Hierarchical Strategy for Simple One-Component 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 an integration of that method into the package ESPResSO++.

To decrease the relaxation time, microscopic monomers are coarse-grained by mapping each subchain with N_{b} monomers onto a soft blob. A polymer chain, originally consisting of N monomers, is replaced by a coarse-grained (CG) chain consisting of N/N_{b} soft blobs linked by a harmonic bond potential, V_{bond}=3 k_{B}T d^{2}/2b_{CG}^2, and an angular bond-bending potential V_{bend}=k_{B}T k_{bend}(1 + \cos(\theta))/2. Here d is the distance and \theta is the angle between consecutive bonds. The interactions between non-bonded soft blobs are taken into account by a repulsive pair potential V_{nb}=k_{B}T
\epsilon U_{G}(r_{ij}). Here r_{ij} is the center-to-center distance between the two blobs, U_{G}(r_{ij}) is a Gaussian function with variance \overline{\sigma}^2 = \sigma_{i}^2 + \sigma_{j}^2 and \sigma_{i} is the gyration radius of blob number i. The gyration radius \sigma is in turn fluctuating. This fluctuation is controlled by the potential V_{sphere}=k_{B}T \, (a_{1}N_{b}^3\sigma^{-6} +
a_{2}N_{b}^{-1}\sigma^{2} + a_{3}\sigma^{-3}).

After equilibrating a configuration at very coarse resolution, each CG polymer chain is replaced with a more fine-grained (FG) chain. In this procedure, a CG blob is divided into several FG blobs. The center of mass (COM) of the FG blobs coincides with the position of the CG blob’s center, and is being kept fixed during the relaxation of the local conformation of the FG monomers within the CG blob.

To develop this module, the following classes have been implemented or modified (and may have been described in more detail elsewhere):

  • A VSpherePair class for calculating V_{nb}=k_{B}T \epsilon U_{G}(r_{ij})
  • A LangevinThermostatOnRadius class for simulating the fluctuations of the radii of the blobs
  • A VSphereSelf class for calculating V_{sphere}=k_{B}T \,
(a_{1}N_{b}^3\sigma^{-6} + a_{2}N_{b}^{-1}\sigma^{2} +
  • A FixedLocalTupleList class for storing the N-tuple of particles consisting of both real and virtual particles
  • A ConstrainCOM class for conserving the COM of N FG blobs with the CG blob

Background Information

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


Explanation of installation of ESPResSO++ can be found at:

After installing this module, an example can be run from hierarchical_strategy_for_one-component subdirectory of the examples folder using the run_example script to be found there.

Source Code

This module was merged into ESPResSo++ in the Pull Request: