Hierarchical Strategy for Simple One-Component Polymer Melts: md-softblob

Reference [1] describes the principles of a hierarchical strategy to equilibrate simple one-component polymer melts described in terms of atomistic or coarse-grained (bead-spring) models. The present module is the central part of our implementation of this method 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 is the central part that controls a suite of programs which realize this method within the framework of the simulation package ESPResSO++.

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. In the last step, microscopic monomers are reinserted into their CG blobs.

The resulting set of FG blobs is set up in such a way that its conformation is consistent with the conformation at the more coarse-grained level. After this setup, the local FG conformation is relaxed into a local equilibrium, again consistent with the (fixed) CG blobs.

The present module implements the actual coarse-graining step, which is therefore described in more detail:

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}). Equilibrated configurations of soft blobs are generated by Molecular Dynamics (MD) simulations based on the above model.

Within the module, the following classes have been implemented or modified:

  • A VSpherePair class for calculating V_{nb}=k_{B}T \epsilon U_{G}(r_{ij})
  • 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 Particle class for comunicating the property “radius” between different cores
  • A LangevinThermostatOnRadius class for simulating the fluctuations of the radii of the blobs

Background Information

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


Explanation of installation:

After installing this module, it can be tested by a Python script found under the following link: