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.