Fulcher fits (see eq. (13)) whose parameters are listed in Table 2.
Figure 6: Temperature dependence of the fitting parameters τ1 (crosses), τ2 (stars) and α for glycerol for model A (left two panels). Temperature dependence of the fitting parameters τ1 (crosses), τ2 (stars), α1 (crosses) and α2 (stars) for model B (right two panels). The solid lines are Vogel-Tammann-Fulcher fits (see eq. (13)) whose parameters are listed in Table 2.