Rheology of fractal networks

TL;DR

This paper models the cytoskeleton as a fractal network of Kelvin-Voigt elements to explain its viscoelastic properties, revealing power-law behaviors similar to those observed in soft biological materials.

Contribution

It introduces a fractal network model with a power law spectrum of retardation times, linking the viscoelasticity to the fractal dimension of the gel.

Findings

Reproduces weak power-law behaviors of moduli with frequency in certain regimes.

Derives different power laws for elastic and viscous moduli in other regimes.

Connects the parameter alpha with the fractal dimension of the network.

Abstract

We model the cytoskeleton as a fractal network by identifying each segment with a simple Kelvin-Voigt element, with a well defined equilibrium length. The final structure retains the elastic characteristics of a solid or a gel, which may support stress, without relaxing. By considering a very simple regular self-similar structure of segments in series and in parallel, in 1, 2 or 3 dimensions, we are able to express the viscoelasticity of the network as an effective generalised Kelvin-Voigt model with a power law spectrum of retardation times, $\cal L\sim\tau^{\alpha}$. We relate the parameter $\alpha$ with the fractal dimension of the gel. In some regimes ($0<\alpha<1$), we recover the weak power law behaviours of the elastic and viscous moduli with the angular frequencies, $G'\sim G''\sim w^\alpha$, that occur in a variety of soft materials, including living cells. In other regimes, we find different power laws for $G'$ and $G''$.