Nonergodicity Parameters

Dependence of the Nonergodicity Parameter in Polybutadiene in the Neighborhood of the Glass Transition. 

Detailed high-frequency (terahertz) dynamical studies of glasses have been probed by inelastic X-ray scattering (IXS) [139], The advantage of this technique is that with reliable measurements it allows determination of the so-called nonergodicity parameter f(q, T) as a function of wavevector q this quantity is defined by the long time limit of the density-density correlation function F(q, t) divided by the static structure factor [15],... 

The quantity e w is the Debye-Waller factor, while Sis(q) is the structure factor due to packing statistics or the inherent structure. The contributions from the anharmonic vibrational degrees of freedom about the inherent structure are denoted by Finei(q,t). In the longtime limit, the nonergodicity parameter becomes [139]... 

In MCT, the nonergodicity parameter fq(T) is a crucial quantity. It describes the long-time limit of the density correlation function or the so-called intermediate scattering function, in the theory denoted (q, t). In its idealized version, MCT predicts a discontinuous change at a critical temperature Tc. 

Close to the crossover temperature Tc, the solution of the idealized MCT equations can be expanded around the nonergodicity parameter fc at Tc and some generic laws can be derived. These leading-order results establish the so-called asymptotic laws of MCT, which can be regarded as generic features of the MCT dynamics at T > Tc. Because almost all experimental tests analyzed the susceptibility spectra in this frame, we are going to briefly summarize the laws [16,17,19]. Examples of tests against experiment are found in Section IV.E. 

The inflection point/of the two-step correlation functions observed in Fig. 13 may be taken as a measure of what is called the nonergodicity parameter within MCT. It quantifies the relaxation strength of the slow degrees of freedom, that is, the a-process. Correspondingly, the quantity 1 / specifies the strength of the... 

Figure 34. Schematic three-step correlation function of type B glass formers close to Tg. Here/ is the nonergodicity parameter, and S is related to the (i-relaxation strength [cf. Eq. (46)]. (Adapted from Ref. 65.)...

The nonergodicity parameter / denoting the relative relaxation strength of the slow relaxation may be introduced phenomenologically to describe a two-step... 

Figure 42. Nonergodicity parameter, (effective Debye-Waller factor) of o-terphenyl as obtained from neutron scattering experiments for different values of the momentum transfer Q. (a) Incoherent, (b) coherent full curves below Tc represent fits to the square root law of MCT yielding Tc = 290 K (From Ref. 201.)...

Figure 43. (a) Nonergodicity parameter 1 /Tcl for toluene, 2-picoline, propylene carbonate... 

Figure 44. Correlation function as obtained from Fourier transformation of some of the dielectric spectra displayed in Fig. 12a [9]. Only the initial part is shown. Dashed lines represent extrapolation to short times providing an estimate of the nonergodicity parameter / at T = 363 K and 295 K, extrapolations were done via a fit by a Kohlrausch decay with = 0.73. Dotted line represents data at 363 K scaled to coincide with data at T = 295 K demonstrating similar stretching.

Given that the asymptotic laws may fail, the only quantity expected to signal the crossover forecast by MCT is the anomaly of the nonergodicity parameter/. 

Concerning the slow dynamics below the crossover temperature Tc, the predictive power of the theory seems to be rather limited. In particular, the emergence of intrinsic slow secondary processes, which seems to be associated with the dynamic crossover in the experimental spectra, is not contained even in the extended versions of the theory consequently, the slow dynamics spectrum is not reproduced correctly. In this respect, the extended theory introducing the hopping mechanism for describing the susceptibility minimum below Tc is misleading. On the other hand, the most prominent prediction of MCT below Tc is the anomaly of the nonergodicity parameter, which, as discussed, is found by different model-independent approaches. However, within the framework of MCT, this anomaly is closely connected with the appearance of a so-called knee feature in the spectral shape of the fast dynamics spectrum below Tc. This feature, however, has not been identified experimentally in molecular liquids, and only indications for its existence are observed in colloidal systems [19]. In molecular systems, merely a more or less smooth crossover to a white noise spectrum has been reported in some cases [183,231,401]. Thus, it may be possible that the knee phenomenon is also smeared out. 

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