Fluctuating Elasticity Mode in Transient Molecular Networks

Transient molecular networks, a class of adaptive soft materials with remarkable application potential, display complex, and intriguing dynamic behavior. By performing dynamic light scattering on a wide angular range, we study the relaxation dynamics of a reversible network formed by DNA tetravalent nanoparticles, finding a slow relaxation mode that is wave vector independent at large $q$ and crosses over to a standard $q^{-2}$ viscoelastic relaxation at low $q$. Exploiting the controlled properties of our DNA network, we attribute this mode to fluctuations in local elasticity induced by connectivity rearrangement. We propose a simple beads and springs model that captures the basic features of this $q^0$ behavior.

Figure 1

Squared intermediate scattering function $f{(q,t)}^2$ measured at $T=27°\mathrm{C}$ (continuous colored lines) in the small $q$ (pink shading), intermediate $q$ (gray shading), and large $q$ (blue shading) intervals, normalized either to $\tau =0$ (a),(c) or to the amplitude of the slow decay (b). Distinct colors indicate different $q$, with $q$ increasing as indicated by the arrows. Dashed lines: stretched exponential fits to the slower component of the correlations [(a)–(c)]; exponential fits to the faster component of the decay in the small $q$ (d) and large $q$ [(f) and inset] ranges. Orange and green marks in pane (b) indicate two subsets of correlations, the latter plotted in pane (e) vs $\tau q^2$. Characteristic times for the fast (full symbols) and slow modes (open symbols) and $q^{-2}$ lines are plot in pane (g).

Figure 2

Relaxation times of the fast mode (${\tau }_F$, empty symbols) and of the slow mode (${\tau }_S$, full symbols) at various T as a function of $q$. Sketches on the right-hand side (where DNA NS are pictured as gray three-arms stars) are aimed to mimic the gel’s aggregation state at the temperatures here considered, where $p$ is the fraction of bound terminals (solid dots: bound terminals, open dots: unbound terminals). At the three $T$, the bond lifetimes are also different, ranging from short-lived (red dots) to long-lived (green dots).

Figure 3

Elasticity fluctuation model. (a) Networks of DNA NS can adopt various topological patterns, characterized by different loop lengths, resulting in different local bulk moduli, as described pictorially in the sketches. Topological relaxation occurring upon binding and unbinding of NS effectively provide fluctuations in the local elasticity, described in the 1D beads and spring model as a switching between elastic constants $K_1$ and $K_2$. (b) and (c) Elasticity fluctuation model for instantaneous relaxation of the bead positions. (b) Sequence of equilibrium configuration of 7 beads at the center of a string of 500 beads and springs. Time expressed in units of ${\mathrm{\Gamma }}^{-1}$, the average spring refresh time. (c) Bead density correlation function $f(q,T)$ (different colors indicate different $q$) and their characteristic times vs $q$ (expressed in units of $1/{\ell }_0$, the mean interbead distance). (d) $f(q,T)$ as in pane (c) calculated in the presence of thermal noise. (e) $q$ dependence of the two characteristic times ${\tau }_1$ (black dots) and ${\tau }_2$ (red dots). Black line: $\mathrm{\Gamma }\tau =1$. Red line: viscoelastic relaxation time ${\tau }_E$ multiplied by $\mathrm{\Gamma }$.