Dynamics of viscoelastic filaments based on Onsager principle

When a polymer solution is uniaxially stretched and held fixed at both ends, the solution quickly separates into droplets connected by strings and takes the beads-on-string structure. The string then becomes thinner by capillary forces. Here we develop a theoretical framework on viscoelastic fluids based on the Onsager principle and apply it to the dynamics of viscoelastic filaments. We show that the beads-on-string structure is a thermodynamic quasi-equilibrium state and derive an equation for the coexistence condition in the pseudo-equilibrium state. Using the condition, we solve the evolution equation analytically and show that the string radius and the tensile stress vary exponentially as predicted by the classical theory of Entov and Hinch [J. Non-Newtonian Fluid Mech. 72, 31 (1997)], but the prefactor for the tensile stress is different from their theory and agrees with the numerical solutions of Clasen et al. [J. Fluid Mech. 556, 283 (2006)].

Figure 1

A cylindrical specimen of initial radius $R_{\mathrm{ini}}$ and length $L_{\mathrm{ini}}$ is uniformly stretched in the $z$ direction.

Figure 2

Formation of the beads-on-string structure due to the phase separation. The enlarged part is for one length period. Fraction $x$ of the initial cylinder becomes the string part, while the fraction $(1-x)$ goes to the beads.

Figure 3

(a) The tensile force as a function of $\lambda$. (b) Phase diagram in the $\alpha \text{−}\lambda$ coordinates.

Figure 4

The string shrinks with time due to the elasto-capillary effect. Only one length period of the beads-on-string structure is shown here.

Figure 5

Comparison between the results from Onsager principle [Eqs. (55) and (56)] and the numerical results of Clasen [14]. The parameters are $\alpha =40$ and ${\eta }_p/{\eta }_s=3.0$. In Ref. [14] the unit of the time is the capillary time ${\tau }_{\gamma }={(\rho a^3/\gamma )}^{1/2}$, while we used the polymer's relaxation time $\tau$. The difference in the time units leads to the appearance of the Deborah number ($\mathrm{De}=\tau /{\tau }_{\gamma }=94.9$) in the $x$ axis.