Effective Perrin theory for the anisotropic diffusion of a strongly hindered rod

Abstract

Abstract. -Slender rods in concentrated suspensions constitute strongly interacting systems with rich dynamics: transport slows down drastically and the anisotropy of the motion becomes arbitrarily large. We develop a mesoscopic description of the dynamics down to the length scale of the interparticle distance. Our theory is based on the exact solution of the Smoluchowski-Perrin equation; it is in quantitative agreement with extensive Brownian dynamics simulations in the dense regime. In particular, we show that the tube confinement is characterised by a power law decay of the intermediate scattering function with exponent 1/2.Brownian motion of highly anisotropic particles is considerably more complex than the diffusion of spherical objects, the basic understanding of which is founded on the seminal works by Einstein and von Smoluchowski. A shape anisotropy results in diffusion coefficients that depend on the direction of motion in the body frame, thus inducing a coupling of translation to the orientation. This anisotropic dynamics has been investigated in recent experiments measuring diffusion coefficients of micrometre sized ellipsoids and rods by single particle tracking [1-3]; in particular, non-Gaussian statistics has been observed [1]. Likewise in dynamic light scattering, the rotational and translational diffusion coefficients were determined simultaneously [4]. For these dilute systems, the ratio of diffusion parallel and perpendicular to the long symmetry axis was limited to values up to D /D ⊥ ≈ 4 in quasi two-dimensional (2D) confinement.Considerably higher values of this ratio have been observed in simulations of semi-dilute suspensions of slender rods, yielding D /D ⊥ up to values of 50 [5,6]. This increase in anisotropy is caused by the steric constraints imposed by surrounding rods; thereby the transverse and rotational motion is suppressed, whereas the longitudinal transport is barely influenced [7]. An intermediate regime of anisotropic diffusion has been derived for ballistic needles within kinetic theory [8] and was observed in simulations [9].For a finite width b, the rods undergo a phase transition to the nematic phase at densities of the order of 1/bL 2 as predicted by Onsager [10]. The dynamics in this ordered phase becomes trivially anisotropic and splits into a fast diffusion along the nematic director axis and slower diffusion perpendicular to it. Such a pronounced anisotropic diffusion has been observed in simulations of nematic elongated ellipsoids [11] and spherocylinders [12,13]. Experiments have also clearly demonstrated orientation-dependent diffusion in colloidal nanorods in the isotropic and nematic phase [14] and in various liquid crystalline phases of fd viruses [15,16]. The phenomena connected with the nematic phase transition are essentially understood and will not be discussed in this work.In the isotropic phase, experiments and computer simulations have been restricted to determine the diffusion coefficients from the mean-square displacements. The complex interplay of tra...

Tracer diffusion inside fibrinogen layers

“…This minimum corresponds to the confinement of a tracer between long, parallel filaments made of fibrinogen molecules. Similar effect has been also detected in models of the anisotropic diffusion of a strongly hindered rod represented by a thin needle moving in an array of hard point obstacles [26].…”

Dynamically crowded solutions of infinitely thin Brownian needles

“…In both theory [15][16][17][18][19][20] and computer simulations [21][22][23][24][25][26] the density-dependent scaling behavior of the long-time rotational and perpendicular translational diffusion coefficients have been established and scale with the number density as n −2 . Computer simulations for two-dimensional toy models have also been performed earlier [27][28][29][30][31] and for a needle in the presence of pointlike obstacles one observes the same scaling laws of the transport coefficients as in three dimensions [29,30]. In experiments, the transport coefficients of a nanowire diffusing through an array of obstacles have been determined only recently, and the drastic slowing down of transport has been observed [32].…”

“…Here, we consider the intermediate scattering function in the case that all segments of the needle contribute to the scattering. For the two-dimensional analog where the needle moves in a planar array of point obstacles, the intermediate scattering function for the geometric center and the entire needle has been evaluated earlier and also compared to the phantom needle [30,59].…”

Intermediate scattering function of an anisotropic Brownian circle swimmer

“…Special cases of the previous equation [Eq. (12)] have already been solved in terms of Mathieu functions for a passive anisotropic Brownian particle (v = 0 and ω = 0) [44], and also for a three dimensional anisotropic passive [45] and active Brownian particle (ω = 0) [41]. Yet, no solution for the ISF of a Brownian circle swimmer has been elaborated up to now.…”