Molar Mass Dependence of Polyethylene Chain Dynamics. A Quasi

Dec 17, 2012 - ISIS, Rutherford Appleton Laboratory, Chilton, Didcot OX11 OQX, U.K.. § Department of Materials, University of Oxford, Parks Road, Oxf...
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Molar Mass Dependence of Polyethylene Chain Dynamics. A QuasiElastic Neutron Scattering Investigation V. Arrighi,*,† J. Tanchawanich,†,∥ and Mark T. F. Telling‡,§ †

Chemistry, School of Engineering and Physical Science, Heriot-Watt University, Edinburgh, U.K. ISIS, Rutherford Appleton Laboratory, Chilton, Didcot OX11 OQX, U.K. § Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, U.K. ‡

ABSTRACT: We present a detailed analysis of the incoherent dynamic structure factor of a series of n-alkanes and high molar mass polyethylene (PE). C30H62 data indicate that two molecular processes are simultaneously active in our experimental temperature and time range but start to disentangle at low temperature. These are tentatively identified with torsion-vibrations and conformational relaxation. Our study fully supports a number of MD investigations and confirms the unique behavior of PE. For the first time, we demonstrate that the molecular processes detected by QENS for alkanes and PE occur on a similar time scale, and we therefore suggest that common molecular mechanisms are responsible for the decay of the intermediate scattering function. Interestingly, the molar mass dependence of the characteristic times and activation energies at the lowest Q investigated (1 Å−1) are comparable to those obtained from rheological measurements and follow a simple trend which can be accounted for by free volume theory. This surprisingly simple outcome suggests that local dynamics (e.g., conformational transitions) are intimately linked to the long-range motion. The surrounding medium has a global effect on the microscopic motion that can be modeled by the friction coefficient.

I. INTRODUCTION As a major commodity polymer, polyethylene (PE) has been extensively investigated. Its rheological properties are of considerable practical importance to optimize processing conditions, and numerous dynamic studies of polyethylene melts have been reported. Experimental rheological1−5 and NMR6,8,7 measurements have been particularly informative and have been supplemented by computer simulations.9−21 Systematic investigations have been carried out to understand the rheological behavior as a function of temperature and molar mass, M, and to test molecular theories of polymer melt dynamics, specifically the Rouse and reptation models.5 Viscosity measurements have revealed a strongly non-Arrhenius temperature dependence;4 the apparent activation energy was found to increase with the molar mass reaching an asymptotic value of 27.62 kJ mol−1 at high M.1−4 Self-diffusion coefficients determined from spin-echo NMR measurements were also reported to vary with M, with values close to those obtained from neutron scattering measurements.6 As predicted by the Rouse model, for unentangled chains, the product of the viscosity, η, and diffusion coefficient, D, was found to be independent of the monomer friction coefficient,5 while above the entanglement threshold, ηD, increased with increasing M. The 13C NMR measurements carried out by Qiu and Ediger8 on unentangled PE melts (C44H90 and a PE sample with weight-average molecular weight, Mw = 2150 g mol−1) gave © XXXX American Chemical Society

evidence of a long time tail described by Rouse modes in addition to a fast segmental motion. For both C44H90 and PE, the activation energy, Ea, for the segmental dynamics was found to be equal to 16 kJ mol−1, in good agreement with results on n-tridecane (C13H18)15 but significantly less than the flow activation energy measured for PE melts.1,2 The large difference between conformational dynamics and flow activation energy suggests that isolated conformational transitions can take place in PE meltsa consequence of the lack of side groups for this polymer. It was postulated that since conformational transitions can occur independently, and do not require cooperation from neighboring chains, this type of motion does not contribute to flow. This behavior differs considerably from that observed for other polymers such as polypropylene (PP) and polydimethylsiloxane (PDMS). For example, we have found very good agreement between the temperature dependence of the characteristic times measured by neutron scattering and that determined from rheological measurements.22,23 Incoherent quasi-elastic neutron scattering (QENS) has also been employed to determined diffusion coefficients.24,25 For example, Rennie et al.24 investigated the diffusion of short nalkanes (C36H74 and C40H82) in a melt of deuterated PE. The Received: September 13, 2012 Revised: December 4, 2012

A

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incoherent intermediate scattering function, computed from the MD trajectories of a short PE chain at 350 K, has provided unambiguous evidence for the existence of two distinct processes.37 Because of its high melting temperature, experimental measurements of PE melt dynamics are usually carried out far above Tg. To be able to access lower temperature, we performed experiments on PE oligomers which have a low melt temperature, Tm, namely C30H62, and by using spectrometers with different energy resolution have attempted to cover a wide temporal window. Despite considerable MD simulations carried out on a range of well-defined n-alkanes, few QENS measurements have been reported. In a recent study, Smuda et al.38,39 have systematically investigated a number of medium chain alkanes using ToF QENS measurements. The authors analyzed the QENS data using two-Lorentzian functions attributed to the long-range diffusion of the whole molecules and local molecular motions. Considerable discrepancy between the pulsed-field gradient NMR (PFG-NMR) and QENS diffusion coefficients was found, the differences increasing with increasing chain length. Here, by analyzing data over a wide time window, we find experimental evidence for the existence of different processes. Furthermore, using a consistent way to describe the dynamics of the hydrocarbon chains, we demonstrate that, within the experimental time window and length scales accessible, there is no substantial difference in the underlying dynamics observed in short n-alkanes and long-chain polyethylene. The paper is structured in the following way. Experimental details on the samples and neutron scattering measurements are given in section II. Our analysis of the QENS data is discussed in section III. We first describe in detail our results for C30H62 and then use a consistent model to extract information on the molar mass dependence of the dynamic processes. Comparison between our results and literature data (from MD and neutron scattering) is given in section IV, followed by Conclusions.

authors detected deviations from the predictions of the Rouse model which they attributed to local chain rigidity effects.26 These measurements were restricted to low momentum transfer, Q, values and to a narrow energy transfer range (−10 to 10 μeV), a regime which is appropriate for the measurement of diffusive behavior of short chain alkanes. More recently, Kanaya et al.27 exploited a time-of-flight (ToF) spectrometer to characterize the local dynamics of a high molecular weight PE melt in the energy range 0.01−10 meV. The authors analyzed the QENS data using two distinct processes (each modeled by a Lorentzian function): a fast, temperature-independent process and a slower motion assigned to conformational transitions. In agreement with molecular dynamics (MD) simulation on PE-like chains, the results of Kanaya et al. suggested that these conformational transitions are localized within three bonds. Over the past 10 years, several attempts have been made to quantitatively compare experiments and predictions from detailed, atomistic MD simulations. For PE, simulations and experiments have been carried out on well-characterized systems, including n-alkanes and low molecular weight PE chains. Quantitative agreement between ToF neutron scattering measurements and MD simulations was reported by Smith et al.28 for n-C100H202 at 504 K. Through the analysis of their MD simulation data, the authors were able to conclude that the decay of the incoherent dynamic structure factor, S(Q,ω), was the result of a combination of torsional oscillations and conformational jumps occurring on a similar (picoseconds) time scale. The relative contribution of these two processes depends on the length scale of observationthe fast process being more pronounced at relatively short distances while the slow process dominates at larger length scales. The results of Smith et al.28 highlight a further anomaly of the PE dynamics. Contrary to the single-step decay observed for PE-like chains, other polymers display a typical two-step decay function consisting of fast motions in the picoseconds regime and a slower diffusive process related to conformational jumps. Evidence of two distinct processes well separated in time has been reported for polyisoprene,29,30 poly(vinyl chloride),31,32 polybutadiene,33 polypropylene,34 and polyisobutylene.27 Indeed, as recently discussed by Arbe and Colmenero, a “two-step decay” correlation function is a general signature of the supercooled liquid state which is supported by numerous experimental data and MD simulations.35 In a previous study,36 we also demonstrated quantitative agreement between the incoherent dynamic structure factors of C44H90 and a PE sample with Mw equal to 2150 g mol−1 from experiments and MD simulations. In that work, our QENS data were analyzed at a fixed temperature (450 K) and at a single resolution. Furthermore, we focused on a refined analysis of the intermediate scattering function, I(Q,t), from MD to disentangle the different dynamic processes contributing to the observed incoherent dynamic structure factor. The present work considerably extends our previous study as we report new QENS data collected over a range of temperatures, at three different resolutions, on a series of samples of increasing molar mass from C30H62 to high molar mass PE. It has been shown by MD simulations carried out by Roe11,12 that separation between the fast and slow processes in PE can be achieved at low temperature, i.e., below the experimental melting point. More recently, detailed analysis of the

II. EXPERIMENTAL SECTION A. Materials. The polyethylene (PE2K and PE108K in Table 1) and n-alkane (C30H62, C44H90, and C60H122) samples were purchased from Scientific Polymer Products and Aldrich, respectively.

Table 1. Molecular Characterization of the Samples Used in This Work sample

formula/code

Mw (g/mol)

triacontane tetratetracontane hexacontane polyethylene polyethylene

C30H62 C44H90 C60H122 PE2K PE108K

443 619 844 2150 108000

PDI

Tm (K)

1.15 1.32

342 362 376 398 375

Melting temperatures (Tm) were determined using a TA Instruments DSC 2010 differential scanning calorimeter with both heat flow and temperature scales calibrated against indium metal. Nitrogen was used as the purge gas, and the samples were scanned at 10 °C min−1. Values of the melting temperature, Tm, are listed in Table 1, alongside their molecular weight, Mw, and polydispersity index (PDI). For the PE2K and PE108K samples, Mw values refer to the weight-average molecular weight. B. Neutron Scattering Measurements. Quasi-elastic neutron scattering experiments were carried out using the high-resolution timeof-flight back scattering spectrometer OSIRIS40 (ISIS, Rutherford Appleton Laboratory, UK) and the direct geometry time-of-flight B

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where β is a shape parameter (0 < β ≤ 1) measuring the deviation from a simple exponential decay and τKWW represents the relaxation time. This function has been extensively used for the analysis of MD simulation results of alkanes, e.g. C44H9046,18,16 and C13H28,15 as well as experimental neutron data.24,35,36 Other models have also been adopted to describe the incoherent dynamic structure factors of short chain alkanes and PE. For example, Buchenau et al.25 fitted the incoherent dynamic structure factor of polyethylene melts using a superposition of two Lorentzians. The narrow component was characterized by a Q2-dependent broadening attributed to free diffusion of the monomeric units/chain segments. The broader component could be fitted equally well by using two different models: (a) restricted diffusion within a sphere or (b) quasi-harmonic vibration. The second model was found to be more realistic, and the results were consistent with studies of drawn deuterated semicrystalline polyethylene at low temperature. Similarly, Kanaya et al.27 used two Lorentzian functions to analyze the QENS data of PE close to 450 K. These authors were able to describe the PE local dynamics in terms of a fast process on the picosecond time scale and a slower one, called the E process, on a 10 ps time scale. The latter was related to a jump diffusion process which omits the connectivity of the chain. Smuda et al.38 and Unruh et al.39 have also used two Lorentzians to model ToF data from several short to medium chain alkanes from C8H18 to C44H90 but have identified one of the two processes with slow long-range diffusion. As we have previously shown for C44H90 and PE2K,36 and reported elsewhere for C30H62,47 the experimental QENS spectra from a wide energy range time-of-flight spectrometer can be fitted equally well by a stretched exponential or a twoLorentzian model. Thus, it is our view that the incoherent dynamic structure factor of PE melts does not provide any clear evidence for the existence of two distinct dynamic processes. In contrast, for other polymers, the local relaxation is split into a fast motion in the picoseconds range and a much slower process.29,32−34 On the basis of the lack of experimental evidence for two distinct processes, as described in the following section, we decided to model the intermediate scattering function using a KWW:

spectrometer NEAT (BENSC, Hahn-Meitner Institut, Germany). A summary of the QENS measurements is given in Table 2.

Table 2. Summary of the QENS Measurements Carried out on n-Alkanes and PE Samples spectrometer NEAT (fwhm = 1.4 meV) NEAT (fwhm = 93 μeV)

OSIRIS (fwhm = 24.5 μeV)

sample C30H62 C30H62 C44H90 C60H122 PE2K PE108K C30H62 C44H90

temp (K) 350, 350, 400, 400, 420, 420, 350, 370,

370, 400, 420, 450, 450, 440, 370, 385,

400 450 440, 450, 480 480 460 450, 470, 480 400 400

Slab sample cells were used for all QENS experiments. The cells used allowed for a sample thickness of 0.18 mm. For the polymeric materials investigated here, such a thickness corresponds to a transmission of ca. 0.9, which minimizes multiple scattering effects; correction for such effects is therefore being neglected during data analysis. In a neutron experiment, the measured quantity of interest is the double-differential scattering cross section, ∂2σ/(∂E ∂Ω), which defines the probability that a neutron is scattered with an energy change ∂E into the solid angle ∂Ω. The neutron scattering cross section, σ, includes both coherent and incoherent scattering contributions. However, for the hydrogenous polymers studied here, the incoherent cross section is significantly larger than the coherent contribution. As a result, any motion observed arises mainly from the incoherent scattering of the hydrogen atoms. On OSIRIS, measurements were carried out on C30H62 and C44H90 in the temperature range 350−400 K. QENS spectra were collected using the PG002 analyzer which afforded an energy resolution of 24.5 μeV (measured as full width at half-height (fwhm)), an energy transfer (ΔE) range from −0.2 to 1.2 meV, and a Q range from 0.5 to 1.8 Å−1. The dynamic incoherent structure factor, S(Q,ω), was computed from the measured time-of-flight data after first subtracting the contribution of the empty cell and correcting for neutron absorption. Data reduction was performed using standard software routines available at ISIS.41 S(Q,ω) was subsequently converted into the time-dependent intermediate scattering function, I(Q,t), using the FURY routine.42 Experiments on NEAT43 were carried out at two energy resolutions, 93 μeV and 1.4 meV (fwhm), by using incident neutrons of wavelength, λ, equal to 5.1 and 3 Å, respectively. These configurations allowed access to Q ranges of 0.5−2.0 and 1.25−3.50 Å−1, respectively. All experimental spectra were corrected in the usual way, i.e., subtraction of the empty cell, adjustment for detector efficiency, normalization using vanadium, and absorption correction. Because of the wide energy transfer ranges covered during the measurements (i.e., −1 to 10 meV and −5 to 50 meV for the higher and lower resolution, respectively), the raw, experimentally measured S(θ,ω) data cannot be simply converted to the incoherent dynamic structure factor S(Q,ω) at constant Q. To achieve this, an interpolation procedure was used (code INGRIDQ), after which the energy transfer range becomes Qdependent with maximum values varying from 1.4 to 50 meV for Q increasing from 0.5 to 3.5 Å−1.

β⎤ ⎡ ⎛ t ⎞ ⎥ I(Q , t ) = A exp⎢ −⎜ ⎟ ⎢⎣ ⎝ τKWW ⎠ ⎥⎦

(2)

where the characteristic time, τKWW, is both temperature- and Q-dependent. The parameter A is a temperature- and Qdependent amplitude factor, related to the atomic mean-square displacement, ⟨u2⟩: ⎡ ⟨u 2⟩Q 2 ⎤ A(Q ) = exp⎢ − ⎥ 3 ⎦ ⎣

III. ANALYSIS OF QENS DATA OF N-ALKANES AND PE Analysis of relaxation processes in polymer materials often makes use of the empirical Kohlrausch−Williams−Watts (KWW) function or stretched exponential:44,45 β⎤ ⎡ ⎛ t ⎞ ⎥ ⎢ ϕ(t ) = exp −⎜ ⎟ ⎢⎣ ⎝ τKWW ⎠ ⎥⎦ (1)

(3)

To account for the changes in the distribution of relaxation time with temperature, we determine the average relaxation time, ⟨τ⟩, which can be obtained from ⎛1 ⎞τ ⟨τ ⟩ = Γ⎜ ⎟ KWW ⎝β⎠ β C

(4)

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Values of the stretched exponent β vary in the range 0.35− 0.63 and display a pronounced temperature and Q dependence (Figure 3), which suggests changes in the distribution of relaxation times. This observation is consistent with our analysis of C44H90 and PE2K at 450 K.36

where Γ represents the gamma function. The dynamic structure factor, S(Q,ω), is related to the intermediate scattering function by Fourier transform and can be expressed in a manner consistent with eq 2: S(Q , ω) = F(Q )[SKWW (Q , ω) ⊗ R(Q , ω)] + B(Q ) (5)

where F(Q) is a temperature- and Q-dependent scaling factor, SKWW(Q,ω) is the numerical Fourier transform of the KWW function, and R(Q,ω) is the spectrometer’s resolution. The parameter B(Q) represents the background which accounts for those fast motions that are not accessible within the experimental energy range. III.1. KWW Representation of the C30H62 Incoherent Dynamic Structure Factor. To date, most of the QENS measurements on n-alkanes and PE have been carried out on time-of-flight spectrometers at low energy resolution but over a wide energy range. It is therefore instructive to compare the dynamic behavior of these materials at different resolutions and energy ranges. As shown in Figures 1 and 2 for T = 350 K, the incoherent dynamic structure factors of C30H62 are well described by eq 5

Figure 3. Momentum transfer dependences of the (a) β parameter and (b) the average relaxation time, ⟨τ⟩, obtained from the analysis of the NEAT S(Q,ω) data of C30H62 at T = 350, 400, and 450 K.

When eq 5 is used to fit the OSIRIS incoherent dynamic structure factors at 350 K, we note considerable discrepancy between β and ⟨τ⟩ values obtained from the two instruments (Figure 4). We note that the OSIRIS β values are higher than those obtained from NEAT and show little variation with Q. At 350 K, the average β is 0.57 ± 0.06, and similar values were obtained at higher temperature. These findings provide experimental evidence in support of the coexistence of different dynamic processes. First of all, we note that the pronounced variation of the shape parameter with T and Q is a distinctive feature of PE-like chains, not displayed by polymeric systems such as polydimethylsiloxane22 and polypropylene48 for which β is found to be independent of the temperature and momentum transfer. MD simulations carried out by Arialdi et al. on short-chain PE37 have shown that, at 350 K, the decay of the intermediate scattering function (and therefore the observed broadening of the incoherent dynamic structure factor) results from a combination of different processes: a fast torsion-vibration motion which is weakly Q- and T-dependent and a slower process related to conformational jumps. Changes in the relative contribution of these two processes with temperature or momentum transfer could be responsible for the observed Q and temperature dependence of the shape parameter. The existence of two processes overlapping in time could also

Figure 1. S(Q,ω) data from NEAT of C30H62 at 350 K and Q = 0.75, 1.50, and 2.0 Å−1 (from front to back). The lines are fits using eq 5.

Figure 2. S(Q,ω) data from OSIRIS of C30H62 at 350 K and Q = 1.0, 1.5, and 1.75 Å−1 (from front to back). The lines are fits using eq 5.

at all resolutions. Similar fits were obtained at other temperatures. For the NEAT data, the background component was found to fluctuate around zero, and so data fitting was performed using a simplified version of eq 5 (with B(Q) = 0). D

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Figure 4. Comparison between (a) β and (b) τeff parameters for C30H62 at 350 K as obtained from analysis of the NEAT (solid squares) and OSIRIS (empty circles) data.

Figure 5. Activation energy extracted from (a) C30H62 NEAT data in the temperature range 350−450 K as a function of Q and (b) OSIRIS data in the temperature range 350−400 K.

explain the discrepancy between the OSIRIS and NEAT fitting parameters. One further observation in support of the existence of two processes is that the temperature dependence of the ⟨τ⟩ values from NEAT varies with the momentum transfer, Q. We note that, for the sample investigated here, measurements are carried out at temperatures that are far above the glass transition. As a result, the segmental relaxation follows a Arrhenius-like temperature dependence. Values of activation energy, Ea, can be extracted from the Arrhenius dependence of ⟨τ⟩ on T, and we observe a systematic decrease of Ea with increasing Q (Figure 5). Once again, this finding seems a unique feature of PE-like chains, which is observed here for short alkanes but also reported for a high molecular weight PE sample.35 For other polymers it is common to separate T and Q dependencies of the relaxation time and use ⟨τ⟩ values at Q = 1 Å−1 to extract information on the activation energy. We note that the OSIRIS Ea values fluctuate around an average of 19.6 ± 3.1 kJ mol−1, close to the apparent activation energy reported from viscosity measurements.4 In summary, the analysis of the S(Q,ω) data from OSIRIS and NEAT strongly supports findings from MD simulations that, within the time scale of QENS measurements, two different processes are simultaneously active, having different activation energies and Q-dependent amplitudes.36,37 Comparison between OSIRIS and NEAT suggests that at this low temperature (350 K) the two processes can be somewhat resolved; the NEAT spectra seem well reproduced by a model that consists of a single process whereas the OSIRIS S(Q,ω) data display a background component associated with fast motions. The KWW parameter obtained from the OSIRIS spectrometer is in good agreement with those reported for the conformational relaxation from MD simulations.36,37

III.2. Time Domain Analysis of the C30H62 Overlapped Data. One advantage of converting the incoherent dynamic structure factor into the time domain intermediate scattering function, I(Q,t), is that, through deconvolution, resolution effects are eliminated, and a direct comparison between experimental data is possible. This approach is model independent unlike the comparison made earlier between fitting parameters extracted using the KWW function. The combined intermediate scattering functions computed from the S(Q,ω) at three different resolutions are shown in Figure 6 for C30H62 at 350 K and selected Qs. The overlap between the three I(Q,t) data sets was found to be excellent.

Figure 6. I(Q,t) of C30H62 at 350 K and Q = 1.25, 1.5, 1.75, and 2.0 Å−1 on scale of 0 to 1 and 0 to 0.2. Symbols correspond to experimental data from the NEAT and OSIRIS spectrometers. Solid curves are fits using eq 6 while the dashed lines are fits using eq 2. E

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⎛ σ2 ⎞ ⟨τ ⟩ = τmax exp⎜ ⎟ ⎝2⎠

Choosing an appropriate model function to describe the I(Q,t) data over the entire time range is not trivial. First of all, as we noted earlier, the experimental data do not provide clear evidence for a two-step decay function; using models such a two-Lorentzian functions is as arbitrary as the use of a single, albeit distributed process. Therefore, we have decided to base our choice of model function on experimental evidence and use a single relaxation function. The discrepancy between fitting parameters obtained from the frequency domain suggests that the KWW function may not be suitable to model the I(Q,t) decay of the NEAT/OSIRIS combined data. This is demonstrated in Figure 6 where the intermediate scattering function of C30H62 at 350 K is seen to deviate from a KWW fit, at all Q values. As suggested by simulations,19,49,36,28 the I(Q,t) data can be expressed as ⎛t ⎞ ⎟ ⎝ τi ⎠

N

I (Q , t ) =

∑ gi exp⎜ i=1

The momentum transfer dependence of the Ea values (Figure 8) is in qualitative agreement with the KWW analysis of the

(6)

Figure 8. Momentum transfer dependence of the activation energy of C30H62. The dashed line is simply a guide to the eye.

where gi corresponds to a log-Gaussian distribution of relaxation times defined as gi ∝ g(ln τi) =

⎛ −(ln τ − τ )2 ⎞ i max ⎟ exp⎜ 2 2 2σ ⎝ ⎠ 2πσ

(8)

S(Q,ω) curves. At low Q, i.e., at large length scale, the activation energy tends to 16.9 ± 1.7 kJ mol−1, close to the Ea reported for C30H62 from rheological measurements (16.78 kJ mol−1). We observe a considerable decrease with increasing Q, until at Q ≥ 3 Å−1 Ea reaches values less than 5 kJ mol−1. These results provide unambiguous evidence of the coexistence of distinct dynamic contributions: a high activation energy process which is likely associated with conformational relaxations and a fast, localized motion. The combined QENS data of C30H62 offer dynamic information over a wide time window, covering more than 3 decades in time. As shown in Figure 6, the I(Q,t) curves decrease monotically to zero within the experimental time range, and it is therefore possible to extract the normalized distribution of relaxation time (DRT), F(ln τ):

1

(7)

Equations 6 and 7 define a function with two adjustable parameters: the relaxation time at the maximum probability, τmax, and the corresponding deviation of τmax, σ. The relationship between log τmax and log Q is nonlinear. As shown in Figure 7, at high temperature and high Q, τmax follows

∞

I (Q , t ) =

∫−∞ F(ln τ ; Q ) exp(−t /τ) d(ln τ)

(9)

50,51

using a CONTIN analysis, in a way similar to that described by Arialdi et al.36 According to eq 9, the I(Q,t) decay is described by a superposition of exponentials, weighted by a DRT. Information on the number of distinct processes and the dispersion of exponential decays describing each process are obtained from the number and width of the peaks appearing in the DRT, respectively. Figure 9 shows the F(ln τ) curves derived from the CONTIN analysis of the combined C30H62 data at 350 K and Q = 1.00−1.75 Å−1. Although we do not observe two separate peaks, the shape of the DRT changes with temperature and momentum transfer. As shown in Figure 9, the distributions of relaxation times at Q = 1.25 and Q = 1.50 Å−1 appear to be symmetric and can be described using a log-Gaussian distribution of relaxation times, but at Q = 1.00 and Q = 1.75 Å−1 the DRT is asymmetric. Although agreement with the DRT data reported by Arialdi et al.37 at 350 K is not quantitative, we note strong similarities: (a) the characteristic time of the fastest process occurs within a similar time range, 0.5−1 ps, and (b) there is a progressive increase in the amplitude of the slow process with decreasing momentum transfer Q. Hence, it is tempting to relate changes in the DRT shape to the disentanglement of two dynamic

Figure 7. Momentum transfer dependence of τmax and σ (inset) obtained from fitting the I(Q,t) data of C30H62 at T = 350, 370, 400, and 450 K using eq 6. Lines are guides to the eyes.

a Q−2 dependence, but we observe deviations at lower temperature. Once again this differs from the findings on simple polymeric systems.29,33,32 The measured σ values (Figure 7) indicate a broadening of the distribution of relaxation times with decreasing temperature and momentum transfer Q, in agreement with our analysis of the incoherent structure factor. As discussed earlier, values of the activation energy can be obtained from the average relaxation times at each individual Q, using the Arrhenius equation. For the log-Gaussian distribution used here, the experimental τmax is related to ⟨τ⟩ by F

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of their molecular weight. I(Q,t) data at different Qs were fitted simultaneously using eqs 6 and 10. Since analysis of the C30H62 data showed that n is temperature-dependent (Figure 11), this was also taken into account when n values were inserted into eq 10.

Figure 9. Distribution of relaxation times, F(ln τ;Q) extracted from the I(Q,t) data of C30H62 at 350 K and Q = 1.00, 1.25, 1.50, and 1.75 Å−1 (from front to back).

processes. However, one should note that these results need to be taken with caution; the CONTIN analysis is highly sensitive to the input data, and a nonsmooth I(Q,t) curve could produce artifacts (in terms of shape and perhaps number of relaxation peaks). Unfortunately, it was not possible to extend the present analysis to Q < 1.00 Å−1 or Q > 1.75 Å−1 and confirm the nonGaussian character of the DRT due to the lack of information at either short or long times. III.3. Effect of Molecular Weight: Time Domain Analysis. To study the molecular weight dependence of the dynamics in PE-like chains, we carried out experiments on C44H90, PE2K, and a high molecular weight PE sample. On the basis of our detailed study of the intermediate scattering function of C30H62, we fitted all I(Q,t) data with a log-Gaussian distribution of relaxation times (eq 6). Examples of fits are shown in Figure 10 for C44H90 at 400 K.

Figure 11. Temperature dependence of the exponent, n, as obtained from fitting I(Q,t) data of C30H62, using eq 6. The continuous line represents a linear interpolation of the n versus T data.

The τ0,max values extracted from the fits of the incoherent intermediate scattering function at 450 K are plotted in Figure 12 as a function of molar mass: τ0,max increases with increasing

Figure 12. Molar mass dependence of τmax at Q = 1.0 Å−1, τ0,max (●), and σ at Q = 1.0 Å−1 (○) and 2.0 Å−1 (□) obtained by fitting I(Q,t) data of various PE-like samples at 450 K using eq 6. Lines are guides to the eye.

chain length, reaching a plateau for Mw values above 2000 g mol−1. This finding is in very good agreement with the MD results of Arialdi et al.,36 who, by comparing the intermediate scattering functions of methylene hydrogens located in the chain core with the predicted I(Q,t) for C44H90 and C154H310 (Mw = 2150 g mol−1), were able to show that chain end effects become negligible at chain lengths of the order of 2000 g mol−1. The other fitting parameter, σ, is temperature- and Qdependent, as observed earlier for C30H62. In Figure 12, σ values at 450 K and Q = 1.0 and 2.0 Å−1 are plotted as a function of molar mass. Interestingly, the molar mass dependence of σ depends on the probed length scale: at high Q, σ is independent of the molar mass while at Q = 1.0 Å−1 the molar mass dependence of σ follows the same trend observed for τ0,max.

Figure 10. Intermediate scattering function, I(Q,t), of C44H90 at 400 K and Q = 1.00, 1.25, 1.50, and 1.75 Å−1 (from top to bottom). Symbols correspond to combined experimental data from NEAT and OSIRIS. Solid lines are fits using eq 6. Inset: same I(Q,t) data but on a log−log scale.

Within the narrow Q range explored in these measurements, 1.0 ≤ Q ≤ 2.0 Å−1, the Q dependence of the relaxation time can be approximated to a power law: τmax = τ0,maxQ−n

(10) −1

where τ0,max corresponds to the τmax value at Q = 1.0 Å . The power law values for C30H62 and C44H90 were found to be very similar, suggesting that n does not depend on the molecular weight. Therefore, to simplify the analysis, the exponent n was assumed to be the same for all samples investigated, irrespective G

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Average relaxation times were calculated as described in the previous section, and, from these, activation energies were computed using the Arrhenius equation. The momentum transfer dependence of the activation energy shown in Figure 13 extends our observations based on the C30H62 data. At the

τ=

⎡ ⎤ B ξ(M , T ) = ξ∞ exp⎢ ⎥ ⎣ Δα(T − T0) ⎦

highest Q explored in these measurements, Ea is essentially independent of the molecular weight and a constant value equal to 11.4 ± 2.0 kJ mol−1 is measured. We note that this is high compared to expectations for torsional vibrations, and indeed our analysis of the C30H62 I(Q,t) data indicates that Q values above 3.0 Å−1 are needed to reach a constant Ea (Figure 5). At the low Q end, Ea values at 1.0 Å−1 should provide a good estimate of the activation energy for the segmental dynamics.

⎛ β⎞ Δα = αl − α0 = α∞⎜1 + ⎟ ⎝ M⎠ T0 =

(11)

where W is a correlation time and b is the statistical segment length.54 The parameter W is related to the segmental friction coefficient, ξ, by ξb

= 3D

N b2

(17)

T0∞ γ 1+ M

(18)

(19)

−1 with α∞ = 5.1 × 10−4 °C, T∞ 0 = 160 K, β = 160 g mol , and γ = 80 g mol−1. Average τ values calculated from eqs 4 or 8 are plotted in Figure 14. We find that the average time follows a similar trend, but there is a considerable discrepancy in the time scale of the observed process. (For the highest molar mass

−1

2

(16)

where ξ∞ is the high temperature limit of the friction coefficient (ξ∞ = 3.7 × 10−11 dyn s cm−1) and B is a constant (B = 0.60). The parameters Δα and T0 represent the difference in the thermal expansion coefficient of the liquid and occupied volumes and the critical temperature at which ξ diverges. They vary with molar mass according to

IV. DISCUSSION ON N-ALKANES AND PE DYNAMICS The internal dynamics of a melt of unentangled polymer chains is usually described by the Rouse model.52 The corresponding scattering law derived by de Gennes53 gives, in the long-time limit:

3kBT

1.65 cm 2 s−1 M w1.98

and ⟨Rg2⟩/Mw = 0.23 Å2 mol g−1 (where ⟨Rg2⟩ is the meansquare radius of gyration) from Mondello et al.17,55 and Boothroyd et al.;55 at 448 K, one expects τ to vary with Mw0.98. At 450 K, estimated τ values for Q = 1 Å−1 vary between 0.22 and 0.76 ps for C44 and PE2K, respectively. These are smaller than the experimental τ values measured by us (e.g., Figure 12). Furthermore, the dependence of our experimental characteristic times on molar mass deviates from expectations based on the above equations. As shown in Figure 12, τ tends to level off for Mw > 2000 g mol−1, rather than increase with increasing Mw. An alternative way to determine τ values is through eq 14. This requires knowledge of the friction coefficient and statistical length. If we use the relationship reported by Pearson et al.5

Figure 13. Molar mass dependence of the activation energy at Q = 1.0 Å−1 (circles) and Q = 2.0 Å−1 (squares). The continuous line represents values from rheological measurements4 while the dashed line is a guide to the eye.

W=

(15)

The calculated τ values depend on molar mass and increase with increasing M. Based on the molar mass dependence of D reported by Pearson et al.4 D=

0.5 ⎡ b2 ⎛ W ⎞ ⎤ I(Q , t ) = exp⎢ −Q 2 ⎜ t ⎟ ⎥ 3⎝π ⎠ ⎦ ⎣

3π Q −4 2 ⟨R ee ⟩D

(12)

kB being the Boltzmann constant and D the Rouse diffusion coefficient, which is equal to D=

kBT Nξ

(13)

If eq 11 is written in a stretched exponential form with β = 0.5, then one can express the characteristic time τ as

τ=

3πξ −4 Q kBTb2

(14)

Figure 14. Molar mass dependence of average τ values at Q = 1.0 Å−1 obtained by fitting I(Q,t) data of various PE-like samples at 450 K using eq 6 (full circles). The green line refers to values calculated from friction coefficients (see text).

Equation 14 can be rewritten in terms of measured quantities, the mean-square end-to-end distance, ⟨Ree2⟩, and the diffusion coefficient: H

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sample the experimental ⟨τ⟩ reach 7 ps, but calculated values level off to 0.8 ps.) Deviations from Rouse predictions are not unexpected and have repeatedly reported in the literature. For example, analysis of MD simulation data of a n-C120H242 melt at 450 K carried out by Padding and Briels56 showed that, for Q > 1.4 nm−1, the decay of the calculated Rouse curves was fast compared to MD predictions. These authors concluded that at length scales shorter than the statistical segment length, the stiffness of the chain becomes important, and this leads to a slower decay of the dynamic structure factor.57 Similar observations were made by Paul, Smith, and Yoon, based on MD simulations of nC100H202.19 Although translational and rotational diffusion of the chains could be described by the Rouse model, Paul et al.19 found systematic deviations in the internal dynamics of the chains. Subdiffusive behavior was observed in the chain dynamics, which was not predicted by the Rouse model. The authors report a value of statistical segment for PE at 450 K, b, equal to 12 Å. This length scale is linked to the Rouse regime: Rouse predictions and MD results are seen to coincide when the mean-square displacements of the atoms have moved more than b2, and this correspond to 100 ps. For times shorter than this, the Rouse model fails. Qualitatively, our QENS results are consistent with MD studies on short alkanes: experimentally, we probe a time and length scale which is outside the Rouse regime. To account for the deviations from Rouse predictions, additional dissipation mechanisms have to be introduced and incorporated in any model that describes chain dynamics.58 The similarity in the time scale of the characteristic times for both n-alkanes and PE suggests that common molecular processes are active and responsible for the decay of the intermediate scattering functions. This is an important result which partly contradicts recent literature studies.38,35 As discussed earlier, Smuda et al.38 have interpreted time-of-flight QENS data of a series of n-alkanes in terms of self-diffusion of the chains. For Q < 2π/Ree, where Ree is the end-to-end vector of the chains, one observes by QENS the overall diffusion of the chain molecules. Using values of end-to-end distance from MD simulation of n-alkanes at 448 K,17 we estimate that selfdiffusion can be observed at Q < 1.1 Å−1 for C6 and Q < 0.18 Å−1 for C66. These estimated values could provide an explanation for the discrepancy found by Smuda et al.38 between QENS and PFG-NMR diffusion coefficients. The lowest Q value explored by the authors was 0.4 Å−1, which converts to Ree = 2π/Q = 16 Å, close to that expected for C16. Thus, within the experimental Q range, 0.4 ≤ Q ≤ 1.8 Å−1, one would expect to be able to extract diffusion coefficients only for the two shortest n-alkanes. For longer chains apparent values would be obtained. The molar mass dependence of the parameters extracted by us from the experimental I(Q,t) data deserves further comment. Pearson et al.4 have invoked free volume theory to account for the molar mass dependence of the apparent flow activation energy. Using the temperature and molar mass dependence of the friction coefficient given by eq 17, the apparent activation energy can be written as Eξ = R

d ln ξ BRT 2 = d(1/T ) Δα(T − T0)2

Assuming the thermal expansion coefficient to be independent of temperature and using eqs 17−19, we can determine Eξ values and compare these to the activation energy from neutron scattering measurements. As shown in Figure 13, at Q = 1.0 Å−1 there is very good agreement between the two trends as well as absolute values. We therefore conclude that, for the slow process, the molar mass dependence of τ and Ea can be accounted for by considering chain end effects. Changes on the chain motion due to these effects are expressed in terms of a single parameter, the friction coefficient.

V. CONCLUSIONS The detailed analysis of the incoherent dynamic structure factor of n-alkanes and PE reported here confirms the unique behavior of polyethylene, giving further support to findings reported by others in the literature: (a) All I(Q,t) data appear to decay via a single process and do not show the characteristic two-step function observed for other polymers. (b) Detailed analysis of the C30H62 data indicates that two molecular processes are simultaneously active in our experimental temperature and time range. Similarly, Arbe et al. have noted that for PE the decaging process (i.e., the fast motion on the picoseconds time scale) practically merges with the microscopic dynamics. (c) Measurements on C30H62 made it possible to access low temperatures (350 K) where, according to MD simulations, disentanglement of the two processes (torsion-vibrations and conformational relaxation) could be achieved. The CONTIN analysis of the intermediate scattering function at 350 K gives evidence of two overlapping processes, in qualitative agreement with MD. (d) Other observations, such as the unusual Q dependence of the activation energy and of the broadening of the distribution of relaxation times, support the idea that the single broad distribution is indeed a composite process resulting from a combination of torsion-vibrations and conformational relaxation. In comparing their PE data with the results of Smuda et al.38 on n-alkanes, Arbe and Colmenero35 commented that “probably all local motions identified for n-alkanes are also active in the long chain PE chains”. Our analysis confirms this: the molar mass dependence of the characteristic times and activation energies at the lowest Q investigated follows a simple trend which can be accounted for by free volume theory. Furthermore, at low Q, Ea values are comparable to those obtained from rheological measurements.4 This surprisingly simple outcome suggests that local dynamics (e.g., conformational transitions) are intimately linked to the long-range motion. The surrounding medium has a global effect on the microscopic motion that can be modeled by the friction coefficient.

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AUTHOR INFORMATION

Corresponding Author

*Fax +44 (0) 131 451 3180; phone +44 (0) 131 451 3180, email [email protected]. Present Address

∥ Esso (Thailand), Thoongsukla, Sriracha, Chonburi 20230 Thailand.

(20) I

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Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank ISIS (RAL, UK) and BENSC (Berlin, Germany) for beam time and acknowledge funding from the EU-TMR (Large Scale Facilities Access) for the measurements carried out on NEAT.

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