Supplementary Information
Role of Short Chain Branching in Polymer Structure and Dynamics
Jun Mo Kim and Chunggi Baig*
Department of Chemical Engineering, School of Energy and Chemical Engineering, Ulsan
National Institute of Science and Technology (UNIST), Ulsan 689-798, South Korea
*
Author to whom correspondence should be addressed.
Chunggi Baig
Email: cbaig@unist.ac.kr
Phone: +82-52-217-2538
Fax:
+82-52-217-2649
1
NEMD Simulation
At microscopic (atomistic) level of description, the linear, the H-shaped, and the SCB,
polymers of the same molecular formula of C178H358 are schematically depicted in Scheme 1.
An SCB polymer molecule consists of 10 branches and 5 CH2 units per each branch, and an
H-shaped chain has 4 branches and 25 CH2 units per each branch. Note that the longest chain
dimension of both the SCB and H-shaped polymers corresponds to 128 CH2 units.
Linear
C178H358
MW
C178
1st Junction
H-shaped
2nd Junction
Arm length
(CH2)25
Backbone length
(CH2)78
MW
C178
10th Junction
1st Junction
Arm length
(CH2)5
SCB
Backbone length
(CH2)128
Scheme 1. Schematic representation of molecular architecture of each polymeric system. The
linear, the H-shaped, and the SCB polymers all have the same molecular formula C178H358.
The NEMD simulations of all the systems were conducted in the NVT Canonical
2
ensemble using the p-SLLOD equations of motion1-3 for an arbitrary homogeneous flow with
the Nosé-Hoover thermostat4,5:
q ia 
p ia
 qia  u
mia
pia  Fia  pia  u  mia  u  u   pia

p
Q
,
pia2
p  
 DNk BT ,
i
a mia
(S1)
Q  DNk BT 2
where the subscripts i and a denote specific molecules and atoms, respectively. The N, V, and
T represent the total number of atoms, the volume of system, and the temperate, respectively.
The qia, pia, and Fia are the position, momentum, and force vectors of atom a in molecul
e i, of mass m i a . The  and p are the coordinate- and momentum-like variables,
respectively, of the Nosé-Hoover thermostat, and Q is the mass parameter of the thermostat
with its relaxation time parameter . kB refers to the Boltzmann constant and D denotes the
dimensionality of the system. For planar Couette flow, the velocity gradient tensor u is
expressed as
 0 0 0


u    0 0  ,
 0 0 0


(S2)
where  is the applied shear rate, and x and y are the flow and the velocity gradient
direction, respectively. The Lees-Edwards boundary condition6 was employed for this simple
shear flow. The set of evolution equations was numerically integrated using the reversible
Reference System Propagator Algorithm (r-RESPA)7 with two distinct time scales for an MD
step: 0.48 fs for three bonded (the bond-stretching, bond-bending, and bond-torsional) and
3
2.39 fs for two non-bonded inter- and intra-molecular Lennard-Jones (LJ) interactions (see
the Supporting Information of Ref. 9 for the r-RESPA formula). A sufficient number of
polymer chains was used for each system (i.e., 216, 162, and 162 chains for linear, H-shaped,
and SCB polymers, respectively) in a rectangular simulation box
(x×y×z) of
263.55Å×65.89Å×65.89Å for the linear and 197.67Å×65.89Å×65.89Å for both the H-shaped
and the SCB systems, enlarged in the flow direction to avoid system-size effects, especially
when exposed to strong flow fields in which chains become highly stretched and aligned. The
flow strength was applied across a wide range (i.e., 0.2 ≤ Wi ≤ 3000), where the Weissenberg
number (Wi) was defined as the product of the longest relaxation time  of the system and the
imposed strain rate;   15.6  1.0 ns for the linear,   33  1.5 ns for the H-shaped, and
  21.7  1.5 ns for the SCB polymer, as estimated by the integral below the stretchedexponential curve10-12 describing the time autocorrelation function of the chain end-to-end
unit vector.
The well-known TraPPE united atom model8 was adopted for linear and branched
alkanes because of their broad usage and accuracy for simulating rheological behavior and
properties of linear and branched alkanes. But, a harmonic potential function was used for the
bond-stretching interaction instead of rigid bond in the original model to diminish problems
related to numerical integrations at small time scales. In the TraPPE model, intermolecular
and intramolecular non-bonded interactions are represented by a 6-12 Lennard-Jones (LJ)
potential:
 
U LJ (rij )  4 ij  ij
 rij

4
12
   ij
  
  rij



6

,


(S3)
where ij=(ij)1/2 and ij=(i+j)/2 adopting the standard Lorentz-Berthelot mixing rules for
interactions between atomistic units i and j. The LJ size parameters  CH ,  CH 2 , and  CH 3
were set equal to 4.68 Å, 3.95 Å, and 3.75 Å, respectively, and the energy parameters
 CH k B ,  CH k B , and  CH k B equal to 10 K, 46 K, and 98 K, respectively. A cut-off
2
3
distance equal to 2.5  CH 2 was used in all the simulations. The intramolecular LJ interaction
was active only between atoms separated by more than three bonds along the chain. The three
bonded interactions (bond-stretching, bond-bending, and bond-torsional) were modeled by as
follows:
bond-stretching: U str (l ) 
2
1
kl  l  leq  ,
2
(S4)
where the bond-stretching constant kl / k B = 452,900 K/Å2 and the equilibrium bond length
leq = 1.54 Å,
2
1
k    eq  ,
2
bond-bending: U ben ( ) 
3
bond-torsional: U tor ( )   am  cos   ,
m
(S5)
(S6)
m0
where the parameter values for each interaction mode can be found in Ref. 8.
The pressure tensor was calculated using the Irving-Kirkwood expression13

1
V
 p ia p ia

 qia Fia 
 mia

(S7)
 xy
,

(S8)
 
i
a
The shear viscosity was calculated as
 
5
BD Simulation
The mesoscopic level of description is depicted in Scheme 2 where each polymer chain is
coarse-grained into beads and springs. The bead-spring BD simulations for the linear, the Hshaped, and the SCB polyethylene melts under shear flow were carried out using Finite
Extensible Nonlinear Elastic (FENE) force law.
FS,i  Khi Qi ,
hi 
1
1 Q
i 2
,
i  1, 2,
, N 1
(S9)
/ bs
Qi
bs
31 beads & 30 springs
1 bead
&
1 spring
Qbi
bs
Qi
21 beads & 20 springs
5 beads
&
5 springs
11 beads & 10 springs
6
Scheme 2. Schematic representation of coarse-grained bead-spring model. The blue circles
and the red coils represent beads and FENE springs, respectively. The Qi and Qbi denote the
ith connector vector in the chain backbone and branch, respectively. The bs represents the
maximum extensibility of each spring.
where Qi and K denote the connector vector and elastic spring constant of the ith spring,
respectively. bs refers to the maximum extensibility of all springs and hi defines specific form
of the spring. N represents the total number of beads in a chain and the number of springs is
thus equal to N1. The number of beads and springs were properly chosen to obtain accurate
rheological properties and structures according to a previous study16,17: As five CH2 units
were coarse-grained into one bead, total 30 springs and 31 beads represented a bead-spring
model of each molecule for all the linear, the H-shaped, and the SCB systems in this work. As
shown in Scheme 2, the SCB backbone consists of 20 springs and 21 beads, and each of 10
branches is represented by 1 bead and 1 spring connected to the backbone. For the H-shaped
polymer, each of 4 branches was made of 5 beads and 5 springs connected to the backbone
ends. Based on the kinetic theory14, the evolution equation of a connector vector
Q
i

 ri 1  ri  has been driven from the force balance on a bead as,
N 1

1
S 
1
i 1
i
dQi    Qi   
,0  Aik F , k  dt    ,0  dW  dW 
k 1


 2

Aik   1
 0

if i  k  0
if i  k  1
otherwise
7
(S10)
The Aik are the elements of the Rouse matrix.18 For more details, you can refer previous
studies.14-16 Eq. (S10) can be made dimensionless with transformations
Qi 

L
L
kBT i
Q , t  t ,  R  0   R , dWi  dt  dWi , FS  KkBT  FS
4 K Vc
Vc
K
(S11)
where the terms with tilde are dimensionless. The resulting dimensionless equation for the
connector vector is found to be

1
dQi    Qi 
4 R

N 1
 A F
k 1
S
,k
ik

1
dWi 1  dWi
 dt 
2 R



(S12)
For junction points in the backbone, the contributions by the branches were added to Eq.
(S12):

1
dQi    Qi 
4 R

N 1
 A F
k 1
 1
A0  
 1
ik
S
,k


A0
1
FS,branch  dt 
dWi 1  dWi
4 R  brfac
2


R


(S13)
if a branch is attached to ri of Qi
if a branch is attached to ri 1 of Qi
where the ‘brfac’ represents branch randomness factor. Depending on the location of branches
in the connector vector
Q
i

 ri 1  ri  of the backbone, the A0 term change its sign. For
branches, the backbone contributions should be added to Eq. (S12) as follows:


1
1
,
S
dQi ,branch    Qi ,branch 
FS,i,branch 
FS,backbone
2  F ,backbone1  dt
4 R  brfac
4 R




(S14)
1
1
1

dWi ,branch

dWi ,backbone
2 R  brfac
2 R
All the evolution equations of the connector vectors were integrated with a modified semiimplicit predictor-corrector method in this work [the original semi-implicit predictor8
corrector method can be found in Ref. 15]. The predictor step was described as follows:


1
Qˆ i ,t  t  Qi ,t    Qi ,t 
 FS,i,t1  2FS,i,t  FS,i,t1   t
4 R




1

dWi 1  dWi
2 R

(S15)

where Qi ,t and FS,i,t are the connector vector and the elastic spring force for the ith
segment at time t. The first corrector step was carried out by
Qi ,t  t 

t S
1
F ,i  t   t   Qi ,t     Qi ,t  ˆ Qˆ i ,t  t 
 FS,i,t1  2FS,i,t  FS,i,t1 

4 R
4 R





1
 FˆS,i,t1 t  FˆS,i,t1 t
4 R

 t
1
dWi 1  dWi
 
2 R
 2

(S16)

Rearrangement of Eq. (S16) gives rise to a cubic equation for the magnitude of Qi as
3
2

t  i
Qi ,t  t  R Qi ,t  t  bs 1 
 Q ,t  t  bs  R  0
4


R 
(S17)
where R is the magnitude of the entire right hand side of Eq. (S16). By solving Eq. (S17), we
can obtain a unique root between 0 and
bs , with the value of Qi ,t  t never going beyond
bs . The second predictor step was given as
Qi ,t  t 

 t S , t  t
1
F ,i  Qi ,t    Qi ,t    Qi ,t  t  
 FS,i,t1  2FS,i,t  FS,i,t1 

4 R
4 R


 t
1
1
 FS,i,t1 t  FS,i,t1  
dWi 1  dWi
4 R
2 R
 2




Rearranging Eq. (S18), we can also obtain a cubic equation for magnitude of Qi ,t  t
9
(S18)
3
2

t  i
Qi ,t  t  R Qi ,t  t  bs 1 
 Q ,t  t  bs  R  0 .
4


R 
Solving Eq. (S19), the results for Qi ,t  t , whose values are again always below
(S19)
bs , can
be obtained.
and Qi ,t  t , we can set the residual ε as,
Given all the information of Qi ,t  t

2
Ns
 Q
i
i 1
,t  t
 Q ,t  t 
i

.
(S20)

If the residual is greater than a specific tolerance   105 , Qi ,t  t was copied onto Qi ,t  t
and the second predictor step was repeated until a full convergence. In the BD simulations,
the stress tensor was calculated by the Kramers-Kirkwood expression14
N
    hi Qi Qi
with hi 
i 1
1
2
1  Q i / bs
.
(S21)
For a linear chain, the maximum extensibility parameter of the entire molecule (bm) and that
of each spring (bs) were calculated as
bm , Linear 
3 R
2
Rete
2
max

eq
3  52259.95 A
2
2
bs , Linear 
 43.61 ,
3595.02 A
bm, Linear
N
2

43.61
 0.04846 ,
302
(S22)
Linear
Qmax
 bs , Linear  0.22 .
where Rmax is the fully stretched chain length of the linear C178H358. For the SCB chain, they
were computed as,
10
2
bm , SCB
3 R2
3  26904.81A
 2 max 
 31.455 ,
2
Rete
eq
2566 A
bs ,SCB 
bm,SCB
N
2

31.455
 0.0786387 ,
202
(S23)
SCB
Qmax
 bs , SCB  0.28 .
where Rmax is the fully stretched length of the chain backbone (C128H258).
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