Sound Dispersion and Structural Relaxation
The above expression for the sound velocity contains the bulk modulus K. If K is frequency independent then the liquid behaves as a linear medium, so that sound propagates without dissipation and without mode coupling. In reality, any liquid shows some dispersion: with increasing frequency, K crosses over from the low-frequency, liquid-like limit to the high-frequency, solid-like limit . In normal liquids, most of this cross over takes place at frequencies between GHz and THz, sometimes called hypersound.
At sub-GHz frequencies, a normal liquid cannot sustain shear waves: the zero-frequency limit of the shear modulus is . This is sometimes seen as the defining property of a liquid. However, just as the bulk modulus K, the shear modulus G is frequency dependent, and at hypersound frequencies it shows a similar cross over from the liquid-like limit to a solid-like, non-zero limit .
According to the Kramers-Kronig relation, the dispersion in the sound velocity (given by the real part of K or G) goes along with a maximum in the sound attenuation (dissipation, given by the imaginary part of K or G). According to linear response theory, the Fourier transform of K or G describes how the system returns to equilibrium after an external perturbation; for this reason, the dispersion step in the GHz..THz region is also called structural relaxation. According the fluctuation-dissipation theorem, relaxation towards equilibrium is intimately connected to fluctuations in equilibrium. The density fluctuations associated with sound waves can be experimentally observed by Brillouin scattering.
On supercooling a liquid towards the glass transition, the crossover from liquid-like to solid-like response moves from GHz to MHz, kHz, Hz, ...; equivalently, the characteristic time of structural relaxation increases from ns to μs, ms, s, ... This is the microscopic explanation for the above mentioned viscoelastic behaviour of glass-forming liquids.
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