Thermal activated relaxation

A second difference from the continuum model is that large stresses near the reaction center should undergo thermally activated relaxation. According to the molecular mechanism of stress relaxation proposed above, such irreversible, or plastic, deformations occur in UP when the two decyl radicals back away from the reaction center by rotational translation along their long axes. In the process of making more room for the two new C02 molecules, each radical chain is driven into the adjacent interface between two layers of peroxide molecules. Introduction of a defect or a hole at the end of the peroxide chain should facilitate this motion and allow efficient relaxation of the stress. 

In summary, our photophysical studies indicate that the thermally activated relaxation pathways of (2E)Cr(III) very likely involve 2E-to- (intermediate) surface crossing. These (intermediates) can be associated with some, not necessarily the lowest energy, transition state (or transition states) for ground state substitution. The Arrhenius activation barriers for thermally activated relaxation are remarkably similar from complex to complex, but they can be altered in systems with highly strained ligands. Some of this work indicates that the steric and electronic perturbations of the ligands dictate the choice among possible relaxation channels. 

Next, we discuss whether the chromophores spontaneous, i.e., thermally activated, relaxation is primarily influenced by the movement of the polymer backbone (a-relaxation), by that of the chromophore ( relaxation, or by... 

Coffey WT, Crothers DSF, Dormaim JL, Kalmykov YP, Keimedy EC, Wemsdorfer W (1998b) Thermally activated relaxation time of a single domain ferromagnetic particle subjected to a uniform field at an oblique angle to the easy axis Comparison with experimental observations. Phys Rev Lett 80 5655-5658... 

A mobile population of hydrogen has also been observed by anelastic spectroscopy measurements that were carried out by Palombo et al. Heating NaAlIij doped with 2 mol% TiCl3 to 436 K introduces a thermally activated relaxation process with a frequency of 1 kHz at 70 K. This denotes the formation of a point defect with a very high mobility ( 5 x lO jumps/s at 70 K). The relaxation involves the reorientation of H around Ti. 

Higher temperature may result in a weaker Rehbinder effect as well. This occurs due to the facilitation of a plastic flow at elevated temperatures. Thermal fluctuations result in the relaxation of deformational microheterogeneities. As a result, at elevated temperatures local concentrations of stresses are too low to initiate the formation of primary microcracks. An increase in temperature thus often leads to a transition from brittle fracture in the presence of adsorption-active medium to plastic deformation. The decrease in the rate of deformation of a solid has an analogous effect slow deformation also results in an increased probability of the thermally activated relaxation of locally concentrated deformations and stresses. 

Here, oo and Sst are the dielectric constants for cu -> oo and for cu 0 and r is the relaxation time. A quantitative evaluation of the temperature dependence of the relaxation time shows a thermally-activated relaxation probability. The value for its activation energy A is the same as the value of A obtained from the dc conductivity. It follows that the mechanism for the relaxation must be scattering of the charge-density wave from free charge carriers [41]. 

In most thermally activated relaxation and diffusion processes, the evolution progresses with time expressed in units of a characteristic temperature-dependent time that elapsed since the beginning of the process. If we apply this argument to the evolution of the yield stress and assume the temperature dependence to follow an Arrhenius law, we can express the actual annealing time t at temperature T by an effective temperature-compensated annealing time 0... 

The mechanical anisotropy of oriented polymers is determined by the following factors, which will be discussed in turn (1) the structure of the molecular chain and, where the polymer crystallizes, the crystal structure (2) the molecular orientation and, in a crystalline polymer, the morphology (3) thermally activated relaxation processes in both the crystalline and non-crystalline regions. 

Since the subject of SCP is an extremely broad one, it is necessary to limit its scope for the present discussion. First, no further consideration will be given to ordinary dielectric response, although electrical behavior arising from the presence of electric dipoles may often be confused with SCP behavior and vice versa, especially in blocking electrode situations and especially when a distribution of relaxation times is invoked to explain putative dielectric response behavior. Incidentally, it is worth mentioning that the expressions frequently used to describe a distribution of thermally activated relaxation times are inappropriate when applied over a range of temperatures and when the enthalpy involved in the thermally activated dipole behavior is itself distributed. ... 

To compare the heat release with other low-temperature properties and to verify the consistency of the tunneling model we have measured also the acoustical properties (sound velocity and attenuation), the specific heat and the thermal conductivity of PMMA and PS in the temperature ranges 0.070 K < T < 100 K, 0.070 K < F < 0.200 K and 0.3 K < T < 4 K, respectively. We show that the anomalous time dependence of the heat release of PS is due to the thermally activated relaxation of energy states with excitation energies above 15 K. 

In a recently published paper Parshin and Sahling [8] showed the complexity of the interpretation of the heat release data when thermally activated relaxation is taken into account within the framework of the soft potential model. Further theoretical work on the residual properties of two-level systems and its dependence on the cooling procedure has been published by Brey and Prados [9]. 

In this paper we present a further example of the complexity in the interpretation of the heat release data and an experimental proof of the influence of the relaxation, probably by thermally activated processes, of excited states that contribute to the heat release of the TS at low temperatures. We have studied the long-time heat release of two amorphous polymers with similar low-temperature specific heats and thermal conductivities, cooled under similar conditions. In spite of those similarities we have found a large difference in the absolute value and in the time dependence of the heat release between the two polymers when cooled from temperatures above 15 K. The similarities and differences in the low-temperature properties, their interpretation within the tunneling model, and the influence of thermally activated relaxation are the main scope of this work. Preliminary results were published in Ref [10]. 

The influence of a constant and thermally activated relaxation rate... 

There are different attempts to link the high-temperature T> 10 K) with the low-temperature properties of disordered solids [13, 14]. In particular, it has been proposed that the maximum in the attenuation of phonons observed in amorphous materials at T > 10 K can be interpreted assuming a thermally activated relaxation of the two-level systems. In this Section we discuss the influence of thermally activated relaxation rate on the heat release for a finite cooling rate. 

For a single activation barrier Vq between the two potential wells and at a measuring temperature Tq the thermally activated relaxation rate is given by... 

At Ti < 1 K the contribution of a thermally activated rate for TS with potential barriers 130 K < V < 200 K is irrelevant, i.e. using t 10 s (from acoustic measurements, see below) we obtain a zta(V= 130 K fe) of several years. The potential barriers which are relevant in our time and temperature range through a thermally activated rate are V/ks < 10 K, e.g. K(10 s (10 s))/ B 7 (9) K. We note, however, that according to our calculations TS with those potential barriers already relax at the beginning of the measurement through the one-phonon process. Therefore, at Ti < 1 K we do not expect that thermally activated relaxation influences appreciably the heat release. 

Figure 4.12 Squares PMMA internal friction as a function of temperature at a frequency of 535 Hz. Full circles indicate the attenuation data taken from the ultrasonic measurements at 15 MHz from Ref. [24]. Solid and dotted lines the calculated values for v = 535 Hz and v = 15 MHz following the modified tunneling model considering a thermally activated relaxation rate. For more details see text.

We assume the simplified relation for a thermally activated relaxation rate given by Eq. 16 and add it to the tunneling rate (Eq. 17). For two well-defined harmonic potentials and within the approximations given by Eq. 10 and with A = V/Eo, it can be shown that the internal friction increases linear with temperature just above the temperature-independent region plateau) [11] ... 

This increase is observed for PS, (Fig. 4.13) applying Eq. 22 to the results below 20 K we obtain a zero point energy Eq= 13 2 K. In the same temperature region and due to the influence of the thermally activated relaxation the relative change of the sound velocity can be written as [11] ... 

A nearly linear temperature dependence of 0d/d is observed in both polymers at temperatures about below 50 K (Figs. 4.14 and 4.15). At T 1 K a crossover, due to tunneling and due to thermally activated relaxation, from the linear T dependence to the logarithmic... 

T dependence can be observed for both polymers (insets in Figs. 4.14 and 4.15). Although the tunneling model with the assumption of thermally activated relaxation of the TS provides a reasonable fit for the linear temperature dependence, its origin is still controversial. We note that a linear temperature dependence of the sound velocity above a few Kelvin is a rather general behaviour observed in several amorphous [26, 27], disordered [28], and polycrystalline metals [29]. Nava [28] argued recently against an interpretation in terms of thermally activated relaxation of the TS for the linear dependence of the sound velocity. However, new acoustical results in polycrystalline materials indicate a hnear dependence of the sound velocity comparable with those found in amorphous materials [29]. 

From the theoretical results, discussed above, and for a finite cooling rate, thermally activated relaxation (Eq. 16) should influence the absolute value of the heat release, but only slightly its time dependence in our time and temperature ranges. In Figure 4.16 we show the results of the numerical calculations with and without thermally activated processes for a charging temperature Ti = 80 K and using the parameters obtained from the acoustical data (Wmin 10 °). We note that the experimental data lie between the two computed curves (2)... 

At higher temperatures T we observe the expected saturation of the heat release (Fig. 4.18), however, no deviation from the r -dependence has been measured within experimental error. Deviations are observed if we charge the sample at temperatures Ti > 15 K (Fig. 4.16). In this case we have cooled the sample from 80 K and measured the heat release at 0.090 K (sample PSl) and 0.300 K (sample PS2). According to theoretical estimates the observed difference in the heat release between the two PS samples is not attributed to the difference in measuring temperatures. The rather abrupt saturation of the heat release at Ti > 7 K is attributed to the depopulation of the excited states through thermally activated relaxation in the cooling process (Figs. 4.5 and 4.18), as it will become clear below. 

The following experiment provides further verification that the non-simple time dependence of the heat release for PS is due to energy states excited above 15 K and depopulation during the cooling process, very likely through thermally activated relaxation. We have charged the sample at 80 K for several hours and cooled it then to 1 K. We have left the sample for 22 h at 1 K and later continued to 0.3 K, the temperature at which the heat release was measured (Curve (1) in Fig. 4.19). We observe now a clear deviation from the r law. That means that even after 22 h at 1 K the energy states excited above 15 K have not relaxed completely. After 60 h at 0.3 K we warmed the sample to 1 K for 17 h, cooled it to 0.3 K and measured the heat release (Curve (2) in Fig. 4.19). The heat release follows the theoretical r -dependence very well. 

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