The information in this section describes ESPResSo++ as a whole. Information specific to the additions in this module are in subsequent sections.

Name: Fine-graining procedure for one component melts
Language: Python (2.7) and C++
Licence: GPL <https://opensource.org/licenses/gpl-license>
Documentation Tool: Sphinx and Doxygen
Application Documentation: http://espressopp.github.io/
Relevant Training Material: https://github.com/espressopp/espressopp/tree/master/examples

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

Purpose of Module
Background Information
Building and Testing
Source Code
References

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.

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.
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 $N_{b}$ .
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 $\tau_{\rm{blob}}=\sqrt{m N_{b}\sigma^2/k_{\rm{B}}T}$ .
After including all interactions, MD simulation is performed during $16\tau_{\rm{blob}}$ .
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.