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.
Read more about this topic: Liquid, Microscopic Properties
Famous quotes containing the words sound, dispersion, structural and/or relaxation:
“Is there no hope for me? Is there no way
That I may sight and check that speeding bark
Which out of sight and sound is passing, passing?”
—Paul Laurence Dunbar (1872–1906)
“The slogan offers a counterweight to the general dispersion of thought by holding it fast to a single, utterly succinct and unforgettable expression, one which usually inspires men to immediate action. It abolishes reflection: the slogan does not argue, it asserts and commands.”
—Johan Huizinga (1872–1945)
“The reader uses his eyes as well as or instead of his ears and is in every way encouraged to take a more abstract view of the language he sees. The written or printed sentence lends itself to structural analysis as the spoken does not because the reader’s eye can play back and forth over the words, giving him time to divide the sentence into visually appreciated parts and to reflect on the grammatical function.”
—J. David Bolter (b. 1951)
“Worst of all, there is no sign of any relaxation of antisemitism. Logically it has nothing to do with Fascism. But the human race is imitative rather than logical; and as Fascism spreads antisemitism spreads.”
—George Bernard Shaw (1856–1950)