Entropic characterization of the coil-stretch transition of polymers in random flows

Polymer molecules in a flow undergo a coil-stretch phase transition on an increase of the velocity gradients. Model-independent identification and characterization of the transition in a random flow has been lacking so far. Here we suggest to use the entropy of the extension statistics as a proper measure due to strong fluctuations around the transition. We measure experimentally the entropy as a function of the local Weisenberg number and show that it has a maximum, which identifies and quantifies the transition. We compare the new approach with the traditional one based on the theory using either linear Oldroyd-B or nonlinear finite extensible nonlinear elastic polymer models.

Figure 1

Dependence of the entropy of polymer extensions on the local Weisenberg number ${\mathrm{Wi}}_{\mathrm{loc}}=\lambda {(\partial V_{\theta }/\partial r)}^{\mathrm{rms}}$ in elastic turbulence. The entropy maximum $S=1.97\pm 0.042$ at ${log}_2\left({\mathrm{Wi}}_{\mathrm{loc}}\right)=-0.39\pm 0.48$ is interpreted as the coil-stretch transition in the interval $0.13<{\mathrm{Wi}}_{\mathrm{loc}}<1.23$.

Figure 2

Determination of the onset of the coil-stretch transition ${\mathrm{Wi}}_c$ using the exponents $\alpha$ of the power-law tail of the PDFs of polymer extensions: ${\mathrm{Wi}}_{\mathrm{loc}}=0.58\pm 0.20$.

Figure 3

PDFs of polymer extension, based on the statistics of about 900 molecules for each ${\mathrm{Wi}}_{\mathrm{loc}}$.

Figure 4

Experimental setup. A von Karman swirling flow is created in the gap of $d=675\mu \mathrm{m}$ between the flat end face of the rotating delrin rod of radius $R=2.25$ mm ($d/R=0.3$) and a cover slip. The plastic rod is glued into the metal shaft that is driven via a belt by an optically encoded dc minimotor with less than $1\%$ rms of velocity fluctuations. Beating of the polished delrin surface due to misalignment is less than 1 $\mu \mathrm{m}$ on a radius of 300 $\mu \mathrm{m}$. The delrin-made cell has an inner radius $R_c=6$ mm.

Figure 5

The dependence of the local Weissenberg ${\mathrm{Wi}}_{\mathrm{loc}}=\lambda {(\partial V_{\theta }/\partial r)}^{\mathrm{rms}}$ on Wi in a swirling flow between two disks.

Figure 6

[(a)–(c)] PDFs of the T4DNA normalized stretching in linear-linear presentation in the main plots and in log-log presentation in the corresponding insets at $\mathrm{Wi}=7.2$ (a), 8.7 (b), and 108 (c) in a solution of $c=25.16 \mu \mathrm{g}/\mathrm{ml}$ with $50\%$ sucrose. The thick solid lines are algebraic fittings to the PDF tails $P(x/L)\sim {(x/L)}^{-1-\alpha }$ with $\alpha =3.4$ at $\mathrm{Wi}=7.2, \alpha =-1.8$ (left tail) at $\mathrm{Wi}=8.7$, and $\alpha =-5.0$ at $\mathrm{Wi}=108$.