Polymer stretching in laminar and random flows: Entropic characterization

Polymers in nonuniform flows undergo strong deformation, which in the presence of persistent stretching can result in the coil-stretch transition. This phenomenon has been characterized by using the formalism of nonequilibrium statistical mechanics. In particular, the entropy of the polymer extension reaches a maximum at the transition. We extend the entropic characterization of the coil-stretch transition by studying the differential entropy of the polymer fractional extension in a set of laminar and random velocity fields that are benchmarks for the study of polymer stretching in flow. In the case of random velocity fields, a suitable description of the transition is obtained by considering the entropy of the logarithm of the extension instead of the entropy of the extension itself. Entropy emerges as an effective tool for capturing the coil-stretch transition and comparing its features in different flows.

Figure 1

Left: Entropy of $\rho$ vs Wi for different flows. In all cases (except for the experimental data) the extensibility parameter is set to $b={18}^2$. The experimental data have been translated vertically by $\mathrm{\Delta }{S}_{\rho }=0.33$, which corresponds to a fit to a dumbbell with $b\approx {30}^2$. Right: Entropy of $y=ln\rho$ for the BK and turbulent flows and for the same parameters as in the left panel. The inset shows $S_y$ together with the entropies $S_y^{-}$ and $S_y^{+}$ associated with the PDFs of $y$ conditional on $Q<0$ and $Q>0$, respectively.

Figure 2

Entropy of the fractional end-to-end separation as a function of Wi for a dumbbell and a chain with $N=10$ in (a) a uniaxial extensional flow, (b) a linear shear flow, and (c) isotropic turbulence.