It is well known that liquids flow readily but at the same time they exhibit elastic response under an instantaneous deformation. Such behaviour can be described by the Maxwell-Debye (MD) viscoelastic model: After a sudden shear, the stress decays exponentially to zero with a relaxation time given by MD = /G(∞), where h is the dynamic viscosity and G(∞) is the instantaneous shear modulus [1]. Thus, on timescales much shorter than MD the system responds elastically - like a solid - while the usual fluid behaviour is retrieved on longer timescales. The MD picture is also useful in describing the transition from a supercooled viscous liquid to a glass. For a glass-forming liquid in the supercooled state the viscosity increases strongly as the temperature decreases. Hence, MD increases several orders of magnitude and finally exceeds 100–1000 s at the glass transition temperature (Tg) resulting in predominantly solid-like response, except on very long timescales.

Fig. 49: Low-temperature dispersion curves. The lines are fits with the model. The inset displays normalised correlation functions (symbols) and fits (lines) at q|| = 7.55 x 10-5 Å-1.

Contrary to the MD picture, a different and intriguing solid-like behaviour was reported in the low-frequency regime for a variety of simple liquids and polymer melts. For instance, an elastic plateau in the storage modulus, the real part of the shear modulus G(), was detected by rheometry [2]. Soft, solid-like behaviour can be modelled by the Voigt-Kelvin (VK) approach where the materials display elastic response to applied stresses at low frequencies, but start flowing at higher frequencies and/or stresses. Such behaviour is often observed in complex fluids, e.g. gels, emulsions, and foams. Typically, these fluids consist of two or more separate phases and hence their structural organisation differs from that of simple liquids and polymer melts.

Fig. 50: The circles show the temperature dependence of the viscosity () and a fit with the VTF model is indicated (solid blue line). The red squares show the elasticity (E) extracted from XPCS and the black crosses indicate E deduced from static scattering data. The dashed red line is a guide for the eye following the VTF behaviour.

To study the response of a supercooled liquid close to Tg, we used grazing incidence X-ray photon correlation spectroscopy (XPCS) [3] from the surface. Liquid surfaces undergo continuous fluctuations due to the presence of thermally-excited capillary waves and their dynamics is governed by the surface tension and by bulk properties like the viscosity and the shear modulus. Grazing incidence XPCS was performed to probe the surface dynamics of the fragile glassformer polypropylene glycol with a molecular weight of 4 kDa (PPG 4000, Tg ~ 200 K). The measured dispersion relations (Figure 49) of the surface fluctuation were extracted by fitting the intensity correlation functions (inset, Figure 49) at various temperatures and momentum transfers (q). Obviously, the dispersion curves are not crossing the origin so they deviate from predictions of both the classical capillary wave theory and the MD model. The data can only be modelled if elements from the VK model are employed accounting for the presence of low-frequency elasticity. A novel model combining MD and VK viscoelasticity was developed and hence the viscosity and elasticity could be extracted as a function of temperature (Figure 50). As expected, the viscosity follows the Vogel-Tamman-Fulcher (VTF) expression and surprisingly the elasticity exhibits similar behaviour (Figure 50). This suggests a connection between low-frequency elasticity and dynamical heterogeneity with density fluctuations in the supercooled state that appear frozen at the time scale of the experiment.


Principal publication and authors

Y. Chushkin, C. Caronna, and A. Madsen, Europhys. Lett. 83, 36001 (2008).


[1] J.D. Ferry, Viscoelastic Properties of Polymers (Wiley & Sons. Inc. 1980).
[2] H. Mendil, P. Baroni and L. Noirez, Eur. Phys. J. E 19, 77 (2006).
[3] G. Grübel, A. Madsen, A. Robert, X-ray Photon Correlation Spectroscopy in Soft Matter Characterization, Eds. R. Borsali and R. Pecora, pp 953-996. Springer (2008).