Thermodynamic Consistency in Variable-Level Coarse Graining of Polymeric Liquids

Numerically optimized reduced descriptions of macromolecular liquids often present thermodynamic inconsistency with atomistic level descriptions even if the total correlation function, i.e. the structure, appears to be in agreement. An analytical expression for the effective potential between a pair of coarse-grained units is derived starting from the first-principles Ornstein-Zernike equation, for a polymer liquid where each chain is represented as a collection of interpenetrating blobs, with a variable number of blobs, $n_b$, of size $N_b$. The potential is characterized by a long tail, slowly decaying with characteristic scaling exponent of $N_b^{1/4}$. This general result applies to any coarse-grained model of polymer melts with units larger than the persistence length, highlighting the importance of the long, repulsive, potential tail for the model to correctly predict both structural and thermodynamic properties of the macromolecular liquid.

Figure 1

Force curves for the $n_b=1$ model (solid lines) and the approximated Eq. (2) (squares) for (top to bottom) $N=100$ (red), $N=200$ (blue), $N=500$ (green), and $N=1000$ (brown), along with force curves for models with a fixed blob length, $N_b=50$, and increasing numbers of blobs (dashed lines, from top to bottom): $n_b=2$ (red), $n_b=4$ (blue), $n_b=10$ (green), and $n_b=20$ (brown). Inset: Normalized virial density [$N^{-2}{r}^3{F}^{bb}(r)$] for the numerical forces (lines) and their analytical forms (squares) for $n_b=1$. Degree of polymerization ($N$) increases from top to bottom at the peak below $5R_{gb}$. All systems are at 400 K and density ${\rho }_m=0.03355{\AA }^{-3}$.

Figure 2

Pressure measured from simulations with input CG forces from Fig. 1, where $n_b=1$ (orange squares connected by dashed guide lines), and $N_b=50$ with $n_b>1$ (open blue circles). Also depicted are UA-MD simulations (green X symbols) for systems which relax fast enough for UA-MD to be feasible ($N\le 200$). Data are collected for increasing degree of polymerization $N$, at the same values of $N$ as Fig. 1. All simulations were performed at 400 K and density ${\rho }_m=0.03355{\AA }^{-3}$.

Figure 3

Numerical force curves for $N=100$, $n_b=1$ models of PE with increasing densities (bottom to top) $0.03226{\AA }^{-3}$ (red), $0.03355{\AA }^{-3}$ (blue), $0.03441{\AA }^{-3}$ (green), $0.03656{\AA }^{-3}$ (black), and $0.03871{\AA }^{-3}$ (gray), along with corresponding mean force curves (dashed lines, also from bottom to top). Insert: detail of the attractive contribution to the force (intercepts occur at increasing separation for increasing density). All systems are at 400 K.

Figure 4

Pressures measured in MS-MD simulations versus density, for $N=44$ (triangles) and $N=100$ (circles), performed with effective potentials where$c_0$ is determined from UA-MD simulations. Also reported are values from the corresponding UA-MD simulations (crosses for $N=44$ and stars for $N=100$). All simulations were performed at 400 K. Connecting lines are added as a guide to the eye.