Conformational properties of blends of cyclic and linear polymer melts

An adapted version of the annealing algorithm to identify primitive paths of a melt of ring polymers is presented. This algorithm ensures that the primitive path length becomes zero for nonconcatenated rings, and that no entanglements are observed. The bond-fluctuation model was used to simulate ring-linear blends with $N=150$ and 300 monomers. The primitive path length and the average number of entanglements of the linear component were found to be independent of the blend composition. In contrast, the primitive path length and the average number of entanglements on a ring molecule increased approximately linearly with the fraction of linear chains, and for large $N$, they approached values comparable with linear chains. Threading of ring molecules by linear chains, and ring-ring interactions were observed only in the presence of linear chains. It is conjectured that for large $N$, these latter interactions facilitate the formation of a percolating entangled network, thereby resulting in a disproportionate retardation of the dynamical processes.

Figure 1

(Color online) Autocorrelation of the end-to-end vectors of linear polymers for $N_L=300$, and the diametrical vector of ring polymers for $N_R=300$. The fraction of linear chains ${\varphi }_L/\varphi =0.5$. Autocorrelation of the diametrical vector of ring polymers $p_R\left(t\right)$ decays faster than the autocorrelation of the end-to-end vector of the linear chains $p_L\left(t\right)$.

Figure 2

Two nonconcatenated rings affixed at points $A$ and $B$ forming a “reef knot.”

Figure 3

(Color online) Mean primitive path length of rings (triangles) and linear chains (circles) for a blend of ring and linear molecules with $N_R=N_L=150$ at different compositions. As the fraction of linear chains ${\varphi }_L/\varphi$ increases, average primitive path length of the rings increases, while average primitive path length of the linear chains remains relatively unchanged.

Figure 4

(Color online) Mean primitive path length of rings (triangles) and linear chains (circles) for a blend of ring and linear molecules with $N_R=N_L=300$ at different compositions.

Figure 5

(Color online) Average number of entanglements as identified by the ILD algorithm on rings (triangles) and linear chains (circles) for a blend of ring and linear molecules with $N_R=N_L=150$ at different compositions. As the fraction of linear chains ${\varphi }_L/\varphi$ increases, the average number of entanglements on rings increases, while the average number of entanglements on linear chains remains relatively unchanged.

Figure 6

(Color online) Average number of entanglements as identified by the ILD algorithm on rings (triangles) and linear chains (circles) for a blend of ring and linear molecules with $N_R=N_L=300$ at different compositions.

Figure 7

(Color online) Snapshots of representative primitive paths of rings at different linear fractions for $N=300$ shows increased infiltration of ring volume by entanglements. For clarity, only a single ring molecule (solid), interacting matrix segments (transparent), and location of entanglement points as identified by the ILD algorithm are depicted. When (a) ${\varphi }_L=0.0$, the individual rings are completely disentangled. As ${\varphi }_L$ increases through (b) 0.025, (c) 0.100, and (d) 0.375, the number of entanglements increases.

Figure 8

(Color online) Fraction of unentangled rings decreases from unity (pure rings) to zero as the fraction of the linear chains increases. The decrease is more rapid as $N$ increases.

Figure 9

(Color online) Simulation box showing only entangled polymer chains. Linears (solid black) thread through multiple rings inducing ring-ring interactions and form a percolating network that spans the entire simulation box. (a) $\text{N}=150$ with ${\varphi }_L/\varphi =0.042$ and (b) $\text{N}=300$ with ${\varphi }_L/\varphi =0.05$. Note: The periodic boundary condition causes some rings to appear as linears with small molecular weight.

Figure 10

(Color online) Plot of distance diffused by rings and linears as a function of time for the slowest system considered in this study, $N=300$ and ${\varphi }_L=0.45$. Horizontal lines indicate equilibrated $⟨R_g⟩$.

Figure 11

(Color online) For well equilibrated linears $R^2\left(n\right)\sim n^{1.0}$, when $n\gg 1$ demonstrating the Gaussian character of the linears at all but the smallest length scales. The solid line has a slope of unity.

Figure 12

(Color online) For equilibrated rings with $N=300$, the shape of the $R^2\left(n\right)$ curve is similar to that of Gaussian rings. The rings are compressed in a melt of pure rings and start to swell as the fraction of linears is increased.