Solenoids and Plectonemes in Stretched and Twisted Elastomeric Filaments

We study the behavior of a naturally straight highly extensible elastic filament subjected to large extensional and twisting strains. We find that two different phases can coexist for a range of parameter values: the plectoneme and the solenoid. A simple theory based on a neo-Hookean model for the material of the filament and accounting for the slender geometry suffices to explain these observations, and leads to a phase diagram that is consistent with observations. Extension and relaxation experiments on these phases show the presence of large hysteresis loops and sawtoothlike force-displacement curves which are different for the plectoneme and the solenoid.

Figure 1

(a) The experiment consists of clamping a rubber rod to a rigid support, which is then stretched by hanging a weight $W$ and is twisted about its axis by rotating a bar attached to the weight. (b) For small to intermediate values of the twist, the rod remains straight while its length increases. (c) Beyond a threshold in the twisting strain, the straight configuration loses stability to a helical form. Depending on the initial extension, we see (d) the plectonemic phase or (e) the solenoidal phase. For the solenoidal phase, as the twist is increased still further, the whole rod is converted to (f) a solenoidal structure, which is then transformed into (g) tertiary structures that are constituted of both (h) higher-order solenoidal and (i) plectonemic structures. In (j) we show our notation and point out the appearance of kinks at the junction between the straight and solenoidal parts of the filament.

Figure 2

Equilibrium values of the solenoid angle $\beta$ [obtained from (8)] vs the stretch $\lambda$ for different extensional loads $W$. The curves 1 to 5 correspond to rubber rod with $2a=3.75\mathrm{mm}$ subjected to $W=0.6$, 1.5, 3.0, 4.4, and 6.4 N, respectively. Curve 6 represents the upper limit of the solenoid angle ${\beta }_m$ when the helix pitch $P=D$ the diameter of the stretched filament. The ${\beta }_m$ data from experiments are also plotted with respect to $\lambda$. Symbols ○ and ● represent rubber rods of elastic modulus $E=1.3\mathrm{MPa}$ and diameter $2a=3.175$ and 6.35 mm, respectively.

Figure 3

Phase diagram. The dimensionless twist density $\psi a$ (filled symbols) at which the plectoneme/solenoidal phase arises are plotted with respect to the initial extension ratio ${\lambda }_0$ of the rod. The solid line represents the theoretical transition curves for $\psi a$ while the dashed line represents the transition curve for $\lambda$ as a function of ${\lambda }_0$ calculated using (4, 5, 6, 7, 8, 9). The “circles” correspond to filaments of modulus $E=1.3\mathrm{MPa}$ and diameter $2a=3.175\mathrm{mm}$, the “triangles” correspond to $2a=6.35\mathrm{mm}$, and the squares correspond to $E=2.6\mathrm{MPa}$ and $2a=6.35\mathrm{mm}$. The vertical dashed line separates the plectoneme and the straight phase, while the solid line separates the straight phase from the solenoidal phase.

Figure 4

The force-extension response of a rubber filament during a cycle or loading and unloading displacements (see triangular ramp). Curve 1 depicts the response of a straight filament which is simply stretched. Curve 2 represents that for a filament with a series of solenoids (initial load $W_0=1.0$); as the load increases, the solenoids unwind leading to sawtoothlike response. Curves 3 represent the response of a single plectoneme (initial load $W_0=0.5$) while curve 4 ($W_0=0.1\mathrm{N}$) represents the response of a multiple-plectoneme filament; here the sawtoothlike structure corresponds to the unwinding of the loop at the end of the plectoneme that is similar to a solenoidal structure when stretched. Observe the marked difference between the unloading curves for the solenoidal filament (2) and the plectoneme (3, 4); the latter are smooth while the former show a sawtoothlike response. All the experiments were carried out keeping the total twist constant, and with filaments of diameter $2a=3.75\mathrm{mm}$ and modulus $E=1.3\mathrm{MPa}$.