Scaling behaviour of relaxations

Relaxation behaviour is associated with two generic features - the scaling and the stretching phenomena. Scaling implies that the reduced relaxation properties, (M/M(max)), ( / (max)), etc. measured as a 

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 

Thus the coupling model is characterized by two constants, n = which broadens the relaxation spectrum and r, the primitive relaxation time, which escapes the influence of coupling. Both n, which represents the coupling strength and Tg, are temperature dependent. 

In most liquids and to a good approximation, y0= (1-n). Therefore n can be evaluated by evaluating p. P itself is given by the ratio WJW, where and W are the Debye and actual widths of the relaxation spectrum. Even when loss curves (dielectric, mechanical or any other) are fitted to other standard analytical functions such as Cole-Cole (CC), Davidson-Cole (DC) or Haveriliac-Negami expressions, (see earlier section) one can determine p using the empirical relations 

P is also found to be sensitive to pressure and it decreases linearly so that p = Pq- yP, where p = P at zero external pressure and y is a constant of the order of unity, when pressure is expressed in GPa. This dependence arises from the fact that Tg and T are all pressure dependent. generally increases by about lOK per kbar. Thus, the stretched exponential function and the linear variation of the KWW exponent provides yet another approach to understand the empirical VTF law. Rault (2000) has proposed that below T, where the a- relaxation is characterized by a continuously increasing effective activation barrier, r can be written as r exp EpipRT). That is the activation barrier for the cooperative motion is effectively increased to Epip. Referring to the VTF 

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