Preparation of Entangled Polymer Melts of Various 

Architecture for Coarse-grained Models

by Yelena R. Sliozberg and Jan W. Andzelm

ARL-TR-5679 September 2011

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Army Research Laboratory

Aberdeen Proving Ground, MD 21005

ARL-TR-5679 September 2011

Preparation of Entangled Polymer Melts of Various 

Architecture for Coarse-grained Models

Yelena R. Sliozberg

Oak Ridge Institute for Science and Education

Jan W. Andzelm

Weapons and Materials Research Directorate, ARL

Approved for public release; distribution unlimited. 
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4. TITLE AND SUBTITLE

Preparation of Entangled Polymer Melts of Various Architecture for Coarsegrained Models

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6. AUTHOR(S)

Yelena R. Sliozberg and Jan W. Andzelm

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14. ABSTRACT 

Most of the practically useful polymers have molecular weight range from 20,000 to 200,000. Mechanical properties of these 

polymers are dominated by topological constraints or entanglements. Direct brute-force equilibration of well-entangled 

polymers is still unattainable because of slow reptation dynamics. In this technical report, we have introduced a new fast 

protocol to prepare well-equilibrated entangled polymeric systems of various compositions and architectures. Our algorithm is 

simple to implement and it is general for any coarse-grained polymer model. The present preparation method exploits two 

programs: gel and Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). This report presents a theory 

overview and a manual how to use the method.

15. SUBJECT TERMS

Ammunition, coarse-grained model, polymer builder, LAMMPS

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Yelena R. Sliozberg

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iii

Contents

List of Figures iv

Acknowledgments v

1. Overview 1

2. Models 1

2.1 Kremer-Grest Model .......................................................................................................1

2.2 DPD Polymer Model with mSRP....................................................................................2

3. Generation of Initial Structure Using gel 3

3.1 Theoretical.......................................................................................................................3

3.2 Software Description.......................................................................................................3

4. Equilibration of the Initial Structures 12

5. Results 15

6. References 17

7. Distribution 17
iv

List of Figures

Figure 1. Schematic representation of bead-spring branched block copolymer. Different 

colors correspond to the different types of beads. The picture corresponds to example 1. .....4

Figure 2. Bead-spring dendrimer. Different colors correspond to the different types of beads. 

The picture corresponds to example 2. ......................................................................................9

Figure 3. Bead-spring dendrimer/comb polymer. Different colors correspond to the different 

types of beads. The picture corresponds to example 3. Some side chains are built to be 

rigid..........................................................................................................................................12

Figure 4. Evolution of mean square internal distances for a melt of chain length N = 500 

throughout our preparation procedure. Blue, red and green symbols correspond to the 

chains after initial 1000 τDPD simulation with the soft (DPD) potential, after fast “pushoff” and after 10000 τLJ simulation with Lennard-Jones potential, respectively. The black 

symbols denote our target function obtained for the equilibrated melt, where simulation 

time is 

LJ t 

6

 610 . .................................................................................................................16
v

Acknowledgments

This research was supported in part by an appointment to the Postgraduate Research 

Participation Program at the U.S. Army Research Laboratory (ARL) administered by the Oak 

Ridge Institute of Science and Education through an interagency agreement between the U.S. 

Department of Energy and ARL.
vi

INTENTIONALLY LEFT BLANK.
1

1. Overview

Equilibration of entangled polymers is nontrivial even for a case of coarse grained chains 

because of slow reptation dynamics exhibited by high molecular weight chains, where center-ofmass diffusivity, D scaled with polymer length, N as 

2.3 D  N . Even with modern day parallel 

computers, brute-force equilibration of well-entangled branched polymers is prohibitively 

expensive. Standard method in which one starts with an ensemble of chains with the correct endto-end distance arranged randomly in the simulation cell and introduces the excluded volume 

rapidly, leads to deformation on the intermediate length scales. Another set of long-standing 

methods includes the chain connectivity altering algorithms. These bridging methods employ a 

set of complex Monte Carlo (MC) moves that can be difficult to implement for branched 

polymer architectures and these MC moves have very low acceptance ratio (1, 2). 

New preparation protocol has been established to overcome the described difficulties. This

protocol is general to any polymer architecture and does not bring any deformation of polymer 

chains. Short simulation runs with soft potential used in Dissipative Particle Dynamics (DPD) (3) 

is performed for an ensemble of polymer chains with the correct end-to-end distance. After a 

gradual increase of the strength of DPD potential, short simulation with target coarse-grained 

potential is performed. 

In this report, we adopt our method for two coarse-grained models 1) Kremer-Grest model (4) 

and 2) DPD model with Segmental Repulsive Potential (mSRP) (5). However, the described 

protocol is general and applicable for any coarse-grained model. 

The present preparation method exploits two programs: gel and Large-scale Atomic/Molecular 

Massively Parallel Simulator (LAMMPS). Gel is an in house written C program of coarsegrained polymer builder, and LAMMPS is a molecular dynamics program from Sandia National 

Laboratories (6). 

2. Models

2.1 Kremer-Grest Model

Kremer-Grest model: The pair interaction between topologically nonconnected particles is 

described by the standard truncated Lennard-Jones pair potential: 

 

































 















 











 













1 2 6 1 2 6

4

c c

LJ

r r r r

U r

   



, where ε is the depth of the potential well and σ is 

the distance where interparticle force is zero. rc represents the cutoff distance. 



1/ 6

rc 2

is 
2

chosen, yielding so-called Weeks-Chandler-Andersen excluded volume potential, UWCA. 

Topologically bound monomers interact according to the standard FENE/Lennard-Jones bonded 

potential, UFENE/LJ. U r U r U r FENE / LJ



FENE  WCA

and 

 

































  

2

0

2

0

ln 1

2 R

r

R

a

U r FENE

, where 

standard parameter values for of R0 = 1.5 σ and a = 30 ε/σ2.

To model different flexibility of polymer chains, a bending potential of 

form

      U r  k 1cos bend

is employed, where



is the angle between two neighboring bonds 

and 



k

is the bending constant. The detailed model description can be found in the original paper 

(4).

DPD model used as the first step. The pair interaction between topologically nonconnected 

particles is described by soft potential 

 

2

1

2

















 

c

DPD

DPD

r

a r

U r

, where adpd = 25 kBT and rc =1 in 

DPD units of length are standard DPD parameters (3). Topologically bound monomers interact 

according to the standard FENE/Lennard-Jones bonded potential, UFENE/LJ. 

U r U r U r FENE / LJ

FENE  WCA

and 

 

































  

2

0

2

0

ln 1

2 R

r

R

a

U r FENE

, where standard parameter 

values for of R0 = 1.5 σ and a = 30 ε/σ2. 

2.2 DPD Polymer Model with mSRP

The pair interaction between topologically nonconnected particles is described by soft potential 

 

2

1

2

















 

c

DPD

DPD

r

a r

U r

where adpd is the maximum repulsion in kBT units and rc =1 in DPD

units of length. Topologically bound monomers interact according to the harmonic potential, 

Uharm.    

2

0

2

r r

k

U r harm   , where k is the harmonic constant and r0 is the equilibrium bond 

distance. In order to avoid bond shrinkage and make a distance between bonded particles close to 

the distance between nonbonded particles, the values of k = 225 kBT/ρ and r0 = 0.85 rc are used.

Chains crossing as a result of soft DPD potential is greatly reduced by introducing the Segmental 

Repulsive Potential (mSRP),

 

2

1

2

















 

c

SRP

SRP d

a d

U r

. The potential is effective for any pair of 

bonds of the distance between midpoints of interacting bonds, d within cutoff, dc with exception 

of neighboring bonds (no repulsion). The repulsion is equally divided between beads composing 

the bond. The values of aSRP = 100 kBT and dc = 0.8 rc significantly decrease the number of 

topological violations. As a result of absence of bond repulsion between two neighboring bonds 

and soft nonbonded potential, the value of average bond angle 



becomes greater than π/2. 

Addition of bending potential 

      U r  k 1cos bend

 resolves this problem. The value 

k  2
3

kBT is enough to compensate the action of mSRP potential and polymer behaves as a flexible 

chain. In depth explanation can found is in the original paper (5)

During all steps of our melt preparation we have used DPD thermostat on pairwise interactions. 

In a DPD simulation of polymers, particles interact with each other via a pairwise, two-body, 

short-ranged force, F, that is written as the sum of a conservative force, F

C

, dissipative force, F

D

, 

and random force, F

R

, as follows: 

   .

  

  

j i

R

ij

j i

D

ij

j i

C Fi Fij F F

For a more detailed description of 

the DPD thermostat, see the original papers by Hoogerbrugge and Koelman (7) and Español and 

Warren (8).

3. Generation of Initial Structure Using gel

3.1 Theoretical

To equilibrate a polymer structure, it is necessary to generate an ensemble of polymer chains 

with the correct end-to-end distance (1). 

Polymer chains are generated inside a periodic cubical box with the correct end-to-end distance 

( )  1

2 2 R N  l CN N  on large length scales (N is large), where N is a number of beads, l is a 

bond distance and 

CN

is Flory characteristic ratio and 

CN C

at 

N . For a Gaussian chain 

with a small bond angle



the average cosine is 

1

1

cos











C

C

 . For example, fully flexible 

Lennard-Jones chains with 



1/ 6

rc 2

have

C1.88

and DPD/SRP polymers have 

C1.32 . 

Chains are built via a random walk in 3D space with either a desired angle or a random

angle

  

, that is sampled around

cos . Bead number density of Kremer-Grest system, 

ρ = 0.85σ–3and of DPD/SRP system ρ = 3rc

–3

, which is a standard density for coarse-grained 

polymeric melts for these models, respectively. 

3.2 Software Description

Program gel generates melts, blends and solutions. Polymer chains may include several blocks of 

different types of beads and/or angles. Different types of beads are used to drive a phase 

separation. Possible architectures include unconnected beads, linear chains, star-branched 

polymers with equal and unequal arm lengths, combs, dendrimers and hybrid star/comb/denrimer 

structures. Chains are modeled as flexible, semi-flexible or rigid. Figure 1 shows the schematic 

picture of polymer architectures.
4

Figure 1. Schematic representation of bead-spring branched 

block copolymer. Different colors correspond to the 

different types of beads. The picture corresponds to 

example 1.

Source code: gel.c.

Arguments: Lx, Ly, Lz, R, Nrep. Where Lx, Ly, Lz are the dimensions of a periodic simulation box, 

R is equilibrium length of the bond and is Nrep a number of replicas. Value of R is taken with 

respect to the bond potential, for example FENE bond has R ≈ 0.9655σ. 
5

Input: input is a topology file (written by a user)

Input file Commentaries

# atom

#polymer1

⁪

 

......... 

sub

Number of molecular species

Number of bead types

Number of angle types

Number of molecules of polymer 1

Number of arms (sidechains) in a polymeric units at the same branch point and number of 

repeated units

Number of blocks, repeated units and an indicator of subbranches in the first arm (sidechain). 

The first arm always represents a main chain (backbone). The indicator equal 0 means that 

there are no subbranches. The indicator =1 means that there are subbranches at the end of 

this chain (branch point). 

Type and number of beads and cosine, case and type of angle, in the first block. Case = 0 

is a random angle (with average cosine), case = 1 is a regular angle and case =2 is a special 

case for barbed wire ( all 3 angles are 120°)

Number of arms (sidechains) in a substar and number of repeated units

Number of blocks, repeated units and an indicator of subbranches in the first substar. The 

first arm always represents a main chain. The indicator is always 0. 

Type and number of beads and cosine, case and type of angles in the first block. Case = 0 is a 

random angle (with average cosine), case = 1 is a regular angle and case =2 is a special case 

for barbed wire ( all 3 angles are 120 °)

Output: data_i.chain and snap_i.xyz files, where 

i  0,...Nrep 1. data_i.chain is topology file 

for LAMMPS software. snap_i.xyz is xyz file for Visual Molecular Dynamics (VMD) software 

(for a visual checking). VMD is a molecular visualization program for displaying, animating, and 

analyzing large biomolecular systems using 3-D graphics and built-in scripting.

To run the code, a user has to have gel.c and input files in a working directory. gcc compiler 

could be used for compiling, as gcc gel.c –o gel –lm. The user has to write input to define a 

desirable topology. 

In the following example the code generates 3 replicas of a blend of linear flexible chains,

composed of 500 chains of 250 beads and 250 chains of 500 beads. The periodic box size is 

66.5×66.5×66.5 and bond length is 0.9655.

#molecules 2

#atomtypes 1

#angletypes 0

#polymer1 500

polymer1 1 1
6

chain1 1 1 0

1 250 -0.303135889 0 0

#polymer2 250

polymer2 1 1

chain1 1 1 0

1 500 -0.303135889 0 0

To run the user has to type ./gel 66.5 66.5 66.5 0.9655 3. This action will produce the following 

output files: data_0.chain, data_1.chain, data_2.chain and snap_0.xyz, snap_1.xyz, snap_2.xyz. 

The following are examples of input files for several possible branched structures:

Example 1. “Barbed wire” polymer (see figure 1).

#molecules 1 

#atomtypes 2 

#angletypes 2 

#polymer1 1 

polymer1 2 2

chain1 1 1 0 

2 2 -0.303135889 0 0

chain2 2 1 1 

1 1 -0.303135889 0 0

1 1 -1.0 1 1

branch1 2 1

subchain1 2 1 0 

1 1 -0.5 2 2

1 1 -1.0 1 1

subchain2 2 1 0 

1 1 -0.5 2 2

1 1 -1.0 1 1

Example 2. Dendrimer. Some of angles are set to 180° to make visualization easier (see figure 

2).

#molecules 1

#atomtypes 5

#angletypes 2

#polymer1 1

polymer1 6 1

chain1 1 1 1

1 10 -1.0 1 1

chain2 1 1 1

1 10 -1.0 1 1

chain3 1 1 1

2 5 -1.0 1 1

chain4 1 1 1

2 5 -1.0 1 1
7

chain4 1 1 1

4 10 -1.0 1 1

chain5 1 1 1

4 10 -1.0 1 1

branch1 3 1

subchain1 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain2 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain3 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

branch2 3 1

subchain1 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain2 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain3 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

branch3 5 1

subchain1 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain2 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain3 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain4 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1
8

3 4 -0.303135889 0 0

subchain5 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

branch4 5 1

subchain1 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain2 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain3 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain4 3 1 0 3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain5 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

branch5 2 1

subchain1 3 1 0

5 1 -0.5 1 2

5 1 -1.0 1 1

5 4 -0.303135889 0 0

subchain2 3 1 0

5 1 -0.5 1 2

5 1 -1.0 1 1

5 4 -0.303135889 0 0

branch6 21

subchain1 3 1 0

5 1 -0.5 1 2

5 1 -1.0 1 1

5 4 -0.303135889 0 0

subchain2 3 1 0

5 1 -0.5 1 2

5 1 -1.0 1 1

5 4 -0.303135889 0 0
9

Figure 2. Bead-spring dendrimer. Different colors

correspond to the different types of beads. 

The picture corresponds to example 2.

Example 3. Comb/dendrimer. Some of angles are set to 180 and 120° to make visualization 

easier (see figure 3).

#molecules 1

#atomtypes 6

#angletypes 2

#polymer1 1

polymer1 7 10

chain1 1 1 0

6 40 -0.303135889 0 0

chain2 1 1 1

1 10 -1.0 1 1

chain3 1 1 1

1 10 -1.0 1 1

chain4 1 1 1

2 5 -1.0 1 1

chain5 1 1 1

2 5 -1.0 1 1

chain6 1 1 1

4 10 -1.0 1 1

chain7 1 1 1

4 10 -1.0 1 1

branch2 3 1

subchain1 3 1 0

3 1 -0.5 1 2
10

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain2 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain3 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

branch3 3 1

subchain1 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain2 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain3 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

branch4 5 1

subchain1 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain2 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain3 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain4 3 1 0

3 1 -0.5 1 2

3 1-1.0 1 1

3 4 -0.303135889 0 0

subchain5 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

branch5 5 1

subchain1 3 1 0
11

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain2 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain3 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain4 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

subchain5 3 1 0

3 1 -0.5 1 2

3 1 -1.0 1 1

3 4 -0.303135889 0 0

branch6 2 1

subchain1 3 1 0

5 1 -0.5 1 2

5 1 -1.0 1 1

5 4 -0.303135889 0 0

subchain2 3 1 0

5 1 -0.5 1 2

5 1 -1.0 1 1

5 4 -0.303135889 0 0

branch7 2 1

subchain1 3 1 0

5 1 -0.5 1 2

5 1 -1.0 1 1

5 4 -0.303135889 0 0

subchain2 3 1 0

5 1 -0.5 1 2

5 1 -1.0 1 1

5 4 -0.303135889 0 0
12

Figure 3. Bead-spring dendrimer/comb polymer. Different colors correspond to the

different types of beads. The picture corresponds to example 3. Some side

chains are built to be rigid.

4. Equilibration of the Initial Structures

Short simulation runs with soft potential used in DPD have been performed for an ensemble of 

polymer chains with the correct end-to-end distance. DPD potential allows chains to pass 

through each other to speed up the polymer dynamics. Simulation with DPD potential deforms 

the chains on the intermediate length scales. However, this deformation is fully resolved after 

500 τDPD. After this step, the following procedures are adopted for two coarse-grained models 1) 

Kremer-Grest model (4) and 2) DPD model with Segmental Repulsive Potential (mSRP) (5)

Kremer-Grest model. Next step is a gradual increase of the strength of DPD potential in order 

to increase distance between nonbonded monomers or “push-off”. Finally, DPD potential is 

substituted to full Lennard-Jones potential. This step does not significantly alter the structural 

properties of the melt. Finally, the melt is simulated with Lennard-Jones potentials for time equal 

to 10000 τLJ to remove DPD alterations of structural properties on the short length scale. 

DPD with mSRP model. mSRP and bending potential (see section 2) are added and the melt 

are equilibrated for time = 1000 τDPD/mSRP to remove DPD alterations of structural properties on 

the short length scale. 

To equilibrate the initial stricture one has to use LAMMPS software. To run LAMMPS, a user 

has to copy data.chain produce by gel program in a working directory. The following are input 

scripts for LAMMPS for Kremer-Grest and DPD with mSRP models, respectively.
13

Kremer-Grest model.

units lj

atom_style bond

special_bonds lj/coul 0 1 1

read_data data.chain

neighbor 0.4 bin

neigh_modify every 1 delay 1

communicate single vel yes

bond_style fene

bond_coeff 1 30.0 1.5 1.0 1.0

dump mydump all custom 100000 chain_un*.xyz id type xu yu zu

fix 1 all nve

timestep 0.01

thermo 1000

pair_style dpd 1.0 1.0 122347

pair_coeff * * 25 4.5 1.0

velocity all create 1.0 17786140

run 50000

write_restart my_restart_dpd

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

pair_coeff * * 150.0 4.5 1.0

velocity all create 1.0 15486120

run 50

pair_coeff * * 200.0 4.5 1.0

velocity all create 1.0 17986120

run 100

pair_coeff * * 250.0 4.5 1.0

velocity all create 1.0 15006120

run 100

pair_coeff * * 500.0 4.5 1.0

velocity all create 1.0 15087720

run 100

pair_coeff * * 1000.0 4.5 1.0

velocity all create 1.0 15086189

run 100

write_restart my_restart_dpd1

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 15086120
14

run 50

velocity all create 1.0 15086121

run 50

velocity all create 1.0 15086111

run 50

write_restart my_restart_push

velocity all create 1.0 15086125

run 1000000

write_restart my_restart_lj

DPD with mSRP model.

units lj

atom_style angle

special_bonds lj/coul 1 1 1

newton on off

read_data data.chain

neighbor 2.5 bin

neigh_modify every 1 delay 1

communicate single vel yes

# syntax for bond-bond pair_style

# use hybrid/overlay with regular dpd

# dpdbb <cutoff> <distance: min=0 midpoint=2> <search for top volations: no=0 ye

s=1> <skin distance>

bond_style harmonic

bond_coeff 1 225.0 0.85

pair_style dpd 1.0 1.0 373692

pair_coeff * * 60.0 4.5 1.0

velocity all create 1.0 17786140

dump mydump all custom 10000 chain_dpd*.xyz id type x y z

thermo 1000

fix 1 all nve

timestep 0.01

run 50000

write_restart my_restart_dpd

velocity all create 1.0 40783012

undump mydump

pair_style hybrid/overlay dpd 1.0 1.0 373692 dpdbb 0.8 2 0 2.5

# syntax for bond-bond pair_coeff

# bondtype bondtype a0 cutoff(optional)

pair_coeff * * dpdbb * * 100.0 0.8

pair_coeff * * dpd 60.0 4.5 1.0

angle_style cosine

angle_coeff 1 2.0
15

dump mydump1 all custom 10000 chain*.xyz id type x y z

restart 10000 periodic_restart

run 100000

write_restart my_restart_1

5. Results

In order to characterize the chain conformations in the melt after push-off, the authors performed 

calculations of mean square internal distances of chains

R n / n

2

is averaged over all possible 

combinations of segments of size 

n  i  j

along the chains, where 

i  j1,N

are monomer 

indices. This metric is an excellent indicator of chain configuration at all length scales. Code 

msid_1.c is used to calculate this function. Arguments are N, a, Ntot, file1, file2, where N is a 

number of monomers per chain, a is a size of the cubical periodic box, Ntot is a total number of 

particles, file1 is an input name and file2 is an output name. For example, ./msid_1 500 66.5 

250000 snap_lj.xyz lj_msid. Note: input files should be in VMD xyz format. To show that our 

preparation technique produced a sample with topology equivalent to the well-equilibrated 

structure, we applied our method to a melt of M = 500 chains of N = 500 beads. Simulation box 

was 66.5 ×66.5×66.5 σ3. We compared a topology of 1) the melt produced by our method or 2) 

the melt equilibrated by running MD simulation for time long enough that the chains moves 3 

times their own size (the run time was 

LJ t 

6

 6.510

). Figure 4 shows the results
16

Figure 4. Evolution of mean square internal distances for a

melt of chain length N = 500 throughout our preparation

procedure. Blue, red and green symbols correspond to the

chains after initial 1000 τDPD simulation with the soft (DPD)

potential, after fast “push-off” and after 10000 τLJ simulation

with Lennard-Jones potential, respectively. The black

symbols denote our target function obtained for the equilibrated

melt, where simulation time is 

LJ t 

6

 610 .

Using geometrical analysis, the authors estimated the entanglement length, Ne by running 

Z1code (9). The authors found that Ne for an equilibrated and prepared system is 83 and 84, 

respectively.

The results demonstrate the excellent agreement between structural properties of the melts 

generated via our method and brute-force equilibration.
17

6. References

1. Auhl, R.; Everaers, R.; Grest, G. S.; Kremer, K.; Plimpton, S. J. J. Chem. Phys. 2003, 119, 

12718–12728.

2. Subramanian, G. J. Chem. Phys. 2010, 133, 164902.

3. Groot, R. D.; Warren, P. B. J. Chem. Phys. 1997, 4423–4435.

4. Kremer, K.; Grest, G. S. J. Phys.: Condens. Matter 1990, 2, SA295–SA298.

5. Sirk, T. W.; Sliozberg, Y. R.; Brennan, J. K.; Andzelm J. W.; Lísal, M. to be submitted

6. Plimpton, S. J. Comp. Phys. 1995, 117, 1–19.

7. Koelman, J. M. V. A.; Hoogerbrugge, P. J. Europhys. Lett. 1993, 21, 363–368.

8. Español, P.; Warren, P. B. Europhys. Lett. 1995, 30, 191–196.

9. Kröger, M. Comput. Phys. Commun. 2005, 168, 209-232.
18

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