Dynamical Effective Field Model for Interacting Ferrofluids: II. The proper relaxation time and effects of dynamic correlations

The recently proposed dynamical effective field model (DEFM) is quantitatively accurate for describing dynamical magnetic response of ferrofluids. In paper I it is derived under the framework of dynamical density functional theory, via which the original ensemble of bare Brownian particles is mapped to an ensemble of dressed particles. However, it remains to clarify how the characteristic rotational relaxation time of a dressed particle, denoted by \(\tau_r\), is quantitatively related to that of a bare particle, denoted by \(\tau^0_r\). By building macro-micro connections via two different routes, I reveal that under some gentle assumptions \(\tau_r\) can be identified with the long-time rotational self-diffusion time. I further introduce two simple but useful integrated correlation factors, describing the effects of quasi-static (adiabatic) and dynamic (nonadiabatic) inter-particle correlations, respectively. In terms of both correlation factors I reformulate the dynamic magnetic susceptibility in an illuminating and elegant form. Remarkably, it shows that the macro-micro connection is established via two successive steps: a dynamical coarse-graining with nonadiabatic effects accounted for by the dynamic factor, followed by equilibrium statistical mechanical averaging captured by the static factor. Surprisingly, \(\tau_r/\tau^0_r\) is found insensitive to changes of particle volume fraction. I provide a physical picture to explain it. Furthermore, an empirical formula is proposed to characterize the dependence of \(\tau_r/\tau^0_r\) on dipole-dipole interaction strength. The DEFM supplemented with this formula leads to parameter-free predictions in good agreement with results from Brownian dynamics simulations. The theoretical developments presented in this paper may have important consequences to studies of ferrofluid dynamics in particular and other systems modelled by DDFTs in general.