Critical acoustic attenuation and the one-scale relaxation of configurational stresses

We develop a self-consistent mesoscopic theory of the frequency-dependent bulk viscosity and acoustic attenuation of a compressible soft-matter mixture near a liquid-liquid demixing critical point. The theory starts from a scalar Ginzburg-Landau free-energy functional, the associated Korteweg configurational stress tensor, and Gaussian conserved order-parameter dynamics. Within this approximation the dynamic bulk viscosity is obtained as an exact Green-Kubo spectral representation: it is a continuous superposition of Debye-type relaxation modes, each associated with one Fourier component of the concentration field. By introducing the critical scaling variable x = kξ, where ξ is the correlation length, we show that the exact response has the universal one-parameter form Δζ(ω) =Δζ(0)Φ(ω/(2Γ_ξ)), with Γ_ξ =Mκξ^-4. The radial spectral density controlling the static weight has a maximum at , showing that the dominant contribution originates from modes with wavelength of order ξ. The classical Mandelstam-Leontovich single-relaxation-time form is therefore not postulated but appears as the leading single-scale approximation to the exact continuous spectrum. We also clarify the regime of validity of the Gaussian Model B treatment, its relation to the asymptotic Model H critical regime, and the acoustic quantity in which the relaxation enters. The theory provides a direct connection between equilibrium configurational-stress fluctuations, frequency-dependent bulk viscosity, and measurable critical acoustic attenuation in soft matter.