Abstract
Experiments have shown that spatial heterogeneities can arise when the glass transition in polymers as well as in a number of low molecular weight compounds is approached by lowering the temperature. This formation of “clusters” has been detected predominantly by small angle light scattering and ultrasmall angle x-ray scattering from the central peak on length scales up to about 200 nm and by mechanical measurements including, in particular, piezorheometry for length scales up to several microns. Here we use a macroscopic two-fluid model to study the formation of clusters observed by the various experimental techniques. As additional macroscopic variables, when compared to simple fluids, we use a transient strain field to incorporate transient positional order, along with the velocity difference and a relaxing concentration field for the two subsystems. We show that an external homogeneous shear, as it is applied in piezorheometry, can lead to the onset of spatial pattern formation. To address the issue of additional spectral weight under the central peak we investigate the coupling to all macroscopic variables. We find that there are additional static as well as dissipative contributions from both, transient positional order, as well as from concentration variations due to cluster formation, and additional reversible couplings from the velocity difference. We also briefly discuss the influence of transient orientational order. Finally, we point out that our description is more general, and could be applied above continuous or almost continuous transitions
Highlights
The appearance of heterogeneities on many length and time scales as the glass transition is approached from higher temperatures has been of interest for a number for years
Using the macroscopic dynamics approach we have studied in this paper a two-fluid model for cluster formation
The experimental results analyzed cover cluster sizes between ∼ 10nm and ∼ 20μm. They have been mainly obtained by optical techniques such as light scattering from the central (Rayleigh) peak and photon correlation spectroscopy as well as ultrasmall angle x-ray scattering for length scales up to ∼ 2000 A
Summary
The appearance of heterogeneities on many length and time scales as the glass transition is approached from higher temperatures has been of interest for a number for years. It is instructive to compare this result with the one previously obtained for the breakdown of flow alignment in nematic liquid crystals (Brand and Pleiner 2021a) In this case one has the same macroscopic variables except for the strain Uyz. Inspecting Eqs. As already discussed in the introduction, extensive light and ultrasmall angle x-ray scattering studies have shown that the spectral weight under the central peak is larger than expected on the basis of the hydrodynamics of simple fluids (Fischer 1993; Kanaya et al 1994; 1995; Walkenhorst et al 1998; Patkowski et al 2000; Patkowski et al 2001a; Patkowski et al 2001b; Fischer et al 2002) These experiments stimulated work on the macroscopic dynamics of cluster formation above the glass transition (Brand and Kawasaki 2003). In closing this section we point out that there is another class of systems for which transient positional order as well as transient orientational order play an important role for the understanding of the macroscopic properties, namely the sponge or L3 phase in lyotropic liquid crystals close to the phase transition to the isotropic liquid phase (Pleiner and Brand 1991; Brand and Pleiner 2002)
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