A mesoscopic model for the rheology of dilute and semidilute solutions of wormlike micelles

TL;DR

The paper develops a mesoscopic, bead-spring model for dilute and unentangled wormlike micellar solutions based on persistent worms as the fundamental units, incorporating semiflexibility through bending potentials and hydrodynamic interactions. It demonstrates that fusion–scission dynamics and ring formation yield polydisperse, polyflexible micelles whose static properties align with mean-field and scaling theories, and whose linear viscoelastic response reveals distinct HI effects and Rouse–Zimm crossovers. The framework reproduces universal length-distribution scalings, characterizes overlap concentrations, and provides first-principles-like insight into G'(ω) and G''(ω) for living micelles, including the impact of rings and HI. Overall, the model offers a computationally tractable pathway to connect molecular-scale micellar structure and dynamics with macroscopic rheology in the dilute to semidilute regime, with potential extensions to semiflexible micelles and more complex flows.

Abstract

The concept of a `persistent worm' is introduced, representing the smallest possible length of a wormlike micelle, and modelled by a bead-spring chain with sticky beads at the ends. Persistent worms are allowed to combine with each other at their sticky ends to form wormlike micelles with a distribution of lengths, and the semiflexibility of a wormlike micelle is captured with a bending potential between springs, both within and across persistent worms that stick to each other. Multi-particle Brownian dynamics simulations of such polydisperse and `polyflexible' wormlike micelles, with hydrodynamic interactions included and coupled with reversible scission/fusion of persistent worms, are used to investigate the static and dynamic properties of wormlike micellar solutions in the dilute and unentangled semidilute concentration regimes. The influence of the sticker energy and persistent worm concentration are examined and simulations are shown to validate theoretical mean-field predictions of the universal scaling with concentration of the chain length distribution of linear wormlike micelles, independent of the sticker energy. The presence of wormlike micelles that form rings is shown not to affect the static properties of linear wormlike micelles and mean-field predictions of ring length distributions are validated. Linear viscoelastic storage and loss moduli are computed and the unique features in the intermediate frequency regime compared to those of homopolymer solutions are highlighted. The inclusion of hydrodynamic interactions (HI) enables the distinction between Rouse and Zimm dynamics in wormlike micelle solutions to be elucidated. While the neglect of HI is justified in the case of entangled wormlike micelle solutions, here the concentration at which the onset of the screening of hydrodynamic interactions occurs is clearly identified.

Figures (24)

• Figure 1: Schematic diagrams of two persistent worms represented by bead-spring chains with $N_{\text{pw}} =3$ and $N_{\text{pw}} =5$ beads, respectively. The natural length of the Fraenkel spring is denoted by $b$ and $\ell_{\text{pw}}$ is the natural length of a persistent worm. The angle $\theta$ between successive connector vectors is used in the calculation of the bending energy of the chain. The beads at the ends of a chain are 'sticky' and can associate with other sticky beads to form pairs of stuck beads, and consequently, wormlike micelles.
• Figure 2: The Soddemann-Dünweg-Kremer (SDK) potential ($U_{\mu\nu}^{\mathrm{SDK}}/k_{\text{B}} T$), defined in Eq. (\ref{['eq:SDK']}), plotted as a function of the radial distance $r_{\mu\nu}$ (in units of $l_H$), for different well depths $\epsilon_{st}$.
• Figure 3: Three step process (A $\longrightarrow$ B $\longrightarrow$ C) for implementing an inter-persistent-worm bending potential between two persistent worms, represented here by trumbbells. The terminal beads (red) of the persistent worms are sticky monomers connected to each other by backbone monomers (blue). The hypothetical bead $3^\prime$ is positioned at the centre of mass of beads 2 and 3. The labels $\bm{F}_\mu^{(\text{b})}$ ($\mu = 1,2,3^{\prime},4,5$) represent the bending forces on each of the beads $\mu$.
• Figure 4: A schematic illustratation of a simulation box (of magnitude $L_{\text{box}}$ in each dimension), displaying an ensemble containing a ring and linear wormlike micelles of various lengths formed by the fusion of persistent worms represented by trumbells. The application of intra- and inter-persistent-worm bending potentials is illustrated by the angles $\theta$.
• Figure 5: Schematic diagram describing the procedure for determining the overlap concentration in a wormlike micelle solution for a fixed value of the sticker energy $\epsilon_{st}$. The black line depicts the dependence of the overlap length $L^*_{\text{c}}$ on the effective concentration $c^{\text{eff}}$, while the blue line models the dependence of the mean length $\bar{L}$ on $c^{\text{eff}}$. The coordinates of the intersection point, ($c^*, L^*$), correspond to the overlap concentration and overlap length, respectively, with $c^*$ representing the effective concentration $c^{\text{eff}}$ at which the mean length of wormlike micelles in the system is exactly the length for which $c^{\text{eff}}$ is the overlap concentration.