- Email: [email protected]
Computer simulation of melting of polymer crystals
N
ELSEVIER
Physica B 219&220 (1996) 420422
Computer simulation of melting ofpolymer crystals Nobuyuki Takahashi a, ,, Masamichi Hikosaka b, Takashi Yamamoto
c
a Hokkaido University of Education, Hakodate, Hachiman-cho, Hakodate 040, Japan b Facully of Integrated Arts and Sciences, Hiroshima University, Hiyashi-Hiroshima 739, Japan
c Faculty of Science, Yamaguchi University, Yamayuchi 753, Japan
Abstract Melting processes of an n-alkane crystal (16 chains of (CH2)Is: initially orthorhombic) with increasing temperature in the vacuum are simulated by a molecular dynamics method. The simulation was carried out using a kind of Lennard-Jones potential. On heating at 1 K/ps, the surface chains melt at much lower temperature than the inner chains. Details of the melting behavior are examined based on the changes in the internal rotation angles and the calculated X-ray diffraction intensity. This result indicates a new concept of surface melting of a polymer crystal.
1. Introduction
2. Molecular dynamics method
Mesophases of polymer crystal surfaces were suggested to be playing an important role in polymer crystallization [1]. Computer simulations have supplied tests of ideas of disorders [2, 3] and phase transitions [4-7] in polymer crystals. In our previous paper [8], we performed a Monte Carlo simulation of free surfaces of an n-alkane crystal to clarify the microscopic state of polymer crystal surfaces. It was shown that increase in temperature gives rise to very pronounced disorder which is very similar to a rotator phase at the crystal surface, two or three layers in thickness. In this paper, we present melting processes of an n-alkane crystal in the vacuum simulated by a molecular dynamics method. We have found that melting of surface chains starts below the melting point of the bulk crystal. We discuss about the possibility of a new concept of surface melting of a polymer crystal below the melting point based on the X-ray diffraction and the internal rotation angles of molecular chains calculated for the simulated structures.
The equations of motion of 720 particles, which constitute atoms in 16 chains of (CH2)15 modeling pentadecane, are integrated by the use of the Verlet's method with a time step of 1 fs. Forces between atoms are calculated by the use of valence forces and the Lennard-Jones (L-J) force. The valence forces for stretching, bending and torsion are derived from the empirical potential functions. The potential functions for the stretching and bending are harmonic functions of the bond length and the bond angle, respectively. The force constants for stretching and the bending are transferred from those used for the vibrational analysis [9]. The torsional force is derived from the simple threefold potential as a function of the internal rotational angle r(/kj. The barrier for the internal rotation about a carbon-carbon single bond is the same as that determined by Scott and Scheraga [10]. The truncated L-J 6 12 forces with the cut-off radius 0.65 nm are calculated between atoms in different chains and between atoms connected by three or more bonds. The depth of the potential energy minimum cij and the distance of the minimum potential r0(/used in this work are listed in Table l. The parameters for the C. • .H pair are conventional averages of the values for the C-.. C and H . . . H pairs. Initial positions of atoms are those of the planar zigzag chain with herringbone packing in the orthorhombic crystal. Initial velocities of atoms are chosen from the Maxwell dis-
* Corresponding author. 0921-4526/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved SSDI 092 1-4526( 95 )00764-4
421
N. Takahashi et aL / Physica B 219&220 (1996) 420-422 Table l Potential parameters
10
I
360 K
I
I~
surface
CU(10-22 J) r0i/(10-1°m) Reference H-..H 8.5 C. - -C 8.9
2.4 3.4
Scott and Scheraga [10] McCullough and McMahon [1 1]
~ ) N ~ insi(te
(70 ps) chain 6 ~ (a) -10 tribution corresponding to the initial temperature (1 K) by the use of random numbers. The system is preheated from 1 K to 290 K with a heating rate of 6 K/ps and then heated at a rate of I K/ps. To heat the system velocities are multiplied by a constant value each time step.
J
n /1 4 ~ 1 1I
I
5 39~i~_~~
I
I
I
I
5
10
-
5o
0
~" 3. Results and discussion
J
10
--5
(b) -10
t -10
-15
t -5
15
(lO-Inm]
The system holds ordered herringbone packing at 290 K. Heating at a rate of 1 K/ps from 290 K causes increase of twisting motion of surface chains and results in orientational disorder about the chain axis and conformational disorder at the chain end (Fig. l(a)). Above 370 K almost surface chains melt (Fig. l(b)), while the inner chains keep crystalline order. The inner chains melt above 420 K. The melting behavior of surface chains can be quantitatively discussed by evaluating the internal rotation angle and displacement of a chain (chain 2) on the surface indicated by a thick arrow in Fig. 1. Fig. 2 shows the root mean square of the internal rotation angles z i y and two components, dp( |I C) and d,.(2_ c) of the vector between the centers of mass of the chain 2 and the nearest inner chain (chain 6). ( V / ~ of chain 2 and dp show distinguishable increase above 370 K (Tm~ ), which clearly indicates melting of the surface chain. Calculated diffraction intensity of (0 0 2) reflection (I(0 0 2), defined on the subcell) disappears after melting as shown in Fig. 3.1(0 0 2) calculated on the surface and that on the inside each decreases with increasing temperature as shown in Fig. 4. The intensity from the surface decreases at lower temperature (Ts ) than that from the inside ( Tm). It is to be noted that the real pentadecane crystal undergoes transition to a rotator phase at 271 K just before melting and melts at 283 K [12]. Therefore, the simulated melting temperature 420 K is significantly higher than the real value of pentadecane 283 K. This discrepancy may be due to the L-J parameters used here. The simulated melting temperature can be decreased to 340 K when we apply another kind of L-J parameters fitted to Williams Set VII potential [13] in a preliminary simulation. At present the discrepancy in the melting tempera-
Fig. 1. Projections of molecular chains along the c axis (H the chain axis) of the inner chains.
T(K)
290 2
21
I
310
330 350
,
,
,
,
I
~
370 390 ch.,.;n6 |
,
chain 2
v
t
E
5o
c 5o
0
20 f
40 60 {ps J
80
100
Fig. 2. Root mean square of the internal rotation angles for a surface chain and an inner chain and two components, dp(He) and dv(±C) of the vector between the centers of mass of the surface chain and the nearest inner chain.
ture is not so important, because the very high heating rate (1 K/ps) used here may cause more or less super heating. More details will be reported in the next paper.
N. Takahashi et al. / Physica B 219&220 (1996) 420-422
422 I
I
melting o f a polymer crystal ". This must be important in the study of polymer crystallization mechanism.
I
1 ~ 20 ps
(002
c
=
3
290 ~ 310 K
2
Acknowledgements
1 7=
0
-~
I
I
[
I
I
[
120 ~ 140 ps 420 ~ 440 K "--
"~
0
0
2
4 s
6
8
[
10
(nm-1 )
The calculation was performed on IBM Power Station 320 installed by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture, No. 03455003. We would like to thank K. Honda, R. Mitsui, and S. Maruyama for cooperation in the early stage of this study.
Fig. 3. Calculated X-ray diffraction intensity profiles along (0 0 ~).
References
T(K) 290
330
370
420 480
(a)
I ¢
I
....
540
[ ....
I
2 1 (b)
Ol
I
tO
80
120
160
200
(psi
Fig. 4. Time dependence of the calculated X-ray diffraction (0 0 2) peak intensities (a) for surface chains and (b) for the inner chains. The distinguishable disordering of the surface chains below the melting point indicates a new concept of "surface
[1] A. Keller, M. Hikosaka, S. Rastogi, A. Toda, PA. Barham and G. Goldbeck-Wood, J. Mater. Sci. 29 (1994) 2579. [2] T. Yamamoto and Y. Kimikawa, J. Chem. Phys. 97 (1992) 5163. [3] G.L. Liang, D.W. Noid, B.G. Sumpter and B. Wunderlich, Makromol. Chem. Theory Simul. 2 (1993) 245. [4] T. Yamamoto, J. Chem. Phys. 89 (1988) 2356. [5] J.P. Ryckaert, I.R. McDonald and M.L. Klein, Mol. Phys. 67 (1989) 957. [6] N. Takahashi, Computer Aided Innovation of New Materials II, eds. M. Doyama et al. (Elsevier, Amsterdam, 1993) p. 299. [7] K. Esselink, P.AA. Hilbers and B.W.H. van Beest, J. Chem. Phys. 101 (1994) 9033. [8] T. Yamamoto, M. Hikosaka and N. Takahashi, Macromolecules 27 (1994) 1466. [9] M. Kobayashi and H. Tadokoro, J. Chem. Phys. 66 (1977) 1258. [10] R.A. Scott and H. A. Scheraga, J. Chem. Phys. 45 (1966) 2091. [11] R.L. McCullough and P.E. McMahon Trans. Faraday Soc. 60 (1964) 2089. [12] G. Ungar, J. Phys. Chem. 87 (1983) 689. [13] D.E. Williams, J. Chem. Phys. 47 (1967) 4680.
ELSEVIER
Physica B 219&220 (1996) 420422
Computer simulation of melting ofpolymer crystals Nobuyuki Takahashi a, ,, Masamichi Hikosaka b, Takashi Yamamoto
c
a Hokkaido University of Education, Hakodate, Hachiman-cho, Hakodate 040, Japan b Facully of Integrated Arts and Sciences, Hiroshima University, Hiyashi-Hiroshima 739, Japan
c Faculty of Science, Yamaguchi University, Yamayuchi 753, Japan
Abstract Melting processes of an n-alkane crystal (16 chains of (CH2)Is: initially orthorhombic) with increasing temperature in the vacuum are simulated by a molecular dynamics method. The simulation was carried out using a kind of Lennard-Jones potential. On heating at 1 K/ps, the surface chains melt at much lower temperature than the inner chains. Details of the melting behavior are examined based on the changes in the internal rotation angles and the calculated X-ray diffraction intensity. This result indicates a new concept of surface melting of a polymer crystal.
1. Introduction
2. Molecular dynamics method
Mesophases of polymer crystal surfaces were suggested to be playing an important role in polymer crystallization [1]. Computer simulations have supplied tests of ideas of disorders [2, 3] and phase transitions [4-7] in polymer crystals. In our previous paper [8], we performed a Monte Carlo simulation of free surfaces of an n-alkane crystal to clarify the microscopic state of polymer crystal surfaces. It was shown that increase in temperature gives rise to very pronounced disorder which is very similar to a rotator phase at the crystal surface, two or three layers in thickness. In this paper, we present melting processes of an n-alkane crystal in the vacuum simulated by a molecular dynamics method. We have found that melting of surface chains starts below the melting point of the bulk crystal. We discuss about the possibility of a new concept of surface melting of a polymer crystal below the melting point based on the X-ray diffraction and the internal rotation angles of molecular chains calculated for the simulated structures.
The equations of motion of 720 particles, which constitute atoms in 16 chains of (CH2)15 modeling pentadecane, are integrated by the use of the Verlet's method with a time step of 1 fs. Forces between atoms are calculated by the use of valence forces and the Lennard-Jones (L-J) force. The valence forces for stretching, bending and torsion are derived from the empirical potential functions. The potential functions for the stretching and bending are harmonic functions of the bond length and the bond angle, respectively. The force constants for stretching and the bending are transferred from those used for the vibrational analysis [9]. The torsional force is derived from the simple threefold potential as a function of the internal rotational angle r(/kj. The barrier for the internal rotation about a carbon-carbon single bond is the same as that determined by Scott and Scheraga [10]. The truncated L-J 6 12 forces with the cut-off radius 0.65 nm are calculated between atoms in different chains and between atoms connected by three or more bonds. The depth of the potential energy minimum cij and the distance of the minimum potential r0(/used in this work are listed in Table l. The parameters for the C. • .H pair are conventional averages of the values for the C-.. C and H . . . H pairs. Initial positions of atoms are those of the planar zigzag chain with herringbone packing in the orthorhombic crystal. Initial velocities of atoms are chosen from the Maxwell dis-
* Corresponding author. 0921-4526/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved SSDI 092 1-4526( 95 )00764-4
421
N. Takahashi et aL / Physica B 219&220 (1996) 420-422 Table l Potential parameters
10
I
360 K
I
I~
surface
CU(10-22 J) r0i/(10-1°m) Reference H-..H 8.5 C. - -C 8.9
2.4 3.4
Scott and Scheraga [10] McCullough and McMahon [1 1]
~ ) N ~ insi(te
(70 ps) chain 6 ~ (a) -10 tribution corresponding to the initial temperature (1 K) by the use of random numbers. The system is preheated from 1 K to 290 K with a heating rate of 6 K/ps and then heated at a rate of I K/ps. To heat the system velocities are multiplied by a constant value each time step.
J
n /1 4 ~ 1 1I
I
5 39~i~_~~
I
I
I
I
5
10
-
5o
0
~" 3. Results and discussion
J
10
--5
(b) -10
t -10
-15
t -5
15
(lO-Inm]
The system holds ordered herringbone packing at 290 K. Heating at a rate of 1 K/ps from 290 K causes increase of twisting motion of surface chains and results in orientational disorder about the chain axis and conformational disorder at the chain end (Fig. l(a)). Above 370 K almost surface chains melt (Fig. l(b)), while the inner chains keep crystalline order. The inner chains melt above 420 K. The melting behavior of surface chains can be quantitatively discussed by evaluating the internal rotation angle and displacement of a chain (chain 2) on the surface indicated by a thick arrow in Fig. 1. Fig. 2 shows the root mean square of the internal rotation angles z i y and two components, dp( |I C) and d,.(2_ c) of the vector between the centers of mass of the chain 2 and the nearest inner chain (chain 6). ( V / ~ of chain 2 and dp show distinguishable increase above 370 K (Tm~ ), which clearly indicates melting of the surface chain. Calculated diffraction intensity of (0 0 2) reflection (I(0 0 2), defined on the subcell) disappears after melting as shown in Fig. 3.1(0 0 2) calculated on the surface and that on the inside each decreases with increasing temperature as shown in Fig. 4. The intensity from the surface decreases at lower temperature (Ts ) than that from the inside ( Tm). It is to be noted that the real pentadecane crystal undergoes transition to a rotator phase at 271 K just before melting and melts at 283 K [12]. Therefore, the simulated melting temperature 420 K is significantly higher than the real value of pentadecane 283 K. This discrepancy may be due to the L-J parameters used here. The simulated melting temperature can be decreased to 340 K when we apply another kind of L-J parameters fitted to Williams Set VII potential [13] in a preliminary simulation. At present the discrepancy in the melting tempera-
Fig. 1. Projections of molecular chains along the c axis (H the chain axis) of the inner chains.
T(K)
290 2
21
I
310
330 350
,
,
,
,
I
~
370 390 ch.,.;n6 |
,
chain 2
v
t
E
5o
c 5o
0
20 f
40 60 {ps J
80
100
Fig. 2. Root mean square of the internal rotation angles for a surface chain and an inner chain and two components, dp(He) and dv(±C) of the vector between the centers of mass of the surface chain and the nearest inner chain.
ture is not so important, because the very high heating rate (1 K/ps) used here may cause more or less super heating. More details will be reported in the next paper.
N. Takahashi et al. / Physica B 219&220 (1996) 420-422
422 I
I
melting o f a polymer crystal ". This must be important in the study of polymer crystallization mechanism.
I
1 ~ 20 ps
(002
c
=
3
290 ~ 310 K
2
Acknowledgements
1 7=
0
-~
I
I
[
I
I
[
120 ~ 140 ps 420 ~ 440 K "--
"~
0
0
2
4 s
6
8
[
10
(nm-1 )
The calculation was performed on IBM Power Station 320 installed by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture, No. 03455003. We would like to thank K. Honda, R. Mitsui, and S. Maruyama for cooperation in the early stage of this study.
Fig. 3. Calculated X-ray diffraction intensity profiles along (0 0 ~).
References
T(K) 290
330
370
420 480
(a)
I ¢
I
....
540
[ ....
I
2 1 (b)
Ol
I
tO
80
120
160
200
(psi
Fig. 4. Time dependence of the calculated X-ray diffraction (0 0 2) peak intensities (a) for surface chains and (b) for the inner chains. The distinguishable disordering of the surface chains below the melting point indicates a new concept of "surface
[1] A. Keller, M. Hikosaka, S. Rastogi, A. Toda, PA. Barham and G. Goldbeck-Wood, J. Mater. Sci. 29 (1994) 2579. [2] T. Yamamoto and Y. Kimikawa, J. Chem. Phys. 97 (1992) 5163. [3] G.L. Liang, D.W. Noid, B.G. Sumpter and B. Wunderlich, Makromol. Chem. Theory Simul. 2 (1993) 245. [4] T. Yamamoto, J. Chem. Phys. 89 (1988) 2356. [5] J.P. Ryckaert, I.R. McDonald and M.L. Klein, Mol. Phys. 67 (1989) 957. [6] N. Takahashi, Computer Aided Innovation of New Materials II, eds. M. Doyama et al. (Elsevier, Amsterdam, 1993) p. 299. [7] K. Esselink, P.AA. Hilbers and B.W.H. van Beest, J. Chem. Phys. 101 (1994) 9033. [8] T. Yamamoto, M. Hikosaka and N. Takahashi, Macromolecules 27 (1994) 1466. [9] M. Kobayashi and H. Tadokoro, J. Chem. Phys. 66 (1977) 1258. [10] R.A. Scott and H. A. Scheraga, J. Chem. Phys. 45 (1966) 2091. [11] R.L. McCullough and P.E. McMahon Trans. Faraday Soc. 60 (1964) 2089. [12] G. Ungar, J. Phys. Chem. 87 (1983) 689. [13] D.E. Williams, J. Chem. Phys. 47 (1967) 4680.