Anomalous diffusion and stress relaxation in surfactant micelles

We investigate the mechanisms of anomalous diffusion in cationic surfactant micelles using molecular dynamics simulations in the presence of explicit salt and solvent-mediated interactions. Simulations show that when the counterion density increases, saddle-shaped branched interfaces manifest. In experiments, branched structures exhibit lower viscosity as compared to linear and wormlike micelles. This has long been attributed to stress relaxation arising from the sliding motion of branches along the main chain. Our simulations reveal a mechanism of branch motion resulting from an enhanced counterion condensation at the branched interfaces and provide quantitative evidence of stress relaxation facilitated by branched sliding. Furthermore, depending on the surfactant and salt concentrations, which in turn determine the microstructure, we observe normal, subdiffusive, and superdiffusive motions of surfactants. Specifically, superdiffusive behavior is associated with branch sliding, breakage and recombination of micelle fragments, as well as constraint release in entangled systems.

Figure 1

Snapshots from simulations depicting the microstructure at the surfactant concentration of $c_D=0.1M$ and various salt to surfactant ratios ($R$): (a) $R=0.2$, (b) $R=0.4$, (c) $R=0.6$, and (d) $R=0.8$. Color scheme: red (${\mathrm{Sal}}^{-}$), yellow (hydrophilic part of the surfactant), and cyan (hydrocarbon tail). For the sake of clarity, water molecules are represented by purple dots in the background.

Figure 2

Hydrophobic-water interfaces of micelles at $c_D=0.2M$ with different Gaussian curvatures ($K$): (a) spherical, $K>0[R=0$ (top), $R=1.2$ (bottom)], (b) cylindrical ($R=1.2$), $K=0$, and (c) hyperbolic ($R=1.2$), $K<0$. Changes in the shape and curvature are induced by the condensation of the counterions as depicted in the top panel. The color code for the various beads is the same as that in Fig. 1.

Figure 3

Charge distribution and energetics of a branched micelle obtained from equilibrium simulations at $c_D=0.3M$ and $R=0.8$. (a) Distribution of net charge $Q=\sum q_i$ over the micelle surface. The color map corresponds to the $Q$ values of the abscissa. Labels: $S$ (spherical); $C$ (cylindrical); and $B$ (branch point). (b) Micelle surface color coded with $Q$ values. (c) The potential energy of a branched micelle as a function of time associated with the sliding motion.

Figure 4

Pathways of branch formation.

Figure 5

Stress relaxation simulations. (a) $c_D=0.2M$. (b) $c_D=0.3M$. (i) and (ii) Microstructures at different $R$ values. (iii) Viscosity $\eta$ as a function of time for different concentrations. (iv) Time evolution of stress at different salt and surfactant concentrations after the cessation of an imposed extensional deformation (semilog $y$ plot).

Figure 6

EA MSD for different molecules and ions in a solution for $c_D=0.1M$ and $R=0.2$ (log-log plot).

Figure 7

First row: Microstructures for $R=0.2,0.4$, and 0.8 at $c_D=0.1M$. The number of branch points increases with increasing $R$ as shown by the solid circles. Second row: TA MSD of representative surfactants showing normal, subdiffusive, and superdiffusive motions. (C-B): TA MSD of individual surfactant molecules in the solution belonging to a micelle that either combines with another micelle or breaks apart into two micelles. (B): TA MSD of individual surfactant molecules belonging to a branched micelle. Thick solid lines (black) represent normal diffusion, i.e., $\langle r^2\left(\mathrm{\Delta }\right)\rangle \sim t$. Third row: Distribution of the diffusion exponent $\beta$.