Non-ergodic parameter

Fig. 4.3 Scaling representation of the spin-echo data at the first static structure factor peak Qmax- Different symbols correspond to different temperatures. Solid line is a KWW description (Eq. 4.8) of the master curve for 1,4-polybutadiene at Qmax=l-48 A L The scale r(T) is taken from a macroscopic viscosity measurement [130]. Inset Temperature dependence of the non-ergodicity parameter/(Q) near the lines through the points correspond to the MCT predictions (Eq. 4.37) (Reprinted with permission from [124]. Copyright 1988 The American Physical Society)...

If the non-ergodicity parameter is fixed to a T-independent valne, an increase of from 0.36 to 0.45 with increasing T in the T-range stndied is reported. 

Fig. 4.35 Right-hand side Monomeric friction coefficients derived from the viscosity measurements on PB [205]. The open and solid symbols denote results obtained from different molecular weights. Solid line is the result of a power-law fit. Dashed line is the Vogel-Fulcher parametrization following [205]. Left hand side Temperature dependence of the non-ergodicity parameter. The three symbols display results from three different independent experimental runs. Solid line is the result of a fit with (Eq. 4.37) (Reprinted with permission from [204]. Copyright 1990 The American Physical Society)...

Crossover Temperature for Various Glass Formers as Reported by the Different Methods From the Temperature Dependence of the Stretching Parameter y(T), Scaling the Time Constant xa — xa(r) [cf. Eq. (42)], Non-ergodicity Parameter 1 —f(T) Obtained from Spectra Analysis, Electron Paramagnetic Resonance (EPR), and from Tests of the Asymptotic Laws of Mode Coupling Theory ... 

Flere (1 - f) and (1 - S) are the losses brought about by the corresponding process and gfastj/3 (t -> oo) = 0 (see Fig. 6). The factor / can be regarded as a generalized non-ergodicity parameter and, hence, it is expected to show a similar anomaly as the Debye-Waller factor /g (see Fig. 5). Such decomposition of the correlation function is useful in spin-lattice relaxation studies, as will be discussed in Section 3.2.4. 

Fig. 5. Non-ergodicity parameter fQ (Debye-Waller factor) of o-terphenyl as obtained from NS experiments for different values of the momentum transfer Q. (Adapted from Ref. 60.)...

In MCT, the non-ergodicity parameter /e(7) is the crucial quantity. It describes the long-time behavior of S(Q,t). In its idealized version, MCT predicts a discontinuous change at a critical temperature T223... 

They also evaluated a fraction of the a-process in 7(Q, t), which corresponds to the non-ergodic parameter/g in MCT, as a function of temperature in the region below and above Tg. According to the MCT/g is given by... 

For this correlator (t) the idealized MCT predicts that it consists of two parts in the / -relaxation regime the time independent non-ergodicity parameter /, and a part for which the wave vector and the time dependence factorize, i.e.,... 

Here,/Qj ax( ) is a generalized Debye-Waller factor giving account for the decay of the correlations at faster times. In the framework of the MCT it is also called non-ergodicity factor. The characteristic timescale r (T) is the structural relaxation time and (3 is the stretching parameter (0[Pg.73]

Whereas the correlation function S(Q, t) decays to zero in the liquid state, this is no longer the case below Tc, where the system becomes non-ergodic. Such scenario results from a generalized oscillator equation with non-linear damping. The slowing down of the molecular dynamics creates an enhanced damping, which in turn slows down the correlation function etc. The density is taken as a control parameter. In Fig. 38a, /(the Q dependence is omitted in the most simple approach) is displayed... 

Entanglement 49,50,57 Entropic forces 24 Entropy, conformational 26 Entropy-driven dynamics 25 Ergodicity parameter, non- 113-115,141... 

From the monotonic increase of the lifetimes with N it follows that in the thermodynamic limit the SK model is non-ergodic (Palmer 1982) below the time average of the order parameter does not agree with the ensemble average q. Monte Carlo simulation yields for the time average... 

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