Diffusion of rod-like particles in complex fluids

TL;DR

This paper addresses how rod-like particles diffuse in complex fluids where the Stokes-Einstein relation may fail. It extends the wave-vector-dependent shear viscosity framework by modeling a rod as a dimer and deriving anisotropic diffusion coefficients from the viscosity function $\eta(k)$ within linear response theory and Smoluchowski dynamics, with the diffusion coefficient linked to mobility via $D = k_B T \mu$. A key contribution is explicit expressions for parallel and perpendicular mobilities that depend on a dominant hydrodynamic term $G_{\mathrm{eff}}$ and on a self-mobility term $\mu_{\text{single}}(a)$, plus a simple phenomenological form of $\eta(k)$ that interpolates between macroscopic and solvent viscosities. The results show the model can reproduce both isotropic, sphere-like diffusion in simple fluids and strongly anisotropic, reptation-like diffusion in crowded media, offering a practical tool for interpreting rod-like tracer experiments in polymer solutions and similar complex fluids. The approach highlights $\eta(k)$ as a central, tunable quantity governing mobility across scales, with clear limitations in gels and in regimes where hydrodynamic interactions are not dominant.

Abstract

Diffusion of particles in complex fluids and gels is difficult to describe and often lies beyond the scope of the classical Stokes-Einstein relation. One of the main lines of research over the past few decades has sought to relate diffusivity to a fundamental dissipative property of the fluid: the wave-vector-dependent shear viscosity function. Here, we use linear response theory to extend this viscosity function framework to rod-like particles. Using a dimer (two-bead particle) as a minimal rod-like probe, we derive explicit expressions for its diffusion coefficients parallel and perpendicular to its axis in terms of the viscosity function. We show that this description captures the full range of behaviors, from nearly isotropic diffusion of the rod-like probe to highly anisotropic, reptation-like motion. The method is based on a microscopic statistical-mechanical treatment of the Smoluchowski dynamics, yet leads to simple final formulas, providing a practical tool for interpreting diffusion experiments on rod-like tracers in complex fluids. We also clarify the limitations of this approach, emphasizing that the present formulation is primarily suited to complex liquids like polymer solutions and only indirectly applicable to gels.

Figures (2)

• Figure 1: Streamlines of the velocity fields from Eq. (\ref{['eq:Velocity=000020field']}) generated by a point force $\mathbf{F}=-10^{-12}\hat{\mathbf{z}}\,\text{N}$ with the viscosity function given by Eq. (\ref{['eq:=000020rational=000020eta(k)']}) for: (a) length $\lambda=10^{-7}\text{m}$, viscosity ratio $\eta_{\text{macro}}/\eta_{0}=5$; (b) length $\lambda=10^{-7}\text{m}$, viscosity ratio $\eta_{\text{macro}}/\eta_{0}=48$.
• Figure 2: Mobility ratio $\mu_{\parallel}/\mu_{\perp}$ for a dimer composed of two identical spherical beads of radius $a=10^{-8}\,\mathrm{m}$, separated by a center-to-center distance $R$. The mobility coefficients are determined by the "dominant contribution approximation" defined by Eq. (\ref{['eq:dominant=000020contribution=000020app']}) in the complex fluid with the viscosity function from Eq. (\ref{['eq:=000020rational=000020eta(k)']}) characterized by macroscopic-to-solvent viscosity ratio $\eta_{\mathrm{macro}}/\eta_{0}$ and length $\lambda$$\left[\mu m\right]$. The single-bead mobilities required in the "dominant contribution approximation" were determined from Eq. (\ref{['eq:phi=000020within=000020Soft=000020Matter']}).