Systematic time-scale-bridging molecular dynamics applied to flowing polymer melts

We present a thermodynamically guided, low-noise, time-scale-bridging, and pertinently efficient strategy for the dynamic simulation of microscopic models for complex fluids. The systematic coarse-graining method is exemplified for low-molecular polymeric systems subjected to homogeneous flow fields. We use established concepts of nonequilibrium thermodynamics and an alternating Monte Carlo-molecular-dynamics iteration scheme in order to obtain the model equations for the slow variables. For chosen flow situations of interest, the established model predicts structural as well as material functions beyond the regime of linear response. As a by-product, we present steady-state simulation results for polymers in general flow situations, including simple, planar, and yet unexplored equibiaxial elongation. The method is simple to implement and allows for the calculation of time-dependent behavior through quantities readily available from nonequilibrium steady states.

Figure 1

(Color online) Components of gyration tensor $\mathbf{x}$ (left) and Lagrange multiplier $\mathbf{\Lambda }$ (right panel) vs shear rate for a FENE polymer melt $(N=20)$. Lagrange multipliers self-consistently enter the anisotropy and stretching of polymer chains. Comparison with standard nonequilibrium MD (NEMD) reference results (left panel) shows that the generalized canonical distribution (3) provides a good description of the nonequilibrium stationary state in shear flow. We use Lennard-Jones units throughout this paper.

Figure 2

Different components of the friction matrix $\mathbf{M}$ as a function of the separating time ${\tau }_s$ obtained in step (iii) of our procedure. Solid and open symbols correspond to the integral formula mentioned in the text and Eq. (6), respectively. Results correspond to a chain length of $N=30$ and a planar shear flow with dimensionless shear rate $\dot{\gamma }=0.00036$.

Figure 3

(Color online) Flow alignment angle $\chi$ calculated from gyration tensor $\mathbf{x}$ (open circles) and $\mathbf{\Lambda }$ (solid squares) as a function of the logarithm of the shear rate $\dot{\gamma }$ for planar shear flow (chain length $N=30$). Diamonds correspond to NEMD reference results taken from [14] obtained under the same conditions.

Figure 4

(Color online) (a) We follow the notation employed in [27] where the transposed flow gradient is written as $\mathbf{\nabla }\mathbf{v}=\dot{ϵ}[{\mathbf{e}}_1{\mathbf{e}}_1+m{\mathbf{e}}_2{\mathbf{e}}_2-(1+m){\mathbf{e}}_3{\mathbf{e}}_3]+\dot{\gamma }{\mathbf{e}}_1{\mathbf{e}}_2$, with shear rate $\dot{\gamma }$, elongation rate $\dot{ϵ}$, special cases $m=-0.5$ (simple), 0 (planar), $+0.5$ (elliptical), and 1 (equibiaxial elongation) when $\dot{\gamma }=0$, and simple shear, when $\dot{ϵ}=0$. Besides shear viscosity $\eta$, the graph shows the properly (cf. text part and [15, 27]) scaled viscosities $H_1\equiv {\eta }_1∕\left[2(2+m)\right]$ and $H_2\equiv {\eta }_2∕\left[2(1+2m)\right]$ vs flow rate for $N=20$, where ${\eta }_1\equiv ({\sigma }_{11}-{\sigma }_{33})∕\dot{ϵ}$ and ${\eta }_2\equiv ({\sigma }_{22}-{\sigma }_{33})∕\dot{ϵ}$. (b) Maximum component of the gyration tensor ${\mathbf{x}}_{11}$ for the same types of flow vs flow rate $(N=20)$.

Figure 5

(Color online) Polymer contribution to non-Newtonian shear viscosity vs shear rate for various molecular weights. Exemplarily, reference results obtained via direct NEMD simulation [14] are shown for $N=30$. The inset shows zero-rate shear viscosity ${\eta }_0$ and first viscometric function ${\Psi }_{1,0}$ vs chain length $N$, both coinciding with data from extensive NEMD [14].