Supplementary material
Coupling between intra- and inter-chain orderings in flow-induced
crystallization of polyethylene: A non-equilibrium molecular
dynamics simulation study
Junsheng Yang,a), b) Xiaoliang Tang,a) Zhen Wang,a) Tingyu Xu,a) Fucheng Tian,a)
Youxin Ji,a) Liangbin Lia )*
a)
National Synchrotron Radiation Lab and CAS Key Laboratory of Soft Matter Chemistry, University
of Science and Technology of China, Hefei, China
b)
Computational Physics Key Laboratory of Sichuan Province, Yibin University, Yibin, China
* Email：[email protected]
Forced Field
All the simulations are performed with the molecular dynamics simulations.
Details of the force field have been provided as follows：
Etotal  Ebond (r )  Eangle ( )  Edihedtal ( )  EvdW (r )
(1)
where Ebond is the 2-body potential for bond length r, Eangle the 3-body potential for
bond angle θ, Edihedral the 4-body potential for dihedral (torsion) angle ø, and EvdW
another 2-body potential for nonbonded distance r. The bond length, bond angle and
van der Waals （vdW） interactions adopted are YPS force filed.1 In the recent work
of Hossain et al2 and Huang et al3, the dihedral angle rotation potential has been made
some changes. The potential functions for amorphous polyethylene as shown in the
equations below:
Ebond (r ) 
1
K b (r  r0 ) 2
2
S1
(2)
Eangle ( ) 
1
K (   0 ) 2
2
(3)
3
Edihedral ( )   Ci (cos  )i
(4)
i 0


EvdW (r )  4 [( )12  ( )6 ], r  rc (5)
r
r
where Kb and Kθ are the stiffness constants for the bond length and bond angle
potentials, respectively, r0 and θ0 are the equilibrium bond length and bond angle,
respectively, and the variable Ci contains the coefficients of dihedral multi-harmonic.
The non-bonded or van der Waals （vdW） interactions are 12-6 Lennard-Jones type,
where r is the distance between two atoms, σ is the distance at zero energy, and ε is
the energy well depth. The cutoff distance rc is taken as 9.5 Å. Table 1 gives the
values for all of the constants used in the interatomic potential.
TABLE 1. Potential parameters used for polyethylene MD calculations
Parameters
Values
Unites
Kb
r0
Kθ
θ0
C0
C1
C2
C3
σ
ε
350
1.53
60
1.911
1.736
-4.490
0.776
6.990
4.01
0.112
kcal/mol
Å
kcal/mol/rad2
rad
kcal/mol
kcal/mol
kcal/mol
kcal/mol
Å
kcal/mol
Simulation Model
In order to investigate the physical mechanism of flow-induced crystallization
（FIC） of polyethylene (PE), three different molecular weight systems (C50, C100
S2
and C200, respectively) are considered. As shown in Figure S1, a uniaxial strain
tensile deformation is performed at a constant strain rate of 108 s-1 with a pressure is 0
kPa in the axes perpendicular to tensile direction. The NEMD results indicate that the
nucleation and growth process have not appeared when the system with low
molecular weights (C50 and C100 systems) until stretched to strain of 4 at 350 K,
while this process happens in C200 system, see the Figure S1. The densities of C50
and C100 systems are lower than that of C200 system during extension, see the Figure
S2d. Thus, the C50 and C100 systems are amorphous states during extension. The
gauche/trans segment ratio is the basic parameter to reflect structure properties of the
system. The C1-C4 distance of PE chains at about 2.9 Å and 3.9 Å are assigned
gauche (g) and trans (t) molecular segments, respectively. All-trans segments of
C1-C5, C1-C6, and C1-C7 can be represent by tt, ttt, and tttt, respectively. Figure S3
display the total radial distribution function g(r)total and inter-chain radial distribution
function g(r)inter of different PE systems with the strain of 4. The g(r)total shown that
C200 system has lower gauche segment fraction and higher trans segment fraction
than that of C50 and C100 systems. The g(r)total of C50 and C100 systems has no
significant difference compared with the beginning also prove that the C50 and C100
systems are amorphous states. The g(r)inter reveal that the intermolecular order degree
for C200 system is higher than that of C50 and C100 systems. The bond length and
bond angle potential energies decomposition of C50 and C100 systems have remained
basically unchanged during extension, see Figure S2a and b. The dihedral angle
rotation and vdW energies are decreasing, while density is increasing with strain for
S3
C200 system. The decreases of the vdW and dihedral angle rotation energies means
the orientation of the chains is dominated by the inter-chain interactions whereas
conformation adjustment is mainly dominated by the intra-chain dihedral rotation of
PE. In order to clarify the coupling between intra-chain conformational and
inter-chain orientation and density ordering, the C200 model is mainly discussed in
this work.
FIGs. 1. Atomic snapshots of the three different molecular weight systems (C50,
C100, and C200) at strain of 4 with 350 K.
S4
FIGs. 2. The evolution of potential energy components (a-c) and average density
ρof the three different systems during extension.
FIGs. 3. The total radial distribution function g(r)total (a) and inter-chain radial
distribution function g(r)inter (b) of the three different systems with the systems are
stretched to strain of 4 under the strain rate of 108s-1 at 350 K
Structure Factor
We analyzed the total radial distribution function g(r)total and the structure factor
S5
S(q) at various times t to clarify the evolution of the crystal nucleus during extension,.
The g(r)total is defined as4
g  r total 
V
NN 2
  r  r  ,
N
N
ij
(6)
i i j
where V and N are the volume and number of particles in the ensemble, respectively,
andδis the Knonecker delta function. The structure factors are defined as (equivalent
to Fourier transform)

S  q   1  4πρ r 2 g  r 
0
sinqr
dr ,
qr
(7)
whereρis the ensemble density, q is the wavevector. Evolution of the structure factor,
S(q), for the C200 system under the stretching has been represented in Figs. 4.
FIGs. 4. Evolution of the structure factor S(q) for the C200 system under the
stretching. The evolution of the second-order Bragg peak location, converted to an
effective distance, d, to the chains.
S6
Frequency count
400
orthorhombic
hexagonal
melt
300
200
100
0
0.20
0.25
0.30
0.35
0.40
0.45
Q4
FIGs. 5 The distribution of Q4 values of orthorhombic, hexagonal and melts structure
of PE with the same size
Figs. 6 shows the spatial density distributions and atomic snapshots of the system at
strain rate of 108, 109 and 1010 s-1. Obviously, the spatial local density of the ordered
structure formed at strain rate of 108s-1 is higher than that in the strain rate of 109 and
1010s-1. This is because of the degree of the ordered structure formed in the strain rate
of 108s-1 is higher than that of 109 s-1. Also mean trans percentage of final structure in
the strain rate of 108s-1 is higher than that of 109 and 1010s-1. When the strain rate
reaching a certain level, the system will fracture. Comparing spatial density
distributions and atomic snapshots of the system at strain rate of 108, 109 and 1010 s-1,
the coupling or decoupling between intra-chain and inter-chain orderings at low or
high strain rate can be understood. This is also the reason why the opposite evolution
trend of EvdW at strain rate of 1010 s-1.
S7
FIGs. 6. Snapshots of spatial local density distributions and atomic snapshots of the
system at strain rate of 108, 109, and 1010s-1, respectively.
REFERENCES
1
M. S. Lavine, N. Waheed, and G. C. Rutledge, Polymer 44, 1771 (2003).
2
D. Hossain, M. A. Tschopp, D. K. Ward, J. L. Bouvard, P. Wang, and M. F. Horstemeyer, Polymer
51, 6071 (2010).
3
L. Huang, X. Yang, X. Jia, and D. Cao, Physical chemistry chemical physics : PCCP 16, 24892
(2014).
4
R. H. Gee, N. Lacevic, and L. E. Fried, Nature materials 5, 39 (2006).
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