The use of horizontal shift factors fo determining the activation energy of a relaxation process (e.g. creep or stress relaxation) is well known from the literature. Usually, if possible, only horizontal shifts are used and the ln a(T) vs. 1/T plots are used to estimate the activation energy. It is usually added that sometimes the temperature dependence of the limiting moduli (or compliances) have to be taken into account too.
It is also well known that by plotting the DMA loss maximum temperature against the logarithm of the measuring frequency also gives an estimate for the activation energy. As the time and frequency dependent measurements are inter-convertible and they describe the same relaxation processes the activation energies obtained by the two methods should be indentical (or at least close to each other). To my surprise, in a concrete case (I cannot share the details) there was a cpnsiderable difference between them. I wonder, whether other colleagues met similar problems.
I suspect that the temperature dependence of the limiting moduli is one possible reason, as in some cases a longer horizontal shift may compensate for the vertical shift and we get a larger shift (and lower activation energy) as compared to the G" maximum temperature vs. 1/T plot. I would appreciate comments.
It is widely accepted that the magnitude of the vertical shift factor can be calculated by multiplying the horizontal shift factor by a constant. However, the magnitude of the vertical shift factor can be calculated using another method. The other method defines the direct energy gap as the energy difference between states in which electrons are filled and empty.
The problem is solved. I developed a relatively simple method, which works if the mechanical relaxation process can be well described by a single Havriliak-Negami (H-N) equation. Using 3 frequencies (covering 2 decades) I measured the G'(T) and G"(T) functions, plotted the Cole-Cole diagram, determined the G(U) /unrelaxed/ and G(R) /relaxed/ moduli. In the next step I calculated a normalized set of functions (G'-G(U))/(G(R)-G(U)) for the real part and (G"/(G(U)-G(R)) for the imaginary part (this depends on the H-N shape factors only), created the Cole-Cole representation of these normalized functions and determined the alpha and beta shape parameters from the slopes at the rubbery and glassy states. Finally, using the alpha and beta parameters I calculated analytically the (G'-G(U))/(G(R)-G(U)) function (which is varied between 0 and 1) and determined the omega*tau values where this function takes 0.1; 0.2 etc. values. Going back to the experimentally found (G'-G(U))/(G(R)-G(U)) functions I determined the temperatures where these 0.1; 0.2 etc. values are reached at various frequencies and plotted the ln(omega*tau) values vs 1/T. It gave the Arrhenius diagram in a fairly broad temperature range (much more data points than the 3 G"(T) maxima at the three frequencies, which allowed to see the temperature dependence of the activation energy, which explained the mismatch. Moreover, much less time is needed for the tests.
@Gyorgy Banhegyi Thanks for the answers. In this sentence that you posted:
"In the next step I calculated a normalized set of functions (G'-G(U))/(G(R)-G(U)) for the real part and (G"/(G(U)-G(R)) for the imaginary part (this depends on the H-N shape factors only), created the Cole-Cole representation of these normalized functions and determined the alpha and beta shape parameters from the slopes at the rubbery and glassy states";
can you confirm the accuracy of this equation: (G'-G(U))/(G(R)-G(U)) and (G"/(G(U)-G(R))? Thanks.
Yes, these are the reduced variables, which depend only on the shape factors. In the Cole-Cole presentation even the absolute value of the relaxation time is not important.
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