Article pubs.acs.org/Macromolecules
Cite This: Macromolecules XXXX, XXX, XXX−XXX
Molecular Dynamics Study of Polyethylene: Anomalous Chain Mobility in the Condis Phase Vadim I. Sultanov,*,†,‡ Vadim V. Atrazhev,†,‡ Dmitry V. Dmitriev,†,‡ Nikolay S. Erikhman,†,‡ David U. Furrer,§ and Sergei F. Burlatsky∥ †
Russian Academy of Science, Institute of Biochemical Physics, Kosygin str. 4, Moscow 119334, Russia Science for Technology LLC, Leninskiy pr-t 95, 119313 Moscow, Russia § Pratt & Whitney, 400 Main Street, East Hartford, Connecticut 06108, United States ∥ United Technologies Research Center, 411 Silver Lane, East Hartford, Connecticut 06108, United States
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‡
S Supporting Information *
ABSTRACT: A sharp (several orders of magnitude) increase of chain mobility was observed at a certain temperature in molecular dynamics simulation of the polyethylene crystal. Chain mobility as a function of temperature in a polyethylene perfect crystalline structure was studied by applying molecular dynamics simulation with COMPASS and flexible Williams force fields. Three crystallographic crystalline phases of polyethylene were observed in the simulations in both force fields: orthorhombic, monoclinic, and conformationally disordered (condis) phases. There is no three-dimensional crystallographic order in the condis phaseonly two-dimensional translational symmetry, which was observed in the plane perpendicular to the chain direction. The simulations show that the chain mobility undergoes a dramatic increase by several orders of magnitude at the transition from monoclinic to condis phase of the crystal. In the condis phase, mobility is almost independent of temperature, which indicates that the diffusion activation energy is less than kT. The calculated mobility in the condis phase is almost the same in both force fields. The calculated kinetic energy of the system per C−C bond at the temperature of the monoclinic → condis transition is approximately equal to the barrier of torsion (trans-eclipsed-trans) around the C−C bond.
1. INTRODUCTION
Monoclinic and triclinic PE structures (Figure 1b,c) were discovered later. In both of them, all chain planes are parallel to each other (have the same setting angle). In a monoclinic PE lattice, every chain may be translated into every other one in the direction perpendicular to c, whereas in the triclinic lattice, one half of all chains are in the “antiphase” relative to the other half, that is, shifted by one methylene group or, equivalently, rotated by 180° around the chain axis. Pierce and co-workers4 were the first who observed odd X-ray reflections in PE films appearing after drawing and intensifying upon successive redrawings (in mutually perpendicular directions) and attributed these reflections to forming crystallites of some new, nonorthorhombic crystalline modification. In 1957, Teare and Holmes5 attributed these extra reflections to the planes of triclinic PE. After Pierce et al., a number of studies obtained monoclinic and triclinic PE subjecting orthorhombic PE to stress beyond the yield point, either by tension6−9 or by compression.10−12 However, monoclinic PE can also be obtained by long-time annealing orthorhombic PE at a temperature that is slightly lower than the melting point with subsequent slow cooling to room temperature.13−15 This suggests that monoclinic PE may be more energetically
There are three known crystalline phases for linear polyethylene (PE): orthorhombic, monoclinic, and triclinic (Figure 1). All of them are formed by PE chains stretched in the same direction (called crystalline direction c). In the so-called extended chain crystal, chains are stretched in the c direction along their whole length. Alternatively, chains can consist of several linear sections packed within the crystal interleaved with bent sections located at ab facets of the crystal (Figure 2). Such a crystal is called a folded chain crystal.1 The most energetically favorable state of the PE chain is that in which carbon atoms form the planar zigzag where all chemically bound C−C−C−C sequences are in transconformation (dihedral angles have value of 180°): that is, the so-called all-trans chain. In all three crystalline modifications, PE chains have an all-trans configuration. The orientation of an all-trans chain plane (plane in which carbon atoms reside) within the crystal is characterized by the setting angle, that is, the dihedral angle between the chain plane and the crystalline ac plane. The first reported crystalline structure of PE is orthorhombic in which one half of the chains have the setting angle near 45° and the other half have it near 135° (Figure 1a). The orthorhombic PE crystals have been known since 19393 and are the most prevalent. © XXXX American Chemical Society
Received: December 20, 2018 Revised: June 12, 2019
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DOI: 10.1021/acs.macromol.8b02710 Macromolecules XXXX, XXX, XXX−XXX
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Figure 1. Lattices of true PE crystals: (a) orthorhombic, (b) monoclinic, and (c) triclinic according to ref 2.
in hexagonal polyethylene are likely to be in a (possibly disordered) helical conformation”. As it is currently clear,2 such helices with modest deviations (20% and the latter as 7%. The calculated fraction of gauche dihedrals in the condis phase agrees with the results of Tashiro et al. On the setting pair histogram, taking into account its periodicity in both dimensions, we now see a high-probability strip along the main diagonal and two low-probability parallel strips. Thus, all values of the local setting angle are equally probable. For a given value of the local setting angle at a certain carbon atom, the value at the next carbon atom in most cases will differ only modestly (as in the monoclinic phase), but sometimes it will deviate by as much as ∼90°. Obviously, such leaps of the local setting angle along the chain happen when both carbon atoms belong to gauche dihedral. On the interchain setting pair histogram, the probability density lies within the narrow range of 0.7−1.3, implying only a weak correlation between neighboring chains. We can see a ridge along the main diagonal, which means that equal orientation of methylene groups in the neighbor chains is more probable. On this ridge, we can discern six maxima, which resemble six preferable directions in the RII phase. For RII, we could expect to observe also five ridges parallel to the main diagonal corresponding to setting angle differences by π/6, 2π/ 6, ... 5π/6. However, for condis PE outside the main diagonal, we see a rather complex pattern of correlation, which is hard to explain. Wentzel−Milner-type histograms look very similar for FW and COMPASS force fields. For orthorhombic and monoclinic PE, our histograms are fairly similar to the histograms for C23 alkane in Figure 9a,b in the paper of Wentzel and Milner.45 On the other hand, the histograms for condis PE look quite different from the histograms for C23 RI and RII phases (Figure 9c,d in paper45). This is not surprising if one takes into account the different nature of rotator phases of alkanes and the condis phase of PE. While most of the dihedrals in rotator phases are close to 180°, in condis PE at least every fifth (and at most every third) dihedral is gauche, which results in other
means that the local setting angle changes slowly upon moving along the chain. Since the spots are distinct and not degenerated into a strip along the main diagonal, these changes provide only fluctuations around the average plane but not chain twisting into a helix. Tilts of these average planes are the coordinates (on any axis) of the peak points of these spots, viz. approximately 135 and 225°. All of these indicate that the phase of the sample is orthorhombic. In the orthorhombic phase, two of six neighbor chains have the same setting angle as a chosen chain, and the remaining four have different setting angles. The interchain setting pair histogram is in accord with this configuration: spots lying off the main diagonal and corresponding to the roughly perpendicular mutual orientation (Δα = 90 or 270°) are twice as intensive as spots lying on the main diagonal and corresponding to the same orientation. Since we are using original bisector vectors, here we have two more peak values at ca. 135 + 180° and 225 − 180° in addition to ca. 135 and 225°. Shape of the spots close to round shows an absence of correlation between deviations of local setting angles of neighbor chains from ideal values. Since in the orthorhombic phase the chains are approximately planar, we can calculate not only the local setting angle at each atom but also the setting angle of the whole chains, which may be compared with setting angles calculated by Tashiro.47 From the histograms of chain setting angles (Supporting Information), we obtain that at any temperature, the average value of the chain setting angle measured relative to the b axis lies within range 43−44°, which is close to 43° as calculated by Tashiro. Standard deviations of chain setting angles are much less than that obtained by Tashiro (see Figure 6 in paper47). At 350 K and higher in Tashiro’s system, the helicoidal chain motion takes place, which results in the second peak at the setting angle histograms. In our systems, there is no helicoidal motion in orthorhombic PE, and thus for all temperatures the histograms remain unimodal. These differences of our results from the results of Tashiro are explained by other force fields and shorter chain lengths (24 CH2 groups) used in the work of Tashiro. In the temperature range between the first and second phase transitions (see the “monoclinic” column in Figures 8 and 9), the dihedral pair histogram has a substantially similar look as at lower temperatures, again meaning temperature-blurred alltrans configuration of the chains. However, the setting pair histogram now looks different. To understand it correctly, we should remember that the angle is a cyclic variable; thus, all types of our histograms are periodic in both dimensions. In fact, we have two spots: one with the center at (180, 180°) and the second with the center at (0, 0°). The choice of range (0, 360°) for both angle coordinates made the second spot split into four parts seen in all four corners of the plot. This means that the average planes of all chains are parallel to each other or coincide, which is not obvious from the visualization of the atomistic system in this phase at high temperatures. These two spots are even more oblong than in the orthorhombic phase (below 440 K), showing that local setting angles have greater variation relative to the average plane, but again the difference between two consecutive setting angles is small. The interchain setting pair histogram comprises the same two spots as the setting pair histogram with centers at (180, 180°) and (0, 0°) (the latter is again split into four parts) but not so markedly oblong in FW and ideally round in COMPASS. This again is evidence that all of the chains are H
DOI: 10.1021/acs.macromol.8b02710 Macromolecules XXXX, XXX, XXX−XXX
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The calculated diffusion coefficient for the chain lengths of 100 CH2 groups in the monoclinic phase near the temperature of the monoclinic → condis transition is 8 × 106 cm2/s. Unfortunately, direct comparison of the calculated diffusion coefficient in the monoclinic phase with experimental data is impossible because the data on mobility in the PE crystal is available in the literature only for the orthorhombic phase.48 Moreover, experimental data for the orthorhombic phase were obtained for temperatures near 350 K, which is substantially lower than the temperature of the monoclinic → condis transition in our simulations (480 K). Therefore, we compare our results for chain mobility in the monoclinic phase with prediction of the model for chain mobility in the orthorhombic phase developed by Milner and Wentzel in ref 43 for PE and long alkanes. The model43 is based on the assumption that chain mobility in the PE crystal is governed by the motion of twist solitons through the chain. The jump rate for the chain (the frequency of longitudinal displacement of the chain by one CH2 group) was calculated as a function of chain length, diffusion coefficient of the soliton, and the energy of soliton formation. The calculated diffusion coefficient of the soliton in the orthorhombic phase is 5 × 10−3 cm2/s, and the energy of soliton formation is 41 kJ/mol. Utilizing these values, the jump rate of 30 000 1/s was obtained in ref 43 for the chain in PE lamella with the thickness of 30 nm at 350 K. This agrees by the order of magnitude with the experimental value for the jump rate in the PE orthorhombic crystal48 for these conditions. The calculated jump rate for the chain of length of 100 CH2 groups near the temperature of the monoclinic → condis transition (475 K) with parameters from ref 43 is 2.5 × 106 1/s. The jump rate in our simulations can be estimated as D/a2, where a is the length of the CH2 group and D is the chain diffusion coefficient. Using the diffusion coefficient calculated in FW for the monoclinic phase for temperature 475 K, we obtain the jump rate of 1.5 × 108 1/s. Jump rate and diffusion coefficient in the monoclinic phase calculated from our MD trajectories are approximately 50× higher than that predicted by the Milner and Wentzel model43 for the same conditions. This can be caused by the following reasons. The model from ref 43 was developed for the orthorhombic phase, and chain mobility in this phase can differ from mobility in the monoclinic phase. The Milner and Wentzel model is based on the twist soliton mechanism, which is likely the major mechanism of PE chain mobility in the crystal near room temperature. At higher temperatures, another mechanism of chain mobility such as the motion of gauche-containing “kinks” like ...TTG + TG − TT..., ...TTG+G+TT..., and ...TTG−G−TT can be involved. We assume that the kink transport is the major mechanism of chain mobility in the condis phase because the fraction of gauche dihedral angles in this phase is high. However, in the monoclinic phase near the temperature of the monoclinic → condis transition, some nonzero fraction of gauche dihedral angles appears. The mechanism of kink transport might contribute to chain mobility at high temperature (475 K), whereas it is frozen at low temperature (350 K) due to high energy of kink formation in the true crystalline phase. Calculated diffusion coefficients in the condis phase for chain lengths of 100 CH2 groups are 3.5 × 10−5 cm2/s. The jump rate of the PE chain in the condis phase estimated from the calculated diffusion coefficient is 2 × 1011 s−1 (for chain with 100 CH2 groups). The jump rate in crystalline phases of alkanes was calculated analytically and simulated by molecular
preferable local environments. Nevertheless, Wentzel−Milnertype histograms for C23 RI and RII and for condis PE have common features. Similar to that in paper,45 we marked the same dotted boundaries over the histogram for condis PE. From four low-probability regions for C23 RI and RII (middleleft, middle-right, top-center, and bottom-center), for condis PE low-probability regions include only two (middle-left and middle-right). Thus, we can infer that in condis PE, the “headto-head” collisions between hydrogen atoms are also avoided but in half of the cases. In our simulations, the condis phase exists in a rather wide range of temperatures. The temperature 500 K for which the histograms in the condis column of Figure 9 were plotted is close to the lower limit of the temperature range for the condis phase. For temperatures closer to the melting point, both the interchain setting pair histogram and the Wentzel−Milner-type histograms become nearly uniform, and no structure can be visually observed on color maps. 3.3. Chain Mobility. We calculated the diffusion coefficients of polyethylene chains of different lengths at the number of temperatures within the range from 200 K to sample melting. In the orthorhombic phase, no chain motion was detected at all temperatures within 10 ns of MD simulation. In the monoclinic phase below 450 K for 50 and 100 CH2 chains and 460 K for 200 CH2 chains, within 10 ns only several chain shifts were registered by one methylene group; these were insufficient for reliable determination of the diffusion coefficient. Mobility data for the monoclinic phase at higher temperatures and for the condis phase are plotted in Arrhenius coordinates (ln(D) vs 1/RT) in Figure 10 for different chain lengths in the FW force field and in Figure 11 for chain length 100 methylene groups in both force fields.
Figure 10. Arrhenius plot of the chain diffusion coefficient as a function of temperature for three chain lengths (50, 100, and 200 methylene groups) in the FW force field.
Figure 11. Arrhenius plot of the chain diffusion coefficient as a function of temperature for chain length 100 methylene groups: comparison of FW and COMPASS force fields.
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Macromolecules dynamics in ref 43. The value of approximately 1.5 × 1010 s−1 was obtained for the rotator RII phase, that is, by the order of magnitude lower than the calculated jump rate in the condis phase. The difference in the jump rate is associated with the different mechanism of chain mobility in the RII phase of alkanes and the condis phase of PE. We do not expect twist solitons within condis PE since there are no planar zigzag chain segments of sufficient length to contain a twist soliton inside it. Gauche dihedrals are too numerous in condis PE. Probably, the motion of gauche-containing kinks like ...TTG+TG−TT..., ...TTG+G+TT..., and ...TTG−G−TT..., instead of twist solitons, along the chain leads to longitudinal motion of chains in condis PE. In the condis phase, the temperature dependences of the diffusion coefficient in the Arrhenius coordinate shown in Figure 10 are the straight horizontal lines. The diffusion coefficient calculated in the COMPASS force field is in remarkably good agreement with the diffusion coefficient calculated in the FW force field for the chain containing 100 CH2 groups (Figure 11). The apparently zero slope of the Arrhenius curve presented in Figure 10 implies that activation energy of relative motion of the chains in the condis phase is almost equal to or less than kT. The motion of the polymer chain in the crystalline phase is governed by diffusion of lattice defects such as twist solitons.49,50 Probably, the motion of gauche-containing kinks instead of twist solitons leads to longitudinal motion of chains in condis PE. Activation energy of chain diffusion is a sum of activation energy of kink diffusion and the energy of kink formation. Interaction of the monomers involved in the kink with the neighbor chains gives substantial contribution into activation energy of kink diffusion. There is no translational symmetry in the longitudinal direction in the condis phase. Geometrical orientation of the monomers is random, and the monomers of the chain feel the random potential of the neighbor chains. Longitudinal displacement of the kink causes variation of the potential energy of the monomers of the kink by different random values with opposite signs. The total variation of the energy of longitudinal displacement of the kink in the condis phase is a sum of random values with zero average and tends to zero. Therefore, the average activation energy of kink diffusion is equal to zero in the condis phase. The fraction of gauche dihedral angles in the condis phase is large and weakly dependent on temperature (Figure 7). This indicates that the energy of kink formation is almost equal to or less than kT. Therefore, the total activation energy of the chain diffusion in the condis phase is almost equal to or less than kT. Geometrical orientation of the monomers is governed by torsion around the C−C bond. This torsion is restricted by interaction with the neighbor chains and by potential energy of torsion for an isolated chain. At the monoclinic → condis transition, the volume of the crystal increases, and restrictions caused by the neighbor chains substantially reduce. In the FW force field, the energy of torsion around C−C bonds for an isolated chain can be calculated in a straightforward way. The calculated total barrier of the torsion (energy between transconformation and eclipsed conformation) is 4.21 kcal/mol. When the kinetic energy per C−C bond exceeds the total barrier of the torsion, the torsion around the C−C bond unfreezes. The temperature that corresponds to the kinetic energy of 4.21 kcal/mol is equal to 470 K. The temperature of 470 K is slightly lower than the temperature of the monoclinic
→ condis transition (487 K). Free rotation results in almost random conformations of the chains in the condis phase. Therefore, the monomer of one chain moves in an effective random potential generated by neighbor chains. This is very different from the periodic potential in true crystalline phases such as monoclinic and orthorhombic. Average diffusion coefficients for chain lengths of 50, 100, and 200 CH2 groups are 7.77 × 10−5, 3.44 × 10−5, and 1.94 × 10−5 cm2 s−1, respectively. These values are approximately inversely proportional to the chain length. The mobility that we observed in the condis phase is several orders of magnitude higher than the results for the true crystalline phase observed in the current paper, modeling literature, and experiment. A similar huge jump in chain mobility was observed at the transition from the orthorhombic to condis phase experimentally in ref 34 by de Langen and Prins. In ref 34, the diffusion coefficient in condis and orthorhombic PE was measured at room temperature and high pressure of 4900 bar by the NMR technique. As expected, our prediction of chain mobility for both phases at atmospheric pressure and elevated temperature above 480 K is higher than experimentally measured at high pressure and room temperature. However, the qualitative conclusion about the mobility jump at the transition point agrees with experimental observations. Our calculation predicts the jump of chain mobility of 3 orders of magnitude, which agrees with the measured jump of chain mobility in ref 34.
4. CONCLUSIONS Chain mobility as a function of temperature in a polyethylene perfect crystalline structure was studied by applying molecular dynamics simulation with COMPASS and flexible Williams force fields. A sharp (3 orders of magnitude) increase of chain mobility was observed in the condis phase of the polyethylene crystal in molecular dynamics simulations. This agrees with experimental results of de Langen and Prins.34 The calculated activation energy of diffusion in the condis phase is less than kT. The monoclinic → condis transition is close to the temperature that is required to unfreeze around the C−C bond. A high fraction of gauche conformations was observed in the condis phase, which might imply contribution of gauchecontaining kinks like ...TTG+TG−TT..., ...TTG+G+TT..., and ...TTG−G−TT... into mobility. It was observed that although the local setting angles are uniformly distributed in the condis phase, there is still a correlation with some complex pattern between local setting angles of the closest carbon atoms from neighboring chains. The results obtained in COMPASS and FW force fields for chain mobility and pair distribution functions in the condis phase are in good quantitative agreement. This increases credibility of modeling predictions.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b02710. Dynamic visualization of orthorhombic polyethylene crystal orthorhombic (AVI) J
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Dynamic visualization of monoclinic polyethylene crystal monoclinic (AVI) Dynamic visualization of polyethylene condis crystal condis (AVI) Dynamic visualization of one layer of chains in the polyethylene condis crystal (AVI) Distribution of chain setting angles in orthorhombic polyethylene (PDF)
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Tel: +7 495 939 08 88. Fax: +7 499 137 82 31. ORCID
Vadim I. Sultanov: 0000-0001-6031-4683 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We gratefully acknowledge the Joint Supercomputer Center of the Russian Academy of Sciences for computational resources granted.
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ABBREVIATIONS condis, conformationally disordered; 2D, two-dimensional; 3D, three-dimensional; PE, polyethylene; IR, infrared; NMR, nuclear magnetic resonance; MD, molecular dynamics; NPT, constant number of particles, pressure, and temperature
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REFERENCES
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DOI: 10.1021/acs.macromol.8b02710 Macromolecules XXXX, XXX, XXX−XXX
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DOI: 10.1021/acs.macromol.8b02710 Macromolecules XXXX, XXX, XXX−XXX
