## How to equilibrate a polymer melt?

**Introduction to polymer physics**

The simplest model to describe polymer chains is the Freely Jointed Chain (FJC) model [FJC]_. In this simple model ([randomcoil]_) a polymer chain is represented as a succession of monomer units that interact $textbf{only}$ by covalent binding forces, therefore the potential energy of the polymer is taken to be independent of its shape. Therefore at thermodynamic equilibrium, all of its shape configurations are equally likely to occur as the polymer fluctuates in time, according to the Boltzmann distribution. As a result, the bond length is fixed, and the internal rotations are completely free [equilibrating]_

Polymer configuration can be described by setting up the positions of each monomer: for example $3N$ Cartesian coordinates $XYZ$ with Respect to the laboratory-fixed frame. Since all of the configurations are random, the end-to-end vector, connecting the first and the last units of a chain should have a zero average [average]_ and a certain variance.

$$\mathbf{R}(N)=\sum_{i=1}\^{i=N}\mathbf{r}_i$$

Evidently [central_limit]_, a long freely jointed chain will obey Gaussian statistics: the probability to observe a certain end-to-end vector $\mathbf{R}$ is given by the Gaussian distribution function with zero average (centered at the origin) and mean squared value.

$$\Big \langle \mathbf{R}(N) \Big \rangle = 0$$

$$\Big \langle \mathbf{R\^2}(N)\Big \rangle \equiv \Big \langle R\^2(N)\Big \rangle = Nb\^2$$ where $b$ is the bond length and $N$ is the number of bonds. Another value used in polymer physics is the radius of gyration $R_g$ which is defined by the:

$$R\^2_g(N) = \frac{1}{N} \Sigma_{i}(\mathbf{r}_i-\mathbf{r}_{CM})\^2$$

where $\mathbf{r_i}$ \- is the position vector of bead number $i$ in the chain and $r_{CM}$ is the position vector of the center of mass of the polymer chain. Schematically it is represented in [figPolymerParameters]_. For ideal chains\index{ideal chains}, the radius of gyration (see [figPolymerParameters]_) can also be calculated by:

$$<R\^2_g(N)> = \frac{<R\^2(N)>}{6}$$

Despite its simplicity, $<R\^2_g(N)> = \frac{<R\^2(N)>}{6}$ is among the most fundamental results of polymer science - it provides an estimate of the length scales in polymer melts, as well as serving as a bridge to connect experimental results to MD.

FIG. [figPolymerParameters]_ Schematic representation of a polymer hain. Two central characteristic values are shown: the end-to-end distance which corresponds to the vector connecting the first and the last beads of the polymer, and the gyration radius which is a characteristic the average size of the polymer chain.

In practice, chains are non-ideal, they interact and have internal stiffness. Moreover, since the number of beads $N$ is usually a large number, no distinction is made between the finite and the infinite number of bonds. In 1969 Flory used these ideas to define the characteristic ratio $C_{\infty}$ of a polymer as:

$$C_{\infty} = \lim_{N\to\infty} \frac{<R\^2(N)>}{Nb\^2}$$

By definition, $C\^{FJC}_{\infty} = 1$. Values of $C_{\infty}$ larger than $1$ occur when some of the degrees of freedom are constrained. The $C_{\infty}$ can be used as a measure of stiffness along the polymer backbone. The value $C_{\infty}$ is experimentally observable therefore one can judge the level of equilibration of polymer melts by how well their Mean Square Internal Distances($R\^2(n)/n$) plot saturate on the value $<R\^2(n \to \infty)> \to C_{\infty} Nb\^2$, explained in [figMSID]_.

FIG. [figMSID]_ Schematic representation of evolution of Mean Square Internal Distances (MSID) for equilibrated and non-equilibrated polymer Melts. Upon equilibration chain, MSID saturates on correct End-to-End distances that can be directly obtained from the number of units in the chain, bond length and chain stiffness

## Theoretical aspects of polymer melt equilibration in MD

The MD scheme takes care of the time evolution of the model used to study a particular system. Still, the first configuration (meaning positions and velocities (for all particles) have to be defined. For the case of short inorganic molecules, the initial positions can be set up ”by hand” on the vertices of a perfect crystal. This is not the case for the long chain high-temperature polymer melt. Therefore, one needs to start with a configuration that closely resembles an equilibrated, disordered and amorphous system. This can be achieved using a Monte-Carlo algorithm that will efficiently decorrelate an artificial configuration to let it acquire equilibrium properties. For instance, in the present work melts were created using self-avoiding Random walk via a chain.f tool provided by the LAMMPS Molecular Dynamics Simulator ([lammps]_) package.

After initial configurations are created, polymer chains need to be equilibrated. Unlike short molecules, long chain polymers require both thermodynamic and configurational equilibration. Configurational equilibration can be achieved when the Mean Square Internal Distance ($MSID$) the parameter is equilibrated and correspond to pseudo-Gaussian chain, see FIG.[figmsid]_. This was done via Kremer-Grest equilibration process 7 using bead-springs polymer representation ([sliozberg]_ )The the method used for configurational equilibration is a fast ’Dpd-push-off’ - It is a commonly used way to prepare well-equilibrated melts. This method is an extension of the slow push-off method developed by Auhl et al. ([auhl]_). The idea of application of soft repulsive potentials for equilibration of polymer melts is effective provided the potential is applied to the initial configurations that closely match equilibrium structures at large length scales. The details and the practical aspects of the algorithm will be discussed below. The MSID plots, alongside plots of the thermodynamic parameters evolution, provide evidence on a fine quality of equilibration achieved using the procedure explained above.

Mean Square Internal Distance Plot ($MSID$). The plot indicates the well-equilibrated character of the polymer melt. Linear growth at the initial stages is followed by saturation on the

## Practical aspects of polymer equilibration

We have already discussed the theoretical aspects of polymer equilibration. We emphasized the importance of creating decorrelated initial configurations and tracking the evolution of MSID parameter as a gauge of equilibration. In this subsection, we will discuss the practical aspects of polymer equilibration.

Now it’s time to “make hands dirty” and cover the steps and code used for equilibration.

a. We start with unphysical Kremer-Grest equilibration using bead-spring model. (no angles in the atom topology only atoms+bonds) b. We add the angle/dihedral parts c. Finish up via physical NVE+NPT run

Below is my LAMMPS code I used for the Kremer-Grest equilibration. Details may be found in papers of [sliozberg]_ or [auhl]_. This is the main part of the equilibration, after this step we expect the chains to be fully relaxed and have Gaussian chain statistics.

``` {.sourceCode .perl}

# Kremer-Grest model.

units lj atom_style bond special_bonds lj/coul 0 1 1 read_data init.data neighbor 0.4 bin neigh_modify every 1 delay 1 comm_modify vel yes bond_style fene bond_coeff * 30.0 1.5 1.0 1.0 dump mydump all dcd 50000 equil.dcd timestep 0.01 thermo 100 thermo_modify norm no pair_style dpd 1.0 1.0 122347 # very soft pair-potential pair_coeff * * 25 4.5 1.0

velocity all create 1.0 17786140

# bonds in init.data are unphysically close

# fix nve/limit doesn’t let the system to explode

# during the equilibration run

fix 1 all nve/limit 0.001 run 500 fix 1 all nve/limit 0.05 run 500 fix 1 all nve/limit 0.1 run 500 unfix 1 fix 1 all nve run 50000

write_data tmp.restart_dpd.data

pair_coeff * * 50.0 4.5 1.0 velocity all create 1.0 15086120 run 50 pair_coeff * * 100.0 4.5 1.0 velocity all create 1.0 15786120 run 50 #…… run 100 pair_coeff * * 1000.0 4.5 1.0 velocity all create 1.0 15086189 run 100 write_data tmp.restart_dpd1.data

pair_style hybrid/overlay lj/cut 1.122462 dpd/tstat 1.0 1.0 1.122462 122347 pair_modify shift yes pair_coeff * * lj/cut 1.0 1.0 1.122462 pair_coeff * * dpd/tstat 4.5 1.122462 velocity all create 1.0 1508612013 # this velocity reset is repeated 10 times run 50 write_data tmp.restart_push.data velocity all create 1.0 15086125 run 2000000

write_data equil.data

```
The init.data that was created using self-avoiding random walk and
imported at the beginning of the script above didn't have any angle
sections. But the actual simulation that we will run will need this parameter. So here I present a short SED/AWK script to update the data
file. Interested read may find a good Introductory tutorial on
<https://quickleft.com/blog/command-line-tutorials-sed-awk/>.
``` {.sourceCode .perl}
LAMMPS FENE chain data file
140 atoms
134 bonds
0 angles
0 dihedrals
0 impropers
...
```

So the data file didn’t have any information about angle, dihedrals in the systems. Lets now consider possible ways of tackling this problem.

``` {.sourceCode .perl} #!/bin/bash

# here is the main file for creating a polymer melt

# input : DATA_file_input = initial melt created by chain.f, or the equil.data - equilibrated melt by in.kremer that has only fene Bonds

# output : DATA_file_output = final melt that has angles ( if required dihedrals as well)

#set input and output datain=”equil.data” dataout=”tmp.data” #get information about number of atoms and number of bonds STR=$(less $datain | grep atoms) LIT=$(echo $STR | grep -o [0-9]*) natoms=$LIT

# calculate number of angles, knowing how many atoms and bonds we have

STRb=$(less $datain | grep bonds) LITb=$(echo $STRb | grep -o [0-9]*) nbond=$LITb

let nangle=(2*$nbond-$natoms) let ndih=0

#write data file in required format cp $datain 2.info

echo “LAMMPS data file

$natoms atoms $nbond bonds $nangle angles

1 atom types 1 bond types 1 angle types “ > 1.info

sed -i ‘1,/bond types/d’ 2.info sed -i ‘1d’ 2.info sed -i ‘/Velocities/,/Bonds/{//!d}’ 2.info sed -i ‘/Velocities/d’ 2.info

cat *.info > temprary_file rm *.info

#creating additional files for generating angle bond topology echo “1 * * * * * “ > angles_by_type.txt

# sed -i ‘/improper/d’ result_without.data

bash gen_all_angles_topo.sh temprary_file $dataout rm temprary_file rm angles_by_type.txt

#replace multiple blanc lines with a single one sed -i ‘/^$/N;/^\n$/D’ $dataout

sed -i ‘/dihedral/d’ $dataout

sed -i ‘/improper/d’ $dataout

sed -i ‘/Bond Coeffs/,/Atoms/{//!d}’ $dataout sed -i ‘/Bond Coeffs/d’ $dataout

```
After applying this script we get the following data file:
``` {.sourceCode .perl}
LAMMPS data file
140 atoms
134 bonds
128 angles
1 atom types
1 bond types
1 angle types
...
```