Linear Viscoelasticity of Semiflexible Polymers with Hydrodynamic Interactions

TL;DR

This work develops a Brownian-dynamics bead-spring model with FENE-Fraenkel springs, bending stiffness, and Rotne–Prager–Yamakawa hydrodynamics to study the linear viscoelasticity of a single semiflexible polymer in infinite dilution. By computing G(t) via Green–Kubo relations and transforming to G'(ω) and G''(ω), the authors show that appropriate spring parameters reproduce bead-rod (SRT) behavior over a wide time window, with intermediate-time power laws $G(t) \,\propto \, t^{-\,\alpha}$ where $\alpha$ ranges from $1/2$ (flexible) to $5/4$ (stiff). Hydrodynamic interactions shift intermediate-time slopes toward Zimm-like values for flexible chains and become less influential as stiffness increases, with a crossover near $L/l_p \approx 10$; HI increase agreement with experimental data across a broad frequency range. Validation against MPCD data and comparisons to PBLG and collagen experiments demonstrate that the FENE-Fraenkel bead-spring model can quantitatively capture the linear viscoelastic response of semiflexible polymers, offering a computationally efficient bridge between bead-rod theories and flexible-bead-spring models. The study lays the groundwork for extending to finite concentrations, entanglements, and networks, where HI screening and additional length scales become pivotal.

Abstract

The linear viscoelastic response of single semiflexible polymer chains in the infinite-dilution limit is studied using Brownian dynamics simulations of coarse-grained bead-spring chains. The springs obey the FENE-Fraenkel force law, a bending potential is used to capture chain stiffness and hydrodynamic interactions are included through the Rotne-Prager-Yamakawa tensor. By calculating the relaxation modulus following a step strain, we demonstrate that the bead-spring chain behaves like an inextensible semiflexible rod over a wide time window with an appropriate choice of spring stiffness and chain extensibility. In the absence of hydrodynamic interactions, our results agree with the existing theoretical predictions for the linear viscoelastic response of free-draining, inextensible, semiflexible rods in the limit of infinite dilution. It is shown that at intermediate times, the stress relaxation modulus exhibits power law behaviour, with the exponent ranging from $(-1/2)$ for flexible chains to $(-5/4)$ for highly rigid chains. At long times, rigid chains undergo orientational relaxation, while flexible chains exhibit Rouse relaxation. Hydrodynamic interactions are found to effect the behaviour at intermediate and long times, with the difference from free-draining behaviour increasing with increasing chain flexibility. Computations of the frequency dependence of loss and storage moduli are found to be in good agreement with experimental data for a wide variety of systems involving semiflexible polymers of varying stiffness across a broad frequency range.

Figures (15)

• Figure 1: Schematic representation of the linear viscoelastic response of a dilute solution of semiflexible polymers. (a) Relaxation modulus $G(t)$ as a function of time $t$ following a step strain. (b) Storage and Loss moduli, $G^{\prime}(\omega)$ and $G^{\prime \prime}(\omega)$, as a function of frequency $\omega$ obtained by Fourier transforming $G(t)$. The exponent $\alpha$ characterizes the power law scaling behavior at the intermediate time and frequency regimes of the relaxation modulus and the dynamic moduli, respectively.
• Figure 2: Force versus extension curves for different spring laws: Hookean (Blue), FENE (Red), Fraenkel (Magenta) and FENE-Fraenkel (Green). $Q_{0}$ denotes the FENE spring stretchability parameter, $\sigma$ is the Fraenkel spring's natural length and $s$ is the spring extensibility around $\sigma$
• Figure 3: Diagram illustrating the labeling scheme for beads, segments, and included angles. The position $\bm{r}_{\mu}$ of bead $\mu$ relative to the center of mass. $\bm{u}_{\mu}$ is the unit vector of the segment connecting beads $\mu$ and $\mu+1$ with length $Q_{\mu}$. Included angle $\theta_{\mu}$ is the angle between unit vectors $\bm{u}_{\mu}$ and $\bm{u}_{\mu-1}$. This figure has been reproduced from Pincus2023 with permission.
• Figure 4: The nondimensional relaxation modulus $G(t)$ as a function of scaled time for $N_b=32$ FENE--Fraenkel chain with $H_R = 1 \times 10^{5}$ and $L/l_p = 3$. The blue curve is a fit to the simulation data using a sum of exponential functions with 7 exponents.
• Figure 5: (a) Internal monomer mean square displacement $g_{1}(t)$ and (b) end-to-end unit vector autocorrelation function $\bm{u}(t)$ for a semiflexible chain with bending stiffness $C=20$ and $N_{b}=48$ beads. with HI (squares) and without HI (circles). The results of Nikoubashman2016, with and without HI, are denoted by asterisks (*) and crosses ($\times$), respectively.