We have successfully predicted the several physical properties (parameters of crystal structure, torsional angles in crystal structure, glass transition temperature, melting temperature, and Young’s modulus) of the PTT polymer using modified Dreding force field. The formation of nuclei precursors in the induction period of isothermal crystallization of polymer from melt is observed in our simulation. Between Tg and Tm, the torsional and van der Waals forces drive the segments to form precursors consisting of parallel segments. The amount of precursors increases rapidly soon after the system is cooled and then fluctuates around some asymptotic value. During this later stage, it is found that the torsional distribution of the polymer backbone for segments within the precursor slowly rearranges to that in the crystalline state. In addition, the temperature variation of the amount of precursor resembles that of the crystallization rate between Tg and Tm. It is likely that the precursors will eventually evolve to compact nuclei for subsequent crystallization processes.
Based on the simulation results, we propose that nuclear precursors are formed in the induction period, prior to the appearance of nuclei during the crystallization process.
The precursor has a high internal orientation order but is loosely packed. The precursors are constantly forming, growing, and disintegrating. The formation of precursors is mainly driven by van der Waals and torsional forces. These two forces may either collaborate or compete, resulting in the segment to reorganization to form a more compact cluster. Therefore, the precursors could eventually become compact nuclei for the subsequent crystallization process. However, the precursor internal structure development (e.g. torsional reorganization) takes place at a much slower rate (compared
to the formation of precursors), even the rearrangement of the trsional angle φ2 would accelerate by stress. As a consequence, there observed (experimentally) a long induction period prior to the crystallization of polymers.
The formation of oriented precursor in the stress-induced crystallization is also observed in our simulation. The torsional and van der Waals forces are the dominant interactions, similar to the case observed in isothermal crystallization. The amount of stress-induced precursor increases in all regions of temperature. The maximum size of oriented precursor is larger than that created only by thermal stimulation. During stress-induced crystallization, the torsional distribution of the polymer backbone for segments rapidly rearrange to the t-t-t-t conformation in bulk phase. Within oriented precursors, the response of the torsional angle induced by stress is faster than that only induced by thermal stimulation, especially trans in φ1(the transition rate of trans state in φ1 is faster than that of gauche state in φ2).
In our simulation, the structures of precursor packing by polymer segments are observed in isothermal and stress-induced crystallization such as many simulation works, which use coarse-grained or bead-spring models. Furthermore, the atomic model supports us to understand more detailed development (such as torsional angle) and to approach the atomic mechanism of the polymer crystallization at very early state.
Appendix A
In this section, we show the value of parameters used in Dreiding force field. In our system, there are three kinds of atom (carbon, oxygen, and hydrogen) defined as different atomic fore field types in Table A.1 (according the atomic position shown as Figure A.1). Then, one can identify eight kinds of bond types (shown in Table A.2), three groups, overall 13 kinds) of angle types, 4 groups (overall 16 kinds) of torsion types, and inversion.
Figure A.1 The atomic position on the PTT polymer repeat unit.
Table A.1 The definition of each atom type in atomic model.
Atomic position Atomic force field type
C1, C in ring C_R
O2 O_R C3,C4 C_3
O5 O_2
H, H end H_
Table A.2 The value of parameters of band energy in our simulation.
Type 1/2Kb (Kcal/mole/Å2) R0(Å)
C_R, C_R 525 1.39
C_3, C_3 350 1.53
C_3, O_R 350 1.42
C_R, O_R 525 1.35
C_R, O_2 700 1.25
C_3, H_ 350 1.09
C_R, H_ 350 1.02
O_R, H_ 350 0.98
Table A.3 The value of parameters of angle energy in our simulation.
Type 1/2Kθ (Kcal/mole) θ0 (deg.)
X, C_R, X
C_R, C_R, C_R 100 120
C_R, C_R, O_2 100 120
C_R, C_R, O_R 100 120
C_R, C_R, H_ 100 120
O_2, C_R, H_(end) 100 120
O_R, C_R, O_2 100 120
X, C_3, X
C_3, C_3, C_3 100 109.471
C_3, C_3, O_R 100 109.471
C_3, C_3, H_ 100 109.471
O_R, C_3, H_ 100 109.471
H_, C_3, H_ 100 109.471
X, O_R, X
C_R, O_R, C_3 100 120
C_3, O_R, H_ 100 104.51
Table A.4 The value of parameters of torsion energy in our simulation.
Type 1/2Kφ (Kcal/mole) n d
X, C_R, C_R, X
C_R, C_R, C_R, H_ 25 1 2
C_R, C_R, C_R, C_R 25 1 2
H_, C_R, C_R, H_ 25 1 2
C_R, C_R, C_R, O_2 10 1 2
C_R, C_R, C_R, O_R 10 1 2
C_R, C_R, C_R, H_ (end) 10 1 2
X, C_3, C_3, X
H_, C_3, C_3,O_R 2 -1 3
O_R, C_3, C_3, C_3 2 -1 3
H_, C_3, C_3, H_ 2 -1 3
H_, C_3, C_3, C_3 2 -1 3
X, C_R, O_R, X
C_R, C_R, O_R, C_3 25 1 2
O_2, C_R, O_R, C_3 25 1 2
X, C_3, O_R, X
C_3, C_3, O_R, C_R 2 -1 3
C_3, C_3, O_R, H_ 2 -1 3
H_, C_3, O_R, H_ 2 -1 3
H_, C_3,O_R, C_R 2 -1 3
Table A.5 The value of parameters of inversion energy in our simulation.
Type 1/2Kψ (Kcal/mole) ψ0
40 0
40 0
40 0
40 0
Table A.5 The value of parameters of Van der Waals interaction in our simulation.
Type D0 R0 ζ
C_R 0.0951 3.8983 14.034
C_3 0.0951 3.8983 14.034
O_R 0.0957 3.4046 13.483
O_2 0.0957 3.4046 13.483
H_ 0.0152 3.195 12.382
H_ (end) 0.0001 3.195 12.000
The cross-type coefficients of non-bond interaction are computed using the mixing rule:
Aij= Aii×Ajj (A.1a)
jj ii
ij B B
B = × (A.1b)
2 C Cij Cii + jj
= (A.1c)
and the non-bond interaction of different atoms are calculated as follow:
6 ij r C ij
ij A e B r
U = − ij − − (A.2)
Appendix B
In Appendix B, we show the precursors which are detected in the double size system (eight chains in the amorphous state) during isothermal crystallization process.
The number and size of individual precursors are shown in Table B.1. The intensity of the RDF, average size (Figure B.1, B.3 and B.5) and distribution of torsional angle (Figure B.2, B.4, and B.6) within the maximum size of precursor are shown at 350K, 400K, and 450K, respectively.
Table B.1 The number and size of individual precursors identified in MD simulations.
(27x8)
Temperature (K) Number of precursors† Simulation length (ns)
350 1(5) 12
400 9(7,6,6,5,5,5,5,5,5) 12
450 8(6,5,5,5,5,5,5,5) 12
The intensity of the RDF
simulation time (ns)
The average size of precursor
0
simulation time (ns)
The average size of precursor
0
Figure B.1 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 350 K. (27x8)
simulation time (ns)
The fraction of the torsional angle
Figure B.2 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols) at 350 K. (27x8)
The intensity of the RDF
simulation time (ns)
The average size of precursor
0
simulation time (ns)
The average size of precursor
0
Figure B.3 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 400 K. (27x8)
simulation time (ns)
The fraction of the torsional angle
Figure B.4 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols) at 350 K. (27x8)
The intensity of the RDF
simulation time (ns)
The average size of precursor
0
simulation time (ns)
The average size of precursor
0 1 Size 2
RDF
Figure B.5 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 450 K. (27x8)
simulation time (ns)
The fraction of the torsional angle
Figure B.6 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols) at 350 K. (27x8)
Appendix C
In appendix C, we show the results of precursor formation in the system contenting 14 chains, each chain having 27 repeat units, during stress-induced crystallization. The number and size of individual precursors are shown in Table C.1, C2, and C3 with draw speed 1x1010s-1, 5x1010s-1, and 1x1011s-1, respectively. Figures C.1, C.3, and C.5 show the time evolution of RDF intensity at 4.11Å, and the average size of precursor at 200K, 300K, and 400K, respectively, with a draw speed of 1x1010s-1; Figures C.7, C.9, and C.11 show the results for draw speed 5x1010s-1; Figures C.13, C.15, and C.17 for draw speed 1x1011s-1. On the other hand, Figures C.2, C.4, and C.6 show the results of the fractions of backbone torsions in the oriented precursors at 200K, 300K, and 400K, respectively, with a draw speed 1x1010s-1; Figures C.8, C.10, and C.12 show the results for draw speed 5x1010s-1; Figures C.14, C.16, and C.18 for draw speed 1x1011s-1.
Table C.1 The number and size of individual precursors identified in stress-induced crystallization with draw speed 1x1010s-1. (27x14)
Temperature (K) Number of precursor Strains (%)
200 10(42,16,14,9,7,7,6,6,5,5) 300
250 13(24,17,10,9,8,8,7,7,6,6,6,5,5) 300
300 17(20,17,14,13,10,9,7,7,6,6,6,6,5,5,5,5,5) 300 350 15(24,23,18,9,8,8,7,6,6,6,6,5,5,5,5) 300 400 15(33,13,11,9,9,7,6,6,5,5,5,5,5,5,5) 300 450 14(29,24,15,13,12,10,8,7,6,6,6,6,5,5) 300
Table C.2 The number and size of individual precursors identified in stress-induced crystallization with draw speed 5x1010s-1. (27x14)
Temperature (K) Number of precursor Strains (%)
200 13(25,12,10,8,7,7,7,7,6,5,5,5,5) 300
250 14(28,15,9,7,7,6,6,6,5,5,5,5,5,5). 300
300 7(35,23,8,8,6,6,5) 300
350 14(27,13,10,10,7,7,6,6,6,6,6,5,5,5) 300
400 12(32,11,11,10,10,8,8,7,6,5,5,5) 300
450 12(27,12,9,8,8,8,8,7,6,5,5,5) 300
Table C.3 The number and size of individual precursors identified in stress-induced crystallization with draw speed 1x1011s-1. (27x14)
Temperature (K) Number of precursor Strains (%)
200 12(28,8,7,7,6,6,5,5,5,5,5,5) 300
250 14(18,10,8,8,7,7,6,6,5,5,5,5,5,5) 300
300 8(31,12,12,11,10,7,5,5) 300
350 11(35,16,10,10,9,8,8,6,6,5,5) 300
400 12(27,12,10,6,6,6,6,6,5,5,5,5) 300
450 8(21,19,13,12,11,5,5,5) 300
200K
The average size of precursor
0
The average size of precursor
0
Figure C.1 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 200 K with draw speed 1x1010s-1. (27x14)
Figure C.2 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols) at 200 K with draw speed 1x1010s-1. (27x14)
300K
The average size of precursor
0
The average size of precursor
0
Figure C.3 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 300 K with draw speed 1x1010s-1. (27x14)
Figure C.4 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols) at 300 K with draw speed 1x1010s-1. (27x14)
400K
The average size of precursor
0
The average size of precursor
0
Figure C.5 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 400 K with draw speed 1x1010s-1. (27x14)
Figure C.6 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols) at 400 K with draw speed 1x1010s-1. (27x14)
200K
The average size of precursor
0
The average size of precursor
0
Figure C.7 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 200 K with draw speed 5x1010s-1. (27x14)
Figure C.8 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols)
10 -1
300K
The average size of precursor
0
The average size of precursor
0
Figure C.9 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 300 K with draw speed 5x1010s-1. (27x14)
Figure C.10 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols) at 300 K with draw speed 5x1010s-1. (27x14)
400K
The average size of precursor
0
The average size of precursor
0
Figure C.11 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 400 K with draw speed 5x1010s-1. (27x14)
Figure C.12 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols)
200K
The average size of precursor
0
The average size of precursor
0
Figure C.13 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 200 K with draw speed 1x1011s-1. (27x14)
Figure C.14 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols) at 200 K with draw speed 1x1011s-1. (27x14)
300K
The average size of precursor
0
The average size of precursor
0
Figure C.15 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 300 K with draw speed 1x1011s-1. (27x14)
Figure C.16 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols)
11 -1
400K
The average size of precursor
0
The average size of precursor
0
Figure C.17 The time evolution of RDF intensity (4.11Å) (diamond) and the time averaged number of parallel segments contained in a representative precursor (triangle) at 400 K with draw speed 1x1011s-1. (27x14)
Figure C.18 the backbone torsions <φ1> in trans (triangles) and <φ2> in gauche (circles) for segments in a precursor (closed symbols) and outside any precursor (open symbols) at 400 K with draw speed 1x1011s-1. (27x14)
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