Pulling-force-induced elongation and alignment effects on entanglement and knotting characteristics of linear polymers in a melt

We employ a primitive path (PP) algorithm and the Gauss linking integral to study the degree of entanglement and knotting characteristics of linear polymer model chains in a melt under the action of a constant pulling force applied to selected chain ends. Our results for the amount of entanglement, the linking number, the average crossing number, the writhe of the chains and their PPs and the writhe of the entanglement strands all suggest a different response at the length scale of entanglement strands than that of the chains themselves and of the corresponding PPs. Our findings indicate that the chains first stretch at the level of entanglement strands and next the PP (tube) gets oriented with the “flow.” These two phases of the extension and alignment of the chains coincide with two phases related to the disentanglement of the chains. Soon after the onset of external force the PPs attain a more entangled conformation, and the number of nontrivially linked end-to-end closed chains increases. Next, the chains disentangle continuously to attain an almost unentangled conformation. Using the linking matrix of the chains in the melt, we furthermore show that these phases are accompanied by a different scaling of the homogeneity of the global entanglement in the system. The homogeneity of the end-to-end closed chains first increases to a maximum and then decreases slowly to a value characterizing a completely unlinked system.

Figure 1

(a) $+1$ crossing and (b) $-1$ crossing. By assigning a sign to each crossing, one recovers information concerning which arc comes over and under.

Figure 2

A system generated by two chains. For the computation of $\mathrm{LK}(I,J)$ we have $\mathrm{LK}(I,J)=L(I_0,J_0)+L(I_0,J_1)+L(I_0,J_2)+L(I_0,J_3)$.

Figure 3

(a), (b) Selected sequences of snapshots for individual chains contained in the melt, artificially shifted vertically with time. The marked head groups experience a constant force in the $x$ direction. Chains tend to either unravel completely (chain A, whose absolute writhe decreases from $2.246$ to $0.130)$, or to tighten existing knots (chain B, whose $\left|W\right|$ increases from $0.237$ to $0.820)$; the shrinkage in the $y$ direction of not yet unfolded chain ends is caused by the invisible surrounding chains. The stretched terminal part containing the force-bearing end group is responsible for an initial increase of the number of entanglements, which are subsequently destroyed upon unraveling. (c) Absolute displacement of the force-bearing bead in the $x$ direction for the two chains shown in (a) and (b), as well as $\langle \left|x\right(t)-x(0\left)\right|\rangle$ obtained as an average over all chains. The initial quick increase terminates when $\langle \left|x\right(t)-x(0\left)\right|\rangle$ exceeds 40, which corresponds to roughly half of the length of a stretched entanglement strand. This regime is followed by a regime of constant speed of the polymer's center of mass, $v=\langle |x\left(t\right)-x\left(0\right)|\rangle /t\approx 0.0078=F/\zeta N^2$ with $\zeta \approx 0.64$ reflecting Stokes friction.

Figure 4

Visualization of the final elongated melt state. Only the parent images of the total 100 chains are shown. Subset of chains with (a) $\left|W\right|<0.5$ and (b) $\left|W\right|\ge 0.5$. As expected, most of the chains with $\left|W\right|>0.5$ contain local tight knots [76].

Figure 5

(a) Square root of the mean squared end-to-end distance, ${\mathcal{R}}_{\mathrm{ee}}={\langle R_{\mathrm{ee}}^2\rangle }^{1/2}$, as a function of time. (b) The corresponding double logarithmic plot.

Figure 6

(a) Number of entanglements computed by the Z1 algorithm. The vertical dashed line is at $t=150$. (b) Double logarithmic plot of $\mathcal{Z}=\langle Z\rangle$ with $t$. (c) Probability distribution of $Z$ in the course of time. (d) Average contour length of the PP, ${\mathcal{L}}_{\mathrm{PP}}=\langle L_{\mathrm{PP}}\rangle$. (e) Double logarithmic plot of ${\mathcal{L}}_{\mathrm{PP}}$ vs $t$. (f) Probability distribution of $L_{\mathrm{PP}}$ in the course of time.

Figure 7

The mean primitive path contour length, ${\mathcal{L}}_{\mathrm{PP}}$, versus the square root of the mean squared end-to-end distance, ${\mathcal{R}}_{\mathrm{ee}}$.

Figure 8

$N_e$ estimators: ${\mathcal{N}}_e^{\mathrm{S}\text{−}\mathrm{coil}},{\mathcal{N}}_e^{\mathrm{S}\text{−}\mathrm{kink}},{\mathcal{N}}_e^{\mathrm{mod}\mathrm{S}\text{−}\mathrm{coil}},{\mathcal{N}}_e^{\mathrm{mod}\mathrm{S}\text{−}\mathrm{kink}}$ defined in Eqs. (5, 6, 7, 8). The latter two estimators exceed $N=100$ at large times. Values for all four estimators in equilibrium $\left(t=0\right)$ had been obtained for a large set of initial configurations in [65].

Figure 9

The entanglement length as obtained by the estimator involving the writhe and $\mathcal{Z}$, Eq. (9), using the equilibrium value $\kappa =2.34$ (black data points). The yellow line is a smoothed version of the data (averaged over $\Delta t=50)$ and the green line shows the values of the $N_e$ estimator obtained using the measured nonequilibrium analog to bending stiffness, $\kappa \left(t\right)=2.34+0.02t$ in Eq. (9). The estimator gives values intermediate between the ${\mathcal{N}}_e^{\mathrm{S}\text{−}\mathrm{coil}}$ and ${\mathcal{N}}_e^{\mathrm{S}\text{−}\mathrm{kink}}$ estimators (Fig. 8), but shows also a jump to values greater than 100 at about $t\approx 450$.

Figure 10

(a) $\langle \mathrm{ACN}\rangle$ vs time. (b) $\langle \mathrm{ACN}\rangle$ vs ${\mathcal{L}}_{\mathrm{PP}}$.

Figure 11

(a) Mean absolute writhe, $\langle \left|W\right|\rangle$ vs time. The vertical dashed line is at $t=150$. (b) Corresponding double logarithmic plot. (c) Probability distribution of $W$ in the course of time. (d) Mean absolute writhe of the PPs, $\langle \left|W_{\mathrm{PP}}\right|\rangle$ vs time. The two vertical dashed lines are at $t=50$ and $t=300$, respectively. (e) Corresponding double logarithmic plot. (f) Probability distribution of $W_{\mathrm{PP}}$. (g) Mean absolute linking number, $\langle \left|L\right|\rangle$ (vertical dashed line at $t=150)$. (h) Corresponding double logarithmic plot. (i) Probability distribution of $L$. We find $\langle \left|L\right|\rangle \approx {10}^{-1.5}{t}^{0.16}$ for $t\in [15,150],\langle \left|L\right|\rangle \approx {10}^{-0.8}{t}^{-0.15}$ for $t\in [150,350]$, and $\langle \left|L\right|\rangle \approx {10}^{1.1}{t}^{-0.86}$ for $t\in [400,700]$.

Figure 12

(a) Writhe versus $\mathcal{Z}$ and the (b) corresponding double logarithmic plot. We find $\langle \left|W\right|\rangle \approx {10}^{2.34}{\mathcal{Z}}^{-3.59}$ for $t\in [25,150]$ and $\langle \left|W\right|\rangle \approx {10}^{-0.64}{\mathcal{Z}}^{0.49}$ for $t>150$. (c) Mean absolute writhe of the PP versus $Z$ and (d) corresponding double logarithmic plot. We find $\langle |W_{\mathrm{PP}}|\rangle \approx {10}^{0.06}{\mathcal{Z}}^{-0.94}$ for $t\in [25,50],\langle |W_{\mathrm{PP}}|\rangle \approx {10}^{-0.85}{\mathcal{Z}}^{0.45}$ for $t\in [50,150],\langle |W_{\mathrm{PP}}|\rangle \approx {10}^{-0.89}{\mathcal{Z}}^{0.51}$ for $t\in [150,350],\langle |W_{\mathrm{PP}}|\rangle \approx {10}^{-1.09}{\mathcal{Z}}^{0.79}$ for $t\in [350,700]$, and $\langle |W_{\mathrm{PP}}|\rangle \approx {10}^{-1.2}{\mathcal{Z}}^{1.46}$ for $t>700$. (e) Mean writhe of an entanglement strand, ${\mathcal{W}}_e$. The line at $t\approx 700$ indicates the transition of $\mathcal{Z}$ to values $\mathcal{Z}<1$. (f) Corresponding double logarithmic plot. We find ${\mathcal{W}}_e\approx {10}^{0.005}{\mathcal{Z}}^{-0.46}$ for $t\in [25,150]$ and ${\mathcal{W}}_e\approx {10}^{-1.64}{\mathcal{Z}}^{0.22}$ for $t>150$.

Figure 13

(a) Mean absolute linking number of the original chains with all surrounding chains (black squares), with images of other chains (white triangles) and with self-images (white diamonds). The vertical dashed line corresponds to $t=150$. (b) Same as (a) for the corresponding end-to-end closed chains, and vertical dashed lines at $t=50$ and $t=300$.

Figure 14

(a) The mean second smallest eigenvalue ${\lambda }_2$ of the Laplacian of the [inking matrix of open (squares) and end-to-end closed (diamonds) chains. (b) The maximum eigenvalue ${\lambda }_{\max }$ of the Laplacian of the periodic linking matrix for open (squares) and end-to-end closed (diamonds) chains. Vertical dashed lines are drawn at $t=50,t=300$, and $t=450$.