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# Viscoelasticity and frequency dependent viscosity

Hydrodynamics is characterized by the region of low (k,omega) defined by the wavenumber k and frequency omega. Low k corresponds to long wavelength 2pitext{/}k which is much bigger than the intermolecular distance, and low omega corresponds to frequency which is much smaller than the reciprocal of collision time. For example consider a viscous fluid to which an instantaneous force is applied, at very small time scales it will give elastic response as the stress will be proportional to strain gamma rather than the rate of strain dot{gamma}, given as

sigma=-E gamma

where E is the elastic modulus. At much larger time scales, the shear stress sigma is proportional to the rate of strain dot{gamma}

sigma=-eta dot{gamma}

where eta is the viscosity. Historically it was Poisson (1829-1831) who first suggested elastic response of liquids to sudden disturbances, while Maxwell (1867-1873) extended this idea mathematically. The Maxwell relaxation time given by the ratio tau_{M}=etatext{/}E defines the time scale below which the liquid behaves like an elastic solid, and for t text{>>} tau_{M} it behaves like a viscous fluid. Thus viscoelasticity is described by

frac{sigma}{eta} + frac{dot{sigma}}{E} = dot{gamma}

where dot{sigma} is the time derivative of stress. This viscoelastic behaviour can be explained by frequency dependent viscosity eta(omega) instead of usual viscosity employed in linearised hydrodynamics. Molecular dynamics is one such method to investigate hydrodynamics and transport properties in the region of finite (k,omega). Using equilibrium molecular dynamics we can evaluate the transport properties ( similar to Green-Kubo relation in zero frequency limit ) as

eta(omega) = B_T int_0^infty dt text{ e}^{iomega t} langle J_{eta}(0) J_{eta}(t)rangle_{ce}

where eta(omega) is the frequency dependent viscosity, B_T is the thermodynamic constant, J_{eta} is the associated thermodynamic current and langle cdot rangle_{ce} denotes canonical ensemble. The viscosity transport function can be rewritten as eta(omega) = eta^R(omega) – i eta^I(omega) consisting of dissipative real part phi^R(omega) and non-dissipative imaginary part eta^I(omega) given as

eta^R(omega)=B_T int_0^infty dt text{ cos}(omega t) langle J_{eta}(0) J_{eta}(t)rangle_{ce} and eta^I(omega)=B_T int_0^infty dt text{ sin}(omega t) langle J_{eta}(0) J_{eta}(t)rangle_{ce}

Consider a simple model where the correlation function langle J_{eta}(0) J_{eta}(t) rangle undergoes exponential relaxation based on relaxation time tau_{eta} described by text{exp}^{-t/tau_{eta}}, for such a case we can derive frequency dependent viscosity as

eta(omega) = B_T frac{langle [J_{eta}(0)]^2rangle}{i omega + tau_{eta}^{-1}}

Assuming that the relaxation time tau_{eta} can be evaluated at zero frequency, such that tau_{eta}=B_T^{-1}etatext{/}langle [J_{eta}(0)]^2rangle. Using tau_{eta} we can approximate the frequency dependent viscosity as

eta(omega) = etatext{/}(1+omegatau_{eta})

When time scales are much lesser than tau_{eta} we get elastic behaviour and when it is much larger than tau_{eta} we get viscous behaviour, similar to Maxwell's treatment of viscoelastic behavior.