Mixing Linear Polymers with Rings and Catenanes: Bulk and Interfacial Behavior

Abstract

We derive and parameterize effective interaction potentials between a multitude of different types of ring polymers and linear chains, varying the bending rigidity and solvent quality for the former species. We further develop and apply a density functional treatment for mixtures of both disconnected (chain-ring) and connected (chain-polycatenane) mixtures of the same, drawing coexistence binodals and exploring the ensuing response functions as well as the interface and wetting behavior of the mixtures. We show that worsening of the solvent quality for the rings brings about a stronger propensity for macroscopic phase separation in the linear-polycatenane mixtures, which is predominantly of the demixing type between phases of similar overall particle density. We formulate a simple criterion based on the effective interactions, allowing us to determine whether any specific linear-ring mixture will undergo a demixing phase separation.

Figure 1

Representative snapshots of monomer-resolved systems at approximately vanishing center-of-mass separation, the coding (A–D) corresponding to the four selected cases from Table 1. For each of the four triplets, the left panel shows typical conformations of two rings, which are rendered in slightly different shades of color and glossiness for better visibility. The middle panels show conformations between chain (black) and ring and the right panel conformations of two chains.

Figure 2

Effective isotropic potentials between two rings (22), two chains (11), and ring and chain (12) as a function of separation between the two molecules. The labels (A–D) on the panels correspond to the four selected cases from Table 1. Points were obtained by monomer-resolved simulations, and lines are fit using eq 6. Distance is normalized by the characteristic length scale of the chain–chain interaction, R₁₁. Analogous figures for additional combinations of parameters can be found in Figure S1 in the SI.

Figure 3

Total pressure of polycatenane (M₂ = 5) and long-chain (M₁ = 5) mixture as a function of ring fraction at total density ρR₁₁³ = 0.025 plotted for the system D from Table 1.

Figure 4

Phase diagrams for mixtures of linear chains (M₁ = 1) and rings (M₂ = 1), plotted on the total density-ring fraction plane, (ρ, x). The labels (A–D) on the panels correspond to the four selected cases from Table 1. Solid golden lines represent binodals, and solid black lines represent spinodals with the critical point highlighted by a black point with white interior. Dashed golden lines are selected tielines, connecting coexisting points on the binodals. Selected points marked by dark red cross are further explored in Figure 6 and selected tielines highlighted in dark red are further explored in Figure 8.

Figure 5

Phase diagrams for mixtures of long chains (M₁ = 10) and polycatenanes (M₂ = 20), plotted on the plane spanned by the total blob density and the fraction of rings, (ρ, x). The labels A–D on the panels correspond to the four selected cases from Table 1. Solid golden lines represent binodals, and solid black lines represent spinodals with critical points highlighted by a black point with white interior. Dashed golden lines correspond to selected tielines, connecting coexisting points on the binodals. Selected points marked by dark red crosses are further explored in Figure 7.

Figure 6

Concentration–concentration structure factors S_cc(k) from eq 31 (main plots) and the number–number structure factors S_nn(k) from eq 30 (insets) for mixtures of linear chains (M₁ = 1) and rings (M₂ = 1). The labels (A–D) on the panels correspond to the four selected cases from Table 1. We show the functions for various ring fractions at selected densities and ring fractions, corresponding to the dark red crosses in Figure 4.

Figure 7

Concentration–concentration response functions Q_cc(k) and their number–number counterparts Q_nn(k) from eq 30 (insets), for mixtures of long linear chains (M₁ = 10) and polycatenanes (M₂ = 20). The labels (A–D) on the panels correspond to the four selected cases from Table 1. We show the functions for various ring fractions at selected densities and ring fractions, corresponding to the dark red crosses in Figure 5.

Figure 8

Interface density profiles for solutions of linear chains (M₁ = 1) and rings (M₂ = 1) as a function of the position z vertical to the interface. The labels (A–D) on the panels correspond to the four selected cases from Table 1. We show the profiles for different values of the pressure, as indicated in the legends, corresponding to the dark red tielines from Figure 4. Dashed curves denote ρ₁(z) and solid ones ρ₂(z). The boundary conditions are chosen in such a way that we have the coexisting phase of the left end point of the tieline at z → +∞ and that of the right end point at z → −∞.

Figure 9

Cosine of the contact angle between a planar wall and a liquid drop (linear-chain-rich phase) in coexistence with the vapor (ring-polymer-rich phase), as a function of the strength βΦ₂ of the wall–ring interaction for systems A, B, C, and D as indicated in the legend. The corresponding pressures are A: 2P_c; B: 2P_c; C: 1.4P_c; and D: 2P_c, where P_c denotes the pressure at the critical point.