Primitive relaxation function

Different relaxation functions are derived assuming that the real (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level the relaxation should be of classical, Debye-like type, whatever the pattern of nonclassical relaxation at longer times. The analysis of diffusion for a Brownian particle, where the assumption that the... 

Figure 22. Dielectric loss spectrum of m-FA at 279 K and 1.69 GPa ( ), 1.60 GPa ( ), 1.52 GPa data ( ), and 1.4 GPa data (A). Dielectric loss spectrum of mFA at ambient pressure and 174 K ( ), 177 K data (o), and 180 K (A). The dashed lines are fits to the data at 279 K and under GPa pressures by the one-sided Fourier transform of the KWW function. The solid fines are similar fits to the ambient pressure data. The vertical arrows indicate the calculated primitive relaxation frequencies, Vo, for all the data sets.

The evolution of the many-molecule dynamics, with more and more units participating in the motion with increasing time, is mirrored directly in colloidal suspensions of particles using confocal microscopy [213]. The correlation function of the dynamically heterogeneous a-relaxation is stretched over more decades of time than the linear exponential Debye relaxation function as a consequence of the intermolecularly cooperative dynamics. Other multidimensional NMR experiments [226] have shown that molecular reorientation in the heterogeneous a-relaxation occurs by relatively small jump angles, conceptually simlar to the primitive relaxation or as found experimentally for the JG relaxation [227]. 

In the previous subsection, we have provided conceptually the rationale and experimentally some data to justify the expectation that the primitive relaxation time To of the CM should correspond to the characteristic relaxation time of the Johari-Go Id stein (JG) secondary relaxation Xjg- Furthermore, it is clear from the CM relation, Ta = ( "to)1 1- , given before by Eq. 6 that To mimics Ta in behavior or vice versa. Thus, the same is expected to hold between Xjg and Ta. This expectation is confirmed in Section V from the properties of tjg- The JG relaxation exists in many glass-formers and hence there are plenty of experimental data to test the prediction, xjG T,P) xo(T,P). Broadband dielectric relaxation data collected over many decades of frequencies are best for carrying out the test. The fit of the a-loss peak by the one-sided Fourier transform of a Kohlrausch function [Eq. (1)] determines n and Ta, and together with tc 2 ps, To is calculated from Eq. 6... 

Since fV(T,t) is equal to fVg(T) for ( <1, the relaxation function will be a simple exponential function for this time regime. For the region co t >1, it is more complex. It has been shown by Ngai and co-workers (Nagai, 1979 Nagai et al., 1984) that KWW stretched exponential function, exp[-(t/r) ], with j3= (1 - n), is a satisfactory solution to the above equation. When coJWg 1, the effective relaxation time r and the primitive relaxation time Vg = 1/Wg are related as... 

In order to check the validity of CM predictions, the frequency of primitive relaxation Vo=(27iro)" was calculated according to Eq. (1), where a-loss peak frequency v was directly determined from the maximum, and n from the KWW function lit. The arrow in Fig. 1 indicates the location of the calculated frequency Vq of the primitive process, which is in good correspondence with the loss maximum frequency of the /0-process, vp. The peak of the JG /0-relaxation is well separated from the a-processes at the selected temperature, so allowing a reliable evaluation without any deconvolution procedure. The same procedure... 

This equation has been used as the basis to explain the T-P superposition of the O -relaxation of a component in mixtures of van der Waals glass-formers and polymer blends as discussed in Capaccioli and Ngai (2005). Concentration fluctuations in the mixture or blend create a distribution of environments /. Each environment, i, has its own coupling parameter, tii, primitive relaxation time, Toi, and the corresponding Kohlrausch function with stretch exponent, (1 — ,), which determines the relaxation time t, by the CM equation rai =. In the same manner as shown earlier for neat glass-... 

Importantly for the structural recovery of glasses, the model predicts an equilibrium decay function which is of KWW form (jS= 1 —n), see equation (81), for even a single primitive species. Thus the requirement of a non-exponential decay function is fulfilled by the model. Although the other models use a broad relaxation function to describe behavior, they neither make the prediction of equation (91), nor can the general equation (89) result from them. In general, n and t can be functions of Tand d however, we treat only the case where t is a function of 8, i.e. t = T (r, 5(t )). Then, rewriting equation (86) in terms of t and identifying the macroscopic variable 0 with the departure from equilibrium 5, we find that, for isothermal volume recovery, equation (89) becomes... 

Henceforth we take the primitive path co-ordinate s=L-z from the free end inwards to the branch point so that t(s) is an increasing function of s. The prefactor Tq is an inverse attempt frequency for explorations of the potential by the free end, and may be expected to scale as the Rouse time for the star arm (in fact this is not quite true - the actual scaling is as [25,26]). The relaxation mod-... 

Since To or Xjg is usually much larger than tc =2 ps and n/( 1 — n) is a monotonic decreasing function with decreasing n, xa(7) of nano-confined liquids decreases on decreasing the size of the pores. Consequently, the difference between xa and To or x/c becomes smaller [298,302,303]. This trend is shown in Fig. 48 by the dielectric relaxation data of PDMS confined in silanized glass pores of various sizes. If in sufficiently small pores n —> 0, then xa —> To or Tjc, and the characteristics of the a-relaxation will not be very different from that of the JG relaxation. The location of the primitive frequency Vq corresponding to tq calculated from the bulk xa and n = 0.48... 

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