Adiabatic Bulk Modulus and Sound Velocity of Nematic Liquid Crystals in the Isotropic Phase

We obtain an expression for the sound velocity \(\varvec{v}_{\varvec{s}}\) in the isotropic phase of nematic liquid crystals from the nematodynamics that incorporates the energetic coupling between \(\varvec{Q}_{\varvec{\alpha \beta }}\varvec{(} \textbf{r}\varvec{)}\) (the order parameter of the nematic-isotropic transition) and inhomogeneities in the local mass density \(\varvec{\rho }\varvec{(} \textbf{r}\varvec{)}\). It differs from the well-known formula \(\varvec{v}_{\varvec{s}}\varvec{=}\sqrt{\varvec{B}_{\varvec{s}}\varvec{/}\varvec{\rho }_{\varvec{0}}}\), in which \(\varvec{B}_{\varvec{s}}\) is the adiabatic bulk modulus and \(\varvec{\rho }_{\varvec{0}}\) is the average value of \(\varvec{\rho }\varvec{(} \textbf{r}\varvec{)}\). Using the measured values of \(\varvec{v}_{\varvec{s}}\) in the regime of low frequencies, we then calculate \(\varvec{B}_{\varvec{s}}\) as a function of temperature for three different liquid crystals. The critical form of \(\varvec{B}_{\varvec{s}}\) that emerges is unexpected, and its physical interpretation is discussed in terms of the Anderson-Grüneisen parameter. An explanation for the critical dip of \(\varvec{v}_{\varvec{s}}\) near the nematic-isotropic transition is provided.