Multiscale approach to equilibrating model polymer melts

We present an effective and simple multiscale method for equilibrating Kremer Grest model polymer melts of varying stiffness. In our approach, we progressively equilibrate the melt structure above the tube scale, inside the tube and finally at the monomeric scale. We make use of models designed to be computationally effective at each scale. Density fluctuations in the melt structure above the tube scale are minimized through a Monte Carlo simulated annealing of a lattice polymer model. Subsequently the melt structure below the tube scale is equilibrated via the Rouse dynamics of a force-capped Kremer-Grest model that allows chains to partially interpenetrate. Finally the Kremer-Grest force field is introduced to freeze the topological state and enforce correct monomer packing. We generate 15 melts of 500 chains of 10.000 beads for varying chain stiffness as well as a number of melts with 1.000 chains of 15.000 monomers. To validate the equilibration process we study the time evolution of bulk, collective, and single-chain observables at the monomeric, mesoscopic, and macroscopic length scales. Extension of the present method to longer, branched, or polydisperse chains, and/or larger system sizes is straightforward.

Figure 1

Visualization of a melt of 1000 chains of 15 000 beads each during the equilibration process. All the polymers are shown on the left hand side of the box, while the same ten randomly selected polymers are shown on the right hand side of the box. Conformation just after lattice annealing (a), after $0.1{\tau }_e$ (b), and $1{\tau }_e$ (c) of Rouse dynamics simulation with the force-capped KG model, and final melt configuration after KG warmup (d).

Figure 2

The pair potential (a) and bond potential (b) for the full KG model (green dashed lines) and for the fcKG model (red solid lines). Shown is also the height of the energetic barrier for chains to pass through each other as function of the force cap (c). The circle denotes our choice of $U_{\mathrm{WCA}}^{\mathrm{cap}}(r=0)=5\epsilon$.

Figure 3

Kuhn length $l_K$ vs stiffness parameter for the KG (green circles) and fcKG models (red boxes). The lines are polynomial fits $\frac{l_K\left(\kappa \right)}{\sigma }=1.795+0.358\frac{\kappa }{\epsilon }+0.172\frac{{\kappa }^2}{{\epsilon }^2}+0.019\frac{{\kappa }^3}{{\epsilon }^3}$ (hashed black line) and $\frac{l_K^{fc}\left({\kappa }_{fc}\right)}{\sigma }=1.666+0.389\frac{{\kappa }_{fc}}{\epsilon }+0.192\frac{{\kappa }_{fc}^2}{{\epsilon }^2}+0.012\frac{{\kappa }_{fc}^3}{{\epsilon }^3}$ (dotted black line). The inset shows the relation between ${\kappa }_{fc}$ and $\kappa$ defined by Eq. (17) (solid black line).

Figure 4

Tube segmental length $a\left(\kappa \right)$ vs stiffness parameter for the KG model (green symbols); also shown is an interpolation given by $\frac{a\left(\kappa \right)}{\sigma }=11.32-2.096\frac{\kappa }{\epsilon }-0.0293\frac{{\kappa }^2}{{\epsilon }^2}+0.1465\frac{{\kappa }^3}{{\epsilon }^3}$ (hashed line). The inset shows Eq. (18) (red solid curve) compared to the simulation data.

Figure 5

Kuhn friction for the fcKG model as function of stiffness parameter ${\kappa }_{fc}$. The line through the data points is the fit $\frac{{\zeta }_K^{fc}\left({\kappa }_{fc}\right)\tau }{m_b}=5.5657+1.4367\frac{{\kappa }_{fc}}{\epsilon }+0.7564\frac{{\kappa }_{fc}^2}{{\epsilon }^2}+0.30372\frac{{\kappa }_{fc}^3}{{\epsilon }^3}$.

Figure 6

Mean-square displacements for the fcKG model (filled symbols) and KG model (open symbols) for melts with $M\times N_b=500\times 10.000$ for $\kappa ,{\kappa }_{fc}=-1\epsilon$ (black circle), $0\epsilon$ (red box), $1\epsilon$ (green diamond), and $2\epsilon$ (blue triangle up). The KG model data were scaled with the Kuhn length and time of the corresponding force-capped KG model to retain their relative positions.

Figure 7

Characterization of simulated annealing process showing (a) total acceptance probability and (b) the angle (blue crosses) and incompressibility (circles) energy contributions. The inset shows the temperature profile during annealing.

Figure 8

Melt characterization during simulated annealing showing (a) $\langle R^2\rangle /\langle R_g^2\rangle \approx 6$ ratio and (b) chain stiffness $c_L$ for a lattice melt with (black circles) and without the angular energy contribution in the Hamiltonian (blue crosses).

Figure 9

Structure factor for initial lattice configurations with density fluctuations (red boxes), and after simulated annealing with the incompressibility term (black circles) averaged over several melt states. Also shown are the power laws expected from the density fluctuations and from incompressibility (red hashed and black dash-dotted lines, respectively).

Figure 10

Evolution of mean-square internal distances during equilibration for the initial lattice configuration (black circle), and after 0.1, $0.2,0.5,1$, 2, 5, and $10{\tau }_e^{fc}$ (red box, green diamond, blue triangle up, red triangle left, brown triangle down, orange triangle right, purple plus, respectively) of Rouse simulation for $\kappa =0$.

Figure 11

Topological evolution of the melt during the Rouse simulation. The entanglement lengths of the force-capped KG model melts for $\kappa =-1\epsilon ,0\epsilon ,1.5\epsilon$, and $2.5\epsilon$ (denoted by red box, black circle, green diamond, and blue triangle up, respectively). Also shown are the entanglement length of melts after the KG warm up for three different times along the Rouse simulation for $\kappa =0\epsilon$ (magenta crosses).

Figure 12

Evolution of structure factor during equilibration process. Structure factor for the annealed lattice (black circle), after 0.1, 1, and $10{\tau }_e^{fc}$ (denoted by green box, red diamond, and magenta triangle up, respectively) of Rouse simulation, and for the final melt state after KG warm (blue cross).

Figure 13

Mean-square internal distances of equilibrated melts (colored symbols) compared to brute force equilibrated melts (dashed black lines) and Kuhn length (dotted black line) for varying stiffness.