Brillouin light scattering from viscoelastic media: a complex-modulus framework for longitudinal dynamics

Brillouin light scattering (BLS) has evolved from a sensitive probe of acoustic modes in simple liquids into a quantitative tool widely used in condensed-matter science. Its interpretation, however, draws on a fragmented theoretical tradition: simple-hydrodynamic treatments rely on a small-attenuation approximation that fails in viscoelastic systems, while relaxing liquids are described within the memory-function formalism, natural in the molecular-hydrodynamics regime but less suited to macroscopic observables. This hampers the extraction of consistent viscoelastic information, especially for the growing community using BLS to probe the GHz mechanics of soft and biological matter. We present a unified description of BLS from the longitudinal acoustic modes of viscoelastic media, organized around the complex longitudinal modulus M*(ω). From the linearised hydrodynamic equations and the fluctuation–dissipation theorem, the dynamic structure factor S(q,ω) is derived directly, without small-attenuation expansion. Simple hydrodynamics, thermoelastic relaxation, and single- and multi-relaxation viscoelastic responses then appear as additive, physically transparent contributions to one M*(ω). We discuss practical consequences – DHO versus Lorentzian fits, the Landau–Placzek ratio, the Mountain peak – and apply the framework to associated liquids, glass-formers, and biological matter, where comparison with mechanical and dielectric spectroscopies confirms the power of a unified viscoelastic picture.