Fine-graining: A Component of the Hierarchical Equilibration Strategy for Polymer Melts

The module is an implementation of the existing hierarchical strategy [1] for the equilibration of simple one-component polymer melts in ESPResSO++.

Purpose of Module

To study the properties of polymer melts by numerical simulations, equilibrated configurations must be prepared. However, the relaxation time for high molecular weight polymer melts is huge and increases, according to reptation theory, with the third power of the molecular weight. Hence, an effective method for decreasing the equilibration time is required. The hierarchical strategy pioneered in Ref. [1] is a particularly suitable way to do this. The present module provides a part of that method described below.

To decrease the relaxation time, microscopic monomers are coarse-grained (CG) by mapping each subchain with N_{b} monomers onto a soft blob. The CG system is then characterized by a much lower molecular weight and thus is equilibrated quickly. One thus obtains a configuration that is equilibrated on large scales but does not provide information about the structure on smaller (i.e. more fine-grained (FG)) scales.

To obtain the latter, the resolution is step-by-step increased by recursively applying a fine-graining procedure to the previous (more coarse-grained) level. In such a fine-graining step, each CG polymer chain is replaced with a more fine-grained chain, by dividing a CG blob into several FG blobs.

The present module provides the python script which performs this fine-graining procedure. The implementation detail is in following below.

  1. The microscopic configuration of N polymers consisted of M monomers is prepared. The system size L is determined by the number of density \rho= (N \times M) /L^3 \approx 0.85. m and \sigma stands for the mass and the diameter of monomers.

    We presuppose that equilibrated CG chain at N_{b} is already obtained.

  2. The softblobs at N_{b} is divided into 2 softblobs at N_{b}/2 under the constraint conditions defined as

    \mathbf{R}_i^{N_{b}} = \frac{\mathbf{R}_{2i-1}^{N_{b}/2} + \mathbf{R}_{2i}^{N_{b}/2}}{2} \equiv \mathbf{r}_{\rm{com}}^i,

    R_{g, N_{b}/2}^{(2i-1)} = R_{g, N_{b}/2}^{2i} = \frac{R_{g, N_{b}}^{i}}{\sqrt{2}},

    where \mathbf{R}_{i}^{N_{b}} stands for \mathbf{R}_{i} at N_{b} and R_{g, N_{b}}^{i} stand for R_{g}^{i} at N_{b}. Namely, the center of mass of 2 softblobs at N_{b}/2 is identical with the position of a softblob at N_{b}.

  3. For equilibrating a local configuration at N_{b}/2, NVT MD simulation is performed.

    In the beginning, a MD simulation takes into account bonding potential V_{\rm{bond}},

    the potential for fluctuating radius of gyration V_{sphere} and the constrain potentials for center of mass described as

    V_{\rm{com}}(\mathbf{r}_{\rm{com}}^i) = k_{\rm{com}}(\mathbf{r}_{\rm{com}}^i - \mathbf{R}_i^{N_{b}})^2.

    Each 16\tau_{\rm{blob}}, MD simulation is including the bending interactions V_{\rm{angle}}

    and non bonding interactions V_{\rm{nb}} in this order.

    Where \tau_{\rm{blob}}=\sqrt{m N_{b}\sigma^2/k_{\rm{B}}T}.

  4. After including all interactions, MD simulation is performed during 16\tau_{\rm{blob}}.

  5. In order to obtain the snapshot which has the ideal mean square internal distance (MSID) <R(n)^2>, MD simulation is continued to carry out. Where MSID <R(n)^2> is defined as

    <R(n)^2> \equiv \frac{1}{M/N_{b} -n}\sum^{N-1}_{j=0}\sum^{M/N_{b} -n}_{i=1}(\mathbf{R}_{i+(M/N_{b})j} - \mathbf{R}_{i+(M/N_{b})j+n})^2.

    This is calculated in each \tau_{\rm{blob}}.

    After obtaining good snapshot at N_{b}/2, fine-graining procedure is finished.

Background Information

The implementation of this module is based on ESPResSO++. You can learn about ESPResSO++ from the following links:

Building and Testing

Explanation of installation:

After installing this module, it can be tested according to the README file found under the following link:

Source Code

This module has been merged into ESPResSo++: