Translation Drag Coefficient of a Self-Similar Assembly of Spheres Immersed in an Incompressible Fluid

Abstract

On the basis of deterministic fractals and the Rotne-Prager hydrodynamic interaction tensor, we confirm the asymptotic as well as the finite size scaling of the friction coefficient lambda of a self-similar structure. The fractal assembly is made of N spheres with its dimension varying from D < 1 to D = 3. The number of spheres can be as high as N approximately O(10(4)). The asymptotic scaling behavior of the friction coefficient per sphere is lambda approximately N(1/D-1) for D > 1, lambda approximately (lnN)(-1) for D = 1, and lambda approximately N(0) for D < 1. The crossover behavior indicates that while in the regime of D > 1 the hydrodynamic screening effect grows with the size, for D<1 it is limited in a finite range, which decays with decreasing D.

Comparison of Approximate Methods for Calculating the Friction Coefficient and Intrinsic Viscosity of Nanoparticles and Macromolecules

“…This is consistent with our arguments above where we noted that weaker shape fluctuations should render the double-sum formula more accurate. Experimental values of R g / R h cluster around 1.5, but values as low as 1.3 and as high as 1.7 are also reported . The so-called “draining effect”, i.e., the sensitivity of R h and [η] to monomer size and shape and the associated slow convergence of R g / R h to its infinite N value, appears to be responsible for this commonly reported experimental disparity.…”

“…The fractal mass scaling characteristics of R h and [η] for flexible polymer chains and other fractal objects are reported elsewhere . The scaling law for R h has also been recently examined by Tseng et al for a variety of model fractal aggregates based on the Zimm bead model method of calculation. (See also ref for a discussion of this mass scaling of R h and the scaling variables that govern the rate of approach of these scaling relations to their asymptotic long chain limits.)…”

“…4 The fractal mass scaling characteristics of R h and [η] for flexible polymer chains and other fractal objects are reported elsewhere. 22 The scaling law for R h has also been recently examined by Tseng et al 54 for a variety of model fractal aggregates based on the Zimm bead model method…”

“…Experimental values of R g /R h cluster around 1.5, but values as low as 1.3 and as high as 1.7 are also reported. 54 The so-called "draining effect", i.e., the sensitivity of R h and [η] to monomer size and shape and the associated slow convergence of R g /R h to its infinite N value, appears to be responsible for this commonly reported experimental disparity. (In a separate work, we will focus on quantifying this draining effect.)…”