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Conformational Transition Behavior of Amorphous Polyethylene across the Glass Transition Temperature Rongliang Wu, Xiongfei Zhang, Qing Ji, Bin Kong, and Xiaozhen Yang* Beijing National Laboratory for Molecular Sciences (BNLMS), Joint Laboratory of Polymer Science and Materials, State Key Laboratory of Polymer Physics and Chemistry, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190, China ReceiVed: December 16, 2008; ReVised Manuscript ReceiVed: March 21, 2009
Molecular dynamics simulations have been used to investigate the conformational transition behavior in amorphous polyethylene with different chain lengths across the glass transition temperature (Tg). In the present study, we examined the barrier height of conformational transition rates in different states. It was found that two lines of the logarithmic rates versus inverse temperature in the melt state and in the glass state are evidently different. The two lines have an intersection, which indicates Tg well. The barrier height in the glass state was unexpectedly observed lower than that in the melt state. For gaining better understanding of the transition barrier reduction, we analyzed motion heterogeneity of the systems and found the torsional transition rate distribution becoming gradually heterogeneous when the temperature went down to the glass state. The result indicates that the motion heterogeneity was caused by the torsion transition being “frozen”. The frozen torsions made the system into a nonequilibrium state and possess a novel transition behavior, which accounted for most of the transitions that started at a location close to top of barrier, and also the enhancement of a small magnitude of transition jumps. 1. Introduction The glass transition is marked by drastic changes in the mechanical properties of the material from a rubbery, viscous, amorphous solid to a brittle, glassy, amorphous solid. The nature of glass transition is still not clearly known and it remains a topic of intense experimental1-8 and theoretical interest.9-11 Common ways to account for the glass transition is based on the free volume effect,12 for example, the mode coupling theory (MCT),10,13 but recent experiments have shown that such explanations are not fundamentally sound2,14 especially at temperatures approaching the glass transition temperature (Tg).2 At high temperatures, the dynamics are nearly homogeneous and the MCT can be reasonably successful, while at low temperatures, the dynamics are strongly heterogeneous and a distribution of local barrier heights can be generated.2 According to the Adam-Gibbs theory,9 the origin of viscous slowing-down of dynamics close to Tg is the decrease in the number of configurations that the system is able to sample. The motions of single torsional bonds in polymer backbones are essential to the dynamics of polymer chains, and are of course closely related to the configurations of the entire system. In addition, Kanaya et al.15,16 found by quasielastic neutron scattering that the slow process of polymer motion on a time scale from several tens to hundreds of picoseconds could be assigned to an elementary process related to the local chain conformational transitions. Thus investigations on the single bond conformational transitions might shed some light on the microscopic mechanism of polymer glass transition. Currently, the rate of the conformational transition around single carbon-carbon bond can still not be easily observed in experiments for it is extremely fast.17 While molecular dynamics (MD) simulations make it possible to provide detailed informa* To whom correspondence should be addressed. E-mail: yangx@ iccas.ac.cn.
tion on the conformational transitions. Takeuchi and Roe18 as well as Boyd et al.19 found the conformational transition rates remain Arrhenius in character through Tg with an activation energy nearly equal to that for a single torsional barrier, implying there is little frictional drag contribution to the conformational crossing.19 Fukuda and Kikuchi20 investigated the difference in the conformational transitions between the melt and the subglass cis-polyisoprene chains. It was reported that 50% and 64% of the bond pairs showed no conformational jumps during 20 ns MD runs at 223 and 173 K, respectively. Karatasos et al.21 showed that the transition rates of torsions four or five bonds away from the chain ends illustrated an enhanced transition rates relative to those within the interior of the chains. Most other works19,22-25 focused on the overall relaxation behavior of polymer melts, which was often found to abide by the superArrhenius Vogel-Fulcher-Tammann behavior. The divergence of the time scales between the various macroscopic relaxation processes, whose complete decay requires the ergodic participation of all bonds,26 and the microscopic conformational transition rate was associated with the increasingly heterogeneous character of conformational dynamics in the melt with decreasing temperature.19,22 It was demonstrated that the greater dispersion in the transition rates lead to longer autocorrelation time and more stretched exponential character.26 The heterogeneous dynamics was supposed to be due to the bulk packing causing bonds to be trapped at and oscillate about torsional angles away from the torsional energy minima.19,22 Since using various criterions for the conformational transition behavior,18-20,27,28 the rates obtained are different as a function of temperature. In our previous work28 we examined all the available criterions for a conformational transition occurring to the system, and found that the shallow jumps (a criterion), which also included those without reaching deep of the potential well ((20° from the minima), were significant in the characterization of the motion of polymer chains across Tg. Using such reasonable
10.1021/jp8110919 CCC: $40.75 2009 American Chemical Society Published on Web 06/15/2009
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TABLE 1: Potential Functions and Parameters Nonbonded Unb(rij) ) A exp(-Brij) -C/rij6 H, H C, H C, C
A (kJ/mol)
B (nm-1)
C (kJ · nm6/mol)
11054.41 17995.14 62803.09
37.39 34.37 30.91
1.15 × 10-4 5.54 × 10-4 2.67 × 10-3
Bond Ubond(rij) ) 1/2Kbond(rij - rij0)2 rij (nm)
Kbond (kJ/(mol · nm2))
0.109 0.153
285348.8 376141.6
0
C-H C-C
Angle Ubend(Φijk) ) 1/2Kbend(Φijk - Φijk0)2 H-C-H C-C-H C-C-C
Φijk0(deg)
Kbend (kJ/(mol · rad))
108 110 111
322.168 359.824 451.872 Torsion
Utorsion(θijkl) ) ∑n3 ) 0Cn cosn(φ) H-C-C-H H-C-C-C C-C-C-C
C0
C1
C3
0.490 0.490 0.607
1.469 1.469 4.665
-1.958 -1.958 -5.272
a The parameters for torsion are equivalently converted for use in GROMACS.
criterion to learn the dynamics of the conformational transition rates of polymer chains through Tg is still open. In the present work, the conformational transition rates have been utilized to investigate the glass transition behavior of polyethylene (PE) chains by MD simulation. It has been found that the local motion behavior can describe Tg well, and the obtained barriers of the conformational transition in the melt are unexpectedly larger than those in the glass state. To get better understanding of the barrier reduction at low temperatures, analysis on the departure dihedrals and the transition dihedrals was made. 2. Methods 2.1. Force Field and Simulation Details. The linear PE chains, CnH2n+2, simulated have n ) 20, 44, and 100 with 216, 125, and 64 chains, respectively. The explicit atom force field developed by Smith and co-workers29 with minor modifications was used. The potential functions and parameters are listed in Table 1. The force field has proven accurate in predicting equilibrium and dynamic properties29-33 as well as the conformational characteristics31 of alkanes and polyethylene chains in melts. The torsional potential of the force field is given in Figure 1, in which the barriers between g and t are located at 120° and 240°. All simulations in the current study were carried out in GROMACS 3.3.34 The initial random coil configurations were generated from MD of all-trans chains at 600 K in vacuum, and then large gaseous cubic boxes with linear dimensions between 10-30 nm were built from these random coils and 500 ps NVT runs were performed at 600 K to allow the molecular structures to relax. The systems were subsequently shrunk in dimension with NPT runs of about 3 ns with periodic
Figure 1. Torsional potential with the t, g+ and g- states as well as the designated specific transitions. Ea is the barrier height of ergodic tg- transition in the melt and Ea′ is the schematic barrier of the trapped torsions in the glass state.
boundary conditions in xyz directions until density equilibriums were reached. After further equilibration runs of 2 ns, the systems were annealed at the same speed, 0.1 K/ps, to lower temperatures. All systems underwent separate annealing processes to the target temperatures from different initial conditions obtained from the equilibration runs. Each system was further equilibrated for 3 ns at an external pressure of 1 bar, subsequent production runs were extended for 2-4 ns, depending on the temperature. The trajectories were kept every 0.1 ps, resulting in more than 20 thousand frames. The leapfrog algorithm with an integration time step of 1 fs was used. All bonds were constraint with the LINCS35 algorithm. The Nose-Hoover36 temperature coupling and Parrinello-Rahman37 pressure coupling methods were used to control the temperature and pressure, respectively. Twin range nonbonded dispersion interactions were truncated at 0.8 and 1.0 nm. 2.2. Conformational Transitions. Conformational transitions are based upon the rotational isomeric state (RIS) model, all transitions between the RIS can be recorded in MD simulations and the transition rate kij from state i to j is defined as
kij )
Nij Nij ) Nit φiNt
(1)
where Nij is the number of transitions from state i to j; N is the total number of backbone dihedrals in the system; Ni is the number of dihedrals in state i; φi is the fraction of dihedrals in state i, and t is the sampling time. The determination of conformational transition varies in literature, for it depends crucially upon the time interval in recording or comparing subsequent conformations and it also gives different results with different definitions of RIS.38 The greater the time interval between sampling configurations, the smaller the number of barrier crossings will be counted since the torsions may cross and recross back without being detected. It was demonstrated that the lower limit on the time required for most recrossings to take place was ∼0.1 ps,38 which is exactly the time interval we keep our trajectories. As for the RIS definition, some claimed that each transition should cross the torsional potential minima of both the before and the after state; Boyd et al.19 proposed the criterion based on the dihedral angle being within a relatively narrow window
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Tg(n) ) Tg(∞) -
Figure 2. Specific volumes obtained from NPT simulations for the three systems. The lines are linear fits and the designated intersections are volumetric Tg.
of (5° or (10°; Roe et al.18 and Brown et al.38 accepted a transition if the dihedral angle crossed the top of the potential barrier; and Sumpter et al.39 counted transitions when the dihedral angles rotated more than 90°. Our previous work28 on the MD simulations of atactic polypropylene investigated the influence of RIS width (W) on the conformational transition behavior. The transitions which include those from just over the top to those reaching the potential well bottom within (20° from the minima were called “shallow jumps”, and those reaching deeper into the potential well bottom were called “deep jumps”. We examined both of the jumps and found the obtained rates as a function of temperature showed evidently discrete turning around Tg for the shallow jumps, while the deep jumps remained nearly unchanged albeit the existence of Tg. It was therefore concluded that the shallow jumps were significant for the property changes of polymers upon vitrification. In the present work, broad RIS was adopted as well as the shallow jumps, and the g+, t, and g- states are defined in the ranges of 10°∼110°, 130°∼230°, and 250°∼350°, respectively (W ) (50°) as shown in Figure 1. Transition rates with other definitions of RIS are also provided in the Supporting Information, and similar behaviors are obtained. The overall transition rate, kt, is defined as the sum over the specific transitions between the above-defined RIS
C n
(3)
where Tg(∞) is extrapolated to infinite chain length (279 K), and C is a constant. The experimental Tg of polyethylene varies from 200 to 250 K depending on the cooling process.41 Our values of Tg are very close to the observation values with a little bit higher degrees. The higher degrees is probably due to the much faster annealing speed (0.1 K/ps) implemented than could be achieved in experiments, but the simulated structures were in the equilibrium states at various temperatures above Tg. There are two reasons for the obtained structures in equilibrium. One is that in our earlier study42 a similar full atomic PE model that reached equilibrium within 1.5 ns at 400 K was concluded, but in the present study our equilibrium duration was even longer, 2 ns. The other reason is that we examined conformational distribution. Despite the high cooling rate, the trans-conformation fraction increases from ∼60% in the melt to ∼70% around Tg. Making a Boltzmann plot, logarithmic fraction vs reciprocal temperature, 2.51 kJ/ mol was obtained for the energy difference between the trans and the gauche conformation in the molten state. This value is consistent with the experimental value 2.09 ( 0.42 kJ/mol,43-46 and also with the calculated potential difference of butane 2.20 kJ/mol in Figure 1, which validates that the above simulated structures are in equilibrium. In the present study, our data show that the structures at various temperatures are in the equilibrium states on one hand, and on the other hand the structures are also in the amorphous state. Figure 4 shows a comparison of the single chain form factor of the three systems calculated from the simulation trajectories using the expression12
P(q) )
1 N2
〈
N
N
sin(qRij) qRij j-1
∑∑ i-1
〉
(4)
where N is the number of backbone carbon atoms per chain; Rij is the distance between carbon i and j within the same chain; q is the magnitude of the scattering vector. The angular brackets refer to average over all chains and time frames of a simulation. The line was calculated as a reference for pure random coils using the Debye function12
kt ) ktg+ + kg+t + ktg- + kg-t + kg+g- + kg-g+ ) 2(ktg + kgt + kgg) (2) where ktg, kgt, kgg are averages of ktg+ and ktg-, kg+t and kg-t, kg+gand kg-g+, respectively. 3. Results and Discussions 3.1. Structure and Glass Transition Temperature. The equilibrium specific volumes obtained from the NPT runs are plotted against temperature in Figure 2. The solid lines are linear fits, and the intersections of the lines are located at the temperatures of 204, 243, and 265 K for the C20H42, C44H90, and C100H202 systems, respectively. These temperatures are believed to be the volumetric Tg for PE at different chain lengths. The molecular weight dependence of Tg can be described by the equation proposed by Fox and Flory40 for intermediate chain lengths as shown in Figure 3:
Figure 3. Glass transition temperatures from specific volume and transition rate against reciprocal chain length n. The constant C is typically in the range 103-104 K mol-1.
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P(q) )
2 [exp(-q2〈Rg2〉) + q2〈Rg2〉 - 1] 2 2 (q 〈Rg 〉) 2
Wu et al.
(5)
where 〈Rg2〉 is the mean-square radius of gyration calculated from the C100H202 system. The P(q) for the C100H202 system agrees well with the Debye function at both 500 and 100 K, implying Gaussian behavior of the chains at these two temperatures. The other two systems also exhibit close Gaussian behavior for the short chain in the random coil shape. 3.2. Conformational Transition Behavior across Tg. Conformational transition is of local dynamics in polymer systems. The transitions between the rotational isomeric states occur to each backbone torsion and the accumulation of such conformational transition is believed to be the very origin of the global changes in polymer chain systems. Earlier studies18,19 came to the same conclusion that there was no evident difference in the conformational transition behavior when the system crossed Tg. In fact, Takeuchi and Roe already found the conformational transition rate at 150 K is substantially higher than the value predicted by the Arrhenius line,18 but the single value below Tg could not offer them more insight into the glass state. Boyd et al.19 had more data below Tg, but the data below 180 K were not shown. Our previous study28 found that the criterion for the transition behavior is important. It was proposed that the shallow jumps offer more evident data than the deep jumps do on showing different conformational transition behaviors in the two states. Based on these findings, we have utilized the shallow jumps to investigate the changes of local chain conformational transitions in the melt and in the glass state. In the present study, the overall transition rates, kt, are plotted in logarithm with respect to inverse temperatures for all three systems in Figure 5. As also shown in many publications,18,23,25 kt have perfect Arrhenius relationships with temperature in the high temperature region,
( )
kt ) A exp -
Ea RT
or
ln kt ) -
Ea 1 + const RT
(6)
where Ea represents the activation energy or barrier for conformational transition and R is the ideal gas constant. When the temperature decreases through Tg in the glass state, significant deviation of kt from the Arrhenius line in the high temperature region is observed. In the low temperature region,
Figure 4. The single chain form factors of polymer chains plotted with the Debye function.
Figure 5. Overall transition rates against inverse temperature of amorphous polyethylene.
TABLE 2: Activation Energies of Conformational Transitions Ea (kJ/mol) Tg
C20H42
overall gft tfg gTg
6.48 6.99 6.39 8.61
12.49 11.09 13.79 17.59
C44H90
overall gft tfg gTg
4.96 5.02 4.93 10.55
12.51 11.11 13.30 17.73
C100H202
overall gft tfg gTg
3.95 3.97 3.94 12.46
12.53 10.98 13.43 17.93
the rates also fit well with the Arrhenius equation for all chain lengths, meanwhile, the temperature dependences get much smaller with increase of the chain length. The intersections of the transition rates take place below the volumetric Tg, locating at 182, 208, and 221 K for the C20H42, C44H90, and C100H202 systems, respectively. These Tg temperatures can also be well fit by eq 3, which yields a Tg(∞) value of 230 K (see Figure 3), closer than the volumetric Tg(∞) to the experimental Tg values41 (200-250 K). Tg obtained from the conformational transition rates is more accurate and lower than that from the volumes. This is because that the two measurements regard different relaxation behaviors. Thermodynamically, the former is based on equilibrium of local conformations and the latter on equilibrium of long scale structures, even the equilibrium duration used here proved reasonable.42 The activation energies (eq 6), or the transition barriers, for the overall and specific transitions between the RIS are listed in Table 2. Results show that above Tg, the transition barriers are very close to each other, which indicates that the conformational transition behavior at high temperatures are nearly the same for all chain lengths. Generally speaking, the t f g barrier is not necessarily the same as the g f t barrier. These barrier heights calculated from a torsional potential curve are an approximation, since the conformational transition does not move along the potential curve but the free energy surface. The barrier heights therefore should be from the measurement of a kinetic process. Table 2 with such measurements provides a chance to compare them with those in the potential curve (Figure 1). The
Tg Characterization through Conformational Transition barrier difference between g f t and t f g transitions are 2.70, 2.19, and 2.45 kJ/mol for C20H42, C44H90, and C100H202 systems, respectively. These values are very close to the potential energy difference, about 2.20 kJ/mol, between the g and t states obtained from Figure 1. The barriers for g T g transitions are the highest of the three, with a value near 18 kJ/mol, which is 2 kJ/mol lower than the one from the potential energy at φ ) 0. These results indicate that the barriers obtained from the kinetic process on the free energy surface are different from those from the potential curve, but close to each other. Below Tg, all barriers in Table 2 were found drastically reduced. This finding is unexpected but truly exists. The barrier height in the glass state is lower than that in the melt state. This phenomenon was found here related to the chain being “frozen”, the breaking down of ergodicity in the glass state. The break down of ergodicity leads directly to the inhomogeneous behavior. 3.3. Heterogeneous Distribution of Conformational Transition. The spatial heterogeneity in dynamics has been evidenced to exist in polymer and super-cooled liquids near Tg.2,8 The heterogeneity means the local dynamics in some regions of the sample can be orders of magnitude faster than dynamics in other regions only a few nanometers away.2 The conformational transition behavior also exhibits such heterogeneous distributions19 since it is of local dynamics. In the present study, we found that the heterogeneity is caused by the chain being “frozen”, which was observed as shown in Figure 6. In this figure, every bond in the system gets along as the abscissa axis. In part a, the conformational transition rates are basically around a fluctuating value. Part b of the figure describes the transition behavior in the glass state. The transition rate behaves in opposite directions: some tremendously increased, even higher than those at above Tg, and some somehow decreased. It is evident that many bonds have kt ) 0 in the curves at 250, 180, and 140 K, as was also found by Fukuda and Kikuchi.20 This means the bonds have been frozen within a RIS. The number of the frozen bonds increases as the temperature decreases. The system thus loses the ergodicity. On the contrary, those activated torsions should make major contributions to the phenomena of larger kt and reduced barrier in the glass state. The details on these transitions might offer more information on how these activated torsions are generated. The above results and discussion showed the heterogeneity of every bond in the system. Now we open another window to see the heterogeneity of a polymer chain. The average kt of torsional bonds along the C44H90 chains are plotted in Figure 7 at various temperatures. In the melt, the kt distributions are nearly homogeneous except the ending 2-3 torsions, which have slightly larger values than those interior of the chains. Analysis shows that the barrier of the ending torsion is 0.6 kJ/mol lower than those of the interior torsions. The higher mobility of the ending torsional transitions causes a barrier height descent from 12.53, 12.51, to 12.49 kJ/mol for the three different molecular weights (Table 2). In the glass state, the kt distribution loses the equilibrium feature and gets increasingly heterogeneous. The ending torsion mobility even disappears (Figure 7). Results in Figure 5 and in Table 2 also show that below Tg the transition rate increases and the barrier height considerably decreases when increasing molecular weight. At this moment, these novel behaviors cannot be interpreted as those above in the thermodynamics equilibrium state. It needs further study. 3.4. Departure Dihedrals and Transition Dihedrals. The details of the conformational transition behavior can be characterized by the departure dihedral and the transition dihedral.
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Figure 6. The transition rate of each torsional bond of the C44H90 system (a) in the melt and (b) in the glass state.
The departure dihedral is the dihedral site, where the dihedral starts a transition. The transition dihedral is the region the dihedral sweeps over during a transition. Figure 8a shows distributions of the departure dihedrals during the g f t transitions at all temperatures for the C100H202 system. At 300-500 K, the departure dihedral distributions have almost the same curves, spanning nearly the entire basin of RIS. While below Tg, significant decrease of population around the basin center (∼69°) and sharp enhancement of population near the barrier top (120°) can be distinctively seen. The transition dihedral distribution given in Figure 8b shows a similar phenomenon. Though a broad definition of RIS (100°) is used in our shallow jumps, the fast oscillations around the barrier top (with transition dihedrals ∼20°) do not prevail at all temperatures. At high temperatures the dihedrals cross over a broad range of dihedral angles from 20° to more than 100°; while below Tg, most dihedrals sweep over 30 ( 10° and many of the larger magnitude transitions from 60°∼100° are inhibited. As already found in Figure 6b, below Tg the local dynamics is substantially activated and some torsions have transition rates orders of magnitude higher than expected. These activated torsions were supposed from small magnitude transitions, and
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Wu et al. the present study, our result of the departure dihedral and the transition dihedral indicates that in the glass state the conformational transition behavior becomes (1) starting very close to the top of the barrier, (2) jumping a very small step, about 30°. This result supports the above mechanism. It should be noted here that the number of activated torsions is still quite minor seen from Figure 6, although they make major contributions to the system dynamics. Most of the torsions are still trapped within the RIS bottom, generally attributed to the “frozen” local conformational structure or environment. 4. Conclusions
Figure 7. Average transition rate distributions along the C44H90 chains at selected temperatures.
Molecular dynamics simulations have been performed on linear polyethylene chains with 20, 44, and 100 backbone atoms in the amorphous state through Tg. The local dynamics, namely the conformational transition rates in our study, follow independent Arrhenius relations for all chain lengths at both high and low temperatures, the intersects of Arrhenius lines at both temperature ranges indicate Tg. Above Tg, the transition rates and barriers are very close to each other, which indicates that the conformational transition behaviors are nearly the same for all chain lengths at high temperatures. Below Tg, the Arrhenius lines diverge from each other and the transition rates appear orders of magnitude larger than expected in equilibrium, and in the meantime the transition barriers get unexpectedly lessened in the glass state. This transition barrier reduction was accompanied by heterogeneity of the transition rate distribution. We found in the present study that the heterogeneity was caused by the torsions gradually being “frozen” (kt ) 0) as temperature descent. The torsions in the system being stochastically frozen made changes in the transition process. We found that the departure dihedrals of the transition enhanced at the location close to the top of barrier in the glass state. This change made an opportunity of easy step jumping over the barrier. On the other hand, the transitions were found with small magnitude of dihedral angle rotations, about 30°. The change in the transition process is probably the reason of the activated dynamics and markedly diminished barriers in the glass state. Acknowledgment. We acknowledge the support from National Science Foundation of China (20474073, 20490220, 20674090, 20874107). Supporting Information Available: Transition rates with different RIS width. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes
Figure 8. The distributions of (a) the departure dihedral and (b) the transition dihedral of g f t transitions at all temperatures for the C100H202 system.
an activated dynamics mechanism was assumed by Boyd and Smith.26 They thought that in glasses where the dynamics get severely retarded, overpopulation of torsions divergent from the potential minima are located, and the residence times at these sites are so long that the bonds can be considered to be trapped. These torsion sites may become eligible centers for the conformational transitions, which results in the reduced effective barrier (Ea′ compared with Ea sketched in Figure 1).26 In
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