Screening confinement of entanglements: Role of a self-propelling end inducing ballistic chain reptation

Synthetic and natural nanomaterials with self-propelling mechanisms continue to be explored to boost chain mobility beyond normal reptation in the crowded environments of entangled chains. Here we employ scaling theory and numerical simulations to demonstrate that activating one chain end of a singular or isolated chain boosts entanglement-constrained chain reptation from the one-dimensional diffusive mobility as described by the de Gennes–Edwards–Doi model to ballistic motion along the entanglement tube contour. The active chain is effectively screened from the constraint of entanglements on length scales exceeding the tube size.

Figure 1

Left panel: Sketches showing the reptation motions of active and passive chains along the entanglement tube. We mark both ends of the passive chain as green arrows to indicate their identical mobility, while the active chain is marked as being oriented by a red arrow, with the red-to-green color contour change representing the decreasing influence of the self-propelling end on the stretching of chain segments closer to the passive end. Right panel: Scaling predictions of mean squared displacements (MSDs), for the self-propelling end (shown in red lines) based on the theoretical discussions proposed in the present paper, and for a passive chain end (shown in green lines) based on the de-Gennes-Edwards-Doi model. The dashed lines are shown to separate different scaling regimes.

Figure 2

The modeled system of a single chain in an entangled polymer melt. (a) A simulation snapshot showing the conformation of an active chain of $N=128$ and bending energy parameter ${\epsilon }_{\mathrm{bend}}=0$, with its self-propelling end of driving force $f_{\mathrm{sp}}=8$ (red). The entangled polymer melt is shown in grey. (b) Simulation result of the stress relaxation moduli of the entangled polymer melt, and the corresponding theoretical fit based on Eq. (12) with the fit parameters $n_{\mathrm{m}}=0.80$, ${\tau }_{\mathrm{mon}}=3.0$, $\gamma =0.50$, $G_{\mathrm{pla}}^{\mathrm{red}}=0.088$ and ${\tau }_{\mathrm{ter}}=4.3\times {10}^6$. The dashed horizontal line indicates the plateau modulus of $G\left(t\right)=n_{\mathrm{m}}kT{G}_{\mathrm{pla}}^{\mathrm{red}}$. Here, the polymer chain length and the bending energy parameter between neighboring bonds governing the stiffness of polymer chains in the melt are $N=256$ and ${\epsilon }_{\mathrm{bend}}=2$, respectively.

Figure 3

Simulation results. (a) Dependence of polymer subchain size, quantified by the mean square end-to-end distance, on the subchain's chemical distance to the respective end. In case of the active chain, the active endmonomer is driven with $f_{\mathrm{sp}}=8$ (red squares), while its second end remains passive (red-green squares). The passive chain has two passive ends (green spheres). Both chains have $N=64$ monomers. (b) MSDs of the corresponding chain ends. (c) MSDs of the self-propelling end for different chain lengths. (d) Comparison between the MSDs of the active and the passive ends of the same chain of length $N=64$ and driving force $f_{\mathrm{sp}}=8$.

Figure 4

Timedependent conformational evolvements of the passive and active chains with $f_{\mathrm{sp}}=8$ and $N=128$. The conformations are obtained by dumping their coordinates in intervals of ${10}^3{\tau }_0$, and a total of 40 conformations is displayed for each chain. Beads are only shown at the chain ends, otherwise only the bond vectors are plotted. Time evolution is shown in terms of color changes of the chain ends: red $\to$ yellow (active end of the active chain and one arbitrary end of the passive chain) or blue $\to$ green (remaining passive ends). The dashed curve is only used for guiding the eyes.

Figure 5

Scaling dependence of the Fickian diffusion coefficients of the active (with $f_{\mathrm{sp}}=8$ and 16) and passive chains, obtained from the MSDs of their respective ends, on the chain length. Note that the MSD curves, for the self-propelling and passive ends of the same active chain, coincide with each other in the Fickian diffusion regime as shown in Fig. 3.